AN INVESTIGATION OF THERMAL CREEP BUCKLING
by
GEORGE ARNOLD WEBSTER
Thesis presented in the faculty of engineering in the! University of London for the degree of Doctor of Philosophy
Mechanical Engineering Department Imperial College of Science and Technology London August 1962 - 1 -
ABSTRACT
A simplified theory is developed for calculating the creep
buckling of components in the form of prismatic bars, and
extended to cover the case of two-member components such as reactor fuel elements. The rate modulus (evolved at the mean stress) approach is used and all phases of creep and elastic strains are considerec. For the two-member components it is
shown that a solution in closed form can only be obtained when
the mean stresses in the components remain constant. In general this will not be the case and a graphical method of solution involving isochronous stress-strain curves is suggested.
Throughout the Enalysis a modified form of the Andrade
tensile creep equation proposed by Graham and Wailes is assumed.
Results of creep buckling experiments on E1C-M commercially
pure aluminium rods and on magnox A.12 rods are presented, together with data for the creep buckling of composite specimens of the two materials, and compared with the theory. 2
CONTENTS page no.
ABSTRACT 1 NOMENCLATURE
INTRODUCTION 7
LITERATURE SURVEY 9 (i) Engineering theories of creep 9 (ii) The buckling phenomenon 11 (iii)Theories of creep buckling 13
THEORETICAL ANALYSIS OF COLUMN CREEP BUCKLING 18 (i) Theory of Hoff 19 (ii) Formulation of proposed theory 21 a)Application to axially loaded column 22 b) Application to axially loaded composite 22 column c)Introduction of elastic deflection 23 (iii) Secondary phase of creep only 25 a)Single member 25 b)Composite member 26 (iv) All phases of creep, single member only 27 (v) All phases of creep, two member component 29 DESCRIPTION OF APPARATUS 33 (i) Small creep buckling furnaces 33 a)Furnace 34 b)Loading system 35 c)Measuring system 36 d)Electrical circuit 37 (ii) Wade Furnace 38 (iii) Tensile creep testing machines 39 (iv) Subsidiary apparatus 41
CALIBRATION OF THE APPARATUS 42 (i) Small creep buckling furnaces 42 a)Alignment of the lOading and measuring 42 systems b)Furnace temperature calibration 43 - 3 page no.
(ii) Wade Furnace 45
(iii)Tensile creep tecting machines 45
EXPERIMENTAL PROCEDURE 45 (i) Preliminary experiments 45 (ii) Manufacture of test specimens 46 a) Manufacture of small creep buckling 47 furnace specimens b) Wade furnace specimens 48 c)Tensile creep specimens 49 (iii) Initial bow of creep buckling specimens 49 (iv) Creep buckling experiments 51 a)Small creep buckling furnace tests 51 b) Wade Furnace tests 53 (v) Control experiments 54
EXPERIMEJ TAL RESULTS 56 (i) Tensile creep results 56 (ii) Creep buckling results 57 a)Single member components 57 b) Two member components 59 c)Wade furnace tests 59 d) Metallurgical. examination of specimens 60
ANALYSIS OF RESULTS 61 (i) Tensile creep data 61 a)Magnox A.12 results 63 b) E1C-M commerically pure aluminium 64 results (ii) Single member creep buckling data 64 a) Initial elastic deflection 65 b) Magnox A.12 creep buckling results 66 c)E1C-M commercially pure aluminium 68 creep buckling results (iii) Two member component creep buckling data 69 a)Small creep buckling furnace results 69 b) Wade Furnace results 71 page no.
DISCUSSION 71 (i) Comparison between tensile and creep 71 buckling data a) Magnox A.12 results 72 b) E1C-M commercially pure aluminium 72 results (ii) Comparison between experiillental and 74 calculated creep buckling times a)Single component specimens 74 b) Two component specimens 77 c)Wade furnace tests 78 CONCLUSIONS 79 ACKNOWLEDGM2,NTS 83 REFERENCES 84
APPENDIX I 89 (i) The Graham-Valles analysis of tensile 89 creep data (ii) Application of Graham-Walles analysis to 93 creep buckling data
APPENDIX II 96 Theoretical creep buckling times for two component columns
APPENDIX III 3.o8 Details of apparatus and specimens
APPENDIX IV 124 Experimental results and calculated curves - 5
NOMENCLATURE
A area
C constant in creep equation
E Young's modulus
E creep rate modulus
I second moment of area
K curvature
K rate of change of curvature
L length of column
M bending moment
axial load
PE Euler buckling load for single column
P' Euler buckling load for composite column
T temperature
A non-dimensional central deflection
a non-dimensional central deflection rate
b b b coefficients in Fourier expansion 1 2 3 ... h height of web of idealized H-section
exponent in creep equation
r, s subscripts referring to rod and sheath respectively
t time
tM time in which primary term in Graham-Walles equation contributes standard strain e*
t similarly for secondary and tertiary terms 1 ) t 3 )
w non-dimensional deflection at section y non-dimensional deflection rate at section y - 6 - y column axial coordinate z lateral deflection of column at section y z central column deflection m initial column central deflection before application z!om of load
om initial column central deflection after application of load
constants in Graham-Walles equation, i is an C. M K. T!1 1 1 integer varying from 1 to n atfl, y functions in creep equation e E creep strain, strain rate e e mean creep strain, mean strain rate e* arbitrary standard creep strain in Graham-Walles analysis e elastic strain o total strain, total strain rate
p radius of gyration a",6 stress, mean stress
aa 0 • * respectively (.54,T ars. 0119 (98°- INTRODUCTION
Creep is the property of a solid to change its shape with time
even when the stresses in it caused by external or body forces or by
changes in temperature remain constant. This property, as a rule, is undesirable in elements of structures and machinery, and for this reason materials possessing it at r'oom temperature, such as lead, have
been largely excluded from engineering applications. Therefore, the stress analysis of metallic structures which function at room temp- erature can generally be carried out using principles of the theories of elasticity and plasticity.
As a result until fairly recently, engineers have been satisfied with a few simple physical properties of the materials that they employ in design. These are Young's Modulus, the ultimate tensile stress, the yield stress (usually specified arbitrarily) and the elongation to failure. The yield stress serves as a rough measure of the range of loading within which no severe permanent set will take place. It also serves as an approximate upper limit for buckling stresses, which, in the elastic range are controlled by Young's
Modulus. However, in aircraft structures, gas and steam turbines and nuclear reactors, metallic structures are required to operate at temperatures much above room tempe-rature and are also often subject to high stresses. At such temperatures most metals, even those specifically developed for high temperature work exhibit mechanical effects additional to those explicable in terms of the classical theories of elasticity and plasticity. The best known of those -8 additional effects is creep.
It has become important,therefore, to study stress-strain-time relations for the alloys used in the design of these high performance structures. First a body of empirical data on the creep of these alloys under different stresses and temperatures is needed, secondly elifixeering anf-ttrnientre-rri-erl-theory of creep must be established which takes into
account this experimental data, and thirdly a creep mechanism has to be developed from the creep theory. It is with the last part that this report is chiefly concerned and in particular with the buckling of composite columns in the presence of creep. A particular aspect of this problem is in nuclear power stations where a reactor fuel element, which can be regarded as a composite column consisting of a uranium rod enclosed in a canning material, usually the magnesium alloy designated magnox, is subject to relatively high axial compressive loads and temperatures and consequently buckles due to creep. An additional effect which occurs in nuclear power stations, namely that due to irradiation, has bean treated by Roberts and Cottrell1. and extended to the creep buckling phenomenon by
Alexander2. and will not be considered here. Uraniim is an extremely o. cufft'ciat• reatex4:02 lb vse anisotropic material and as a result is not euitab±cAfor laboratory creep buckling experiments. As a preliminary to investigating the thermal creep buckling behaviour of uranium reactor fuel elements in magnox cans, experiments have been carried out on simulated fuel elements in which the uranium has been replaced by commercially pure 0 I- erruree orrotai elleuiC our aluminium which is a laeti, 1)eirave-4 materiay. In this way it. has been possible to develop a sLeplified theory which does not account
for the effects of anisotopy and irradiation. LITERATURE SURVEY
As mentioned iu the introduction a prerequisite of developing enshieenims a creep buckling theory is to develop amfundamentalAtheory of creep.
The study of creep buckling is further complicated by the great diversity of analytical expressions that have been proposed to describe the creep phenomenon. However a brief synopsis of some of the more useful engineering theories of creep is given as a preliminary to the creep buckling problems. The reasons for the particular choice of creep equation which is used are also indicated.
(i) Engineering theories of creep
Creep tests are usually
performed as uniaxial tensile
tests at constant load or
stress (the results of the
two are practically identical
for small strains) and tem-
erature. The elongation of
the test piece is measured
Fig 1. at consecutive times and plotted as creep strain e versus time t. These plots usually resemble the curve shown in fig 1., where OA is the primary, AB the secondary and BC the tertiary phase of creep. The classical 3,4. experiments were performed by Andrade who found that the portion of the creep curve OAB could be represented by the equation
e = a t + /9t • • • • • • e • • • • • ea, 100 (1) for small strains. a and pare in general functions of stress and
- 10 - arrearr t hate temperature„, and their determination irfteDeen the main study in this field in recent years.
A number of simple methods of extrapolating tensile creep data, mainly applicable to the secondary phase only, have been proposed.
The methods differ in detail, but in principle extrapolate either;
a) time to stress with temperature constant, e.g. Norton'',
Bailey6., Nadai7. or b) time to temperature with stress constant, e.g. Larson & Miller8.
Manson & Haferd9., Dorn et a110', Heimerl & McEvity11.,
All the above methods effectively apply to the secondary phase 12. l4. of creep only. Libove , Odqvist13. and Hoskin have suggested modifications to allow for the primary phase, but all these methods are essentially simplifications of the problem and are suspect if extrapolations are required over long periods of time. 15. Graham has suggested an equation of more general form than any of the above, which represents creep strain as the sum of a number of terms of the form, Qi 0. • • 0 0.0 0. where /A. is a function of time and temperature and
P. K. are constants. ca.. a. a.
After examination of much creep data II .0
Graham and Wailes16 "1718.have been able to expand the above equation in the form, i=n Ki• 1-20Ki e = Z "'" Ti - T • • • • • • • • • C.1 cr t i=1 where iti takes in fact only three values, namely , 1 and 3 and the xi/ ratios-.take values from the 'standard' series 1, 1•6 etc. Equation (3) can therefore be written in the form,
7 3 +13t • • • • • • • • • • • • e= at yt • • • •!4) which is the Andrade creep equation modified to include tertiary
creep. Equation (3) therefore applies to all phases of creep and is preferred to the simplified equation for this reason and because
it gives more reliable results when extrapolated. It is used to evaluate the functions a , f3 and y in equation (4). It is common practice to assume that these creep laws are valid
for variable stress as well as for constant stress, although they are
based entirely on constant stress experiments. This is not
necessarily logically consistent. A mathematically consistent equation can be obtained if it is assumed that a mechanical equation of state exists, i.e. that creep rate at constant stress is a function only of stress and strain and not of time
i.e. f(e, g, T) = 0 0.0 .00 40. 0.0 44. 040 (5) 19,20,21. Unfortunately many tests have shown that such an equation does not correspond to reality (especially when unloading takes place) and that creep behaviour is dependent on previous history of loading and temperature. However, here it is assumed that creep rate is a function only of the current stress and time and not of the path by which these current conditions are reached (an assumption which 22. Johnson et al found reasonable). With this assumption in mind equation (4) can be applied directly to variable stress systems.
(ii) The Buckling Phenomenon
Previously, theoretical studies of column behaviour were concerned 444 kokavivr ActLo,L. 4 dso 4VALAI NA4OL% Wi.14re egi.4 44,444.444 ta„,pg,t6;*„ (4,44 iA.z.e.44I - 12 -
with perfect columns and primarily with the load which causes the
columns to fail instantaneously. The theoretical critical load for
the elastic buckling of perfect columns was derived by Euler. The
case of inelastic buckling has been treated by many authors, in 23. 24. particular Shanley and Hoff . A perfect column is defined as one which is perfectly straight, homogeneous and axially loaded.
It will reidain absolutely straight until the critical load is reached;
at this load it becomes unstable and deflects suddenly causing failure.
When a load is applied to a practical column it behaves quite
differently from a perfect column because of imperfections such as
non-homogeneity, initial curvature and loading eccentricities. As
soon as the load is applied a practical column will deflect laterally
and will continue to deflect because of creep and will ultimately
fail. This 'ultimate' failure may occur in a few minutes or a matter
of year: depending on the loading conditions. In order to calculate
the lateral deflections that occur instantaneously a value of initial
curwture (bow) is assumed to represent the effects of all imper-
fections. As the load is applied the ordinate's of the "initial"
curvature are "instantly" increased.
The time dependent, increasing creep component of the total lateral deflection may be visualized as follows: After the column
has deflected to its initial position immediately after loading, the
stress-distribution across any cros-section is not uniform. This
causes non-uniform creep of the longitudinal fibres and hence increased
lateral deflection. As the lateral deflection increases so does the'
stress differential across the cross-section and more creep is - 13 -
obtained. This is clearly an unstable condition with respect to
the classical theory of buckling as any compressive load, however
small, will ultimately cause collapse if applied for a sufficient length of time. Critical loads therefore, as defined in elastic or inelastic buckling theory, have no meaning. However, an unstable situation as defined above may be quite acceptable to the engineer
provided that collapse or excessive deformation does not occur during
the lifetime of the column. Consequently the significant factor in creep buckling is the time required for the column to collapse under a given load. This time is termed the 'critical time' and is defined in a number of ways, but when applied to imperfect columns is usually regarded as the time required for the Column deflections to become infinite.
(iii) Theories of Creep Buckling
The problem of creep buckling appears to have been first treated by Marin25' in 1947 who considered secondary phase creep only and by
Libove12. who attempted to allow for the effect of primary creep.
In recent years progress has evolved along two main lines.
•In a convenient mathematical representation of the problem, material response is described by the use of visco-elastic elements such as elastic spriw!,is and viscous dashpots. These can be arranged in such a way that the model can be made to exhibit all the effects
(such as instantaneous strain, viscous creep and permanent and recover- able strains) which occur during column buckling. Such a treatment 26. 27. was considered by Rosenthal and Baer and Hoff and resulted in mathematically solvable equations. The principal disadvantage is - 14 -
that many materials do not obey these idealized response models.
The second approach which has attracted by far the most attention,
has consisted of attempts to devise .Methods that use real rather than
simulated column behaviour. This is obviously more satisfactory
from the engineering point of view. Unfortunately these methods usually result in complicated mathematical equations which can only
be solved if simplifications are introduced into the analysis or
lengthy step-by-step iterative solutions are employed. Some of these
approaches are discussed briefly below;
Interest initially was concentrated on the linear viscoelastic
creep law, which can be expressed in the form
tl • 0 • • • • . • • • • • ... (6) because it permits immediate application to a column of any cross- 28. section. Kempner obtained a solution in closed form and showed
that for any material obeying the linear visco-elastic creep law the
critical time is infinite, i.e. infinite deflections are only obtained
after infinite time.
Eon-linear viscoelastic creep laws as typified by the following equation .• ( cr_y TE C) • • • • • • • • • • • • • • • • • • (7) cannot be readily applied to columns of general cross-section and most papers therefore deal with columns of idealized H-section.
These are statically determinate, and the flange stresses can be calculated directly from the given load and bending moment. An idealized H-section is one consisting of two concentrated flanges of equal area connected together by a shear resistant web. The depth - 15 -
of the web and areas of the flanges are chosen so that the area and
moment of inertia of the actual column are conserved. 24. Such an approach was used by Hoff , who neglected the elastic term in equation (7) and assumed a value of n = 3. He further assumed the deflected shape to remain a half sine wave throughout
buckling and obtained an expression in closed form for the total deflection against time. The differential equation he obtains for deflection rate can be written in the form29.,
a= Ra • • • • • • • • • • • • • • • • • • (8) WL 4 where a is the non-dimensional total deflection at the middle of the i . column (a = Zmf p), L is the length of the column,
p radius of gyration,
creep rate corresponding to the mean stress ;(= P/A) on the column.
The assumption of a half sine wave deflected shape is clearly inconsistent with the differential equation. However Hoff showed that the shape can be represented by a Fourier series and that the amplitude of the third harmonic and higher orders can be neglected for small deflections. Equation (8) when solved gives a finite value for the critical time.
Odqvist30. extended- Hoff's analysis to allow for irrecoverable primary creep, which he considered to occur instantaneously while 31. Hult further extended it to include elastic deflections. It is found that the effect of both these refinements is to reduce the critical time. 32,33. Kempner solved the problem for general values of n (from 34. 1 to 14) and also included the elastic term while Hult applied the analysis to idealised H-sections with unequal flange areas.
Kempner although he assumed a half sine wave deflected shape, solved the problem for the mid point only instead of using the averaging process involved in the Fourier expansion. All these solutions give similar results and conclude that for n> 1, a critical time exists which is finite.
In the above theories so far the critical time has been defined as the time for the column deflections to become infinite. However an alternative definition based on the instantaneous stress-strain curve of the material has been proposed by Hoff. As the deflection of a column increases due to creep, its bending rigidity progressively decreases until a deflection is reached at which the buckling load becomes equal to the applied load. This deflection is defined as the critical deflection and depends only on the instantaneous elastic- plastic properties of the material. The critical time is therefore the time to reach the critical deflection. This is analogous to defining the critical time as the time at which the velocity of the column lateral deflection becomes infinite. A numerical semi- graphical method for this case has been suggested by Chapman et a135•.
The static stress-strain curve is followed and the creep strain at certain stress levels, reached after small intervals of time, is found from the relevant creep strain-time curve for that stress level.
The numerical procedure is followed until the critical deflection is reached, and hence the critical time evaluated.
- 17- 23. Shanley has :Cis° discussed a- the principles of creep buckling with the aid of a simplified
column. He proposed a step-by-
step integration procedure for the
calculation of the deflection of a
column with initial eccentricity. t Fig 2. Two basic assumptions were stated,
a) The material behaves elastically during instantaneous change of stress nor, Acr
i.e. • • • 0 • • • • • • • • OIDO 000 (9) Leo E b) Strain rate at constant stress is a function only of the current stress and strain. i.e. /e = f(a-te)A t .. • • • • • • • • • . • . • ... (10) A general expression for the strain rate under these assumptions can be formulated by considering the stress-time curve in any fibre of a beam to be made up of small steps as shown in fig. 2, so that,
AeT = Ae+ 6eo = + f(a-, e) At or in the limit
= + f(a, e) • • • • • • • • • • 0 • • • ... (ii) Such a method, which involves considerable computatior, was used by Higgins36' and Lin37. for rectangular column cross-sections.
Shanley also suggested a method for the treatment of initially perfect columns. He began by establishing the isochronous stress- strain curves of the material and denoting the slope of these curves - 18 - as the time dependent tangent modulus. The value of this modulus decreases with increasing time at constant stress so that eventually the applied load will become equal to the buckling load and collapse will occur. For perfect columns this gives a conservative estimate 38. of the critical time, but Carlson has shown that this is not necessarily the case for imperfect columns.
Other treatments of the problem of creep buckling have been given by Johnson et al22' who considered the case where the bending stresses were large compared with the direct stress; creep variational theorems have been used by Schechte39. and Piank0'411. Schlechte compared his solution with some of the simplified solutions previously discussed and concluded that refinement of stress distribution across the cross-section does not always improve the correlation between theory and experiment and does not therefore appear worthwhile.
Hence the assumption of either a linear stress distribution, idealized
H-section or use of the isochronous tangent modulus method are usually adequate.
THEORETICAL ANALYSIS OF COLUMN CREEP BUCKLING
Although, previously, considerable emphasis has been placed on the determination of the critical time for creep buckling, a column ceases to be of use structurally long before 'infinite' deflections are reached. Usually an engineer can specify the maximum allowable lateral deflection he can tolerate in a structure, and this is frequently very small compared with the length of the column. As a result the theory in this section is concerned primarily with the - 19 - determination of the rate at which a column deflects with time due to creep. The theoretical analysis suggested by Hoffis24. presented first so that a comparison may be made with the proposed theory.
(i) Theory of Hoff As previously mentioned Hoff considers a column of idealized H-section, with the same area and second moment of area as the actual column. Let each flange, of area A/2, be separated by a shear resistant web of height h. Then assuming plane sections remain plane, the rate of change of curvature (i) at a section distance y from one end is given by, 2 - el• h are respectively the strains in the inner and outer where 2 flanges. But for small lateral deflections (z), d (d2z) 2 dt dy 2 dC z e2 - el ... 00. *MO 0.. .00 000 (12) ••• - dt dy2;) - h and for an axial load PI the flange stresses are given by,
cr = cy (1 - -E) ... ORO ...... 11. 0.0 (13) 1 P = V(1 + .E) ...... • • • ... (14) 0- P Thus assurding a creep equation of the form of equation (7), but with the elastic term omitted,
••• ••• ••• ••• •••
= (01\n1 Fia/ zyl *el = 7.1
z = • 0 • • • • • • • • • • ;(/ -7 )n Similarly,
) • • • • • • • • • • • • e2 = -p (17) Substituting in equation (12)
d (d2z) n w) - (1 - • • • • • • dt dy2 wnere w = z- . Assuming the deflected shape to reain a half sine wave
z = z sin 11 • 0 • • • • • • • • • • 000 m (19) 2 2 .2 d z z ••• = .00 64,4 00. ... (20) 2 - 2 2 P w dy L L Substituting in equation (18) 2 + w)" (1 Tt h which when expanded gives,
n(n-1)(n-2) 3 n(n-1)(n-2)(n-3)(n-4) = enw + w + W5 + movi(21) 3! 5! This equation connects deflection rate with tensile creep rate at the mean stress and can be solved for relevant values of n. For n = 3 and w = a sinL , the equation reduces to,
A sin -1- = -11L '5 13a sin SI a3 sin3 • • • • • • (22) r 3 By expanding sin Z into a Fourier series
Ira 3 a 1 sin 2EX i.e. sin3 = sin L and neglecting the contribution of the third harmonic, equation (22) can be further reduced to, - 21 - (1,\\2 ,;(a _ 3 .1_ a3) a • • • • • • • • • • • • • • • IT2 4 (8)
This is the form of Hoff's equation quoted earlier. It can be
integrated, as ve is constant for constant axial load, to give the
time to reach a given deflection.
(ii) Formulation of Proposed theory or i;
0'2 tan In this analysis a creep
rate modulus E, (analogous 61 to Young's modulus for simple
elastic bending), suggested 2. by Alexander is used.
t e 2 It is defined as the slope of the stress (or) against
strain rate (&) curve at the
Fig 3. mean stress.
i.e. • • • • • • • * • • • • • ... (23)
Consider a column of arbitrary cross-section as shown in fig 3. Assuming plane sections remain plane during bending, the distribution of both strain and strain rate across a section will be linear. In general this will result in a non-linear stress distribu-
tion (also as shown in fig 3), but it is assumed that this can be . replaced by a linear stress distribution of slope tan E passing
through the mean stress ca. This will be a reasonable approximation
provided the deflections remain small. - 22 - a) Application to an axially loaded column
In the spirit of simple bending theory, for a linear stress-strain
distribution across a section, the relationship between applied bending
moment and curvature is given by the expression
• • • • • • K = EI • • • • • • • • • ... (24) Hence, by analogy, for a linear stress-strain rate distribution in
creep buckling, the relationship between rate of change of curvature and applied bending moment is given by
= • • di • • • • • • • 0 • • • • ... (25) EI But for a half sine wave deflected shape, from equation (20), 2 ..fr K = dz 2 dt •• • •• • • • • • • • — • ... (26) L
. M 02 dz • dt E I _- L2
and substituting M = Pz for an axially loaded column results in,
dt 112 dz 0 IP • • • • • • • • • • • 0 • 0•0 (27) EI PL2
Therefore at the column mid-section where z = z , equation m (27) becomes z 1 §1. 021 m = --7 logg • • 0 • • • • • • • • • • • • ... (28) E PL om
where z om is the initial central deflection immediately after application of the load. b) Application to an axially loaded composite column
A composite column is defined as one which is made up of more than one member (e.g. a rod enclosed in a concentric sheath). The above analysis has been performed for a column composed of one member only. - 23 -
However it may be extended to cover a composite column by a method 29. employed by Young .
For a composite column, assuming all members are constrained to buckle together, the rate of change of curvature at any section must be the same for all members, so that
M M 1 M2 n i = . =. . • • • • • • • • • ... (29) E I E I E I 1 1 2 2 n n where subscripts 1, 2 n refer to the various components of the column.
Now the total bending moment at the centre of the column is,
M = - Pz m = M + M 2 + M 1 n
Pz = K + E I + I m 1E111 2 2 n ni
= g $(iI) ... (30) where = + I + i I . E1I1 2 2 n n
Thus substituting for K in equation (26) gives
Pz m .77-2 dzm E(EI) - L2 dt
dt m = log --- • • • • • • • . • • • . • • ... (31) e z f (L) PL2 om This equation, as would be expected, reduces to equation (28) for a column composed of one member only. c) Introduction of elastic deflection.
So far no mention has been made of elastic deflections, except to say that zom is the initial central deflection after application of the load. This allows for the instantaneous elastic deflection occurring on loading, but neglects subsequent elastic deflections - 24 - which occur as a result of progressive creep deflections.
By simple elastic column theory, the initial central deflection of a column (zo ) is magnified on application of an axial load by m the ratio,
z om 1 = • • • • • • • • • • • zl 1 - P/P • • • • ... (32) om E where P E is tie Euler elastic buckling load and is
2 EI P = -- • • • 0 • • • • • • • • • • • ... (33) E L2
This gives a method of allowing for progressive elastic deflections as follows,
After time t, let the total column deflection be z and time t + At, let the total column deflection be z + AZ
= pz + Az • • • • • • • • • • • • • • • ... ( where LIZ c e 34) and Az = incremental increase in deflection due to creep c Az = corresponding increase in elastic deflection e .% From equation (32) z 1 A'zi 1 - P/P c E
so that Az 0 • • • • • • 0 • • • • • • • OS c = (1 - P/PE) Az (35) But from equation (27), previously developed for creep deflections only A t17I = • • • • • • • • • • • • (36) pL2 z
.'.1 Substituting for z from equation (35) gives
At . 1721 `1 • • • • • • • • • • • • • • • ... (37) pL2
-25-
In the limit At->dt. 2 dt = IT 1 ( P zm 1 - 707- log • • • • • • • • • • P ... (38) E PL om
This equation, therefore, gives the relationship between time and
column deflection allowing for both progressive elastic deflections
and creep deflections.
For the case of a composite column, again
Az 1 Az 1 - P/P1 c But in this case the Euler buckling load is given by v2 = z cm) • • • • • • • .• • .• .• • ... (39) L2
:. The relevant expression for a composite column is
z dt 172 m (1 - P/11) loge I • • • • • • • • • • 0 ••• (4o) /2(ii) om
which again expresses the relationship between column deflection and
time.
In general equations (38) and (4o) will be used to calculate
buckling times. The remainder of the theoretical analysis is concerned primarily with evaluating E, the creep rate modulus.
(iii) Secondary Phase of Creep Only a) Single Member
Assume • • • 0 • • • • • • • • • 0 •
• •
• • • • • • • 9 • • • • • • • - 26 - As 5- is constant for constant applied load, i is also constant and therefore equation (38) can be integrated as follows, 1 2/ zm i fdt = 21-2. (1 - P/PE)1°g e z PL om q zm t = pit2 ( - P/PE) loge z • • • • • • • • • • • . ... (42) PL om which gives the time to achieve a given deflection immediately. If substitution is made for E from equation (41), the resulting equation is, z m t = 17n (-1.5 (1 - P/P) loge z • • • • I • • • • .., (43) om Consider Hoff's equation (8), for small deflections the cubic term can be neglected, and integration of the resulting equation gives, 1 z 17202 m t = • • • ••• • • • • • • • • • • *5. (44) 3 L 7e loge z om It can be seen that for n = 3 equations (43) and (44) are identical, except for the inclusion of the term to allow for progressive elastic deflections in equation (43). b) Composite Member For a composite column of two components (e.g. a rod enclosed in a concentric sheath), assume the creep equations to be, )nr ) ) Er Cr = ) ) • • • • • • • • • • • • • • • ... (45) nu ) = ) s Cs) ) resulting in,
-27- Er = ar/zr n ... • • • • • • • • • • • • ... (46) E8 = 85. /S S
where subscripts r and s refer to rod and sheath respectively.
Provided no relative movement occurs between the two members,
they must shorten together. Thus the equilibrium and displacement
conditions are,
P=pr + Ps
=i5. A +Cr- A • • • • • • • • • • • • • • r r s s • ... (47)
and • • • • 0 • • 9 • • • • • • r = es •... (48)
• • • • • • • • • • • • • • • ... (49)
Hence from equations(47) and (49), 6-r and 7ce can be determined,
and for constant applied load are clearly constant. "Therefore Er and E will be constant and equation (40) can be integrated immediately s to give, 2 z ir m t = (iI). 1 - loge •• • • • • • . • . • • (50) PL om
where Z(EI) = (inr (iI)s
112 and PI =(EI), (EI)_i L — and the time to achieve a given deflection consequently obtained.
(iv) All phases of Creep, Single Member only
It is possible to obtain a simple solution in closed form, including all phases of creep, for a single member using the tangent
modulus approach. In a recent paper (which is bound in at the back 42. of the thesis) Alexander and Webster considered this case, but apart
- 28 - from inclurling the initial instantaneous elastic deflection, neglected
the elastic term in the buckling equation. Here the same procedure
will be adopted with the elastic term incluued.
Consider the modified Andrade creep equation, 3 e = a t 3 +13 t + yt 41110 000 .00 00111 060 SO. (4) Differentiating at constant stress and temperature with respect to
time results in 2 s' =sat- 3 g + 3yt ... • • ... 006 . . - (51) Thus for constant temperature the creep rate modulus may be defined
by the equation, (a9 2 a01" r,„ = 1cbt 3 + X + t2 ••• ••• ••• . . (52) Cr9.4
(acc a where x 7 x = (- - cr, T a c520,11 Therefore unlike the previous case E is now a function of time
and must be integrated with respect to time in equation (38) 2 2T z .1. (10t-3 + X+ 3v,t2)dt = E t (1 - f loge 71 11 - ...... (53) f PL E om Now a, g and y are all functions of stress and temperature, but
for a constant applied load, the mean stress will remain constant
during buckling. Thus for constant temperature, 0, x and l'i will also
remain constant and equation (53) can be integrated to give, 12 T z 0t75 + xt + kt:2) = /21-± , (1 - -1 log -111 ...... (54) PE e z PL` om from which the time to achieve a given bow can be determined. This 42. is the same equation as obtained by Alexander and Webster , but
with a term included to allow for progressively increasing elastic
deflections. - 29 -
(v) All phases of creep, two member component
This problem is complicated by the requirement that both members shall shorten as well as bucxle together, which necessitates that the division of load between the two meLlbers shall alter during creep.
The modified Andrade creep equations allowing for elastic strains for the rod and sheath are respectively, car . 3 ) er = • + art3 + fl t + Er • • • • • • • • • • • (55) 3 ) e = • + a t' + fi t + y t s E; s s For the two ma.zibers to shorten together under an imposed total load P, the equilibrium and displacement conditions must be satisfied as before,
i.e. P = 0' A + a' A • • • • • • • • • • • • • • • ... (47) ✓ r s s
e = 0 ••• ••• ••• • • • • • 6 • • • (48) • r 0 These equations may be written in incremental form as follows,
P = 6.6 r Ar + 6;'S AS
= 0 for constant applied load •00 600 006 660 (56)
andai = 09. 0166 9.0 0410 0.0 ..0 r s (57) where these increments are considered to occur in incremental time At.
Hence, for a tensile creep equation of the form of equation (55), in order that both equations (56) and (57) shall be satisfied at all
and a must alter during times, it is clear that the values of a'r 's deformation. Therefore the creep buckling equation (40) cannot be integrated directly as a l j3 and y (ar.d so therefore 0 , X and it) are now time dependent. However equation (40) can be written in increaental form as follows: ISoci-teoroos STRESS- SW-AM CUENtE, F012 2.01c7 .
to t.,.
150C KRON0 0 5 STRESS - ST IzA114 GU RYE- Fc. R. SHEATH
FIG -4 . - 31 - ,ir 2 P Az t m • • • • • • 000 000 (58) • • • (iI)r + (iI)s PIJ2- (1 )z. I r I where (I)r + (iI)s = 2 i 2 2 (59) +- t 4t 3+ k 3t t 'rt - 3 Xr 5*r 's s s
and integrated numerically.
It is suggested that a convenient method of solution is to plot
the tensile creep curves for the rod and sheath materials in the form
of isochronous stress-strain curves, as shown in fig 4, with suitable
increments p t t etc. time 1' pt 3 = t - t0, At = t - t At t - t2 etc. where At1 1 2 2 1, 3 3 and t = 0. (i.e. the 'to' line is the elastic line for the material). 0 Let values of a: and 6 at time t be respectively iji,T and alsr.
and o' are known from compatibility of elastic Initially cy.ro so strains, since
ro so • • • • • • • • • • • • • • • ... (60) Er and equation (47) must also be satisfied,
p
• • • • • I • • .0. • • Cr . 0 • • • • • (61) ro Es A + . A r Er s
and 0- - • • • • • • • • • • 0 • • • • ... (62) so E r ;A_ +. E . Ar a
of cr and cr are chosen, For the first time interval At1, values rl 51 by trail and error, so as to satisfy both equations (56) and (57).
Equation (57) will be satisfied when, 7
AB = CD 400 06a (63)
AB and CD are respectively the mean strains occurring in the rod and sheath during time interval At1, and can be measured directly from the relevant graphs in figure 4.
The values of andci: rl sl may then be substituted in equation (59) and hence equation (58) evaluated fortime interval Ati, and the corresponding increase in deflection Aznai obtained.
The procedure can then be repeated for time interval At2, and values of 0- and 07 obtained for which equations (56) and (57) are -r2 -62 again satisfied. But in this case equation (57) will be satisfied when, Alis2 A'B' - C'D' + —E • • • • • • • • • • • • • • ... (64)
where AO-2 = j rl
= s2 ers1 since A'B' and C'D' are the creep strains occurring in timePt2 and equation (57) requires compatibility of total strains.
Again substituting in equations (59) and (58), the increase in deflection A zm2 is obtained. Repeating the occuring in time pt2 procedure again for 6t3, A t4 .... A to as often as necessary, and using the compatibility relationship of equation (64) with the relevant AB and CD, the relationship between deflection and time for a composite column may be obtained. Inherent in this method of solution is the assumption that strain rate at constant stress is a function only of the current stress and time; an assumption previously stated. — 33 —
DESCRIPTION OF APPARATUS In the previous section a theoretical analysis is proposed for predicting the creep deflections of an axially compressed column from tensile creep data with the aid of a creep rate modulus. The theory considers both simple and composite columns.
In order to check the theory and assumptions single and composite columns of circular cross-section were subjected to compressive axial loading at temperature, and their deflections recorded with time.
The particular materials tested were E1C-M commercially pure aluminium and a magnesium alloy, magnox A.12. As sufficient data was not available in the literature to define the tensile creep equations of these materials adequately, 'control' tensile creep tests were mounted in an attempt to obtain sufficient data to predict these tensile equations.
The apparatus used for this experimental investigation is described in some detail below,
(i) Small creep buckling furnaces From consideration of the objects of the investigation and the preliminary calculations a convenient size for the test specimens was
18" long by 32" to 1" diameter. It was required to test these at 0 pd geilL lie 1.4.,,,,,, w...1 e...,,A61 r6 A. -4.4eA.i.i.-...r % eclo6C.. between 250 and 350 C, ' - ::_ v n - u • ; . - ; : : - .
It was also necessary to maintain the temperature uniform over the length of the specimen and to control it accurately. cio.seo The ranrJe of stresses involved was i:_lvica;ed to be 0 - 1000 to gztte^re le bur.,h6:d f 1" lb/in% giving a maximum load of 250 lb. As stains of the order of
0.1 to 0.3o were required the measuring system had to be capable of measuring deflections of 0.010" to 0.500" with an accuracy of 0.001".
- 34 -
The apparatus was designed to satisfy the above essential
requirements and in all six furnaces were built. A description of
the components follows. (See Appendix III for details).
.a) Furnace:
The dimensions of the furnace were decided from heat loss and
space requireiiients. In order to obtain the 18H constant temperature
zone for the specimens, the furnace tube was made 30" long. The 6"
overlap at each end was used for wrapping extra heating coils round
to counteract the large heat losses through the ends. The bore was
5Y2" so that there was room for the measuring probes and loading system
as well as for placing the specimen inside the furnace.
Having decided upon the dimensions of the furnace tube, it was
possible by comparison with other furnaces capable of attaining similar
temperatures and fro(..1 size considerations to decide the heat rating
of the furnace. This was estimated at 2.5 k.w.
The tube was made of heat resistant stainless steel. (A
refractory material could not be used as this would prevent free
movement of the loading system when hot, due to expansion). The tube
was accurately machined and provided with flanges at the ends to which
the 'Sindanyo' furnace lid and base were bolted. Two sets of five
holes were also drilled at right angles along the central length of
the tube. These holes took the silica sheaths for the probes.
Asbestos cloth and paper were wrapped around the tube to form an
electrical insultation between it and the windings. The heating
elements were wound in three separate phases directly onto the asbestos
insulation and were held in position by alumina cement. The windings
were then further insulated with asbestos rope. - 35 -
Details of the insulation and of the staggering of the windings in each phase to obtain a constant temperature distribution over the'18"
specimen 'gauge' length are given on page 115 of Appendix III. (It should by noted that the approximate winding distribution was calculated (-ham. 604406,410.... 4 0441mA Atuoji-,,,A dm,40 It by assuming that the heat loss through the ends of the furnace was twice
as great as that from the central section. This afforded a very good approximation.)
19 S.W.G. 'Brightray' series 'C' resistance wire was used for the
heater windings as this gave the best combination, of heater resistance
whilst still being capable of taking the rated current at temperature.
The windings were connected in parallel, but each phase was provided
with its own variable series resistance which was capable of reducing
the power to each phase by nearly 50%. As the power to each phase
could be varied independently, the series resistors were used as a
sensitive control to bring the temperature constant along the specimen
gauge length at different specimen temperatures.
The furnace tube and windings were placed inside a sheet metal
container and bolted to the 'Sindanyo' base. The winding leads were
then connected to their terminals and the silica sheaths inserted
into the holes before the container was filled with “Vermiculite" (2*(failded In,Zce) insulating materia . A 'Sindanyo' lid completed the insulation and
the whole furnace was bolted securely to the support framework.
b) Loading System:
So that as little as possible heat was lost through the lid of
the furnace no components of the loading system extended out through
the top. The specimen was loaded directly by placing slotted weights -36-
on the loading arm protruding through the base of the furnace.
The loading system consisted of two spiders each with three arms.
The spiders were joined together by three connecting rods so that the
complete assembly could slide up and down the furnace tube like a long
piston in a cylinder. The bore was accurately machined to prevent
the spiders from sticking or wobbling in the tube. A three legged
stool was placed around the lower spider (as shown in fig 9, Appendix
III) and stood on the base of the furnace. The loading arm was
connected by a universal joint (to eliedinate the effects of the weights
being placed eccentrically on the loading arm) to the centre of the
lower spider.
The specimen was supported on stainless steel conical ends between
the upper spider and the stool, thus leaving the lower spider 'free'.
A load applied to the loading arm was thus transmitted through the
three connecting rods to the upper spider and hence to the specimen.
Initially care was taken to sat up the furnace vertically and align
the loading system, otherwise eccentric loading could not have been
prevented.
All coanonents of the loading system were manufactured from heat resistant stainless steel to prevent buckling and rusting. c) Measuring System:
Electrical, mechanical and optical systems were all considered.
An optical system, alteeugh probably the most accurate, was eventually eliminated because of its excessive cost. A direct mechanical method with an electrical warning system was eventually chosen.
As the plane of deflection of the specimen inside the furnace -37-
was not known, two measuring systems at right angles were required,
and the actual deflectiorls computed from the components. Five probes,
spaced equally along the length of the specimen, were used in each
system to obtain the deflected shape. The probes, which were made
of NO dia. stainless steel rod, could slide freely inside their silica sheaths. Because the speciLlens were of circular cross section flat
'T'-pieces were screwed onto the ends of the probes so that a lateral disQiacement of the specimen did not affect the probe reading. Out- side the furnace a ISindanyo' end piece wee clamped onto each probe so that it was insulated electrically from its micrometer head.
The probes were screwed in and out by these micrometer heads which were capable of reading correct to within t 0.0005". These were fixed to a rigid reference framework which was itself securely bolted to the furnace support frame. Any deflection of the specimen could then be measured relative to the reference framework by screwing in the probes just to touch the specimen. A small voltage (3 volts) was placed across the specimen and the probes so that when contact was just made a bulb lit. This eliminated any excessive contact pressure from being exerted on the specimen and thus affecting the micrometer reading or the creep behaviour of the specimen. As soon as a reading had been made the probes were retracted to allow free deflection of the specimen. d) Electrical Circuit:
The electrical system consisted essentially of a power supply which was automatically controlled to maintain the furnace temperature constant at the required pre-set value. An outline of the arrange- - 38 - ment is shown in the circuit diagram on fie 12, Appendix III.
The 'Variac' transformer provided manual control of the power
supply to the furnace. If this control was set accurately enough the
effectiveness of the 'Ether' temperature controller was increased.
By careful adjustment of the 'Variac', the 'Ether' controller was
capable of maintaining the furnace temperature constant to within
± 1°C at 300°C. The furnace temperature was measured by a Ichromel- alumer thermocouple and recorded on the 'Ether' controller. A mercury relay switch was tripped (and hence the power supply to the furnace) when the required pre-set temperature was exceeded.
Conversely the switch was automatically closed when the thermocouple temperature fell below the required value.
The electrical circuit for the measuring system is also shown on the circuit diagram.
(ii) Wade Furnace
This furnace was manufactured by A.R. Wade Ltd. of Aldershot to English Electric Co. Ltd. specifications and was kindly loaned by the latter for this investigation. The furnace was rated at 15 k.w. three phase, and designed to test full size reactor fuel elements, that is specimens of the order of 3' to 4' long and 1" to 2" in diameter. A constant axial temperature distribution was maintained by a fan with rapid internal air circulation (or inert atmosphere if desired). Power control to the furnace was effected by a saturable reactor, with a chromel-alumel' thermocouple as the sensing element.
The specimen was again supported vertically on stainless steel conical ends and a support stool. In this case it was loaded by - 39 - means of a 4 to 1 lever arm through a loading bar protruding through the top of the furnace. (For this reason as good a temperature distribution was not obtained as with the smaller furnaces.)
Four port holes were provided diametrically opposite each other at the central section of the specimen so that its mid-point deflection could be observed with the aid of telescopes mounted on rigid tripods.
These tripods were separate from the furnace to eliminate any movement due to. vibration from the fan.
A photograph of the furnace is shown in fig 15 with the furnace door open showing a specimen inside prior to testing. A diagrammatic sketch is also given in fig 14.
(iii) Tensile creep testing machines
Four standard X+ ton Denison Creep Testing machines (Type 47D) were used with mechanical make and break temperature control for the furnaces. The furnaces were approximately 714P long and 5" diameter with their windings in three zones. The power to each zone could be controlled with a series resistor to maintain a constant axial temper- ature distribution along the gauge length of the specimen. Each furnace was suitable for a specimen of 1" gauge length with a cross- sectional area of 1n th of a square inch. Loading was by means of a ten to one lever arm.
The dial gauge type extensometers With tappet rods, supplied with these machines and shown on fig 17a), were used on the specimens giving extensions on a 1" gauge length of .0001" to 0.10". The dial gauge was attached to a tube which was in two halves and which fitted round the specimen, resting in grooves at its base. The tappet rod - 4o -
fitted into a small hole in the top of the specimen. Any extension
of the specimen was transmitted via the tappet rod to a lever arm which
gave a magnification of two to one at the dial gauge
These machines were essentially stress-ruture machines and were
not really suitable for high sensitivity stress-strain experiments.
After several trial experiments slight modifications were carried out
as follows:
In order to reduce the teelperature fluctuations obtained from the
controllers, saturable reactors were fitted onto furnaces numbers
1 and 2 and the mechanical type controllers dispensed with. This
resulted in much better temp,eature control (t 1°C at 300°C as against
+ -o C previously) with less rapid fluctuations.
As a result of the improved temperature control on furnaces
1 and 2 more sensitive and reliable extensometers were attached to
the specimens when these became available. These extensometers could
be used on the original specimens and were clamped directly onto a
7/ioq gauge length as shown in fig 17b). Any extension of the specimen
was relayed outside the furnace by means of a coaxial rod and tube
(one on each side of the specimen to allow for bending) and recorded electrically using 'C.N.S.' transducers. The transducers, which were
precision wound solenoid coils on quartz formers, operated as differential transformers so that any axial displacement of one coil within another, caused by extension of the specimen, altered the electrical output. The extensions on each side of the specimen were automatically averaged, amplified to a suitable scale and fed into an autograph recording machine. The extensometer could be balanced - 41 -
at any given time against a micrometer head reading correct to 10-5
of an inch and the extension obtained. There were six sensitivity
ranges giving extensions from 0.00025" to 0.1" for full scale deflection
with a stability of ± 5.10-6 inches. In order to minimise the effects
of fluctuating thermal gradients all components were made of a steel
with a low thermal coefficient of expansion. Even so fluctuations of
the laboratory temperature could be recorded when the extensometer was used on its most sensitive ranges.
As the stresses required for these experiments were anticipated
to be within the range 50 to 1000 lb/in2, necessitating very small 1 loads on the loading pan (i.e. from 0th1 lb to 2 lb), the loading system was also somewhat modified. The ten to one lever arm was dispensed with and direct loading employed. This allowed the load on the specimen to be ascertained more accurately and also larger weights (1 to 20 lb) to be used with consequent increase in accuracy.
(iv) Subsidiary apparatus
The subsidiary apparatus was mainly standard equipment, available in the laboratory, which was used as required. The most frequently used pieces of apparatus were a three point bending rig, loading jacks, potentiometers, thermocouple switches and a thermocouple welder.
Facilities were also available in the workshop for iaeasuring the deflected shape of a specimen. -42-
CALIBRATTON OF THE APPARATUS (i) Small creep buckling furnaces
Before tests could be carried out on any specimens using this apparatus it had first to be accurately assembled. The loading and measuring systems had to be aligned correctly to prevent eccentric loading of the specimen and erroneous deflection measurements. A dummy specimen was also placed in the furnace to check the temperature distribution before actual test specimens were inserted. The following calibration procedure was adopted;
a) Alignment of the loading and measuring system.
The furnace support frame was levelled using three levelling screws before the furnace was placed on it and bolted down. The furnace tube was then mounted vertical and co-axial with the measuring system reference framework. The loading frame when inserted into the furnace tube was then automatically vertical and it only remained to position and clamp the stool centrally in the tube. This was done with the aid of a special plumb line which was screwed into the centre of the upper spider where the top loading cone would normally be.
In this way the support stool was placed vertically below the upper loading cone and axial loading of the specimen ensured.
The alignment of the measuring system was critical as the two sets of probes had to be at right angles, with their own probes co- planar and horizontal. The 'T'-pieces on the probes had also to be horizontal when recording a measurement. This was achieved by having two spirit levels in the initial setting up stage; one inside the furnace across the 'T'-piece and the other across the flat side of the - 43 -
' Sindanyo' end piece outside. hen both spirit levels were horizontal
the 'Sindanyo' end piece was clamped. This allowed the outside spirit level to be used to maintain the 'T'-piece horizontal when recording readings during a test.
When hot the furnace tube expanded axially relative to the casing and hence tipped the probes out of the horizontal. This was rectified
by moving the silica sheaths vertically in slots provided in the casing until the probes were again horizontal. The sheaths were then clamped. This procedure had to be repeated whenever the furnace temperature was altered.
When the micrometer heads had been positioned in line with the probes the measuring system was ready for use.
b) Furnace temperature calibration.
To obtain a constant temperature along the 'gauge' length of the furnace to within t 10C a dummy specimen, of the same dimensions as the actual test specimens, was inserted into position inside the furnace. This specimen had small holes 0.040" dia. A" deep drilled in spiral fashion along its length 1" apart. 0.010" dia. glass covered copper constantan thermocouples were peened into these holes and taken out througla a hole in the furnace lid to a tiiermocouple switch. Four more thermocouples were secured to the surface of the epecien and also taken out to the switch., The E.M.F.'s across these thermocouples were measured using a Negretti and Zambra "quickreading" poteJltiometer a melting ice cold junction.
The procedure for calibrating the furnace temperature distribtuion was as follows:- The dummy specimen was placed in the furnace and all the thermo-- ccuple connections made. The power supply was switched on and the
'Ether' temperature controller set at the required temperature. The
'Variac' was then adjusted to give a suitable current (about 10% above the minimum required, to allow for mains fluctuations) to maintain the furnace temperature at the required level.
When steady conditions were attained the thermocouple readings were taken and a graph of temperature variation along the furnace length plotted. Using this graph as a guide adjustments were made to the resistances in series with the three heating elements to give a better temperature distribution. This process was repeated until,by trial and error, the required constant temperature distribution was obtained over the length of the specimen. (See graph on fig 13
Appendix III). It should be noted that the potentiometer was allowed to attain a steady state before readings were taken. It was also frequently standardized. If these precautions had not been taken incorrect results could have been obtained due to drift of the potentiometer zero.
When the above calibration experiments were completed the dummy specimen was removed and the furnace was ready for test.
The reason for the surface thermocouples was so that the surface (144.41- rieLLI tor wene4 and body temperatures of the dummy specimen could be compared This was necessary because thermocouples could only be placed on the surface of actual test specimens and the body temperature must be known.
If the thermocouples had been peened into the specimens, stress concentrations would have been obtained at the holes which might have affected the creep buckling behaviour of the specimen. - 1+5 -
(ii) Wade Furnace.
A similar calibration procedure to the previous case was employed.
Initially the furnace and loading bar were aligned vertically and
coaxially. The support stool was again placed vertically below the
upper loading cone with the aid of a special plumb bob which just
fitted into the recess in the top of the stool. Care was also taken
to ensure that the deflection measuring telescopes were positioned
mutually at right angles and coincided with the mid-height of the specimen.
The power to each furnace winding could not be adjusted and the
temperature distribution could only be altered by varying the speed
of the fan. It was found that a speed of 3,000 r.p.m. gave a suitable
temperature distribution at 300°C (i.e. constant to within 3°C from
top to bottom).
(iii)Tensile creep testing machines
These were proprietary machines which had previously had their loading and measuring systems calibrated and aligned. A constant
temperature distribution also could not be established prior to testing as this depended on the quantity of asbestos wool stuffed in the top and bottom of the furnace to prevent loss of heat. Consequently the temperature distribution had to be re-calibrated for each test.
EXPERIMENTAL PROCEDURE
(i) Preliminary Experiments As it was desired to test composite columns, each in the form of a round rod enclosed in a concentric sheath, it was necessary to - 46 - choose two materials of comparable creep strengths. It was decided
to use a magnesium alloy, magnox A.12, as the materiaA the sheath, because it is also used as a canning material for uranium fuel rods in nuclear reactors. Initially creep buckling experiments were performed in the small creep buckling furnaces at different temperatures and loads to determine a suitable material for the rod. A rod of comparable creep strength to uranium being preferable.
From tensile creep data on aluminium and some of its alloys 8. reported by Inglis and Larke43°, and using the Larson-Miller type of temperature extrapolation, it appeared as though an aluminium alloy of British Standards specification HE 30 would be suitable to use with magnox at 300°C. However, on test it was found necessary to raise its temperature to 500°C before it became sufficiently weak. At this temperature magnox practially buckled under its own weight.
Clearly some weaker alloy must be chosen. Eventually commercially pure aluminium E1C-M was found to have the desired creep strength when tested at 300°C.
(ii) Manufacture of test specimens
Initially when ordering material with which to make specimens for the small creep buckling furnaces, extruded bar of 1" diameter was ordered. This was then turned down to 3" diameter to obtain a constant grain size across the cross-section. This was quite feasible in the case of aluminium alloy HE 30 and magnox as they are both readily machinable. However in the case of E1C-M commercially pure aluminium this was not so and considerable difficulty was experienced in trying to machine it. As a result it was decided to use 140 diameter bar in - 47 - the as extruded condition although this would obviously not havq a constant grain size across its cross-section.
In order to obtain as consistent a batch of E1C-M commercially pure aluminium as possible it was requested, from the suppliers that they supply the material all from one ingot and from the middle of the extrusion. It will be shown later that the suppliers did not conform with this request and that in fact material was probably received from near the end of the extrusion.
The magnox A.12. was supplied by English Electric Co. Ltd. and was machined down from either 1", 11/t or 2" diameter bar, depending on the size of the specimen required.
A chemical analysis of the two materials resulted in the following compositions:
E1C-M Aluminium Magnox A.12
Cu < 0.01% Al 0.86%
Si 0.15% Be 0.0073%
Fe 0.60% Mg remainder
Mn 0.09%
Zn 0.06%
Al remainder a) Manufacture of small creep buckling furnace specimens
The following specimens were machined to the specifications shown in figures 18 and 19 for testing in the small creep buckling furnaces;
24 34" dia x 18" long E1C-M rods.
24 magnox rods. -48-
/ 15 1, M dia x 1E-0 long E10-M rods with 7 thick magnox sheaths.
15 II to II It n NM It n tt n ft 11 n 11 It -- n It It 7 7 7 tt It It 11 n Iv n n tt
The single specimens required only threads drilled and tapped in the ends to take the stainless steel end fittings. The composite specimens, however, were fabricated in the following manner:
After machining the aluminium rods were shot blasted to produce a rough surface onto which alumina was sprayed in an even layer approximately 0.004" thick. This had the effect of preventing chemical reaction, between the aluminium rod and the magnox sheath, which can take place slowly at 300°C over prolonged periods of time. It also has the added advantage of providing a rough surface interface between. the rod and sheath, sufficient to prevent any relative movement from occurring between them during buckling. The rod was then placed inside the sheath and the end plugs welded into position. The whole assembly was then evacuated and sealed to prevent the formation of gas pockets. Next the assembly was placed inside a pressurising 2 rig and subjected to a pressure of 12,000 lb/in at a temperature of
230°C. This collapsed the sheath onto the rod and so ensured that they shortened and bent together during the buckling process.
Finally the end plugs were removed by machining.
This method of fabrication was repeated for all the composite specimens independent of the thickness of the sheath. b) Wade furnace specimens.
The following specimens were made to check the effect of size -49- and were geometrically similar to the previous creep buckling specimens,
5 1" dia x 36" long E1C-M rods
5 'I I I 11 + 34" thick magnox sheaths
5 11 11 I 311 11 11 11 These specimens were manufactured in a siilar manner to the smaller specimens. A layer of alumina was again sprayed onto the aluminium rod before the rod and sheath were pressurised together. c) Tensile creep specimens.
Forty E1C-M commercially pure aluminium and forty magnox Al2 specimens were machined to the dimensions shown on fig 20 for testing in tension in the Denison machines. Care was taken to keep the 1" gauge length free of all tool marks and so prevent the formation of stress concentrations. This was difficult to achieve with the aluminium specimens and some tool marks had to be occasionally accepted.
(iii) Initial bow of creep buckling specimens
Before testing all the creep buckling specimens were given an initial bow or permanent set to minimise the effects of machining inaccuracies. If the specimens were not given an initial bow they would have an arbitrary shape and it would be difficult to estimate in which direction they would bow when subjected to an axial load.
They would also deflect into a non-symmetrical shape which could not be represented by a half sine wave.
The permanent set was obtained by subjecting the specimens to three point bending in a bending rig. Increasing loads were suspended from the centre of each specimen and the load against central deflection plotted until a permanent set (f approximately 0.0501) cp.+44A ibe s44424_ was obtained. The load at which this occurred could be determined
by loading the specimen until the load-deflection curve deviated from the elastic line by 0.030", as on removal of the load the curve traversed an almost elastic line.
This procedure could be repeated equally well for the sheathed specimens as well as for the rods alone, to give the same permanent set.
In the case of the large specimens for the Wade furnace, these were too large to bend in the bending rig. A similar procedure was repeated for these, but using a 10,000 lb Buckton Universal testing machine to apply the load.
After the specimens had been given an initial permanent set, the stainless steel end fittings were screwed into place. Graphite grease was smeared on the threads to prevent sticking and facilitate removal after test. With the composite specimens care was taken to ensure that the rod and sheath fitted flush at the ends, otherwise one or the other would take a larger proportion of the load than if they fitted flush in the ideal case.
With the stainless steel end caps in position the specimen was mounted between centres in a lathe. For the small creep buckling specimens the initial deflection was recorded every inch, (with a dial gauge attached to the lathe cross-slide), by rotating the specimen and recording the difference between the maximum and minimum readings.
The actual deflection was then a half of these readings. In this way the initial deflected shape of any specimen could be obtained.
For the large Wade furnace specimens the deflection was recorded at 3" 'intervals. - 51 - (iv) Creep Buckling experiments
From the preliminary experiments it was decided that the best
operating temperature would be .e:bot-t-f.Mir-e-r 302°C was ehocons (S75°1'<,.
As a result all the remaining experiments on the different types of
specimens and materials were performed at this temperature. The same
procedure was adopted for all the single rod specimens and with the
composite specimens with one slight modification.
a) Small creep buckling furnace tests
Before insertion of a specimen into a furnace four copper-
constantan thermocouples were fastened to it, equally spaced along
its length. These were bound tightly onto the specimen with W
diameter asbestos string which held them in place. It also prevented
them from being affected by direct radiation from the furnace wall
so that they would read the correct surface temperature of the specimen.
The specimen was then placed into position between the loading cone
and support stool in the furnace so that it and the loading system
could rotate freely. The thermocouples were arranged to be on the
outside of the bow and the specimen placed so that.it would bow away
from the measuring probes. In this way damage to the probes was avoided if the specimen collapsed completely, and also the thermo-
couple leads did not interfere with the measuring probes. The furnace lid was placed into position and the controlling thermocouple inserted.
When the thermocouple and measuring system connections had been
completed the furnace was switched on and allowed to heat up
gradually by continuously increasing the power from the 'Variac'.
The specimen was then allowed to soak at temperature for 24 hours under
no load. - 52 -
After soaking the no load readings of the probes were recorded in turn by screwing them in with the micrometer heads until the red light, signifying contact, just came on. The load was then applied by means of a hydraulic jack and the hour meter started. The probes were immediately read when again just touching the specimen and the readings recorded in an attempt to determine the initial elastic deflection. The readings of the four thermocouples and cold junction were also recorded to check that the specimen was at the correct temperature. As the deflections increased due to creep the probes and thermocouple redings were repeated at frequent intervals initially, but less frequently as the experiment progressed until just prior to collapse when the readings were recorded more frequently again. When the central deflection had reached a value of the order of A", or when the specimen was collapsing too quickly, the load was removed and the final no load readings taken. On some specimens readings were continued for several hours after removal of the load to determine the extent of creep recovery. A few specimens were allowed to collapse completely.
Final collapse occurred suddenly and the weights came up against a stop (to prevent indefinitely large deflections from occurring and the specimen hitting the furnace wall) with a bang.
Finally the furnace was switched off and allowed to cool down.
The specimen was then removed and its final deflected shape measured in a lathe as previously.
This test procedure was repeated for all the single rod specimens of both E1C-M commercially pure aluminium and magnox A.12. However for the composite rod-sheath specimens the following slight modifi- - 53 -
cation was adopted. After soaking for 24 hours at temperature each
specimen was allowed to cool and removed from its furnace. The initial deflected shape was then checked before the specimen was replaced into its furnace and subjected to the same test procedure as the single rods. In each case the initial deflected shape was
found to have altered. This was probably due to stress relieving of the residual stresses caused by the pressurising process during fabrication. Subsequent soaking failed to change the shape further indicating that 24 hours annealing was adequate to remove the residual stresses, or reduce them sufficiently.
The following precautions were observed when taking micrometer readings to ensure reliable results. Before each reading was taken the probes were screwed to within 0.025" of the specimen and left there for 10 minutes to eliminate any effects of change in length of the probes due to thermal gradients through the furnace and insulation.
The average of three readings was taken, making sure that the 'Sindanyol end piece (and hence the flat 'T' piece inside the furnace) remained horizontal. b) Wade Furnace tests.
These experiments were performed primarily to check geometrical similarity. After the initial bow of the specimen had been recorded the specimen was inserted into its correct position in the furnace. Two types of thermocouples were used to measure its temperature. Four chromel-alumel and four copper-constantan thermocouples were each attached to the specimen in an idential manner and position to those -54-
in the small furnaces, The chremel-eiumel thermocouples were
connected to a recorder in the control panel and provided a continuous
vccord of temperature so that any fluctuations could be immediately
detected. The copper-constantan thermocouples were connected to a
potentiometer and allowed a more accurate value of the specimen
temperature to be obtained. With the thermocouples in position the
furnace door was closed and the oil pump started to lubricate the fan
bearing. The furnace and fan were then switched on and the furnace
allowed to attain stable conditions.
After 24 hours at temperature, in the case of a single rod
specimen, the specimen was loaded after first recording the no load
readings of the two telescopes. The new readings were then recorded
immediately after loading and repeated at suitable intervals as previously.
In each case the telescopes were focussed onto a mark on the specimen surface or the edge of the specimen, and hence the deflection obtained.
The tests were stopped when the specimen began to buckle too quickly
or when a central deflection of 16." was obtained.
With a composite specimen a 24 hour preheat test was carried out
in order to relieve the residual stresses formed in the pressurising
process. The new initial deflected shape was measured before the specimen was replaced into the furnace and loaded.
(v) Control experiments
To obtain the necessary standard tensile creep data to use with
the theory in which to predict the bowing behaviour of a column in
compression, the following tensile creep tests were performed in the
Denison tensile creep testing machines. The test procedure was as -755 - far as possible in accordance with B.S.S. no's 1686 - 1688. The experiments were performed on both E1C-M commercially pure aluminium and magnox A.12 specimens.
The diameter of each specimen was first measured before it was screwed into position between the loading bars. Graphite grease was again smeared on the threads to facilitate removal after testing.
Three copper-constantan thermocouples were attached along the specimen gauge length with diameter asbestos string and the extensometer fitted into position. The furnace was then slid into place round the specimen (and extensometer) and each end stuffed with asbestos wool. Finally the power was switched on and the furnace allowed to attain stable conditions.
Two types of extensometer were employed to measure the extension of the specimens; the standard Denison dial gauge type supplied with the machines and a C.H.S. extensometer which recorded the extension electrically. If the latter was used only two thermocouples were attached along the gauge length because lack of room prevented the attachment of three.
When stable conditions were attained the power to each zone of the furnace was adjusted until a constant temperature distribution was obtained along the specimen gauge length. The amount of asbestos wool stuffed in each end could also be varied to achieve the same result. The specimen was then allowed to soak at temperature for 24 hours before the load was applied. After application of the load the thermocouple and extensometer readings were recorded as necessary, and a plot of extension against time obtained. 56 -
With the C.N.S. extenso_letere an autographic record of extension against time was obtcined, but occasionally the readings were recorded and the instrument re-balanced against the micrometer thimbles. The thimbles were also used to calibrate each sensitivity range on the extensometer.
EXPERIMENTAL RESULTS
The results of the experiments just described are shown plotted graphically in the figures in appendix IV together with the relevant graphs of calculations.
(i) Tensile creep results
These tests were performed, as described, in the Denison tensile creep testing machines at constant load and at a temperature of 302°C.
The magnox A.12 specimens, which were machined from 1" diameter bar and the E1C-M commercially pure aluminium specimens from Ih" diameter bar were all tested in the 'as extruded' condition. The tests were allowed to continue for a maximum of 1000 hours or until a predetermined amount of strain was reached; 1.0% strain in the case of the magnox A.12 specimens and 0.5% strain for the E1C-M commercially pure aluminium soecimens, as these were the maximum fibre strains anticipated in the rods and sheaths respectively in the buckling experiments. The former were loaded in the stress range 100 to 1000 lb/in2 and the latter in' the range 500 to 1500 lb/in2.
The results of the tests are shown plotted in figures 21 to 34 as log creep strain against log time. Unfortunately this data suffers from several limitations because of the size of the specimens tested, -57- method of loading and measurement of extension.
As a specimen of 1" gauge length with a cross-sectional area 2 of To-i n only Could be accommodated in the furnace, this severely
limited the accuracy of the loading and measuring systems. Neither
extensometer could be attached to the gauge length of a specimen 1" with an accuracy of greater than (i.e. 5c/0 error), and the weight
of the loading system,approximately 21b, was sufficient to cause appreciable creep in the magnox A.12 specimens during the 'soaking'
period before application of the load. There was also the possibility of introducing slight bending strains into the specimens during loading.
These could be allowed for using the 'C.N.S.' but not the 'Denison'
type of extensometer. In fact the 'Denison' extensometer tended to introduce bending stresses because its centre of gravity was not co- axial with that of the specimen. It also required frequent tapping
to prevent sticking of its dial gauge needle. The severity and duration of this tapping also appeared to affect the readings. Further fluctuations of 0.001" in the dial gauge readings could be obtained with a ± 2°C change in laboratory and/or furnace temperature. These temperature variations did not affect the 'CNS' extensometer to any marked extent although slight fluctuations could be observed when it was used on its most sensitive ranges.
(ii) Creep buckling results a) Single member components. As with the tensile tests all the specimens, were tested in the as extruded condition at a temperature of 302°C. The tests were
performed in accordance with the procedure described on page 51 - 58 -
and were allowed! to continue until a central deflection of aprroximately
14" was obtained, or for a maximum of 1000 hours. In several cases
readings were recorded for a period after removal of the load to batats.4. X4 v., determine the extent of creep recovery4so that the final deflected
shape, measured in the furnace using the measuring probes, could be
compared with the deflected shape measured in a lathe afterwards.
In all cases the agreement was within 2%.
The deflected shape of a specimen was computed as the experiment
progressed, from the readings of the two banks of probes placed
mutually at right angles, using Pythagorus' theorem. However,
in some cases, before this could be done the readings required
correction for tilt and/or rotation of the loading system which could
occur when the specimen was loaded because the spiders were not a
perfect fit inside the furnace tube due to manufacturing tolerances.
These could be as much as 0.010" so that tilt of this amount could be
obtained. As the specimens started with an initially symmetrical
shape, for axial loading any tilt could be immediately detected from
the micrometer readings and allowed for as these should also be symmet- rical. Similarly any rotation of the specimen or loading system
could be determined and allowed for as the ratio of the deflections
recorded from each bank of probes should remain constant for buckling
in one plane. The extent of these corrections was small but
significant when the deflection was small.
From these corrected readings the results of the initial elastic
deflections occurring on loading are shown in figure 35 for the magnox
A.12 specimens and figure 36 for the E1C-M commercially pure aluminium - 59 - specimens. The creep deflections recorded at constant load as each experiment progressed are shown plotted on log-log scalee in figures
37 to 50. As only the central column deflections are recorded on these creep curves the deflected shapes of two specimens at specific time intervals are shown on figures 51 and 52 for comparison. The range of loading for the magnox A.12 specimens was from 10 to 65 lb and for the E1C-M commercially pure aluminium specimens from 55 to 155 lb. b) Two member components.
For these tests E1C-M commercially pure aluminium was used for the material of the rods and magnox A.12 for the sheaths. The rods 1" 1" 3" 1" were all 1/2P diameter and the sheaths IT , E, , or thick. Again the materials were tested in the as extruded condition at a temperature of 302°C. The range of loading was from 80 lb to 250_1b.
The same procedure and precautions in interpreting the results was employed as with the single member component experiments. In this case the results are plotted in the form of time to reach a given zm ratio of the initial deflection (i.e. z l - 1.5, 2.0, 3.0 and 5.0) om for a given load and are shown in figures 53 to 56. Only the central deflection ratios are plotted although the deflected shapes of the specimens were recorded from the probe readings. However, typical shapes at specific time intervals are shown in figures 57 to 59 for specimens of different sheath thicknesses. c) Wade Furnace tests.
Although it was intended to test both single and composite specimens in this furnace it was in fact only possible to test - 6o - composite specimens with a sheath thickness of AP because of difficulties; experienced with the furnace air circulation fan and loading system.
These took a considerable time to rectify leaving only sufficient time to test one type of composite specimen. The results of the experi- ments are shown, in figure 60, plotted as time to reach a given ratio of the initial central deflection for a given load. The deflected shape could not be ascertained during test as only the central deflec- tion could be measured using the travelling telescopes. d) Metallurgical Examination of specimens.
The chemical analysis of the two materials has already been given on page 47.
Samples from both tested and untested specimens of magnox A.12 and E1C-M commercially pure aluminium were polished on successive grades of silicon carbide paper and finally on a polishing wheel using 6,1 and 1/1 diamond paste. They were then subjected to a metallographical examination with the specific object of defining the grain boundaries and measuring the grain size of the materials. The magnox A.12 specimens were etched using a 5% vital solution and E1C-M commercially pure aluminium specimens using Tucker's reagent.
Each sample was taken from the middle of a specimen.
The initial and final grain sizes of the magnox A.12 3" diameter specimens varied between 0.0035" and 0.0045" with negligible grain growth occuring during test. The grain size across the cross- section was also found to be constant. Photographs of cross-sections through specimen nos 15 and 121 are shown in figure 61 for comparison.
With the E1C-M commercially pure aluminium specimens considerable - 61 - difference in grain size was observed across the cross-section of the
16." diameter specimens, as well as from specimen to specimen. (see figure 62). Without exception the grain size increased towards the centre of the cross-section. This, of course, is to be expected from a section through an extruded bar, as the surface layers undergo more plastic straining during the extrusion process than the centre.
On close examination of figure 62b) it will be noticed that a metallur- 1" gical fault at a radius of approximately exists in specimen no Y. 12 This is known as a 'back-end' defect and occurs near the end of. an extrusion. In.the area bounded by this defect the grains are considerably larger than elseWhere and also considerably larger than in the specimens without a fault. Possibly about half the specimens contained the 'back-end' defect to a greater or lesser extent.
Examination of the large Wade furnace specimens showed that the magnox A.12 sheaths possessed a grain size of approximately 0.006", while that of the E1C-M commercially pure aluminium rods could not be estimated as it varied across the cross-section in much the same manner as for the 16." diameter specimens. However inspite of this variance it can be seen from figure 63 that the grain size is consider- ably larger than in the ih" diameter specimens.
ANALYSIS OF RESULTS
(i) Tensile creep data.
This data on the whole is considered to be not sufficiently accurate and consistant (because of the limitations already described) to determine reliable values of a, # and y to insert into the creep — 62 _ equation (4) for either magnox A.12 or Ele-M eemmercially pure aluminium. However inspite of these liaa+ntiuns the Graham-Wallesi6,q12 method of analysis of tensile creep data, described in detail in
Appendix I, has been applied to the results for comparison with the creep buckling results.
Briefly the method is to rewrite the functions a, f3 and y in terms of definite measurable physical quantities ti, t and t so 1 3 that equation (4) becomes
eftV ft 1, t 3 • • • • • • • • • • • • • • • (6.5 \ 3 . 3 where and t are the times in which each term in equation -5 t1 3 (1+) contributes an arbitrarily specified standard strain e*. A convenient method of determining these quantities is to plot log straii; against log time. The values of log ti, log t1 and log t3 may then be evaluated from these creep curves by a method of curve fitting and cross plotted separately against log stress. Finally the functions a, ft, and yare obtained from these cross plots by further curve fitting K and by use of the Graham-Wailes standard slopes (namely - 1 = 1, g!, Pi
A, 01,41 etc.) Before experimental data can be analysed by this method it usually requires interpretation, because the extensometer reading for which the creep strain is zero is not directly observable due to the indeter- minate amount of strain which occurs on loading. The shape of the early part of a creep curve on log-log scales is extremely sensitive to this initial extenecueter reading as illustrated in fig.24 by comparison of the circles and crosses. Hence a small correction will alter the shape of the curve considerably. Such corrections - 63 - have been applied when necessary and chosen throughout so that equation (65) is a very close fit to an individual curve.
In general it was found necessary to apply a greater correctiens to the results obtained using the 'Denison' extensometers than to those using thetC.N.S.t type of extensometer. However this is hardly surprising in view of the limitations of the former. In fact the maximum correction applied was 0.05%, equivalent to an error of 0.001" in the dial gauge reading, which is consistent with the magnitude of the fluctuations of the dial gauge needle obtained due to temperature effects. With the tC.N.S.1 extensometers the maximum correction employed was 0.015%. a) Magnox A.12 results
The experimental results are shown plotted in figures 21 to 28 with the Graham-Wailes curves superimposed on the corrected points.
0.2.% was chosen as the arbitrary standard strain e* and the corresponding values of ti and tl, at this strain read from the graphs. 3 The t3 component was not obtained as the experiments were stopped before the tertiary phase of creep was reached. The values of log ti 3 and log t1 are both plotted against log stress in figures 64 and
65. Because of the extent of the scatter of the data,which becomes immediately obvious from these graphs particularly for the t1 term, 3 no attempt was made to evaluate a or /I in the creep equation.
Also included on these figures are some results obtained at the same temperature by the General Electric Co. Ltd. and English Electric
Co. Ltd. on as extruded magnox A.12 but with a grain size of approx- imately 0.016". The increase in grain size can be seen to have no - 64 - noticeable effect on the valua, of ti but to increase the value of 3 for t1 a given stress. Attempts were made to correlate other data obtained from English Electric, General Electric, Parsons and Central.
Electricity Research Laboratories at other temperatures but without success, the chief reasons being that the specimens tested varied M grain size from 0.003" to 0.050" and in their prior heat treatment before test, resulting in vastly different creep strengths for nominal:; the same testing conditions. This data is not included as it cannot be used and simply serves to show the marked effect of grain size and prior heat treatment on the creep strength of a. material. b) E1C-M commercially: pure aluminium results.
The experimental results are shown plotted in figures 29 to 34 with the Graham-Walles curves drawn through the corrected points.
In this case the values of ti and t were obtained at a strain of I 0.1%. As before the t3 component was not obtained as the tertiary phase of creep was never reached. The cross-plots of log ti and log 3 t are snown in figures 66 and 67. In both cases there is a consider- I able amount of scatter which is thought to occur because of the lack of homogeneity of the material caused by the 'back-end' defect which was present in some specimens. This will affect the creep strength of the material and the consistency of the results. Again no attempt was made to evaluate cc and in the creep equation.
(ii) Single Member creep buckling data.
The Graham-Walles method of analysis of tensile creep data previously described may be applied with slight modification to creep buckling data of simple columns. The functions 0 , 7: and * in the - 65 - c,:.eep buckling equation (54) may be determined in the same manner as a t f3 and yin the tensile equation, as explained in detail in appendix I. The creep buckling curves are split up into their pri.ry secondary and tertiary terms by the method of curve fitting previc;.,Aly outlined for the tensile data and values of log t., log t1 and log 0 p \ z obtained for an arbitrary standard value of (1 - -.=.- loge z m in PE / om this case. From the separate cross plots of each of these terms against log stress 0, x and * may be evaluated with the aid of the
Graham-Wailes standard slopes.
Experimental creep buckling data suffers from the same drawback as tensile data as the column deflection for which the creep deflectio:,!. is zero is not directly observable. It may be estimated from the amount of initial elastic deflection which occurs on loading but again a small error can alter the shape of the early part of the buckling curve on log-log scales considerably. The experimental points were therefore corrected to give the best correspondence with the shape of the Graham-Wailes creep curve.
For an initial central column deflection of 0.030" the maximum correction applied to the magnox A.12 results was equivalent to 0.001" and to the E1C-M commercially pure aluminium results 0.003". Since the alignment of the loading cup and cone at each end of a specimen and the axiality of the loading system could not be guaranteed to better than 0.003" these corrections are reasonable. The weight of the loading system will also affect the value of the initial deflection. a) Initial elastic deflection The initial elastic deflection which occurs on loading is
- 66 -
represented by equation (32) for a single member column. However
the measured initial elastic deflection is given approximately by,
Zom 1 09,0 (66) z •• • • • • • . • .• • • • • om 1 - (P - 57 PE
since the loading system itself weighed 51b. Hence rearranging rad
substituting for FE gives,' z' 2 om 1 - (P - 5)L •• • • • • • • • • • • • • • ... (67) om trEI zt 2 Therefore a graph of against (P- 5)L gives a method of el 2 om fr I determining Young's modulus for the material of the column at the
test temperature. Such graphs are shown plotted in figures 35 and 36 for the magnox A.12 and E1C-M commercially pure aluminium specimens
respectively. For the former a value of Young's modulus at 302°C
of 3.2.106 lb/in2 was obtained and for the latter 4.34.106 lb/in?. 6 The value for magnox A.12 compares favourably with a value of 3.26.10 2 44. lb/in obtained by Thomas from hot elastic tensile tests.
b) Magnox A.12 creep buckling results
Having determined Young's modulus E, it is now possible to
and plot the creep buckling curves as calculate PE z m against time. The results are shown plotted on e zom log-log scales in figures 37 to 42 with the Graham-Walles curves
fitted through the corrected points. The values of ti and t1 for
equal to 1.0 are shown respec- a standard value of (1 - yr 1 log -a- Ef e z tively plotted against log stress on figures 66 and 69. It is seen
from these graphs that a straight line adequately represents the
-67-
data in each case indicating that both the function 0 and x can be
represented by a single term in stress only, i.e. equations (1.12)
and (1.13) in Appendix I may be modified to read,
cifli /171
= • • • • • • • • • a • • • • • • • • )
x = Cl -m 0 • • • • • • • • • • • • 0 • ... (69)
Analysing this data by the Graham-Walles method of standard
slopes gives a value of -A = - for the primary term and --m = - pi flin for the secondary term. (These slopes are also confirmed by statis-
tical analysis of the data). Hence substituting in equations (68)
and (69) gives -8 0 = 1.72.10 ry • • • • • • • • • • • • • • • • • • (70::
X = 5.54.10-10 Er- • • • • • • • • • • • • • • • • • • (71) and inserting into the creep buckling equation results in
for magnox A.12 zm - --) log - 1.72.10-8 &4"t3 + 5.54.10-10 t • • • ( 72 ) 2 e PL om where Ef' is stress in lb/in2 and t is time in hours.
The times obtained at different loads from this equation for zm z om to reach respectively 1.5, 2.0, 3.0 and 5.0 are compared with the experimental results in figure 72.
Having determined the values of 0 and x in the creep buckling equation it is possible to work backwards and determine what a and
flshould be in the tensile creep equation, since by definition a and x = () c—r , T
-68- Substituting for 0 and integrating given,
ym -8 4;6
a= 1.29.10 • • • • • • • 0 • • • • • 0 (7) and similarly,
r.1 • 2.77.10-1040' 2 • • • • • • • • 0 • • • • • • • • •
Hence the tensile creep equation for magnox A.12 at 302°C bec:.ges
_8 4/3 -10 2 e = 1.29.10 . to + 2.77.10 0- t • • • .75) From these values of a and f3 values of ti and t have been calculae,1 1 for a standard strain of 0.2% and plotted in the form of dashed lines on figures 64 and 65 for comparison • the tensile data. c) E1C-M Commercially pure aluminium creep buckling results
Again the creep buckling results have been plotted in the form z m of (1 - T- loge z against time. The results are shown on log- E om log scales in figures 43 to 50 with the Graham-Walles curves super-
imposed on the corrected points. From these curves the values of log z m ti and log t are obtained at a standard value of 1 - --- 1 loge s 3 om equal to 1.0 and plotted against log stress in figures 70 and 71.
As before it is found that the data can be adequately represented by
straight lines so that and x can again be expressed by equations
of the form of (68) and (69). The Graham-Wailes standard slopes, which are confirmed by statistics, are found to be - 14 for both the
primary and secondary terms so that; -12 - 5/3 0 = 6.40.10 c. • • • • * • • • • • • • • • • ... (76)
x = 2.04.10-27 7 • • • • • • • • • • • • • • • (77) and the creep buckling equation for E1C-M commercially pure aluminium
becomes - 69 - z --5221 (, - -ir )„(30 , m _ 6 .40 .10 -12 oi:513 tI + 2.04.10-27 (7- 7 t (78 ' PL zom whereis stress in lb/in2 and t is time in hours. z From this equation the times to reach values of m of 1.5, , zom 3.0 and 5.0 at different loads are obtained and compared with the experiental results,on figure 73. Working backwards from 0 and x as previously, a and /3 are obi- -12 i.e. a= 2.40.10 6 ••• ••• ••• ••• ••• •., K79' -28 8 . p= 2.55.10 a •• . • • • • • 0 • • • • • • (8 0 and the tensile creep equation for E1C-M commercially pure aluminium at 3020C becomes; -12 8// 3 i -288 e= 2.40.10 cr t + 2.55.10 t (81) and /3 values of t3 and t have been From these values of a I obtained for a standard strain e* of 0.1% and plotted on figures 66 and 67 for comparison with the tensile data.
(iii) Two member component creep buckling data. a) Small creep buckling furnace results These results are shown plotted on figures 53 to 56, in the form of load against log time to reach a given deflection ratio, for the different thicknesses of magnox sheaths. Also shown on these figures are the calculated curves which have been obtained in the manner suggested in the theoretical analysis.
In order to obtain the theoretical curves, the tensile creep equations (75) and (81) previously derived for the two materials, have been used with the elastic strains included, so that for the rod
-70 - P , 2.40.1O-2(rj;13 + 2.55.10- 28- 8 • 4.34.10`' 1 `r r and for the sheath, 4/ -8 3 -10 2 1.29.10 o- t3 2.77.10 cr t • • • 3.26.106 s
where e r and es are the total strains in the rod and sheath respeo:L- 9.1.: These equations are plotted as isochronous stress-strain curves 0.1,.
figures 75 and 76 (i.e, stress against strain at constant times
tl, t2, t3 ...) at suitable time intervals. For a given load and
specimen dimensions the values of csr and for each time interval
may be determined by trial and error for which equations (47) and
(57) are satisfied. A convenient method is to use dividers. Three approximations are usually adequate. to satisfy both equations. Having
obtained 4 rancl(5 -- the relevant values of0 x , and x may also s' r, r 0s s be determined for each time interval and hence Z(PI) since from equation (59)
Z (EI) - 2 • • • • • • • • • • • • ... (84) 1 95 t + X - t 3 + x r r 3 3,* as in this case the contribution of the tertiary terms is nil. (i.e.
* r and *s are both zero). Therefore substituting in equation (58) gives A z 2 m PL At 2 • • • • • • . • • • • • • • • ... (85) qr (1 ^EPL (LI ) so that the incremental increase in deflection in each time interval is obtained. - 71 -
The results of these calculations for each thickness of sheath
and for different loads are tabulated in Appendix II and expressed
in terms of the initial central deflection z' before application o OM the load.
The theoretical curves plotted on figures 53 to 56 were obtai—.2 from these tables.
b) Wade Furnace results
Consider equations (b4) and (85) for geometrically similar
specimens. If the stressestFr and a 's in both the rod and sheath for all geometrically similar specimens are maintained the same, then AZ Z m will be the same for a given time interval since Es, and Ps m are functions only of stress and time at constant temperature. Thus
geometrically similar specimens subjected to the same stresses should
deflect identically.
The experimental results of the tests in the Wade furnace on
36" long rods of 1" diameter with A" thick sheaths are shown in figure
60 and compared with the theoretical curves for the 18" long rods of
'WI diameter with /" thick sheaths. The theoretical loads, however
have been increased fourfold as the area of the large specimens is
four times that of the smaller ones, and consequently the loads four
times greater for the same component stresses.
DISCUSSION
( 1 ) Comparison between tensile and creep buckling data
The value of this comparison is somewhat limited in view of the
doubts expressed regarding the accuracy of the tensile creep data. - 72
However some of the more salient features will be considered with
these limitations in mind.
a) Magnox A.12 results
The tensile creep data and the creep buckling data may be coml,aae ,
immediately on figures 64 and 65 for the primary and secondary pha. • •..:.
of creep respectively. The dashed line on each graph represents for a standard strain of 0.2% caD the expected values of t and t1 using the values of a and fl determined from the creep buckling data.
For the secondary phase (i.e. the t1 term) the calculated line passes
through the experimental points indicating good agreement.
For the primary phase (i.e. the t* term) the calculated line is
considerably to the right of the experimental points indicating that
more primary strain was apparently obtained during the tensile tests
than the buckling tests in which comparatively little primary strain
was obtained. Even if the nebulous results obtained using the
'Denison' type of extensometer are neglected the calculated curve
still lies to the right of the experimental points. This can only
be ascribed to the fact that the apparatus was not sufficiently
accurate to measure the very small amounts of primary strain involved.
Additional effects could also arise from the smallness of the specimens
tested as the weight of the loading system, approximately 2 lb.,
was sufficient in itself to cause strain during the soaking period
before application of the load.
b) E1C-M Commercially pure aluminium results
The comparison between the experimental values of ti and t1 obtained from the tensile creep tests and the calculated values from - 73 - the buckling tests are shown in figures 66 and 67. Again consider-
able scatter of the experLiental data can be observed. However in this case the contribution of the primary term in the creep equatiK,
is greater than in the creep equation for magnox A.12 and sufficier
large to be measured.
Good agreement is obtained between the experimental and calclate:
primary strains, but somewhat less secondary creep strain was meac,
than anticipated. This, however, can be explained by considering
the physical mechanism of creep.
From a metallurgical point of view microscopic creep is generail
considered to be a combined effect of slip within the crystal grains
and grain boundary movement. The ease with which these phenomena
can occur in a given material for specified conditions appears, among
other factors, to be a function of grain size. Generally the larger
the grain size the more resistant is a metal to creep, that is for
given conditions progressively less creep strain would be expected
with increasing grain size in a given time.
If now the grain size across the cross-section of the extruded
1411 diameter E1C-M commercially pure aluminium bars is considered, it will be seen from figure 62 that it increases considerably towards
the centre and particularly within the area bounded by the 'back-end' defect, if this is present. The average grain size of the tensile specimens, which were machined from these bars and were only 0.178" diameter, will therefore be considerably larger than for the buckling specimens. Consequently the creep strength of the tensile specimens will be expected to be greater than that of the buckling specimens, -74- and explains why less secondary creep strain was measured in the tensile creep tests than was predicted from the creep buckling data.
It is interesting to note that, as good agreement was obtained between experimental and calculated primary strains (see figure 66): grain size does not appear to have any marked effect on the magnit: of primary creep strain obtained in a given test. This indicateF that for E1C-M commercially pure aluminium at least, increasing try:-. grain size does not affect primary strain but reduces the magnitud.:. of the secondary strain obtained. This is undoubtedly because the mechanism of plastic flow during the primary phase of creep is very similar to 'instantaneous' plastic flow in which movement of dis- locations occurs within the grains. Thus grain size has no effect.
(ii) Comparison between experimental and calculated creep buckling
times a) Single component specimens.
The experimental results of the tests on the magnox A.12 and
E1C-.M commercially pure aluminium specimens are compared on figures
72 and 73 with the calculated curves based on the creep buckling equation (54). It will be seen on careful observation of these graphs that although the creep buckling equation predicts adequately the rate of increase in deflection up to a value of central deflection equal to approximately 0.100" (i.e. - 3.0) it underestimates it om beyond this deflection. (This may be more readily observed from the actual creep buckling curves as departure from the fitted curves based
( P zm = 1.0, particularly on equation (34) occurs about 1 - P-- loge z E om for the E1C-M commercially pure aluminium specimens subjected to - 75 - high loads). The radius of gyration of the specimens is 0.125" so that departure occurs at values of the central deflection approachir. the radius of gyration of the specimen. 29,45. This is corroberated by Young who has estimated that fm a specimen of rectangular cross-section the assumption of a creep rate modulus E as defined in the text, is reasonable for deflect up to a magnitude of the radius of gyration of the specimen. It was previously stated that the creep rate modulus will be a valid assumption when the stress distribution across the cross-section is approxiAntely linear, as will be the case when the bending stresses are small compared with the direct stress. However for deflection: of the order of the radius of gyration of the specimen, the bending stresses will not be small and the stress distribution will no longer be approximately linear. In such circumstances the assumption of a creep rate modulus cannot be expected to be a good approximation.
Although Hoff's equation (b) cannot be compared directly with the creep buckling equation (54) used, as the latter applies to all phases of creep and the former only to the secondary phase, it can be used to indicate what is to be expected at deflections approaching the magnitude of the radius of gyration (i.e. a = 1 in this case). For
'a' approaching unity, the a3 term, which is incorporated into equation (8) to allow for the fact that a half sine wave deflected shape is not a rigorous solution of the differential creep buckling equation, becomes significant. Hoff's equation (8)therefore predicts the increasing deflection rate obtained when the deflection approaches the radius of gyration of the specimen. - 76 -
So far no mention has been made of the deflected shape of a
specimen, except to assume that it remains that of a half sine wave.
This assumption can be tested as follows by writing the deflection
in the form of a Fourier series Iry i.e. z = b sin + b2 sin - 21-1 + b (86) 1 3 sin 21 + ... and determining the relative magnitudes of the coefficients bl b? b3
But from symmetry b2 = bk = 0, therefore considering only the
first two terms in the expansion the deflection can be expressed as 3/17 z = b sin + b sin 440 400 040 000 000 1 3 (87) This equation has been fitted to the deflected shapes of the two
specimens shown in figures 51 and 52 and result in, at the maximum
deflection for magnox A.12 specimen no. 122, W*I7 z = 0.313 sin - 0.007 sin 27i2
and for E1C-M commercially pure aluminium specimen no 101,
z = 0.155 sin - 0.007 sin 221
where z is in inches.
The error in assuming a half sine wave deflected shape is there-
fore less than 5% even for deflections considerably larger than the radius of gyration of the specimen.
When the load was removed from a specimen at the end of a test elastic as well as creep recovery was recorded. In all cases the ratio of the elastic recovery was found to be the same as the initial elastic deflection occurring on loading. This is to be expected from equation (32). The recovery in creep deflection occurred rapidly
but soon ceased. When this was measured the final deflected shape of
the specimen, computed from the measuring probes, agreed to within
2% of that measured in a lathe. -77- b) Two component specimens
The experimental results are compared with the calculated curves on figures 53 to 56. It will be noticed that good agreement is obtained even as far as a central deflection of 0.150" (i.e. z m = 5.0) in the case of the specimens with the thicker sheaths om compared with 0.100" for the single specimens. This can be explained by the fact that the radius of gyration of these composite specimens is greater than that of the single specimens so that the assumption of a creep rate modulus may be expected to hold for larger central deflections.
On close examination, however, the calculated curves predict a slightly faster rate of increase in deflection than the experimental results. This may be more clearly seen when the graphs are plotted in terms of the real variables (as is shown for example on figure
74). Possible explanations of this are as follows:
The creep strength of commercially pure aluminium, as is readily observed from data obtained by Inglis and Larke43., is extremely susceptible to small changes in temperature. Although care was taken to use the same type of thermocouple wire throughout the tests to obtain consistent temperature measurements, it is possible that the temperature of the E1C-M commercially pure aluminium rods in the composite specimens was less than 302°C. The specimen temperature, of necessity, was always recorded at the surface of the sheath and the layer of alumina (which is a good thermal insulator) between the rod and sheath could cause the temperature of the rod to be signifi- cantly less than that of the surface. The creep strength of the - 78 - rod could then be sufficiently increased to explain the discrepancy
between the experimental and calculated results.
Throughout the calculations recorded in appendix II a time (or
age) hardening theory was assumed when applying the constant stress
creep equations to a variable stress system. This theory states
that strain rate is a function only of the current stress and time.
An alternative assumption is that of strain-hardening in which creep
rate is assumed to be a function only of the current stress and strain.
Both assumptions are idealizations. The former will predict less
total creep strain in a given time for increasing stress and more 0 for decreasing stress. Since the mean stress in the rod increased
and that in the sheath decreased with time, a strain-hardening theory
would have produced a less rapid change in mean stress in the components
than the former. Hence as the creep strength of E1C-M commercially
pure aluminiuM varies more rapidly with stress than that of magnox
A.12, this would result in a less rapid increase in creep deflection,
and possibly closer agreement with experiment.
Again as shown in figures 57 to 59 the assumption of a half sine
wave deflected shape is accurate to within 5% over the range of
deflections considered. c) Wade Furnace tests
In this case the discrepancy bstween the calculated curves and
the experimental points is large (see figure 60). The reason becomes
clear when a cross-section through the specimens is examined metallurgic-
ally as shown in figure 63. The grain size across the/ cross-section
varies considerably and is in fact masked in regions which have been - 79 - heavily cold worked. This indicates that the 'back-end' of an extrusion was again obtained and that the extrusion temperature was too low. The net result is that the experimental results and calculated curves cannot be compared, making a check of geometrical similarity impossible.
CONCLUSIONS
Initially it was intended to determine the tensile creep equations for the rod and sheath materials experimentally from tensile creep tests and from these predict the creep buckling behaviour of both single and composite columns using the theory developed. However, in view of the inadequacy of the tensile creep data it was thought best to determine the creep buckling equations of the two materials from the tests on the single specimens. From these equations it was possible to estimate what the tensile creep equations should be and using these derived tensile equations predict the creep buckling behaviour of the composite columns.
It has been shown in all cases, except where very small strains are involved, that the derived tensile creep equations predict the order of magnitude of the tensile creep strains measured in the tensile tests. Better agreement could not be expected in view of the inaccuracie involved in the tensile apparatus. An added factor with the E1C-M commercially pure aluminium material was the lack of homogeneity of the grain size in the specimens, caused by the extrusion process during manufacture, which introduced further inconsistencies.
Considering the predictions of the creep buckling behaviour of - 80 -
single and composite columns, in general the experiments were continued
beyond what are commonly considered to be small deflections, and in
fact to collapse in some cases. In this way the range of applicability
of the analysis could be demonstrated. Unfortunately, due to metallur-
gical faults in the material, it was not possible to check geometrical
similarity. However, it is shown that for small deflections (i.e.
deflections of less than the radius of gyration) the creep buckling
behaviour of composite columns may be adequately predicted from tests
on single columns, certainly to within the experimental scatter of
the data. Usually the time to reach a given deflection ratio may be predicted with an accuracy of better than 50%. For deflections in excess of the radius of gyration of the specimen departure of the experimental and calculated results occurs indicating that, the assumption of a creep rate modulus, as defined, is no longer valid. This is to be expected however, as the stress distribution across the cross-section for large deflections can no longer be considered to be approximately linear. Thus as most engineering applications are concerned only with small deflections, the use of a creep rate modulus for predicting the creep buckling behaviour of single and composite columns is justified.
The advantage of this method of solving the creep buckling problem over the more exact analyses discussed in the literature survey is that the latter require lengthy computations which do not necessarily give greater accuracy39. This method is also to be 24. preferred to that of Hoff as it considers all phases of creep and
elastic strains. Although 0dqvist30. and Hult31. have extended - 81 - Hoff's analysis to allow for primary creep and elastic strains, the
creep equation used in the above analysis
i.e. e = a + t + yt3 • • • • • • • • • • • • • $ • ... (4) is of more general form and can be expected to predict strains with
greater accuracy when extrapolations over long periods of time are
required.
In the numerical semi-graphical analysis used to predict the
creep buckling behaviour of composite columns, in which the mean
stresses in the component vary with time, a time-hardening creep
theory was assumed. This appears to predict slightly pessimistic
answers and will consequently be a safe assumption to use in engineering
design.
Further work may well be directed as follows. Here the simple
problem of an axially loaded column with pin joints and no lateral restraint has been satisfactorily solved assuming a half sine wave
deflected shape. However, in practice often it will occur that the column will be 'built-in' to some extent at the ends. In such circumstances fixing moments will exist at the ends and the range of application of the creep rate modulus may be expected to be limited.
This could be examined both theoretically and experimentally.
Another problem that sometimes occurs in a structure is when the magnitude of the lateral deflection of a column is limited by a retaining wall. The column will experience a lateral force which may cause it to bend back upon itself, or in a plane at right angles.
The assumption of a half sine wave deflected shape will then not be adequate and an arbitrary shape will have to be assumed. In such - 82 - circumstances a numerical method of solution will probably have to be used even for a single column and it may be convenient to develop a computer programme.
A still further problem is that concerned with a column subjected to pure bending, that is, a column with zero mean stress. In this case the creep rate modulus ap2roach cannot be expected to be a good apprbximation. Single columns (or beams) have already been considered ,25,46. in the literature2 but composite columns have still to be attempted.
Although it has been shown above that the assumption of a time hardening theory of creep is adequate for most engineering purposes as it is pessimistic, it may be worthwhile to develop a strain hardening theory for coulparison. This will of necessity be more complicated particularly if changing elastic strains are to be considerec',
There is also the related topic of superimposed thermal stresses.
When an anisotropic material such as uranium is subjected to thermal cycling in a tensile creep test accelerated creep results. The same effect could also be obtained with a composite column of dis-similar materials. The former case has been treated adequately in the. literature47,48,49,50. but its application to creep buckling behaviour has still to be considered. 83 -
ACKEOWLEDGMENTS
Firstly I should lie to express my sincere thanks to my supervisor, Dr. Alexander, for his continued keen interest in the project and fcr his most valuable help and guidance so freely offered at all. times. Secondly to the English Electric Co. Ltd. for sponsoring the research and in particular to the Fuel Group of the Atomic Power Division for providing some of the test equipment and for manufacturing the specimens: also to Mr. E.C.
Larke of I.C.I. for providing experimental tensile creep data for various aluminium alloys and to G.E C., Parsons and C.E.R.L. who similarly provided data pertaining to magnox A.12.
I would also like to acknowledge help and advice from Mr.
A. Graham of the, particularly for providing facilities to stay at the Establishment to learn the techniques involved in analysing creep data by the Graham-IValles method.
Finally I should like to thank my wife for her continued help and encouragenent and for typing this thesis. - 84 - REFERENCES
(1) Roberts, A.C. and Cottrell, A.H. "Creep of Alpha Uranium During Irradiation with neutrons". Phil. Mag., vol I, p 711, 1956.
(2) Alexander, J.M. "Approximate Theory for the thermal and irradiation creep buckling of a uranium fuel rod and its magnesium can". J. Mech. Eng. Sci., vol I, no. 3, 1959.
(3) Andrade, E.N. da C. "On the viscous flow of metals and allied phenomena". Proc. Roy. Soc. series A, vol 84, p 1, 1910.
(4) Andrade, E.N. da C. "The flow of metals under large constant stresses". Proc. Roy. Soc., series A, vol 90, p 329, 1914.
(5) Norton, F.H. "The creep of steel at high temperature". McGraw-Hill Book Co., New York, 1929.
(6) Bailey, R.W. "The utilization of creep test data in engineering design". I.Mech.E., vol 131, p 131, 1935.
(7) Nadai, A. "The influence of time upon creep. The hyperbolic sine law". Stephen Timoshenko Anniversary Volume, The Macmillan Co., New York, 1938, p. 155. (8) Larson, F.R. and Miller, J. "A time-temperature relation for rupture and creep stresses". Trans. A.S.M.E., Vol 74; no. 5, July 1952, p 765.
(9) Manson, S.S. and Haferd, A.M. "A linear time-temperature relation for the extrapolation of creep and stress-rupture data". 1ACA TN 2890, March 1953.
(10) Dorn J.E., Sherby, 0.D., and Orr, R.L. "Correlation of rupture data for metal at elevated temperatures". Trans. A.S.M., vol 46, p 113, 1954.
(11) Heimerl, G.J. and McEvity, A.J. "Generalized Master curves for creep and rupture". N.A.C.A. TN 4112, October 1957. -85-
(12)Libove, C. "Creep buckling of columns". J. Aero. Sci., vol 19, no. 7, p 459 Jtay 102. (13)Odqvist, F.K.G. "The influence of primary creep on stresses in structural parts". Trans. Roy. Inst. Tech., no. 66, Stockholm 1953. (14)Hoskin, B.C. "Phenomenological theories of time effects in metals at high temperatures with special reference to primary creep". Brown University. Office of Naval Research, T.R. No 7, Sept. 1958.
(15)Graham, A. "The phenomenological method in rheology". Research, vol 6, p 92, 1953.
(16)Graham, A. and Wailes, K.F.A. "Regularities in creep and hot fatigue data". A.R.C. current papers C.P. 379, C.P. 380, published by H.M.S.O., 1958.
(17)Wailes, K.F.A. "Summary of a quantitative presentation of the creep of Nimonic alloys". N.G.T.E. Report no. R. 232, March 1959. (18)Wailes, K.F.A. and Graham, A. "On the extrapolations and scatter of creep data". N.G.T.E. report no. R. 247, October 1961.
(19)Dorn, J.E., Goldberg, A. and Tietz, T.E. "The effect of thermal mechanical history on the strain hardening of metals". Am. Inst. Min. and Met. Eng., Tech. Pub. no. 2445, Sept. 1948.
(20)Orowan, E. "Creep in metallic and non-metallic materials". Proc. 1st U.S. Nat. Cong. App. Mech., p 453, 1951.
(21)Johnson, A.E., Henderson, J. and Mathur, V.D. "Note on the prediction of relaxation stress-time curves from static tensile test data". Metaliurgia, p 215, May 1959.
(22)Johnson, A.E., Mathur, V.D. and Henderson, J. "The creep deflection of magnesium alloy struts". Aircraft Engineering, p 419, Dec. 1956. -86- (23)Shanley, F.R. "Weight strength analysis of aircraft structures". Chapter 18, McGraw-Hill, New York, 1952. (24)Hoff, N.J. "Buckling and stability". J. Roy. Aero. Soc., vol 581 p 3, 1954. (25)Marin, J. "Deflection of columns". J. App. Phys., vol 18, p 103, Jan 1947. (26)Rosenthal, D. and Baer, H.W. "An elementary theory of creep buckling of columns". Proc. 1st U.S. Nat. Congr. App. Mech., June 1951, ASME 1952, p 603. (27)Hoff, N.J. "A survey of theories of creep buckling". Dept. Aero. Eng., Stanford University, Sudaer, no 88, (AFOSR-TN-60-382) June 1958.
(28)Kempner, J. "Creep bending and buckling of linearly viscoelastic columns". N.A.C.A. ET 3136, Jan 1954. (29)Young, A.G. "Creep bowing of the Hinkley Point fuel element". English Electric report VAT 353, Nov 1959. (30)Odqvist, F.K.G. "The influence of primary creep on column buckling". J. App. Mech., vol 21, p 295, 1954.
(31)Hult, J.A.H. "Critical time in creep buckling". J. App. Mech., vol. 22, p 432, 1955. (32)Kempner, J. "Creep bending and buckling of non-linearly viscoelastic columns". i.A.C.A. TN 3136, Jan 1954. (33)Kempner, J. and Patel, S.A. "Creep buckling of columns". N.A.C.A. TN 3138, Jan 1954.
(34)Hult, J.A.H. "Creep buckling". Inst. HR11fasthetslara Kungl. Tek Htigskolan, Stockholm 1955.
(35)Chapman, J.C., Erickson, B. and Hoff, N.J. "A theoretical and experimental investigation of creep buckling". PIBAL report no 4 6, Oct. 1957. p-8'7- (36)Higgins, T.P. "Effect of creep on column deflections". Chapter 20, "Weight-Strength analysis of aircraft structures" by Shanley, McGraw-Hill, 1952.
(37)Lin, T.H. "Creep stresses and deflections of columns". J. App. Mech., vol 23, no 2, p 214, June 1956. (38)Carlson, R.I.. "Tillie dependent tangent modulus applied to column creep buckling". J. App. Mech., vol 23, p 390, 1956.
(39)Schlechte, P.R. "Theoretical analysis of the creep collapse of columns". N.A.S.A. TN D.95, Sept. 1959. (40)Pian, T.H.H. "Creep buckling of curved beam under lateral loading". M.I.T. Tech. report 25-26, Jan. 1958.
(41)Pian, T,H.H, and Chow, C.Y. "Further studies of creep buckling of curved beams under lateral loading". M.I.T.'tech. report 25-28, Dec. 1958.
(42)Alexander, J.M. and Webster, G.A. "The creep buckling of composite members". J. Mech. Eng. Sci., vol 2, no 4, p 342, 1960.
(43)Inglis, N.P. and Larke, E.C. "Strength at elevated temperatures of aluminium and certain aluminium alloys". Proc. I.Mech.E., vol 172, p 991, 1959. (44)Thomas, G. "Tensile properties of magnesium alloys A.12 and ZA". English Electric Co. Ltd., Report no W/AT 656, 24th January.1961.
(45)Young, A.G. "Moment/curvature rate relations during steady creep for a beam under combined bending and direct load". English Electric Co. Ltd., Report no W/AT 282, 20th March 1959.
(46)Marin, J. "Theory of creep deflections in bending". Proc. A.S.T.M., vol 40, pp 944-946, 1940. (47)Young, A.G., Gardiner, K.M. and Rotsey, W.B. "The plastic deformation of alpha-uranium". J. Nuclear Mat., vol 2, no 3, p 234, 1960. -88- (48)Gardner, L.R.T. and Miller, W.N. "The evaluation of creep tests on alpha-uranium under isothermal and thermal cycling conditions". Symposium on uranium and graphite. Inst. of Metals, 1962, paper no 4. (49)Anderson, R.G. and Bishop, J.F.W. "The effect of neutron irradiation and thermal cycling on permanent deformations on uranium under load". Symposium on uranium and graphite. Inst. of Metals, 1962, paper no 3. (50)Alexander, J.M. "Creep buckling of anisotropic material under cycling conditions". Chapter 14, "Thermal stress". To be published by Sir Isaac Pitman & Sons Ltd. APPENDIX I
(i) The Graham-Walles analysis of tensile creep data. 6 17 18. The type of creep equation developed by Graham and Wailes' ' ' reprefients creep strain e as the sum of a number of terms of the form,
i=n R. K• 1+ I -2K i e c .0" 1, IT! - • • • • • • • • • 1 1 i=1
where C. 0 i, x. and T! are constants which may differ from term to term. 'For Ti >T the indices in the equation are negative and for
T! < T positive.
A detailed investigation of individual creep curves (see figures
5a)and5b)overleaf)revealsthat,followingAndradel ic.has a limited number of values, namely 1, 1 and 3, representing respectively what are commonly called the primary, secondary and tertiary phases of creep.
A further study of the experimental data reveals that the curve K i in figure 6b) may be compounded of straight lines of slope - . K. 13± The ratios -- take values from the sequence 1, 3h, *1 A etc. t3 At constant temperature equation (3) can be written in the form,
i=n Q. K.
• • • • • • • • • • • • • • • ... (1.1) i=1 ±20Ki where C! = C. T! -
To determine the values of the constants in equation (1.1), the formula, which involves three variables at constant temperature, is resolved into two-variable component equations. The resolution described below uses two types of granh, namely: strain-time, and stress-time. 90
£
FIG. 5 a) FAMILY OF EKFER.ImENTAL CREED Goizve_S .
4
E' 01.€ faL .11
et
...... •••• ...' Cii.t ...• ..., ...... ,,..,... 0.0 1•00 dm. .., •0.. aw• .••• ...... "" I ...... , •••• ..," ... .., .... I' ...... ' 0.. / ..•' ...... / ..... '...... , .0.. .'" ..-
F:ic, 5 b) PRIMARY SECONDARY AND Ir-WriAR..y CoMPON6441-
OF ANY ONE CREEP
9I
FIG. 6a) PR imAZy COMPONENT AT VARACVS STRESSES
I
(3
t ift
F14 61?) . caoSC.- PLOT OF FIG 6cil AT STANDARD MAIM C..,t
-92-
Alltermsintheformulawhichhavethesame/c.may be grouped
into one compound term so that equation (1.1) may be rewritten to
give, I ) 131 a132' (C'c. + C' + ... )t3 1 2
fl n s = + (CI or m + CI o- • • • • • • 0 • • (1.2) n + ... )t g R„ o ) + (Cf 0- 1- + CI a -1. + ,.. )t3 N p q ) Comparing this equation with the modified Andrade equation, 1, e = at° +, (9t + yt3 •• • • • . • • • •• • • •• • e (4) i9 1 a = ct cr + CI crfl2 + • • • . • • • • • • • • • • . • • • T (1.3) .. 1 2 #,, /5in fl 0- — + 0, 0- • • • • • • (1.4) •• = cim n + • • • • • • . ••• .•!i /3, i3, •• Y = C' er ' + CI o- '1 + .• .• • • 0 • 0 • • • • • • • • • • (1.5) p q The Graham-Walles analysis of creep data may therefore be used
to evaluate the functions a l f3 and yin equation (4) in the following
way. These functions may be expressed in definite measurable physical
quantities by rewriting equation (4) in terms of an arbitrary standard
strain e*.
Equation (4) then becomes, ku
t ) t (t)3 = • • • • • • • • • • • • • • ... (1.6) * (t.i/ \T 1/ e - -4 e* 3 3 461 2 ; -3 t• = -- = CI cr + CI 43-19 + •• • 1 • • where a 1 2 •... (1.7) 4_ -1 Rin gin, = (51) = e* GIC ,N. + CI 0- + • • . • • (1.8) t1 m " n ••• ...... 1 1 7 .. 3 3- = e*(.) 13 P /3 I t = e* CI + 6 + . • • 3 Y ....P O' C' -93- Hence ti , t and t are the times in which each term in equation (4) T 1 3 • contributes the standard strain e*. These quantities are constant
for ,ny one creep curve but depend in general on stress and temperature.
The manner of this dependence for ti is obtained as follows. (A 3 similar procedure is employed for t and t ). 1 3 The creep curve is split into its primary, secondar!fand tertiary
terms as shown in figure 5b), and graphical analysis of the primary
component performed as shown in the succeeding figures by a method of
curve fitting. The curves are compounded of the standard slopes (as shown in figure 6b) and are fitted to the experimental points. The
equation of the curve in figure 6b) may therefore be expressed by
equation (1.7). The equation of the straight line component of slope
- -- is then Pi
ti= • • • • • • • • • • • • • • • ... (1.1o)
K 1 and of slope - -- is fl2 3 7 • • • • • • • • • • • • • • • (1.11) ti e* 1 CI2 a''2 ...
Therefore from the slopes and positions of these lines the constants CI C/ , and /3 may be determined and hence a formulated. 1, 2' 1 2 By a similar procedure fl and y may be obtained and the complete tensile creep equation at constant temperature determined. (ii) Application of Graham-Walles analysis to creep buckling data.
For a column having a tensile creep equation of the form of equation (4) and consisting of one member only, the creep buckling equation is given by equation (54)
- 94
v2i2zm ( 3 i.e. 1 - - 95 t 3 X t lit • • • • •• ... (54) PL )loge om But from equation (1.3), (1.4) and (1.5) 13-1 ^1 fl2-1 = CIR + CI fl + • • • • • • 1 1 c 2 2 (1.12)
-1 16 -1 p 41 F3-- n • • • • • * = (a50- = CIm m + n n (1.13)
_a 0-1 Q-1 (-9 = ct p Jr' P + CIq q + • • • q • .., (1.14) acr a", T P
•• Substituting in equation (54) gives, J1, -1 02-1 i ) (c 1 cr 2 2 1 ) ) z 16 -1 0 -1 ) 1..I 7 P m pm a..:m + qfln & n ---- I = + (CIm + ...)t ) (1.15) PL2 ( - pE ) loge zom ) ) /3 -1 /3 -1 0 (C' #9 & P C'fi & q + ...W) P P cl cl , )
and multiplying both sides by & results in, 1 , 6 , +...)t3 (c (31 a 1 +c ( 1 2 2 ) ) ) 2 z fl _011 IT 1(1 _ P log m = + (01 0 a. m + ci 0 0- 4, ...)t ,/) (1.16) P 1/ e z m m n n AL E om )
(C1 /3 A + cl fi cr q + . ..)t 3 ) P P q By comparison it can be seen that this equation is identical to
equation (1.2) if, -95- z_ /721AL (a. - e is replaced by Plil loge z om 3 andCfloyC!i9.3_.11.1solc.iffill again have values of 1, 1 and K. 3. 1 and - (J-7 have values from the series 1, 34, etc. Therefore the Graham-Walles analysis can be used to determine
0 , x and y in equation (54) in exactly the same manner as previously
described for determining a, 0 and y in the tensile equation provided z P m an arbitrary standard value of 1 - --)loge ---z is chosen. ( PE om • OA APPENDIX II
Theoretical creep bucklin times for two component columns
The following calculations are performed for composite columns each composed of a round rod of. E1C-M commercially pure aluminium enclosed in a concentric sheath of sagnox A.12. The nominal dimensions of the rods are )" diameter 18" long and the 1" 1" 3" 1" sheaths are nominally F ' 16 and T thick. Equations (64) and (65) are used throughout to determine the incremental increases in central deflection for each time interval and equation (32) for the initial elastic deflection on loading z 4 . OM 1 z, 1 - P/P' om' 1" a) Composite column with 7 thick sheath.
Assume average specimen dimensions as follows:-
Rod 18" long x 0.486,0 dia.
Skleath 18" long x I.D. 0.496" x 0.D. 0.6215"
A = 0.185 in2 I =2.73.10-3 ink r r . 4 A = 0.110 in2 I =4.35.10-3 s s 2 P' = 8.32.10 lb -97- TABLE 1. Load P = 180 lb.
E (EI) At 467 r 6 zi 2 s 2 10 ziom om (hrs) (hrs) (1b/in )(1b/in-) 0 0 i 674 506 1.000 0.285 0-0.2 0.1 668 1 515 1.306 1.285 0.149 P.2-0.5 1 0.35 i 68o 496 1.934 I 1.434 0.169 10.5-1.0 0.75 697 i 467 2.277 1.603 0.267 1-2 1.5 721 426 2.513 1 1.80 0.564 ; 2-3 2.5 1 735 402 2.655 2.434 0.694 1.712 3-5 1 4.o I 748 j 380 2.77 3.128 5-7 6.0 755 368 2.(5 4.840 2.573
TABLE 2 Load P = 155 lb. T- Z AZ (EI) m m At t a 6 z, 71 sr ' 104 o m om (hrs) (hrs) ,I (lb/in2)(lb/i 2 o 0 580 436 1.000 0.236 0-0.2 0.1 580 436 1.599 1.236 0.097 0.106 p.2-0.5 i 0.35 591 416 2.373 1.333 0.5-1.0 0.75 609 387 3.20 1.439 0.141 1-2 1.5 630 350 3.69 1.580 0.269 2-3 2.5 647 321 3.85 1.849 0.294 0.640 3-5 4.0 665 291 4.20 2.143 i 0.806 5-7 6.o 675 275 4.33 2.783 7-10 8.5 681 265 4.47 3.589 1.509
-98- TABLE 3. Load P = 135 lb. z Az At t & , a E(BI) m m ,r 2 /L12) z--1-- z' (hrs) ' (hrs) (lb/in )1(lb om om i i 0 0 505 380 1.000 0.20 0-0.2 t 0.1 508 374 1.90 1.20 0.067 0.2-0.5 0.35 t 518 358 2.914 1.267 0.069 0.5-1.0 0.75 !, 535 329 4.21 1.336 0.084 1-2 1.5 554 296 5.17 1.420 o.146 2-3 2.5 569 27o 5.80 1.566 0.143 3-5 4.0 i 589 237 6.23 1.709 0.291 5-7 i 6.0 599 220 6.57 2.000 0.324 7-10 8.5 610 202 6.79 2.324 0.546 10-15 12.5 617 190 7.04 2.870 1.083 15-20 17.5 621 183 7.21 3.953 1.456
TABLE 4. Load P = 115 lb z m Az E(k) m At t cr IcT 4 z1 r s 10 zom om (hrs) (hrs)(1b/in ) (lb/in2) 0 , 0 430 323 1.000 0.164 0-0.5 i 0.25 443 305 3.52 1.164 0.073 0.5-2 1.25 471 255 6.93 1.237 0.118 2-5 1 3.5 500 206 9.44 1.355 0.189 5-10 I 7.5 523 167 10.09 1.544 0.336 1 10-20 1 15 540 138 12.21 1.880 0.675 , 20-35 , 27.5 549 123 13.08 2.555 1.283 35-60 47.5 554 114 13.65 3.838 3.08 -99- TABLE 5.
Load P = 95 lb
z Az , - 1 E(L) m At t 6- 6 Z1 z' s 2 1 ---4-lo om om (hrs) (hrs) . (lb/i )1(1b/in-)
0 k o 355 1 267 1.000 0.132 0-0.5 0.25 368 245 4.61 1.132 0.043 0.5-2 1.25 388 : 211 1 9.70 1.175 0.064 2-5 j 3.5 413 170 14.7 1.239 0.089 5-10 7.5 435 132 1 19.1 1.328 ' 0.123 10-20 i 15 1 452 1C2+ 23.0 1.451 0.222 , 20-35 27.5 462 87 1 26.1 1.673 0.339 35-60 47.5 i 470 74 28.7 2.012 0.618 60-100 80 473 69 30.6 2.630 1.213 100-150 125 475 65 i 32.1 3.843 2.113 I
TABLE 6. Load P = 80 lb y z A z cr. Z(EI) m m At t ar s 4- Tr- l -) 2 -----10 om zom (hrs) (hrs) (1b/in`) ;1 (1b/in )
0 0 299 i 225 1.000 0.110 0-0.5 0.25 311 205 ti 5.74 1.110 0.028 .5-2 1.25 326 • 180 12.79 1.138 0.039 2-5 , 3.5 , 346 144 20.5 1.177 0.050 5-10 1 12.5 367 111 28.7 1.227 0.062 10-20 : 17.5 382 ' 86 37.2 1.289 0.100 20-35 I 27.5 392 68 45.9 1.369 0.131 35-60 I 47.5 400 55 53.3 1.520 0.206 60-100 80 405 48 59.8 1.726 0.333 100-1501 125 408 42 65.7 2.059 0.453 150-200 I 175 409 , 40 69.5 2.512 0.522 200-300 i 250 410 38 73.0 3.03 1.200 300-400 050 411 36 75.9 4.23 1.610 - 100 -
b) Composite Column with thick sheath Assune average specimen dimensions as follows:- Rod 18" long x 0.486" dia. Sheath 18" long x I.D. 0.496" x 0.D. 0.7475"
• • Ar = 0.185 in2 Ir = 2.73.10-3 in4
As = 0.246 in2 Is = 1.23.10-2 . 4 P = 1.582.103 lb.
TABLE 7 Load P = 207 lb.
zm 6zm At ar 6-s z' z' om OM (hrs) i (hrs) (lb/in2) (lb/in 1 0 0 558 419 1.000 0.150 0-0.2 0.1 56o 418 3.43 1.150 0.053 0.2-0.5 0.35 578 405 5.21 1.203 0.056 0.5-1.0 0.75 6o6 384 6.28 1.259 0.076 1-2 1.5 640 359 6.96 1.335 0.151 2-3 2.5 666 339 7.36 1.486 0.156 3-5 4.o 693 319 7.68 1.642 0.336 5-7 6.0 708 308 7.95 1.978 0.390 7-10 8.5 718 301 8.23 2.368 0.676 10-15 12.5 724 296 8.28 3.044 1.440 15-20 17.5 727 294 8.48 4.484 2.067 - 101 -
TABLE 8 Load P = 180 lb. z Az Cr E(ii) m m Qt 4 zit 2 2 10 om ZI (hrs) (hrs) (lb/in ) (lb/in 0 0 486 366 1.000 0.113 0-0.2 0.1 492 36o 4.03 1.113 0.036 0.2-0.5 0.35 507 349 6.26 1.149 0.036 0.5-1.0 0.75 535 329 7.68 1.185 0.051 1-2 1.5 566 305 8.90 1.236 0.091 2-3 2.5 592 286 9.60 1.327 0.091 3-5 4.0 624 262 10.24 1.418 0.185 5-7 6.o 640 250 ln 58 1.603 0.200 7-10 8.5 64 24o 10.89 1.803 0.331 10-15 12.5 666, 231 11.23 2.134 0.633 15-20 17.5 675 224 11.48 2.767 0.800 20-25 22.5 676 223 11.47 3.567 1.032 25-35 3o 678 222 11.52 4.599 2.655
TABLE 9 Load P = 125 lb. z Az' z(ii) m At t z/ z. r 10 OM om (hrs) (hrs) (1b/inc ,(1b/in2 ) 0 0 338 254 1.000 0.086 0-0.5 0.25 360 238 8.6 1.086 0.028 0.5-2 1.25 392 213 14.94 1.114 0.050 2-5 3.5 435 180 , 20.16 1.164 0.077 5-10 7.5 475 150 23.61 1.241 0.117 10-20 15 507 126 25.95 1.358 0.233 0.382 20-35 27.5 526 112 27.79 1.591 35-60 47.5 536 104 29.15 1.973 0.753 60-100 80 542 100 30.3 2.726 1.603 100-150 '125 546 97 ' 30.7 4.329 3.14
- 102 - TABLE 10 Load P = 95 lb a z Az At iE (L) m m r 4 z' z' (hrs) (hrs) (lb/i )1(1b/7n 10 OM OM 0 0 257 193 1.000 0.064 0-0.5 0.25 272 182 11.07 1.064 0.016 0.5-2 1.25 296 164 21.54 1.080 0.025 2-5 3.5 330 138 31.4 1.105 0.035 5-10 7.5 362 114 40.3 1.14 0.047 10-20 15 390 92 49.2 1.187 0.080 20-35 27.5 41 2 76 56.4 1.267 0.118 35-60 47.5 428 64 63.2 1.385 0.182 60-100 80 436 58 67.4 1.567 0.309 100-150 125 442 53 71.2 1.876 0.437 150-200 175 446 50 74..3 2.313 0.516 200-300 250 448 48 76.9 2.829 1.22 300-400 35o 450 47 78.8 4.049 1.708 c) Composite column with It thick sheath Assume average specimen dimensions as follows:- Rod 18" long x 0.486" dia. Sheath 18" long x I.D. 0.496" x 0.D. 0.875" 2 -3 4 Ar = 0.165 in Ir = 2.73.10 in 2 -2 . 4 As = 0.409 in Is = 2.59.10 In P' = 2.93.103 lb.
- 103 - TABLE -11 Load p = 250 lb.
A z Lt t z' Z 01. om CM (hrs) (hrs) (1b/in2)(1b/in 2 0 0 507 382 1.000 0.082 0-0.2 0.1 516 378 7.20 1.082 0.027 0.2-0.5 0.35 536 368 8.58 1.109 0.034 0.5-1 0.75 570 354 12.42 1.143 0.041 1-2 1.5 613 335 13,88 1.184 0.076 2-3 ?.5 65o 318 14.73 1.260 0.076 3-5 -.0 68_ 304 15.57 1.336 0.152 5-7 6.0 698 298 16.00 1.488 0.165 7-10 8.5 710 292 1c,43 1.653 0.268 10-15 12.5 7, 289 16.74 1.921 0.510 15-20 17.5 72 387 16.91 2.431 0.638 20-25 22.5 722 287 17.02 3.069 1 0.800 25.35 >0 72', 286 17.1 3.869 2.01
TABLE 12 Load P = 200 lb. z A z E(EI) m m t 0-* 4 z/ zI r 2 /8 2 10 om om (hrs) (hrs) (1b/in ) (lbtin ) 0 0 406 306 1.000 0.073 0-0.5 0.25 432 294 12.0 1.073 0.032 0.5-2 1.25 4go 268 18.3 1.105 0.064 2-5 3.5 558 237 21.9 1.169 0.113 5-10 7.5 606 215 23.6 1.282 0.191 10-20 15 635 201 24.7 1.473 0.420 20-35 27.5 65o 195 25.6 1.893 0.780 35-60 47.5 654 193 26.0 2.673 1.810 - 104 - TABLE 13 Load P = 160 lb.
I z Az Ts m m Lt t r z' zt 2 om nm (hrs) (hrs) (1b/in2)i(lb/i 0 i 0 324 244 1.000 0,059 0-0.5 0.25 350 232 14.9 1.059 0.020 0.5-2 1.25 391 214 24.1 1.079 0.037 2-5 3.5 448 188 30.5 1.116 0.061 5-10 7.5 501 165 3L.5 1.177 0.095 10-20 1; 545 '45 37.0 1.272 0.191 20-35 ..7.5 136 39.1 1.463 0,312 35-60 47.5 578 131 39.9 1.775 0.618 60-100 I 80 536 127 +1.2 2.393 1.29 100-150 1125 125 42.1 3.684 2.43
TABLE 14 Load P = 12 lb.
z Azm At E(iI) m (77-r 6 Z z' 2 10 om , om (hrs) (hrs)(1b/in 2 ) , (1b/in3 ) 0 0 244 1 :64 j 1.000 0.042 1 0-0.5 0.25 262 i 176 19.2 1.042 0.011 .5-2 1.25 I 291 162 ! 33.64 1.053 0.018 2-5 3.5 335 i 142 45.38 1.071 0.029 5-10 7.5 3o0 122 54.7 1.100 0.041 10-20 15 422 103 62.9 1.141 0.074 20-35 27.5 453 89 69.2 1.215 0.108 35-60 47.5 477 78 74.7 1.323 0.182 60-100 8o 486 74 ► 76.65 1.505 0.281 00-150 125 491 72 81.35 1.786 , 0.450 50-200 175 496 69 82.86 2.236 , 0.543 00-300 250 498 69 82.9 2.779 1.376 oo-400 350 500 67 84.9 4.155 2.01 - 105 -
d) Composite column with Alt thick sheath Assume average specimen dimensions as follows:- Rod 18" long x 0.48611 dia. Sheath 18" long x I.D. 0.496" x 0.D. 0.9943" -3 . 4 •• • Ar = 0.1b5 in2 Ir = 2.73.10 in -2 . 4 As = 0.584 in2 Is = 4.50.10 in pt = 4.83.103 lb.
TABLE 15 Load P = 250 lb.
L(EI) zm Azm At t k,7.: Er'r -. ---7 10 z/om zoT m i (hrs) (hrs) I (lb/in ) lb/i 2 0 0 i 400 i 301 1.000 0.054 0-0.5 0.25 1 428 292 19.53 1.054 0.023 0.5-2 1.25 496 270 28.9 1.077 0.048 2-5 i 3.5 575 i 245 33.6 1.125 0.087 5-10 1 7.5 629 228 35.9 1.212 0.146 10-20 15 659 219 37.8 1.358 0.311 20-35 27.5 670 215 38.9 1.669 0.556 35-60 1 47.5 675 214 39.4 2.225 1.222 60-100 1 80 i 678 213 39.7 3.447 3.00 - 106 - TABLE 16 Load P = 210 lb. z z z(E) m m At 6s z 2 10 om z,om (hrs) (hrs) (lb/in (lb/in ) 336 253 1.000 0.046 0-0.5 0.25 365 243 22.9 1.046 0.016 0.5-2 1.25 418 226 34.9 1.062 0.033 2-5 3.5 488 204 42.0 1.095 0.056 5-10 7.5 55o 184 46.2 1.151 0.090 10-20 15 595 170 49.4 1.241 0.181 20-35 27.5 613 164 51.5 1.422 0.298 35-60 47.5 623 161 52.9 1.720 0.586 60-100 j 8o 626 160 53.3 2.306 1.248 100-150 1 125 628 160 53.9 3.654 2.447
TABLE 17 Load P = 180 lb.
z (El) m 4 zm 6 t j a's 4 zt z, r 10 om om (hrs) (hrs) in2)'(lb/in/in2) 0 0 288 1 217 1.00o 0.034 0-0.5 ! 0.25 317 208 25.9 1.034 0.012 0.5-2 1 1.25 358 195 40.8 1.046 0.024 2-5 j 3.5 418 176 51.1 1.070 0.038 5-10 7.5 478 15Y 57.2 I 1.108 0.059 10-20 15 530 141 61.8 1.167 0.115 20-35 27.5 558 132 65.5 j 1.282 0.180 35-6o 47.5 572 127 67.7 1.462 0.330 60-100 80 581 124 68.8 1.792 0.637 100-150 125 586 123 70.5 2.429 1.052 150-200 :175 588 122 71.0 3.481 1.498 - 107 - TABLE 18 Load P = 150 lb.
E(EI) zin La At a' 1 5- m rs zi zl 10 (hrs): (hrs) (1b/in2)1(1b/in2) om om 0 0 240 180 1.000 0.032 0-0.5 i 0.25 261 173 30.2 1.032 0.009 0.5-2 1 1.25 296 162 49.6 1.041 0.016 2-5 3.5 347 146 63.83 1.057 0.025 5-10 1 7.5 402 129 74.33 1.082 0.037 10-20 15 453 113 84.4 1.119 0.067 20-35 27.5 489 101 89.75 1.186 0.101 35-60 I 47.5 515 93 96.17 1.287 0.170 60-100 1 80 523 91 98.32 1.457 0.301 100-150 125 529 89 100.3 1.750 0.445 150-200 i 175 532 88 101.5 2.203 0.551 200-300 1250 535 87 102.6 2.754 1.363 300-400 !350 538 86 103.6 4.117 2.02 -1o8
APPENDIX III
DETAILS OF APPARATUS AND SPECIMENS
Figs. 7 - 20. log
FIG. 7. SMALL CREEP BUCKLING FURNACE ASSEMBLY. 110 14 DiA
SIM() Amy° LIDS
SILICA SHEATHS
/ASuRIING P ROSE S
SIN DA KO END I> ECES'
•
SIN DAN`(() BASE.
hetEASUR1144 SYSTEM WEIGHTS REPERENCE Fap.t../Ewo RA.
A P CottcriC 1'4.. Fu.. L Si iv_
FlG S. SECT-100 TI-liZolIGH •CRE.E.P locKLIN4 F R.NACE- AssemeLy SPIDER
SPECIMEN
THERMOCOUPLES.
CONNECTING ROD
SUPPORT STOOL
SPIDER
FIG.9. LOADING SYSTE.M WITH SPECIMEN IN POSITION 112 MEASURING SYSTEM RePeRSNci.
FRAMEWORKS
.11
FURNACE
MEASURING
""• ..••• •••*".
/
)••
0
;-( ix
SC,ALEt- FULL st-z_.
FRAME — I1/4.x /4x 4 ANGLE . FIG.10. LINE DIAGRAM OP FUR-NALL SUPPOWT FRAMEW012.K. S 4"
54 0.
V"
ec2 ///////, Y • 6.13.A. THREAD. S I ND AN sf ENT).- PIECE. r/
SECT' ot,1 Ohl ‘A.-Al
— SJRPACES MA C141 NIES:. FLAT FO f2 SPIRIT LEVELS
• 34 DIA
VS" DI A
WIDE SLOTS •
346 01q GRUB scizaw
SCALE FULL SCLQ..
MATERIAL !-- HEAT RESISTANT STAiNALES.S. STEEL.
FIG. 11.DEFLECT‘Oc-,1 MEASURING PROBE_ (10 PER FURNACE) > NEATINC ELEMENTS-19 &M6. > 1 laktsv-irkAy' SERIEs ‘C.1 WIRE H.H.I. t-II=2671., H.2=267!1 1-137.2733-
7 SLIDING, RE SASTANc ES •
ti R.a.= to.51-4 6.5 AhriP s
H.2. 7
- • TC
6 • J
0-2.0 AMP
SPE C itAEN
DE PLE CT 10 N.2.1EASOIKCI CIRCUIT,
•-•"" • 0- 13b V 15.A. E • L. N • ili- KVA, TRANSFORMe(Z.. a FornP 230 V. A.c. SUPPLY . 2 L 8E kg.
- 115 -
TABLE 19 Details of Furnace Windings.
Insulation: 2 layers of Asbestos cloth_ 2 layers of Asbestos paper. 19 S.W.G. Brightray series ICI wire wound as in table. 1 layer of Alumina cement. 2 layers of Asbestos paper. 1 layer of Asbestos yarn WI dia. The space between the furnace tube and the casing is filled with "Vermiculite". Winding Distribution: Perimeter of tube = 20". 19 S.W.G. Brightray series 'C' wire (0.415 Ohms/ft.)
Phase I (resistance 26.7 ohms)
I Distance from furnace - 3 3 - 6 6 - 7 ' bottom in ins. 0 i 7-736 Windings per inch 6 i 5
Phase II (resistance 26.7 ohms)
Distance from furnace 10 - 14 I 14 - 19 19 - 21 bottom in ins. - 10 Windings per inch 3 2 3 i 4
Phase III (resistance 27.3 ohms)
Distance from furnace 21 - 23 23 - 27h i 2734 - 29 bottom in ins. Windings per inch 4 i 5 6
LIG.
A
3 SMbel) Al2ER REPQ.Q50-A-S SENscrIVETN OF kET atZ
TENIPZWyruite. CoNTRDE.LER of S coNTa_oi- 3c2°C. 1°c.
6
U
i 4 - I6 r - 01 1 - 1 - 1 PoS
Ls1
297 • 3D7 °G TEMPI RAT 0 CaE °C-
TEMPERATURE_ DISTRIBUTION AL-otkt4 LE1441-14 OF DUMMY &PEC-)WIEFA
13. 117.
LEVER ARM 4 :1 o f
FURNACE INSUI-ATIC44
TELE.Sc.OPE. TEL =__
FAIN
7._z/
MDTbR. LOAD
EL, 14. 1)1A.C4RAMMATiC SKETCH Or WADE FURNACE. ASSEMBLY; 118
FIG. 15. WADE FURNACE WITH DOOR OPEN
SHOWING SPECIMEN IN POSITION PG 16 DENISON 3/4-TON TENSILE CREEP
TESTING MACHINES. SPECIMEN
EXTENSOMETER
SPECIMEN
EXTENSONIETER
FIG.I7a). 'DENISON' EXTEN5OMETER. FIG. 17b). `C.NS1 EXTEN5OMETER. 2.- HoLES . DRILL 5/;3. DtA x 944: DEER TAP a.aA. t/d.. DEEP. aoRe.. /44..1 DIA Y S/sx DEEP. C.44.1AF.
MAGNIOX A. CAN 2 MO- INA ALUMIhftUM ROD 1WDn \\\\\ \\\ ONI
IS N7- a3c:
It 5 NV3V\ 'I METHOD OF MANUFAc.1URE. I. PREPAQE R.vt) mir TUBE To DETAIL. 2. SPRAY Rot) WITH ALUMINA 0.004r THICK .
3. ASSEMBLE izOb a- TUBE.. Fir Et.* PLI1S WEL.Z. 4-. EVALUATE 4,- SEAL, END PL-04s• i. Pre.eSSt1gt2E. Tz=, 12,000 RS:1 . PtiT alo MAC Ui E. To kEt.4 0. yE_ END TN_ v4s Erc
//// ,////// ////// /////// rn \\\ 5t. 77; \\
////////// 0 c op o 3 n O 0Y` ".
(7 114
I'. 1" DRILL /.53.
=
11 14A1)• 0.1755 -.0015" O O
9 11 945?0,. •\ O
B5F
FIG. 20. TENSILE CREEP SPECIMEN. -124-
APPENDIX IV
EXPERIMENTAL RESULTS AND CALCULATED CURVES
Figs 21 - 76 3.86.5 IASI 218.5 il
V
if Z too < 100 CC 1^ in
co
10
1.0
X X
1 A
I
a. 5 2 5 1 5 2 5 2 TIME lars>
ft EY+PF—fltMeNTAL POV-ITS cD C-ORteCrED F-0,Z zn1 0 EARGits
FIC.1 . I. TENSILE C.R.E.EP COR.VCS Fop. MAGNOK A.11. AT C USING, DENISOPA 1W•ENSOW1ETEES 10 2040 i".s.i. 19S.0 1).51 196 0 p.
143
100 . 0.5 7100 3- 100 § ct) oa. x 10
0I .e ....0...... 5...... 00* 10 to PC X —.-/ • .05 X K
0
.01 2. 5 a 5 2 s 1 s 1 TIME (1..."S x E.XPERIME,MTAL POINTS COR.R.E.C-Ttb POR. zeito ERictoR.
Ftc% 11 TENsu-.E. CREEP CURVES FOR MA4140X A.11. AT Zol.°C USING
DENtSONII EXTG.NISOME.TER.S. L.0
lgo.5 1›.0.. 1795 j).5.L. 154-0 p.s.i. 1 0 .
4 .
.ge 100 V X (00 100 0 .. 0.2. x
0 .9 10 . $ 1 K 3 X X 10 ...... e...... "4.#<."'
OS if 4/
01
)1 . .- . r . .. . — T istiE (krsS x ExPE.P.ME4TAL. 114,11-11-1. 0 c.oRa.F-c-rto PoR. zeko siutek .
FIT . 23. TENsu_s_ CREEP CORmEs E MACINOK A.12 AT "Icar C uSiNtc.) DENISON 1 ..1(,TENSONIETE ZS . 1.0 •
119.0 1:ks.L. 137.1 tms.._ .0 11701P.si. 9 1155 Psi- ;'‹ * X
r X X r 0.5 K X— a X .., 2 x x x 0 1$30 X X 0 too too too o . 'et 0.1 X x A . a o
. e 0.4 * to to x Ix x x 10
.05 K X K
02
01 2 S 2 - 5 2 5 2 5 2 TttAL tkg'S X EyPERtMENTAL POINTS fl toRLECTED Foit. ZERO S42R.OZ..
TENSILE CREEP CUR /ES FOR. MAGMOK A.M. AT 102.° C. USINCi OSNISONt EATENSOME.TER.S• 3.0
II 10
OS
.i.. . 111.1004 ac -.--e
Fro . i111 0I
. . •0
TIME (Lat) X EXIPERINIENVAL. NANTS. 0 Co0S.T.C.-TET:. PoQ Z6120 E-12R.P..
F1 TENSILE CiZE EP, CAMNES FQR MACNITIC A . 1'1 AT 302'C. DEN (S0Ni C-XT EN S0141E.T
a • 67E, Asti 594. tts.L Slo la ,.. (Do I 1.0 too ,
0.5 Z g . loo tn . 0 . ----.0 0 . x10
1 x o Od . 174 - x
5 x 0 ...... e...... 0.00...... ,...0"" 1 x
1 af 1 1- x x X X
II r X
)5. 1 2 5 i - 2 r ,. 5 i 1 _ I 1 me. • • Ey.PERIMENTAL. PooNaTis. • CoReer-T6D Foe. xuto Ft c..26 TENsILE- CREEP CURVES Fog. MAC04014 A.O. AT sox. c F4S. EXTemSONIETE2.S • S"?. 3.331,4045t43-LX-a '‘DNIsc1 "D.,-Got AONI3VI.N1 210.4 S3hzn3 d.-a-a•eD '3-11SN'Al ‘',1d
—4c' cesz a41 a3.L,37t21O3 0 - to _ -1 _, _ _ _ - --... - */ - S S S r s t r zoo-
sac) Y
io I t X o I x X I to
x
al SO 01
)(
1•I
_ . ti Tis• /01 V% 00I
Z S'
is4 10 0 • sd bbt 1•
19S•5 s • i 56 .3 Os
p 100 . 0-1
0
1.1 x
10
•0
.x . . . • _
K
I .
• . .00 - . S 1 Time. (WI • x CAPtitlwiENTAI- 1, 0 NTS. 0 Cold% trzr Et) Foa. 7..6•41.0 fittitok.
F14 .2?. TENSILE CZE EP CUVESP FOP. 1•441.4NO% kia.. AT 302."C. LA01'44 e.v.treNsommT6.k.s
sal.L.W40SNa.i.3.3 osiplact 9n11SQ e) et 01 lit walp0poly .3•40,1 •- o3 s 3/0 0'7 43e2i2 "3-115N 5.1 '6'C'
"ttraVa obarC "ao3 Ciz-e-yOrece 0
S.LNioti 'vase tiara scri.z -av41.3. 5 C I S t 1 .S "C 5 X i 5 t 10. . .
lc X
K
A •
a) X X X NC
X A x V 1.0 I
>c N ' 001 0 X X X e, , X 01 CO X 14 x 0 04 1p X 0 x x OM K X i X 001 % X S X K 59 % X X X x , "C L 54. LSOl a 01 11.4 ostt .•,,i 4 05t1
Ot 5'tlasZvi9st-rsa_vs3 rotis)imaca 'PT•4150 Ov %)) --40 53 hv cr) craw-) • —D. o'rac.. Itvost4two-tV "rand .A-T-Pqn 0tr5V4V40-3
51.H104 141N3V41133 (PAS It V41 I I 1 5 t I 10
ra
_ 5(
1 X . 01 011 . 1
001 0 1 k X re L 001 x Tsi a 9S
1.54 gst. * -Y1.4. L+01 (
.... 1 2.0
S 48 1,.s.t.. 0.6 735 psi. 2 736 ps.i. g . . x x x x .. rse• x. cc 03 x )( . x loo c X x too x• ' too 01 ,, x to to x •O Y * x lipx x
o x x _
.01 1 g 1 2 g / 2. f . 5 2. 5 i TIME (I t) 6.11.PGRIreiro4 Via- 1701Kr .
r-tc4.31 • TF-NSIL.E. custvzs , commv_itcsAutzt P1QL ALOMINI1N1 AT 301°C, UsIMC.. kttRN‘SO EY-TENS orim-r las 2 .0
D-
O S X A A 1 621 1,. CZ% Z X )4, +96 t4 le
100 . r x X 4,.. 1 , 1 ex g k )( 11 X 0 k C; 0 X 100 X tO 100 )S )4 oX)( X 0 K K 0 0 x 74 10 10 0 14)6 X .04....- .
611 c i A 5 ! a_ ,. r a s a (14.5) af-PERimat4-rAL PoINTs 0 eoie.t.crs.) 2.c Salapk,
Ftc., .32. TENSILE CREED CARVES FOR. E1.C.—M . COtAME.RCAALLH Pugs, A%-umtmlut.4 wekr uSikel % 1> Ei4iSor4' e.,gro.NisomF.:reas . Ftsi... L•ao Mi-o - I
.o . 1521 to.s L ioo
OS . Z X . , VI • r". .
10 1506 -Ixs.i. I:4 4 ) I loo
X ID
x
X x
01 c 1 S n .e ( krs . . EY-PECitme.t4TAL. Polt-A-rs
FICA n. TERSH.,E. GREED RYES Etc..-bn CAommitgc‘Ai.-1.4 PORSE Al...0miNtqw1 AT 301.• C. 1)51144 EVT-ENSomezrak$ . 5123.1.731740sr*3..1- 3 % S. t4--0 1 "T•lisn "-Dtt%0%, VICNIN11,4011 '2irrOci A-T-P9TD•e3wV40) V4-,13 0.3 53N210") <131/9V 3-11SNAL - 714
• S,1WIOd n•tf.ISSIAIMgctla X -2v41,L c\ 1) •••• s Tr I V I at v ar v IQ
I Y. VO X X )4 x CI I -
001 01 x x x )( x 1 t•1
s4.'.04.1 001 x °Q
Z 5 -1:s4 91E1
c
a •t 0
x x % X
A X x
1 K I k
x 1.$
07
1 (.,
0.5 r% -- 10.0 (P-s)e %to-c (H4-) tc° IT1 L INITIAL ELASTIC DEFLE_CTiot4 RATIO 0t4 LADING FOR. bt1Ac.,K10X A.11. AT 302° Cr
FIG.35. 0
)9 x x . E f x . lit4° x x )4 • x x x x x x
K
140
Ell mill
14 0 S•0 CPO IS•0 2.01) (12.-5)°.1. 10-s ( 161 Az) 1r2 1
ELAsnc DEFt_Ecx Low, kATto 014 LOADINC% FOR COMMEVAALLY POIZE. ALUMINIUM AT 302.°C.
F14. 36 otos id to/ xlareavv4 1304 S'S CI, 1,INWTA,Cria 2.S "Dia -wawa crarsz Tod •traLORra07 5.33.11aci --rdi.praiweacrri . I 1 %a.
to. x
7 SO-
1 0 1
...... 'C-0 1 •NriCI) In...... • 7 S•0 i-i 4-. . I 01 01 • 01
i-5.4 c..st 01 iv4 Sgt -rs-4 '-c-c I O•t PM ps-i. i-79-5 P.s.1- 0
D to g . \ E N .. $ o...0 10 110 O
i ..-_, Z
, 1
DS
x
DI
Ott i S / R_ _ r. • EKPERINIENTAt. POINTS O c.Ac2.lits6re.0 rots zeR.c. EeRDR. Fl G. 36. CREEP CURVES FOR MAGNOK A.m. AT 3O2°C. aot iv W ›komvIN ziod saran, r)141-11-3oill ci-33210 "6E"'314 --Ilowera 01E32 1504 4:231,1*3111107 0 si.y.nod -WANavireacrfta- , 31nta. L 1 S r 1 S' r 9 "C 1 Sr -c
. .
Y
1 i''
..---, 01 ellu co f; 1.1 .....„ts, g
X
-r c'ele.r4.1 13-4. is! . - - . 149 'W.A. CC X X 1 00 100 100
I
0
\. E 0 5- 144 d 04, 0
I 04 1 . 0 L '...1.0' 10
•10 10
i 0• L
x
6 3.
‘ r
a.
• i1 C 2. 5 2 , $ . I 2 5 2 TIME. (Virs x EakpERtmENTAL Penr4T 0 C.oStit.6.CCILD FOR ZaSSo
FIG .40. CREEP I&UCKLUsIC; CURVES FOR MACINOX. A.12 AT 109.° c. V oTAZE. .1.‘e 'try -XoT4704 1:10.3 S3/1(1•3 ''aN11)1,-)C1i3 cr332)0 oirea 0-1437. 110A Cra.1.-$3**1302 0 sa.sk0e1 -1YAN31P•111t34045 , % 1 S t S‘ t S t S -z 1 k
A X t<
'). 0
Si 01 , x ). X 01 1,
x >. 01
(1 t - -(
. x j i , c ) t o l
001 L
-N.,.5 ' .. k X N 001 3 X . . , , c x 0 001 Y 0 ...• 0 r k YY 0 '1' 5" o6 52 p.si. 14
to
cC ...,IA 0-5 4 J a-. o --sr I00 .--6 o :.'' 0 0.1 •:-.'".
0-t X
_.... -05 10
X
0 L
3. 5 1 5 .. S 2. 5 l 2. TIME. . X EXPERIMENTAL P01t4TS O CORSLECTE.0 FOIL z ezo e
FIG.42. CP.EEl BkICKLIN Ca CURYES FOR. NIAC040X A.a AT 301° C .0 1
TS7 1, o S.- 611 17-S.L 5-,6 p-si £ 0 .4,.•J ia /C f N 0.00000...... i...... 00.....x.eve1 ...... )...... 00.0"...... „,.....r (107 .:::.0) x .
ew x I X a ..•• X /
X X X
X X
II X X X
DS
)).
DI D. S I. S i, _S 1 S a_ TIME • X EV.PgRAMENTiett- PoINTS O c.-oe.a.Eci-ELD Poit. -seito Cape
FIG.43. CREEP INUCKUNC CURVES FOR Elc-Ni commEaciALLy PURE ALUMINUM AT 302:c.. La 521 p.si.. WI t..s..i_ sl D Q g •••..N.1N. £ 5 a, x io )( x ‘` x •*.0 ...---„, x rac" i x x lc x x I I.
4 x k
X
•I
DI
a l 1 c I I S 2 _s t I s t 2 TIME x Ev.PEAUMESTAL PoirCrS aoctizZeareab Poe. 2:40 BAWL .
F16. . 44 CREEP liSLICN.1.44C: CURVES POD. Etc-t4 COmmeactiku-Y POEM ALUMINIUM AT 303.°C_, rnsi 1,40—N -we o ci vwstonsc) -e04 sa 74r7vos aa-av vir•91 J
orsz -pod sr431.1731113.1) - sitnod -rwsakparadwa. - - aim r - u r. ..S Z S. ' -C 1 S Z S t 10
to
X
x St
X x 0 x fro ...c I x . . x . ..—... 'EC x _ i • \-ti K - x i m1:1 al x — ....0 K k co ' N 51 )( .. \ ks31 X N ._. 3' CH wto "1r4 191, 1 . 15-40s4 01 0 400 SI 445 IP.S i.
4-11 t.si..
1 0 e E - x 4 9ID,, x too x x 2' 4-- IA x .. erg 0 _ A X
6%. e1 d I A x t _ ID X a .7.• .•••jr....A...° ...... lire....ir...... °.41.****°".": F- x x X x A X X
le I-1 x X x A li -e
X
0),
X . e I 2 5" 1 .. _ if __ I 2 s I r 2. I- 2—— t Evezimet.srpd... poiKTS 0 c..oaameztvb Foe 2.647.0 coupe.
PIG .4-6. CREEP aucgum.c,, Wastes Put. vc-m coMmeaciALLY PORE titt-umlw.tem AT 3o1.0 C.- . 041. V,4 OINM(1-1V --erc14 X-1-1,91743 %Avon 1,4-013 tos saw 11,1,0% d 3311-3 918
•Iscrents om 'Tod cras..ienracso o *s-ltozict "Itcaw3wrd1•1 • "3 .5 t ..s' , , -t 5 t / 5 t l0
tO
'-5C X x
I x x
X - X X X e-N V X x T x oi go.co x a x ter"e...... e>e...) ...... e:r'#*...... 'e'a°....'Ig.**....-..#.4'.°.'..'''r...... *..-*jzv''e- V 3 x am x 1 cum x 01 A' -.1.5.4 sis '714 6Lc 4Y1 2 0 351 t.sk. • 130 t:0S x x xx x x 119 t•S..i. • O t000
. 0 moo -.....N too 5 4 too Y: 01 4 x too x to
24 tv, x 10 x x x
.1
oil 2. S 1 2. 5 2,. 5 1 i IS g. TIME. . x SxPtinwismmi. POINTS coassaite.b Poit 2sem StRa01Z
FIG. 49. GRE_E P Bt1CICLING C..uk.vE5 FOR Etc.- INA COMmERC tAt..t...X PURE At..UmNit) m AT Iol° . ) .
V ) 115 1,..s.L. 115 tm.L.
....LI I"" loot) 0 r4 x X it. loo0 r
..-Thal, .. at wo I 1 zz. too
10 04 4 10 0 x
)5
.1Ca
01 / 1_ 5 2 5 2 _.... _ /. ...5 1 2 5 2 VI-PeZIMENTA10. POwirs. 0 C.01).11SATIAD Pait. 2.0.1L0 M.49. CREEP 6UCK1-1NG CURVES roe. EIC-K1 ComMERGiALLY PURE ALUNItt41014 . AN VitoNivony Sand 3.-riV1763IN1NOD V4-12 110A 53MT) (DNI-17110ca *Q9'491.4
Ozem mad cr5.1-782racr) S.» ad "TV.1.143Wle8d1.3 x •(54-1) 3vi11 C S .. . t t 3 t 1 5 t 1 S - I li
'I
J x x (A x x , x x x x ool -at „ x 3 't 01 7', x .. 0 x fii.° 0 )6 X , 0 0001 m...p • 001 . ' X N -r94 9Lt x c. di It , 1
lb -114 oot 155
SHAPE. AFTs$ 50 keS. CVaci wAVE)
SHAPE AFreZ, too kri. L Y2. SINE Wow E,.
SHAPE AFTEP_ fgo tsouRaZ SV.Ztas).
0
Lorkt> 20 It..
3
0
LL 0 a 6 0 -
X
IS
1 1% 03 0 t OEPLE.crtoN (11.4s.
DER-EC-TED SVEA46.. OF MA41401c. A.12 . SPEt1tivE.t4 1k10*
FICI I56
Si4APE. AFTER I •tts. ge. WAVE
swkpe. AF-rait 4v LArt. C 6.5t4E WAvs.\
SHAPE kFTEE S.7 1.4- LFDOQIED. SCeiss.
Loikt, iss
IS
6 2
a. 0
g 9 5 0 ci LL
I
511
15
l4 i 0 1 iks) O)-
DEFIAcrED SHAPE. comhisz2opi.1-•4 poka ht.,ewto.kkOm gPE-CAMEN. 1•0' lot
PIC.. 52. a "
So Z wi r: 1.5 Vim .i." V S ■
a _A AWNIKIthil Rob AuoRE ■ X 1, %. .1 ii Tvit(v( SR% VI • 1. 4 IP 1% t DO • . 6 V • • 4 I" • 4 4....iz III • in. 0 X 2 O 0 O V So
100 v X o A x A x A A
So
-. 0 2 S 10° 2 S m'" 2. to; TtR1E. tics) •
COKIPAWS0t4 BerVEGIA el.PRI1t1F-14TAL AND THEZZ.MicAL CREEP a0C.KLINC, TIMES Fok COmPoSATE. ca.A.IstiSiS
Ftq. 53.
25b Z on -.". 7. So Zi,,,,,
.1." 111 A mm41000141 kot, ALot4E. 8 X ii n + t: Tb4tC..K SHEA-v-14 O .L., .". ,, , .1.. Q" f t 200 ter e 1.P V el 4 4 VI S II is. I tb • • 4. •LP 4 il X o 1 4
I
A. x o 0 V
x 9 A A o x A A. 1( 0 100 A . . 4 x & e. .. ,
A 4 tk, h. 4 50
0 1 s le 2 5. Iv 1 s 114- 2 5 1.3. 2 TIME .
CompAzisot4 te-TWEER MrsPeitIMENITAL. AND ME012.E.TicAl. Cte.e, BMW-ANC, TIMES Voi2. COMP05.1TE. C-01-1314Thlt•
Fig. 54.
1" lb
2Sb IP, -.-4 .5•0 Vona v II V:
A At.thaltAtUgt P.06 14PME .2 0 X e. h 4- Ltoit THM K. 54 EAT i4 16 a . l i 2oo u .: 4 4,, "I. * 0 4 • U 2) II Iii
0
A X
K V
A K o th V
A. o loo x A. O A A )( )( A X A A A A A
So
0 t;1 a S 10° '3. S Id 1 S a/ 2. 5 10' 2 TIME (LrS •
COMPARISON fIF-TWEeN EICPERAMENTAL AND THE.DIZETICAL CREEP tUCK.I..1 NC% TIMES F0. ceetApoSITE- COLt3tAKS
FICI. 55.
LOAD (lb).
w 35, _ w 6, V40 4 Z2Y I 0$ N
Po U' .1› T1--, x
a
cos— O P.
x
0
xlJ / _. • 4
X fi,. p 14, x I> o si
:x 0 fli • 11, ol
4 ri, P. x 0
• 4 0 x P.
g .. 4 .1 g 0 2 NI _N 5 0 1.I I • • • : 7Ja $ V el 4 4. $ 4 ...„ A
091 lel
V. cuss. WAVE.
( 2. SIN 11'4. — 0•151)1 Si.14
0 741r
Lob 135 lb.
/3' •••••./ 11.1111111111L 6
I-9 1 0 at u.
4 us
LS Airpor or
A SHAVT. 44FTEZ 10 Lon . 0 L% % 3t 4 .S. L. . 11. .
1 oi Oa. 03 o •
0,Eri-BATE.b. SNAPE 0F ctn./Pi:KATE_ SPer-tsCr..14 wrrs b1 14Lcg. SOCIATt4.
S7 16a.
E "a 0•DoLW st:pAlTrid
0
LOAD 105 (b.
4
.
lc
x Swtrs. APr 15 k 0 4, 4 1/c 4, A 44 s4 448 4
I I IS 04 DWLst-Tiom (14s). 6.1
DEFLEc-mt, 5141.,? E OF compost-re. SPEr4wag WITH -If: Tome_ SKEAT4
F1 4. SS
16'1
Y2 SIRE WAVE
FOORtE.P. Si.44kac. c = 0-)-29 — 0.0065 Caw%
6
LOPO )90 Li>.
ts- 0
X SuAPE. Arn-P, 75 Lo-c 0 ti t% 3001, la i. ** ago "
I I I 18 0. DER.ECTIOs-1 1.4.c) oa
pe.PLEcrel:. OF COMP SPS.CIMEM Wrr14 MACK 5HENTH •
Fick. 59 LOAD 00 • —40 0
I>
•
. 1
rs› . g Ni. O •
P-
0 -3 3 x a V G
0
x 4
x O . 43 0 x tk
c .: :-- •-0 :4 A Tr) , 1 , GV. •-k vt -n r .1t rT2
, . r 0NI,1 N 1 . ow il 11 I) i. C?, W. e ,-01 0 0 0 p
1P91
FIG. gl . SPECIMEN N°. IS 200
FIG . 616). SPECIMEN N°. 121 x2.00
GRAIN SIZE ACROSS Ya"DIA. MAGNOX A.12. SPECIMENS
I6G
FIG. G2 co. SPECIMEN N° 105 x 6 5
FIG. 626). SPECIMEN N° x 6.5
GRAN SIZE ACROSS 1/aR DIA. E1C-M. COMMERCIALLY PURE. ALUMINIUM SPECIMENS I G7.
F1G.G3 a). SPECIMEN N° x6.5
FIG G3. bZ. SPECIMEN N° 5.
GRAIN 5IZE. ACROSS I" DIA EIC-M COMMERCIALLY
PURE ALUMINIUM SPECIMENS
v 1GS • , • • %0 Do • • I Do 9
Soo ' \ IA. —.1- LIRE PREDI4TEt. PROWS GREED Buc43.114.4 DATA. \t. 4. \ TOO 0 \
6rto o
.s 0 D)o
II aL 4.1,
.0e. 0
, -
100 . 0 . x,
Lco- A . x x
Loo• A o X x
X x
0 X co , ) .. A itC I i
X X X
100 1. 2 C 10 1 S !DS 2. S icfr O. TIME tkrs, • 0 EXPT OWL-TS tistry4 F.4TENSOMEMR., X it k tDEMS.1:Xqi I S1441-.1Sti ELAIWRAC.1 DATA. tc..E.c...1 ZOP..C_ • C4toss- PI-arS, OF t.i AT 0.1% STRAIN FOR MAC .&O%. A.a. F14.64. 169
\ 1 )00 \ \ loo \ V \ goo \ \\ LINE PRE bte.TMD ;MOM Ot2.E.P-P Bitt.X.1-11.44 DATA 700 \ 0
\ 600 \3 • \
Soo \\ 0
\ \ \ 4-oo \ co
\ \ - -
300 \ o X\ \ \ 25o 4 \ X „ \ X \
abo X x \ 0 a \ X X \ \ \ K 'co o\
xx \ . \
X x K \ \
100 5 lo' a S iol• . a c \ In- .., Tiklg {..cs 0 Ei.p.rRestx-TS. ustrACI •S tumnomarez. 3( St bE341SON It A ENc•LISA- Li.Nc--ritic: DATA. 0 'C4...(_'DPTh 4%. CROSS- PLOTS OF kt AT STfcr.1t4 Fok. MAGHOx A.12.. AT 3C:113C. F14. GS. 2000 '..k...... , ••...... N., -...., 14 NE P REINCC ay FROMCREEP BUCKLING DATA -..... \` -...., -..... CO ISO 0 ...... 0 ..--.,, ...., A ••••• 1 •••... .4%. .43 ••••,. 0 C ....—/ \ X — I.t.,. Vt %as, tal - -.., Ed ...... HI-- 1,11 ...... XX '..%...... 1000 •,.., -...... -.., ••••.... qoo -.., Xx x \-...... go %...,3/4. 7. 1 • -..,, 0 , N.., K— -3 - I ' •••., 7 . %.„. •...., X A
. ii
SotI . X
4o 10 2 S to* 2 S Io' a 5 101 2 1 TIME 04-Sst) O ev.PT RESULTS US It,14 CA-S` E1tT Et-tS014ETER. )( u I t It D EtilSo 1.4
CROSS- PLOTS OF to AT £14-= 0-I% STRAIN E.IC-M COMMERCIALLY PURE ALUMINIUM TEUSILE CREEP DATA AT 3 cec. a FoR F1G.GG. 11
2 boo
PREDICTED LINE FRO.a CREEP BOGY...LING DATA.
...„, 0 . ioo 0 %., ...... , 1,:... r \ .....„ "... \ 0 ..: ) \` ....,
tn \ id •...... 1- ...... x x 100 ...... -..,, .... 100 .... -..., ...... -.... -.. x oe .... Soo -... • -... -...... , ...... _ x x Too • . -...... , . .., ..., ...... )1 x boo ...... k \\\ \-..., SOO •••••.\ K. .... rs ••••
&No tom S to a 5 l01 , \ to" a 5 Lb3 TIME ) • T. O EV.P. RESULTS LI 5114G EKVEN.$014tETE.
x .4 DEMI SOISi 1 li o, C PLOTS OF AT E.+ 0.1 STRAIN FOR M COMMERCIALLY PURE ALUM(NtOM TENSILE. CREEP DATA AT 02t' FIG. 67. 300 . . X .
1S0
K
10
x x
K ti ISO •
J1 w CL x
x loo
90 ..<
x 1 . • 7
x .
5 . .6.... _ .1. oti 4-
x 5ei . - to 5 2. 10 TIME 1.14-s) . cRoss—PLOTS OF Et AT 0-2. )10Q 3.-PA 1-0 FOR MAGNOx A.12. CREEP BOCK1-1N,C DATA AT 30a%.. 1 PE Jt or Ftc% . 655. 30
. ) ] 4
2oo5
• •
to •-• 4.\\% . .
9 5
5
5
\ 6 5
(1 2
x ) a 5 tat 2 s it-?• a , .s ...1 1 —4. TIME Los . CROSS— PLOTS OF E, AT (I—Pe)1.0o zw% — 1.0 FOR NIACNOY. A.12. CREEP Z"ion : BUCKLING DATA AT 302°C. FIG.69. 200
loo
• 100 K -
TOO
. x 600
X x 500
J1 X >• X ,
X x
X
x X , X 300
X
• K.
300 + S. lo' 2 5 10 1 5 10 TIME.. (1"rs)
c Ross -PLtrr5 OF t. AT (I-- =1.O FOR. E I C-141 COMMERCIALLY PORE ALUMINIUM AT 302° G. I PE FIG. 70.
'IL .V13 C ( .a —r— •-D o•toz 174011i1l740-tV -a-dod ),-1-1V1-3 -3V1V40-D k14-13 210A 01 w oz 2s4-,01 "<3 -4 s-iona-Lsso'n -31N11
0o0(7 000) 001 01 S ....,..
. 13-51-
...... h.,,,......
x
X
X X
X .., X x x Tu X . 00S X
00 X
Ot
X OC
0( 6
0C 01 V
300
A zh x 0
E4.1> Resuirs FOR Z-2-" 1.5 250 Zorn
A X 0 = s.0 s.o
xs4 0
7
V
100
A
a 0
Sb A
u 1.S 2.0 s ZOO' 20,„ z Zona 0' 10° 5 2 S 2 TIME (krs S. 103 3. t.AA4 gox A.12. CREEP ES QC-K L.114(i DATA AT lo 2 •C WIT H CALCUL Are.c) CURVE%
Ft .7R. A
t•S 2.6 z'n=30 zn, 5.0 "ogr, zov, zoo. 700 4 E&P-1: RES‘31-TS FoR z..., _ 1.s Z OVVI.
X % I it le I. r. 2.0
• • 0 i* at = /.0 A x "-.;Th V • • ii = 5.0 600 Z.1)
of
Soo A. 4
A x
400 A
A 0
lop A
Zoo to S a S le • Io'a , S 16' a S It•I TIME 041) .
El t-141 COMMERCIALLY PURE. ALO 14111.•410 NI CR.E.E. Bt./C.41414(i DATA, AT 302°C WITH CA4..C..ULATE-t, CARVES. FIG. 73. CoNWDS.ri. SkE-c-IME-44 wirt-1-Ft; THICK S14E-A1-14 .-
/ , / LOAD 9516,
/ k /
/ , ./ ." .
/ . / .00 / .0*
..0 ... I. ... A" .- .. .
. ' i . _
. —.-. 50 0 TtmE. EY-Pa ia ireeNTAL F°114-r • 0 CAL.c.41- ATZ.D P0IKTS .
FIG.-74 TIPICAL CREEP li,t3C-KLII.4 coavE. FOk. COmPoSITF SPECAmR-N1 30 as fir o 07l 0-s I-0 kr 3 S .7 10 kr
35
45
6o
7S
(00 r
JAP/:0••"'"' ' Ias ISO
g.o0
loo
400 maal Gob ,ey - too ....0#:„.._-,-00,•••••_,.. -- loon
STRAIN t. s FIG .7S iSOCIARONOUS STRESS - STRAIN CURVES FOR IslIAcilvO)C A.12, AT 302.6 1%0 f=0 OS 1.0Inr Z 3 5 7 to Inr• hr
boo go0g0.°I°°°I0°S 1 11111011
, 1
Soo
, at
Soo
• FP.rP-
4 STRA try Sr 10. 14- lb • FIG. - 76 ISO CHRONOUS STRESS STRAW CUP.VES FOR E1C-M COMME2C-1,4LLY PURE ALUM(NI)NI AT 302° G. 342
1
THE CREEP BUCKLING OF COMPOSITE MEMBERS By J. M. Alexander* and G. A. Webstert
The simplified theory put forward in an earlier paper (t)t is extended and generalized to cover the case of the thermal creep buckling of prismatic two-member components such as reactor fuel elements, all phases of creep and elastic strains being considered. The tangent modulus approach is used and small deflections are assumed; it is shown that even with these assumptions a solution in closed form can be obtained only in the case of a single rod. Approximate methods of solution are then discussed, in which primary and secondary creep phases only are considered, but with certain simplifying assumptions. Throughout most of the analysis a modified form of Andrade's creep equation is used, namely e = at113 -1-13t1-7/3, where a, p, and y are functions of stress and temperature, c and t are creep strain and time. Typical curves are presented, based on data for an alnminiurn alloy, illustrating the contribution to buckling deflection likely to be expected from primary creep, for the single rod.
INTRODUCTION mid-height or central section. These assumptions are con- IN A RECENT PAPER (i) an approximate theory was put for- sidered reasonable for the small deflections permissible in ward for the thermal and irradiation creep buckling of a structural situations of the type under discussion. nuclear reactor fuel element. It was suggested in that paper that experimental work should be carried out on the thermal Notation creep buckling of both single and canned rods made from A Area of cross-section, also constant in Larson— common engineering materials for which there exists much Miller equation (2). basic creep data, unlike fuel element materials. In this way B Constant in Larson—Miller equation (3). comparison could be made with the approximate theory, a, b Functions of stress and temperature in approxi- and a firm basis established for predicting the buckling mate creep equations e = a+bt. behaviour of such components. C, n Constants in equation a = Preliminary experiments are being conducted on CiflociTi Constants in Graham and Walles analysis, i aluminium rods and magnesium alloy cans, and it has ranging from 1 to n. become evident from this work that it is not sufficiently E Young's modulus. accurate to neglect the primary phase of creep. As a result E' An equivalent Young's modulus, equal to 1/ac/aa. the theory proposed here has been developed, to provide E Creep rate modulus. a basis of comparison for the experimental work. In general, I Second moment of area. elastic strains have been neglected. L Length of rod and can, or sheath.in. The main assumptions made in this analysis are that [ ( xr ( ± plane sections remain plane during bending of the prismatic tii members, and that they are secured together in such a way P Applied load. that they are constrained to shorten and buckle together. Activation energy for creep. It is also assumed that the buckled shape of the combined Q R Gas constant. component is sinusoidal with maximum deflection at its T Temperature. The MS. of this paper was first received at the Institution on 30th Time. May 1960, and in its revised form, as accepted by the Council for to Load intensity. publication, on 30th September 1960. * Reader in Plasticity, Imperial College of Science and Technology. y Distance along rod from one end. Member of the Institution. Deflection at any point. f Research student at Imperial College of Science and Technology. Function of stress and temperature in Andrade English Electric Company Ltd. * A numerical list of references is given in the Appendix. creep equation. JOURNAL MECHANICAL ENGINEERING SCIENCE Vol 2 No 4 1960
THE CREEP BUCKLING OF COMPOSITE MEMBERS 343 Strain, strain rate. that the ratios K./fl= take values from the 'standard' series 1, Mean strain, mean strain rate. 1163. 8The various constants are determined from Stress, mean stress PIA, for example). systematic analysis of cross-plots of the data, a procedure which it is not proposed to discuss here. The main reason for drawing attention to the Graham and Walles analysis, ( -bcra1 5, T (96 a, T ar 5, T • Suffixes referring to rod and sheath respectively. is to indicate that by its use it is possible to establish the Suffixes referring to 'central' and 'initial central' functions a, /3, and y of equation (1). respectively, for example Zc Central deflection, Zoe Initial central deflection at y = L/2 after CREEP BUCKLING elastic deflection, Z'oe Initial central deflection Secondary phase only (or bow) before elastic deflection due to appli- cation of load. Considering the problem of thermal creep buckling, if the Larson-Miller analysis is applied as a first approximation, THERMAL CREEP the relation between a and E can be shown to take the form, The Andrade creep law, modified to include tertiary creep, at constant temperature: is a = an • • • • (5) E = at1/3 1-Pt-Fyt3 . . . (1) This was the functional relation used in the earlier a, 13, and y are functions of both stress and temperature, analysis (x), for the can material, but •it should be noted and their determination has been the main study in this that only the secondary phase is, in fact, being considered. field in recent years. The formulation of theories of creep It was shown in that paper that an elastic type of analysis may be termed either 'physical' or 'phenomenological'. could be used for the creep problem, provided that the Typical of the physical approach is that used by Larson `creep rate modulus' E was defined as the slope of the and Miller (2). This analysis is based, as have been many against E curve at the mean stress. Thus for a creep law of others, on a 'rate-process' theory which leads to an the form given by equation (5) equation of the form, WRT E = aa) = -1 M . . (6) = Ae- • • . . (2) T where A is a constant and T is the absolute temperature. If two members such as a rod and can of different This theory effectively applies to the secondary phase of materials, both obeying creep laws of the form given by creep only, so that E = c/t, where t is the time taken to equation (5), buckle together, then they must also shorten reach a given creep strain e. together, so that
Thus Er ES (Since E = Eft (11)a = T(B-Fln t) . (3) if only the secondary phase is considered, equation (7) is equivalent to equating Er and is.) Also, the where t is the time to rupture or to a specified strain. sum of the loads equals the total load, so that B is a constant for a given creep strain E (B = log A — P = ar Ari-ErsAs . . . (8) log €) and (QIR)„ indicates that the activation energy Thus, from equations (7) and (5) (for rod and can), depends on the stress level. T(B-Fln t) is commonly designated the `Larson-Miller parameter' and checks creep fa \inr data over a wide range, leading to master creep curves in k), which a or log a is plotted against T(B-Fht t)*. In spite of that is, this apparently ordered behaviour it should be remembered a)nsmr that a small amount of scatter on a log-log plot may involve as = (7, Cs • • (9) a considerable error in terms of the real variable, and it is r therefore inadvisable to extrapolate to large values of time, Substituting equation (9) into equation (8) there results: outside the range of available data. arAr-1- (FL )nsinrCsAs —P = 0 . . (10) Graham and Walles (3) have undertaken a phenomeno- C r logical study of creep data, mainly for the Nimonic alloys, proposing a general relation between creep strain and the It is possible to determine 5r from equation (10) and hence other variables of the form the way in which the load is• shared between the members. i= n Now the members buckle together as well as shorten E = 2 cio-PirilT/-71±20K1 . . (4) together and, as discussed in the earlier paper (30 a side i= reaction must be induced between the members. If this is Following Andrade, the Ki take in fact only three values, assumed to be distributed sinusoidally, of intensity namely, 1, and 3, and it is found from analysis of the data TT * It is generally found that the value of B is somewhat arbitrary, and W = sin —y . . (11) can be chosen to give the best fit to the available data. L JOURNAL MECHANICAL ENGINEERING SCIENCE Vol 2 No 4 1960
344 J. M. ALEXANDER AND G. A. WEBSTER
then it is easily shown that the bending moments at the Thus for constant temperature #, x, and /s will also remain central section of the rod and can are given respectively by constant and equation (20) can be integrated to give • d ir 2 er,2 PL2 Erl — Z = — PrZc + (12) In Zo— = --(01/3i-xt-F0t3) . (21) A r dt L2 ) IT Z„ r2/ EsI cTtde.. /1.2 zc) = _pszc_zon.cL2 2 from which the time t to achieve any given bow can be (13) determined. It may be noted here that there exists a similar relation between In Zc go, and t for creep buckling, as is Summing these equations and integrating, gives given by the Andrade creep equation (equation (1)) between e and t in the tensile creep test. In fact c must be replaced by 1 2 ft dt = j*z` d c (14) PL2 (Oa PL2 (E4-1-(67). PL2 zoc Zc In Zc g oc , a by .77.7 -a;13, T, 13 by 77.27(Tir ) T, and y by Therefore, PL2( -)Y . Also the initial elastic deflection of the rod can 77-22(E/) Z, 71-21 ea 6,-, T t - In . . (15) dearly be allowed for by assuming z, to be the deflection PL2 Zcc after the initial elastic bowing. Thus if Z',„c was the initial where central deflection of the rod before application of the load P, apl).(EI )r+(EI)s . . . (16) Z, may be obtained from the equation 1 Doubtless equation (15) will hold for any number of mem- Zoc = Z'c,c/ (1 PL 22) bers, within the framework of the assumptions involved. ir2E/j . . ( For this case of a single rod, the above results can be derived immediately by noting that, in equation (19), the All creep phases, single members only quantity dt/E can be replaced by d(defda), so that the It is possible to obtain a simple solution in dosed form Z PL2 de including all phases of creep, for a single member, using final solution is given by In — = --). If the elastic the tangent-modulus approach. Differentiating equation (1) Z' , 77.21 ( do. with respect to time gives strain is induded in equation (1), so that it becomes E = 3at-2/34..fl+3yt2 (17) e = at1/3-1-Std-yt3-1-01E then the final result is PL2 1 This equation represents the slope of the basic creep curve — (441/3+x/1-'43+-)• In '„c ir2/ E of c against t, for a given stress and temperature. Thus for Z constant temperature the creep rate modulus may be It is easily shown that this is equivalent to the solution given defined by the equation by equations (21) and (22), provided PL2/7r2E/ is small, an assumption implicit in the whole analysis. 1 ae\ ±X±302 . (18) E — Ve;h5 T = 0-213 All phases of creep, two members tace•\ pl ay This problem is complicated by the requirement that both where and 0 = (To) members should shorten together, which necessitates that 7? X = 5 5, T3 1:5;16., 1 the division of load between the two members should alter Thus unlike the previous case Pis now a function of time during creep. This also requires that elastic deformation and must be integrated with respect to time in equation (14). of the members should be taken into account, although this To do this equation (14) must be rewritten in the form: contribution is likely to be small in the range of interest. ft dt 772 rzc dze The modified Ancd.ra=deuretquua3±ti:msrt+, ayrtllo 3win+ggfor elastic strains . (14a) and tertiary creep, are for rod and sheath respectively: Jo I(E.0 PL2 Jzoc zc Cf r which becomes, for a single member, - . (23) ('t dt fzc dZc (19) Jo Jzo, zc es = ctst1/3+Pst+Yst3+g Substituting for E from equation (18) gives Initially, let it be assumed that the two members simply compress or shorten together, there being no buckling, ta 0-2/3±x dZe ELI Jo +302) dt (20) under the imposed total load P and constant temperature T. fro, Zc = The equilibrium and displacement conditions are Now a, 13, and y are all functions of stress and tempera- . . (24) ture, but for a single member, the mean stress will remain P = are1 usAs •
constant during buckling, given by the equation Er = PIA. Er = Es . . (25)
JOURNAL MECHANICAL ENGINEERING SCIENCE Vol 2 No 4 1960
THE CREEP BUCKLING OF COMPOSITE MEMBERS 345
At constant temperature the a, f3,y terms in equations (23) are known, Sc = SEs from equation (25), and from equation functions of stress only, and in order that equations (24) (24), d (25) should both be satisfied at all times it is clear that A r pie values of ar and Us must alter during the deformation. Bas = —7f4 8ar . . (29)
/ IUs Equating (27) and (28), and making use of equation (29), / / the following expression for Sar is obtained: / r2/3(ar — as)+ — fis)+ 3 t2(Yr — ys)l5 t aar — . (30) V/ (96,.+1:1:08 )013+ (xrd-Ar- xs ) t As As 1 A ) 2Er — 2 ± frr+1108 )t 3+(fr+its Es [AD t i/34-13t+%1 3 t In this way the change of ar can be found for an assumed time increment St, all other quantities being known at any st given instant. Initially, at zero time, equations (23) reduce to the simple elastic equations, and the division of load between the members is given by equations (23), (24), and (25), leading to stresses: Fig. 1. Basic creep curves for both rod and can material at constant temperature for different stress levels = P 1[Ar teAs . (31) This can best be illustrated by reference to Fig. 1, which shows diagrammatically basic creep curves for both rod as = PI{As-Fis Ar . (32) and can material at constant temperature for different stress levels, those for the rod being indicated by full lines, those These values of the stresses define the initial creep curves for the can by broken lines. For the sake of simplicity, to be used, and during the subsequent creep compression elastic strains have been neglected in the diagram. If, at the variation of a can be followed by using equation (30), time t the strain is represented by the point 1, such that mean values of the various quantities being used for each ler = res, and the creep curves for the rod and can are such interval St*. that equation (24) is satisfied by the stresses icr, and las, then To estimate the buckling of the composite member, it after a small interval of time St, the strain will be represented seems reasonable to assume, as before, that the members by the point 2, such that 2er = 2es and ear and 20.5 satisfy will share the load in the same manner as if there were no equation (24). buckling, so that the analysis just described can be used to For the example shown, ar has increased and as has give or and irs at all times. For the members to buckle decreased and it is clear that any stress changes in the together also, some side reaction must be introduced members must always be of opposite sense. Clearly, 'a between them and the differential creep buckling relation numerical method of solution must be adopted, even for becomes, from equation (14), the simple compression problem, and a suitable procedure SZc PL2 St. is as follows: (33) For constant temperature, Zc 77.2 (E/),--F(E/). OE OE In this equation the term (E/),-1-(E/)s is given by SE = N.+ (26) at Oa The relevant equations for the two members are there- (N)5+ (E/)1 2+ fore Yrt-2/3+ Xr+3 Ort iOst-2"+ Xs+ 3Cbst2 . . . (34) (+ art-2/3+/,+3yrt2)8t which is a function of t, or, and ers, since the 9s, x, and & are (27) defined as the derivatives evaluated at the relevant mean (i.t1/3 Xrt Ott' + °Grr stresses. Thus the whole problem can in principle be solved Ses = ast-2/3+19,3-1-3yst2)8t numerically, mean values of the variables being used for 1 each interval, the relevant values of the mean stresses to + (skt1/3-1-xst+ Ost3+1 -), Sas (28) * As a check on the numerical procedure it should be verified that equation (25) is satisfied, by equating equations (23) at each of the in these equations the a,13, y, 0, x, b, are all known func- assumed increments initially, and less frequently as the solution tions of the current stresses, Er and Es are known, t is progresses. JOURNAL MECHANICAL ENGINEERING SCIENCE Vol 2 No 4 1960 346 J. M. ALEXANDER AND G. A. WEBSTER be used in evaluating the cd, x, and of equation (34) being found from the numerical integration of equation (30). The where initial elastic deflection of the composite member can be m= [1(LrC41)/(+.1!)11/23 XrX, X3Xr r Xs/ found from the equation If = = 0, that is, neglecting primary creep, Zor 1 . . (35) lion (15) is recovered, since L'+:1:2 = aso in this case. Z'o, 1 PL2 Xr Xs An alternative way of allowing for primary creep in an 772[(Enr±(El)si approximate way, still neglecting elastic strains, is to assume and provides the initial value of Zc in equation (33). There does not seem to be any simple way of incorporating the elastic changes of deflection during creep buckling of the composite member due to the stress changes, but they will in any case be small.
APPROXIMATE METHODS OF SOLUTION The main reason for the necessity for a numerical method of solution as just discussed, is that the mean stress in each of the members will vary during creep. If the materials of 40 is Os Ir is Or 4212 (Clearly the initial elastic buckling could also be included T 4._ by writing Xr Xr Xs Xs t1/3 1 Xr Xs \ Xr Xs/ tan-r. tin r i / 3 1/..4...1212m m ' 1 = !Oar ) 4.1.. F-1- 1. E r kearlar Er Xr Xs kx, xs/ . (43) = 2 fir +/:1\ in Zc 1 = pas\ 1 'T . (39) PL2 kx,. x..) zos Es .Ecrs1 as Es JOURNAL MECHANICAL ENGINEERING SCIENCE Vol 2 No 4 1960 THE CREEP BUCKLING OF COMPOSITE MEMBERS 347 DISCUSSION [provided the initial values of &r and as had been determined by satisfying equation (24) and the following equation: In the small deflection theory of creep buckling, primary creep will obviously play an important role; its neglect = as±ps . . (44) leading to serious errors in the early stages of buckling. In r Er this analysis methods of varying complexity have been put forward for allowing for this factor in the buckling of a [This is equivalent to equation (41)' but with the elastic prismatic two-component member such as a nuclear reactor strains included.] The additional complexity is unjustified for this approximate solution.) =1000 o=500 Ibl ns The subsequent creep buckling can then be determined by a similar analysis to that first described, the value of the , initial central deflection being that found after the 'instan- taneous' buckling. The creep rates Er = br and ES = b, are 2 1 then made equal so that the stresses az. and a, will alter i ii from their initial values to give br = b,, equation (24) still rit having to be satisfied*. Thus, depending on the functions br and b,, Zir and o, will be determined, and remain constant r, during the ensuing buckling. The buckling time will be given by equation (15), where (EI) is given by the equation r„--- = (45) ZEO r 0 5.0,c10' hOx I05 • x x • x (ab 1 01,8 f—hours \ bar/ er \aO•s/es a At 227°C. a=500 (blunt —1000 Iblun1 , ESTIMATED CREEP BUCKLING OF A i ./ SINGLE ROD OF ALUMINIUM ALLOY H.E. 30 l ,..., From data supplied by Inglis and Larke (4) it is found that l // the coefficients a and 13 of the modified Andrade creep / 2 - equation can be expressed in the following form, as pro- /l,/ posed by Graham and Wailes (3): / , / / / a = F722 .1011. a4131472 — T1-20/3 . . . . (46) S / ," 1 ," = 1-15.10-570741T-1-65120+145.10-57.u2I TA-77120 / .., / , ‘' . . . (47) , .., Insufficient data were available to enable evaluation of . _ . _. .._.... 0 2.5x II' • x • x • 101 the y term, or to allow any more terms to be included in the f—hours expressions for a and 13. b At 277°C. In the above expressions, a is the stress in tonfin2, T the temperature in °C and t is time in hours. Fig. 3. Estimated creep buckling behaviour of aluminium Differentiating equations (46) and (47) with respect to alloy H.E. 30 stress as Specimen 0.5 in. diameter, 18 in. long. = 2.29 .1011 .51131472—T1-2013 . . . . (48) lnzo = PL2—(01/3+xt). c 772/ x = 4.6 .10-57 . I 7712o (49) .01 T+6512°-1-2.9 .10-57 1n. P-77114(X1)* By substituting these values of ck and x in equation (21) (the term being neglected), graphs of In 4/Zoc against t fuel element. Clearly, in practice the method of analysis may be constructed. These are illustrated in Fig. 3, for a used will depend on the material data available and the rod of aluminium alloy H.E. 30, in. diameter 18 in. long, success with which the various methods proposed predict subjected to end loads of 98 lb and 196 lb at temperatures buckling times. Such a comparison demands carefully con- of 227°C and 277°C. (These end loads represent mean trolled experimental work which is at present being under- stresses of 500 and 1000 lb/in2 respectively.) The initial taken by the authors. elastic deflection has been neglected. Fig. 3 shows the substantial contribution to deflection likely to be expected from primary creep, and it is obvious L * These changes in the values of the stresses or and as will necessitate that even a very approximate theory of the type discussed is changes of primary creep terms ar and as, clearly of opposite sign. This effect will be small and has been neglected in this approximate at the end of this report would be preferable to one in analysis. which the primary phase is neglected altogether. Such a JOURNAL MECHANICAL ENGINEERING SCIENCE Vol 2 No 4 1960 348 J. M. ALEXANDER AND G. A. WEBSTER theory will always give slightly pessimistic predictions and APPENDIX therefore would be safe to use in design calculations. REFERENCES (I) ALEXANDER, J. M. 1959 5. mech. Engng Sci., vol. 1, No. 3, p. 211, 'Approximate Theory for the Thermal and ACKNOWLEDGEMENTS Irradiation Creep Buckling of a Uranium Fuel Rod and The authors would like to acknowledge helpful discussions its Magnesium Can'. with Dr. A. G. Young, of the English Electric Company (2) LARSON, F. R. and MILLER, J. 1952 Trans. Amer. Soc. mech. Engrs, vol. 74, No. 5, p. 765, 'A Time—temperature Ltd, in particular for his pointing out the existence of the Relationship for Rupture and Creep Stresses'. summation term in equation (14). (3) GRAHAM, A. and WAILES, K. F. A. Aeronaut. Res. Comm They also acknowledge help and advice from Mr. A. Current Papers, C.P. 379 and C.P. 380, Parts I and II, Graham of the National Gas Turbine Establishment, par- aegularides in Creep and Hot Fatigue Data' (H.M. ticularly for providing facilities for one of the authors to stay Stationery Office, London). (4) INGus, N. P. and LARKS, E. C. 1959 Proc. Instn mech. at the Establishment and learn the techniques involved in Engrs, vol. 172, p. 991, 'Strength at Elevated Tempera. analysing creep data by the Graham—Walles method. tures of Aluminium and Certain Aluminium Alloys'. JOURNAL MECHANICAL ENGINEERING SCIENCE Vol 2 No 4 1960