Creep Buckling and Rupture of Columns with Arbitrary Symmetric Cross Section

0020-7403(94)00102-2

CREEP BUCKLING AND RUPTURE OF COLUMNS WITH ARBITRARY SYMMETRIC CROSS SECTION

VICTOR BIRMAN t and MARK G. MAGID ~ ~University of Missouri-Rolla, Engineering Education Center, 8001 Natural Bridge Road, St. Louis, MO 63121, U.S.A. and ~MICO, St. Louis, MO 63141, U.S.A.

(Received 23 May 1994; and in revised form 3 November 1994)

Abstract--A practical method of analysis of simply supported columns subjected to a steady-state creep is developed. The method can be applied to the columns of an arbitrary symmetric cross section manufactured from materials that follow Norton's law. Two modes of failure are considered, i.e. creep buckling and the failure associated with allowable stresses. Numerical examples presented for actual materials illustrate the effects of various parameters on the creep buckling.

NOTATION

Aj cross-sectional area of the jth pair of layers a 0 non-dimensional amplitude of an initial deviation of the column from the perfect straight shape al non-dimensional amplitude of the total deflection from the perfect straight shape B constant in Norton's law h depth of the column ki, kj(n) coefficients L column length n power in Norton's law Per Euler's buckling load of a simply supported column Po axial force applied to the column (positive in compression) Pj fraction of the axial force applied to the jth couple of layers S safety factor (for creep critical time) t time tall allowable time of loading (for a column subjected to creep) tcr creep critical time creep critical time with the correction for a safety factor tf creep failure time (based on allowable stresses) W total transverse deflection WO initial deviation of the column from the straight shape X axial coordinate Z] distance from the neutral axis to the eentroid of the jth layer corresponding to undeformed shape (positive number) e strain a stress ac constant in Norton's law

INTRODUCTION

The problems of creep buckling and failure represent a challenge for designers of structures. In particular, columns designed to carry constant compressive loads are often subject to creep. The significant interest in this subject can be explained by its practical implications. The problems of creep buckling of columns have been considered since the beginning of the fifties. The classical analytical solution was presented by Hoff [1, 2] who idealized a column by two flanges and neglected the area of the web. The closed form solution was obtained by assumption of a steady creep with a particular form of the Norton-type strain rate-stress relationship. The creep buckling was associated with an unbounded transverse 831 832 V. Birman and M. G. Magid

deflection. Another approach to the creep buckling was based on the concept of a time-dependent tangent modulus similar to the theory of Shanley. An example of early studies utilizing this method is the paper of Carlson [3]. Note that the paper [1] presented a comprehensive list of early studies of creep buckling, including the work of Freudenthal [4], Libove [5] and Kempner and Patel [6]. The effect of a geometric nonlinearity on the creep buckling was considered by Zyczkowski [7] who obtained the conclusion that a finite critical time for viscoelastic columns does not exist as a result of large deformations. However, Huang [8] showed that if large deformations are accompanied by plastic effects, the latter result in a finite critical time. In his paper, Huang considered the creep buckling of two models, i.e. two rigid bars connected by a spring and an I-beam idealized as in the paper of Hoff [1]. Both geometric and physical nonlinearities were included in the analysis, the latter being modeled by the generalized Ramberg-Osgood law. The problem could not be solved exactly and the author used the collocation method, similar to that employed in [7]. An important observation made in [8] was that creep failure associated with high local stresses can occur prior to buckling. A variety of analytical models have been used to characterize creep of the column material, including steady-state creep [1, 8], a hereditary constitutive law [9, 10], and an incremental approach [11]. Other examples of recent studies of the creep buckling of structures can be found in Refs [12-14]. In particular, Minahen and Knauss used hereditary integrals to trace creep of initially imperfect columns [14]. The solution of the governing Volterra integral equation was obtained in a closed form for a standard linear solid. The authors illustrated that elastic kinematically linear models can be sufficient for the solution of the creep buckling problem. In the present paper, the analysis of Hoff is extended to columns of arbitrary symmetric cross section. The solution is applicable to materials characterized by the Norton law without a restriction on the power of the stress (as was the case in the solution of Hoff [1]). These two generalizations should enable a designer to apply the closed form solution presented in the paper to the majority of practical columns.

ANALYSIS Consider a steady-state creep of a simply supported column of an arbitrary cross section that is symmetric with respect to the centroidal axis perpendicular to the plane of buckling. The column is compressed by a constant force, Po. The solution for a simplified case where the cross section was idealized by two concentrated flanges separated by a web of a negligible cross section was given by Hoff [1, 2]. This analysis was limited to the cubic power of stress in the Norton's constitutive equation. Although, as illustrated in the section "Numerical examples", some steels can be characterized by this form of Norton's law, it is important to extend the solution to the general case. According to the approach adopted in this paper, the cross section of the column is subdivided into pairs of symmetric layers. Deformations of the column are assumed sufficiently small so that the ratio of the transverse deflection to the coordinate of the centroids of each of the layers adjacent to the neutral axis w/zl ~ 1. Obviously, the same is correct for other layers, i.e. forj > 1. This assumption enables us to apply thc geometrically linear theory of beams to each couple of layers. The steady-state constitutive relation for the material was taken by Hoff in the form:

= (1)

where a~ and n are constants. As indicated by Hoff, this law is meaningful only if n is a positive odd intcger. If n is an arbitrary number, Eqn (1) can be generalized:

O" n . = sign a. (2) O" c Creep buckling and rupture of columns 833

An alternative expression known as the Norton law can be written in a general case in the form: = Blurt" sign a. (3) Axial strains in the jth pair of layers are represented by: et, c = to +_ zj(w,xx - Wo,x,) (4) where the subscripts "t" and "c" denote the convex and concave layers, and eo is a uniform axial strain. Note that an initial deviation from the perfect shape, Wo, in Eqn (4) includes both an initial imperfection and an additional deflection received by the column at the instant of application of a compressive load. According to [15], if ao and aoo are the amplitudes of an initial deviation and an initial imperfection, respectively,

aoo ao -- 1 -- Po/Pcr" (5) Note that Eqn (5) is applicable only if the column material remains within the elastic range. The stresses in the concave and convex layers forming the jth pair are:

(Te A~ zjAj (6) Pj Mj

respectively, where the moment generated by a compressive force Pj is equal to mj = -Pjw. The equilibrium problem for the jth pair of layers can be formulated by differentiating a difference between the strains given by Eqn (4) with respect to time and substituting the strain rates from Eqn (2). Then, using expressions (6) for the stresses, one obtains:

~(w/zA (7) gtc~ (x/L ) 2 +2z}\a~AjJ 1+~ - 1-~ =0.

Note that as deflections in a convex layer exceed the distance z i, the term (1- w/zj) becomes negative. Then the analysis presented in this paper has to be modified. However, in the present analysis, the approach is similar to that employed in the linear buckling theory. This implies that infinitesimal deformations are assumed throughout the solution until the phase where the buckling conditions are formulated based on infinite deflections of the structure. Obviously, this means that the limitations of the linear buckling theory are applicable to the present solution. If the cross section is subdivided into 2N layers, the number of Eqns (7) is equal to N. The additional condition that must be satisfied is

(P j) = Po. (8) J Obviously, if creep buckling occurs at the same elapsed time, t,, for all layers and condition (8) is satisfied, the compatibility of deformations is assured. The solution is obtained for a sinusoidal initial deviation from the straight position and the corresponding buckling mode shape: h nx Wo = ao ~ sin -~- (9) h. nx w = al(t) -~ sin "-~-. 834 V. Birman and M. G. Magid

1. "Cubic" constitutive law Hoff [ll presented the solution for such constitutive law modeling the column by two flanges. In this section, Hoff's solution is extended to columns with complex cross sections which cannot be accurately modeled by two flanges. If the power in Eqn (1) is n = 3, substitution of the second Eqn (9) into Eqn (7), application of the Galerkin procedure, subsequent separation of the remaining variables (a and t) and integration yield:

,,, I t = -- (lO) p~ln[k~o/ l+l(h)2a2 where I(.,A: ks = 6 \ L ,I (acAs)a" (11)

The ratio al/ao can be evaluated from Eqn (10): al exp (P~ t/2ks) ao 1 + (h/zs) 2 a ~ [1 - exp (P~ t/ks) ]

The condition of the creep buckling, i.e. the requirement that the ratio al/ao must approach infinity, yields the critical time for the jth pair of layers:

cr =-~ln 1+16 . (13) Pi The critical time should be the same for all layers. Therefore, evaluating Pj from Eqn (13) and using condition (8), one obtains: to, = (Io ~ {k, In [1 + 16(~)2 ~-oz]j11l"3V }. (14)

In the case of a column consisting of two layers, Eqn (14) is reduced to the solution of Hoff [ll. Note that this solution remains valid, even if the amplitude of a transverse deflection (alh/2) > zj. 2. The solution for an arbitrary constitutive law (n > 1) As explained in the previous discussion, the ratios w/zj were assumed small. Therefore, the terms in the square brackets in Eqn (7) can be represented by the series

1+ =l+n-+-- -t +... (15) - - zj 2! - 3! where higher-order terms are negligible, within the framework of a geometrically linear theory. Accordingly, Eqn (7) becomes

~t~x 2 + Z-~i \trcAs j w + 6 ~ = 0. (16) The substitution of, the second Eqn (9) into Eqn (16) and the Galerkin procedure result in the following equation: do1 (~z/)2(P • )n (n -1) (n - 2) (h) 2 = "J n [ a~ ] = 0 (17) dt ae As al -t 32 which, upon integration, yields the time-displacement history ks(n) [(all zl t= p--Tin -- l . (18) L\ao,, 1 Creep buckling and rupture of columns 835

In Eqn (18),

ks(n) = 2n (erdAj)" (19) fJ = (n -1)232 (n - 2) (h)

Eqn (18) can be solved for al/ao. Then the buckling condition yields:

t~{) = kj(n)In (1 +ffl ao2). (20) P7 The requirement of the identical critical time for all layers in conjunction with Eqn (8) yield:

t. = P--~o [kj(n) ln(1 +ffl ao2)]1/.. (21)

3. Design for allowable stresses The critical time found using a geometrically linear theory is obtained from the condition of unlimited transverse deflections of the column. Analysis in the vicinity of the critical time would require one to consider both geometric nonlinearity and plasticity. In addition, as has been indicated above, the stresses and strains in convex layers become tensile when the column experiences significant deflections. This emphasizes limitations of the solution based on a linear theory. Nevertheless, the latest findings indicate that a geometrically linear theory can provide a reliable estimate of the column endurance 1-14]. This is probably due to a relatively short time interval when columns experience large deformations prior to buckling. In practical design, a creep buckling safety factor should be used, i.e.

t'cr = Stcr (22) where s < 1. Deformations of the column within the interval 0 < t < t'cr usually remain sufficiently small to justify the application of the linear theory. In addition to the critical time, a failure time (tf) should be considered to keep the stresses in the column within allowable limits. Therefore, the allowable loading time for the column subjected to steady-state creep is:

tal 1 : min (tf, t'cr ) . (23) Note that the failure time can be evaluated based on the stress-deflection relationship and the time-displacement history. The allowable stresses will be first reached in one of the extreme layers of an isotropic column. A larger absolute value of the stress is observed on the concave side, but tensile stresses on the convex side can be even more dangerous because of possible creep rupture. Mentioned here are the papers of Bostrom 1-16] and Zyckowski and Zaborski [17] who considered the tensile brittle rupture lifetime based on the Kachanov-Rabotnov-type damage model [18]. Let's assume that an allowable stress, aaU, is associated with compression in the extreme concave layer. Then the allowable deformation ratio can be found from Eqns (2), (4) and (9):

al -- n2 gN h \ ac / (24)

Eqn (24) can be substituted into Eqn (17) yielding the value al = aa11 corresponding to the allowable stresses. The failure time tf can be found by substituting aan into Eqn (18). Then, applying the requirement that the time necessary for all pairs of layers to reach this deformation must be the same, together with Eqn (8), yields

1 (~ {kj(n)lnf(aal_Ai~21+fia2 ll/a}) ". (25) tf = p~ 1\% J 1 --}-fjaa211 836 V. Birman and M. G. Magid

NUMERICAL EXAMPLES The following examples illustrate the creep critical time obtained for a number of materials that follow Norton's law. Numerous references present data on the values of constants in Norton's law for various materials 1-19-22]. The material data for a number of steels used in this paper are listed in Table 1. The modulus of elasticity for all steels is E = 200 GPa. The column considered in the examples was an I-beam with the moment of inertia about the neutral axis I = 995 x 10-6 m 4. The column was subdivided into two pairs of layers. The first pair was formed by the sections of the web divided by the neutral axis (A1 = 10750.2 mm 2, zl = 141.45 mm). The second pair of layers was formed by the flanges (A2 = 8132.8 mm 2, z2 = 293.95 mm). The length of the column and the height of the cross section were L--6.0 m and h = 0.61 m, respectively, while the amplitude of the initial imperfection was chosen as aoo --- 0.001 m, if not indicated otherwise.

Table 1. Properties of selected steels 1-19]

Number Chemical Thermal Test B n composition treatment temperature (m2/MN) "/h°ur (%) (°C)

1 0.15C Annealing 0.50Mn 0.23Si 840 °C 538 1.43 x 10-11 3.04 0.032S 593 3.10 × 10- lo 3.18 0.025P 649 9.04 × 10- 9 3.03 2 0.60C Air Hardening 0.46Si at 1050 °C 0.28Mn t6.9Cr" Tempering at 0.22Ni 800-820 °C 2.00Mo 600 3.03 x 10- a 1 2.59 0.013S 0.030P

3 0.48C Hardening at 0.68Si 1,100 °C, 0.47Mn 600 3,34 × 10 -28 10.30 0.012P Air tempering 2.24W 13.6Cr 14.5Ni 0.54Mo

4 0.45C Hardening at 600 2.00 x 10- lo 3.00 0.60Si 1,175 °C 650 1.71 × 10 -9 2.93 0.76Mn Stabilization 700 1.24 x 10- s 2.90 13.9Cr at 13.8Ni 750 °C during 1.75W 5 hours 0.40Mo 0.008S 0.024P

5 0.46C Hardening at 600 2.24 × 10- 26 9.15 1.15Si 1,100°C, 14.9Mn 2.25W 0.042P Air tempering 13.9Cr Creep buckling and rupture of columns 837

The critical time is shown as a function of the test temperature and the ratio of the compressive force to the Euler's buckling value for two different steels in Figs 1 and 2. The logarithmic scale for the critical time used in these figures was selected to cover a larger range of the compressive force. As follows from both figures, the critical time is very

"')<. ~, -)(

",. I "'g'"~,..g.~...~ fcr "'-aR,. \ ~',,. "~"-~...~..~...~ ~i x "+, q...

"o.. ~L

0 0,05 0.1 0.15 0.2 0.25

,-t-- 593°C.8oc Po/P cr ~_ 649°C Fig. 1. Critical time as a function of compressive forces for steel 1.

",,

'\ ,~( b

\ \

\ ",.~, fcr ""b.,

"'b, ",i~.

"'b--,,t.,,~

~I ~k

-6 0 0.05 0.1 0,15 0.2 0.25

--~¢ 600oc

"~-" 650°C --n- 700°C Po/P CF

Fig. 2. Critical time as a function of compressive forces for steel 4. 838 V. Birman and M. G. Magid

sensitive to the test temperature. In addition, a very significant drop in the critical time is observed with an increase of the compressive force. Note that the results corresponding to a very short critical time are not reliable because the assumption of steady-state creep becomes invalid. A similarity of the results for two steels in Figs 1, 2 raised a question on the effect of the chemical content of the material on its critical time. This question is addressed in Fig. 3, which presents the results for five steels at the same test temperature (600 °C). The only exception in Fig. 3 is the curve for steel 1 that corresponds to the temperature 593°C, but this difference in the test temperatures should be considered negligible. As follows from Fig. 3, the difference between the critical time for different materials can be dramatic. Depending on the material, a column can resist the creep buckling for a very long time, while a change of the material could result in collapse within minutes. Note that the very high sensitivity to creep demonstrated by steels 3 and 5 is due to a significant difference of the properties of these materials from other steels (see Table 1). Although the results for steels 3 and 5 are shown in Fig. 3, they are unreliable because the assumption of a steady-state creep utilized in this paper cannot be applied to such rapid processes. The effect of the amplitude of the initial imperfection on the critical time is illustrated in Fig. 4. Note that, contrary to the previous figures, the axis of ordinates used in Fig. 4 shows time, rather than its natural logarithm. A dramatic effect of the magnitude of the compressive force on the critical time noticed above is reflected again, in Fig. 4. The creep critical time appears ~o be sensitive to initial imperfections, particularly at a relatively low compressive force. This conclusion justifies a strict control of imperfections in compressed members subjected to creep. Finally, the effects of the constant B and the power n in Norton's law on the critical time are shown in Fig. 5. Three curves in this figure correspond to B = 10-lO, 10-2o and 10 -3°, while the range of the power is from n = 2 to n --- 10. An inspection of data in Table 1 yields an immediate conclusion that the values of B and n in Fig. 5 correspond to the typical steels.

q"----b...~..,je-

-~,_ 0 ~ 0-~0~ .

El. "12~'la

0.05 0.1 0.15 0.2 0.25 - P0/Pcr -÷~ Steel 2 --~ Steel 3 -o Steal4 -~- Steel 5

Fig. 3. Comparison of the critical time for five different steels (test temperature is 600 °C). Creep buckling and rupture of columns 839

60 K. ,.,,

L,,,

40 /er (hours) 30

20 •1-......

...... + ...... -"1" ...... ""'1'

0 0 0.002 0.004 0.006 0.008 0.01 0.012

'~ Po/Pcr=0.1 -t-Po/Por--0.15 ao 0 -o- po/Pcr=0.2

Fig. 4. Effect of the amplitude of initial imperfection and compressive forces on the critical time for steel 1 (test temperature is 538 °C).

m'--. k "IL . 40

"l-.,"--., "n... ..

20 "", ,. ~ t ...

(21" .....÷, ......

0 """ ~..,,. '"" ,.1~,." L,,,

L,.,. q'-,, ,

"'--)~,.. ""..g. .

2 3 4 5 6 7 S 9 i0

'J¢ LOG03) = -10 1"1 --÷- LOG03) = _20 -a- LOG03) = .30 Fig. 5, Effect of the constants in Norton's law on the critical time.

As follows from Fig. 5, an increase of the power in the Norton law results in a reduction of the creep critical time. An increase of the constant B results in a similar effect. The latter conclusion becomes obvious, if one analyzes the data in Table 1. Indeed, the values of B increase at higher test temperatures (see steels 1 and 4), while the change of the power

HS 37-8-D 840 V. Birman and M. G. Magid

n remains limited. Obviously, creep buckling occurs sooner, if temperature increases (as shown, for example, in Figs 1 and 2). This reinforces the conclusions available from Fig. 5. The conclusions obtained at other values of the compressive load were identical to those illustrated in Fig. 5. Note that Fig. 5 clearly indicates the limitations of the analysis based on the assumption of steady-state creep. The materials with small values of the constant B and the power n in Norton's law are better candidates for the steady-state analysis. This implies that the validity of such analysis becomes questionable as temperature increases. Another interesting observation is related to the fact that even if the magnitude of the compressive force represents a small fraction of the Euler's buckling force, creep buckling occurs within a very short time. These results obtained for columns with a small imperfection (aoo = 0.001 m corresponds to just 0.164% of the column height) may seem strange in view of a traditional understanding of the effect of initial imperfections. However, they become obvious if one accounts for the high temperature sustained during the process. As follows from the results presented above, a decrease of temperature results in a dramatic increase of the critical time corresponding to the creep buckling.

CONCLUSIONS A simple but accurate user-friendly method of analysis of simply supported columns subjected to creep with cross sections symmetric about the neutral axis is proposed. The method represents a generalization of the previous solution of Hoff that was limited to columns idealized by two flanges with the cubic power of stress in Norton's law. The results are presented for the critical time of columns manufactured from five different materials. Three of the materials are suitable for the analysis, but creep buckling of the columns manufactured from two other materials occurs too fast to justify a solution based on the steady-state creep. Analysis of the numerical results illustrates that the critical time is very sensitive to the chemical content of the material. A test temperature that affects the constants in Norton's law has a significant effect on the critical time, i.e. an increase of temperature accelerates the creep buckling. The magnitude of the initial imperfection is another important factor, i.e. an increase of the imperfection results in a decrease of the critical time, although this effect is less evident as the magnitude of the compressive force increases. Although a decrease of the compressive forco is beneficial for the critical time, some materials can collapse even at a relatively low compression (the latter conclusion is in agreement with the results of the previous studies of creep buckling, e.g. [-1]).

Acknowledgment--Discussions with Professor George J. Simitses of the University of Cincinnati are warmly appreciated.

REFERENCES

1. N. J. Hoff, Buckling and stability, 41st Wilbur Wright Memorial Lecture. J. Roy. Aero. Soc. 58, 1 (1954). 2. N. J. Hoff, Creep buckling. Aero. Quart. 7, 1 (1956). 3. R. L. Carlson, Time-dependent tangent modulus applied to column creep buckling. J. Appl. Mech. 23, 390 (1956). 4. A. M. Freudenthal, The Inelastic Behavior of Engineering Materials and Structures, p. 518. Wiley, New York (1950). 5. C. Libove, Creep buckling of columns. J. Aero. Sci., 19, 459 (1952). 6. J. Kempner and S. A. Patel, Creep Buckling of Columns, NACA TN 3138 (1954). 7. M. Zyczkowski, Geometrically nonlinear creep buckling of bars. Creep Mechanics, p. 307. (edited by N. J. Hoff). Academic Press, New York (1962). 8. N. C. Huang, Creep buckling of imperfect columns. J. Appl. Mech. 43, 131 (1976). 9. A. M. Vingradov, Buckling of viscoelastic beam columns. AIAA J. 25, 479 (1987). 10. J. M. Stubstad and G. J. Simitses, Creep analysis of beams and arches based on a hereditary visco-elastic- plastic constitutive law. J. Engng Mater. Technol. 112, 210 (1990). 11. Y. Song and G. J. Simitses, Elastoviscoplastic buckling behavior of simply supported columns. AIAA 3. 30, 261 (1992). 12. M. A. Souza and T. N. Bittencourt, Viscoelastic effects on the vibration of structural elements liable to buckling. Thin- Walled Struct. 12, 281 (1991). Creep buckling and rupture of columns 841

13. V. D. Potapov, Stability of viscoelastic rod subject to a random stationary longitudinal force. J. Appl. Math. Mech. 56, 90 (1992). 14. T. M. Minahen and W. G. Knauss, Creep buckling of viscoelastic structures. Int. J. Solids Struct. 30, 1075 (1993). 15. J.C. Chapman, B. Erickson and N. J. Hoff, A theoretical and experimental investigation of creep buckling. Int. J. Mech. Sci. 1, 145 (1960). 16. P. O. Bostrom, Creep buckling considering material damage. Int. J. Solids Struct. 11, 765 (1975). 17. M. Zyczkowski and A. Zaborski, Creep rupture phenomena in creep buckling. Proc. IUTAM Symp. Mechan- ics of Visco-Elastic Media and Bodies, Gothenburg, edited by J. Hult p. 283 (1974). Springer Berlin (1975). 18. L. M. Kachanov, On the time to failure in creep conditions, Izv. Akad. Nauk SSSR, Otd. Tekhn. Nauk. 8, 26 (1958). 19. N. N. Malinin, Applied Theory of Plasticity and Creep, p. 248. Mashinostroenie, Moscow (1975) (In Russian). 20. H. Kraus, Creep Analysis, p. 196. Wiley, New York (1980). 21. J. T. Boyle and J. Spence, Stress Analysis for Creep, p. 251. Butterworths, London (1983). 22. H. H. Pan and G. J. Weng, Determination of transient and steady-state creep of metal-matrix composites by a secant-moduli method. Composites Engn9 3, 661 (1993).