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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.

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