Pergamon Int. J. Mech. Sci. Vol. 37, No. 8, pp. 831-841, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7403/95 $9.50 + 0.00 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.