NON-LINEAR ELASTIC-PLASTIC ANALYSIS OF SKELETAL STEEL STRUCTURES

Abstract

The linear elastic theory though being successfully used by engineers for analysis and design of selected steel structures is neither entirely rational nor economical. The efforts to develop a rational basis for analysis of these structures have led to rapid and significant research advances in recent years in areas such as non-linear elastic-plastic i analysis of frame structures. The non-linear behaviour of a frame structure arises from effects of both material and geometric non-linearities. Material non-linearity is considered by idealising the momentcurvature relationship as bilinear. Geometric non-linearity includes the effects of instability caused by the presence of axial forces in the members, bowing of the deformed members and of finite deflections. These effects lead to prediction of a lower value of collapse load without the formation of a mechanism. Presence of axial forces may also alter the sequence of hinge formation. The consideration of effects of finite deflections, instability and curvature essentially forms the basis for elastic-plastic analysis. Saafan (4) presented the non-linear analysis of plane frames considering geometric non-linearity. Jennings (24) and Powell (25) assumed cubic variation of transverse displace ments along the chord while deriving stiffness for beam-column elements. Majid and Anderson (5?) gave an iterative procedure (ii) for the analysis of large multi-storey frames in the elasticplastic range ignoring finite changes in geometry. Wood, R.H. (22) stressed that the failure of aframe may take place before a mechanism is formed. Tezcan and Ovunc (26) proposed asearch technique for determining buckling loads indicated by excessive defor mations. Bruinette and Penves (67) considered axial forces to • the interaction equation for plastification of arectangular cross-section of .space frame member in addition to bending moments in the principal planes and torsion, but they neglected the effects of non-linearities in the analysis. Alower bound step by step second-order method of analysis has been presented in the present study using iterativeincremental technique based on displacement approach. Equili brium equations have been formulated by taking into account the deformed geometry of the structure. The tangent and instan taneous secant stiffnesses for the members are used leading to faster convergence. In establishing the member secant and tangent stiffnesses, stability functions, bowing functions and curvature coefficients have been employed to include the effects of instability and bowing of members. The method of analysis commences by assuming axial forces in the members to be zero and assembling the structure stiffness[K] with the help of acombined rotation matrix [B] and member secant stiffness [ks ]. The Joint equilibrium (iii) equations are solved for nodal displacements {X} corres ponding to the loads {W] at unit load-factor by Cholesky's method. The member end forces are estimated from the member end displacements {E} determined by transforming nodal displacements {X] in two stages from frame axes to member axes for the deformed geometry of the structure. The non linear member end forces are obtained after revising the member secant stiffnesses by using stability functions and curvature coefficients. The unbalanced loads {/\Vi } and the equilibrated forces are found by transforming these member end forces to those in frame axes. The unbalanced loads are applied as incremental loads on the deformed structure and the structure tangent stiffness is compiled by using member tangent stiffnesses. The incremental nodal disppacements { dX] , found by solving the resulting equilibrium equations in the incremental form, are transformed from frame axes to those in the deformed geometry. The geometry of the structure is revised for further analysis. The incremental member end forces are obtained and added to the previously determined estimated member end forces. This process is repeated till out of balance nodal load vector {&W } disappears and the equili brium between applied loads and actual member end forces is practically achieved. The convergence is checked by the following criterion / jCurrent axial force , IPrevious axial force )^-tolerance specified (iv) The elastic-plastic analysis is canied out by carrying out the above non-linear analysis at two loadfactors (initially x± =1.00, and Ag =1.10). Thus two sets of values of bending moments (M'Al and M' ?), axial forces (P'n and P'A2) and the reduced plastic hinge moments (M'pja and M'pA2) become known. The load-factors at which various ends of different members reach the value of respective plastic hinge moment are determined by a prediction procedure using the following expression. UV a2-^'PAl-^> (M'PA1-M,PA2) +(M*A2-M'n) A minimum value of load-factor K^ is searched from the above load-factors and the corresponding member and end numbers are noted. The loads corresponding to A ° min above the previous loads are applied in a number of steps PSTEP (10 in the present study). The appropriate incremental laod-factor, API is thus determined as under APL = (Amin ~ V/FSTEP The value of load-factor ^ is successively increased by AJ!L and non-linear analysis is carried out till the bending moment at any end of any member is found to be greater than the reduced plastic hinge moments. This load-factor is designated as Ag. The actual value of the load-factor at which a plastic hinge would develop is determined and improved by following an iterative procedure proposed in the (v) study. The convergence for load factors is examined by the following criterion; / |, A current , l k . • A previous ) -^tolerance chosen The plastic hinge is introduced in the plane frame at the noted location. In space frame structures, members are subjected to bending moments in the two principal planes and a torsion. The plastification of space frame members are checked by interaction equations proposed for members having rectangular and I-sections. A plastic hinge is then introduced at the recorded location and the degree of freedom is increased by three allowing three free rotations at the plastified section of a space member. The frame with augmented kinematical degrees of freedom is checked for instability. The above process is repeated till the collapse load is reached as indicated by the onset of instability or when the convergence for unbalance load is not achieved. Two separate programs for non-linear elastic-plastic analysis of plane and space frames have been written. The validity of proposed formulations for analysis of plane and space frames has been established by solving examples originally analysed by Home, and Mao id end Anderson (Plane frame), and by Harrison (space frame). Purther examples of both plane and space frames have been chosen and essential characteristics of the response (vi) of these structures have been studied from the results obtained. It is found that the load-displacement and lord factor bending moment relationships with increasing loads are practically linear till first plastic hinge develops in the frame.Load-displacement diagrams become increasingly non linear with increase in loads as second and subsequent plastic hinges are formed. Axial forces carried by coluums are almost uninfluenced by non-linearities, whereas for some of the beam members, these forces do vary non-linearly and the nature of these forces may also change from compression to tension. Failure of frame structures occurs in general, due to the onset of instability. Collapse loads for the frame structures found by the proposed technique as a lower bound solution by following a second-order iterative-incremental procedure may be considered safe, accurate and reliable.