ELASTIC INSTABILITY

As mentioned previously, the most significant nonlinear influence in the elastic behaviour of frames is the influence of axial forces on the flexural stiffness of members. Tensile axial forces can be considered as increasing the flexural stiffness while compressive forces decrease the flexural stiffness. If a set of compressive member axial forces is increased to the extent that the bending stiffness of the frame as a whole reduces to zero, the frame becomes unstable.

There are a number of ways in which the elastic instability of frames can be analysed, many of which reduce to the solution of the same basic set of equations after simplifying assumptions are made.

2.5.1 Vanishing of the Second Variation of Total Potential Energy

The most general approach to the analysis of the elastic instability of structures is based on the vanishing of the second variation of total potential energy, defined by eqn (2.5) (Roberts and Azizian, 1983). Assuming that the nodal displacements qi are linear functions of displacement variables, δ²qi vanishes and critical conditions are defined by the equation

(2.46) Substituting the nonlinear expression for the axial strain ε defined by eqn (2.27) gives

δ²VP={δq}^T[[KL]+[KGA]]{δq}=0

(2.47) δ²VP is identical to the right hand side of eqns (2.31) and (2.32) and is a complete quadratic form which changes from positive definite to zero, indicating critical conditions, when the determinant of [KL]+ [KGA] vanishes. Hence critical conditions for the complete structure occur when

(2.48) An alternative way of interpreting eqn (2.48) is that critical conditions occur when the incremental or tangent stiffness matrix becomes singular.

Although eqn (2.48) is of general applicability, it requires a knowledge of the axial and flexural deformations at the critical points on the loading path. The analysis can be simplified if it is assumed that prior to the frame becoming unstable, only axial

deformations occur. The axial forces t are then simply equal to ux and eqn (2.46) reduces to

δ²VP={δq}^T[[KL]+[tKGB]]{δq}=0

(2.49) δ²VP is now identical to the right hand side of eqn (2.37) if {q} is replaced by {δq}.

Hence, critical conditions for the complete structure occur when

(2.50) In solving eqn (2.50) it is usually assumed that the critical set of member axial forces can be related to a base set, determined for example from a preliminary linear elastic analysis, by a scalar load factor λ, which is given a negative sign to denote compression. Equation (2.50) then takes the form

(2.51) Equation (2.51) represents a standard eigenvalue problem. The lowest eigenvalue defines the critical load factor λcr and the corresponding eigenvector defines the buckled mode.

2.5.2 Singularity of Secant Stiffness Matrix

The secant stiffness equations defined by eqns (2.39) and (2.45) are indeterminate when the secant stiffness matrix becomes singular. Critical conditions occur therefore when

(2.52) (2.53) Equation (2.52) is identical to eqn (2.50) and can be reduced to the standard eigenvalue problem defined by eqn (2.51). Solution of eqn (2.53) is accomplished by assuming a load factor λ and evaluating the determinant. The process is repeated until the load factor at which the determinant vanishes is found.

2.5.3 Horne’s Method

Horne (1975) proposed an approximate method for determining the elastic critical loads of plane multi-storey sway frames, the only analytical requirement being that of performing a standard linear elastic analysis of the frame. The method can be illustrated by considering the instability of the column of length a shown in Fig. 2.4(a).

Assuming that the column buckles into a state of neutral equilibrium, i.e. zero kinetic energy, the potential energy remains constant and the loss of potential energy of the external forces is equal to the increase in strain energy of the column. If, due to buckling, the end A of the column sways by with a corresponding axial shortening δū, the governing energy equation can be written

P.δū=UE

(2.54) in which UE represents the change in strain energy of the column.

To determine UE it is assumed that the buckled shape δw is the same as that produced by a concentrated horizontal force H as shown in Fig. 2.4(b). UE is then equal to

and eqn (2.54) becomes

(2.55) If the critical load Pcr is related to a base load P0 by a scalar load factor λcr and H is assumed equal to nP0 (n being a scalar load factor

FIG. 2.4. Buckling of a column.

less than unity) eqn (2.55) can be rewritten as

(2.56) or

(2.57) Two extreme cases are now considered for the buckled shape, as shown in Fig. 2.4(c) and (d). The first, which is a simple rigid body rotation, represents the case in which the columns of a frame are stiff compared with the beams. The second, which is a pure sway mode, represents the case in which the beams are stiff compared with the columns.

Assuming simple polynomials to represent the buckled shape δw, the axial shortening is given by

(2.58)

For the two extreme cases δū is given by

(2.59) Since these two values differ by only approximately 20% it is convenient to take the average, and substituting in eqn (2.57) gives

(2.60) The procedure can now be summarised as follows. Apply a horizontal force of nP0 to the end of the column and determine . The critical load factor is then given by eqn (2.60).

Now consider the generalisation of this procedure for frames. The frame shown in Fig.

2.5 has N=4 storeys. P0i represents the vertical loads applied at the ith storey and Hi=nP0i

are the corresponding horizontal forces assumed applied to the frame. The governing energy equation for the frame (see eqn (2.56)) is then

(2.61) Assuming that the rth storey of the frame becomes unstable before all

FIG. 2.5. Multi-storey sway frame.

the others, eqn (2.61) can be simplified as

(2.62) The summation can now be cancelled to give

(2.63)

and proceeding as for the simple column

(2.64) The procedure for frames can now be summarised. Load the frame with horizontal forces at each storey Hi equal to nP0i. Determine the sway deflections and locate the storey for which is a maximum and assume equal to . The critical load factor is then calculated from eqn (2.63).

2.6 SECOND ORDER EFFECTS AND ELASTIC CRITICAL