NONLINEAR ELASTIC ANALYSIS

Nonlinear elastic analysis of frames can be performed either incrementally or iteratively.

Incremental analysis involves the determination of the incremental or tangent stiffness matrix relating small increments in external forces and corresponding displacements; this depends on the current geometry and state of stress. Complete solutions for the entire loading history can then be obtained by incrementing either forces or displacements (Roberts, 1970; Roberts and Ashwell, 1971). Incrementing displacements has the advantage that solutions do not break down at horizontal tangents on load-deflection curves, which is one form of critical load condition.

Iterative solutions are based on the determination of a secant stiffness matrix, which is derived assuming that the current geometry and state of stress is known (Majid, 1972).

The first cycle then gives new values for the current geometry and state of stress which are then included in the second cycle, and the sequence is repeated until the assumed values are consistent with the calculated values.

In general, incremental analysis is theoretically more sound than iterative analysis.

Incremental analysis follows the complete loading history and is able to detect bifurcations or branching points along equilibrium paths. This is not true of iterative solutions which may not converge to the lowest equilibrium path which is of interest in practice. However, for multi-storey frames, such complexities seldom exist and either form of solution appears satisfactory. Incremental analysis is considered first since it illustrates the full interaction between bending and axial displacements for individual elements and will help to indicate the approximations often made in iterative solutions.

2.4.1 Incremental Analysis

The incremental stiffness matrix for a finite element of a frame can be derived as follows (Roberts, 1970). It is assumed that all displacements are small so that nonlinear strains can be related to the initial geometry; this is a satisfactory assumption for the majority of practical frames. For an element having an initial transverse imperfection w0, the nonlinear expression for the axial strain ε is (Timoshenko and Gere, 1961)

(2.27) The first and second variations of ε with respect to displacements u and w are

δε=−zδwxx+δux+wxδwx

(2.28) δ²ε=0·5δwx.δwx

(2.29) Substituting eqns (2.27)–(2.29) into eqn (2.4) and noting that δ²qi vanishes, since the nodal displacements qi are assumed to be linear functions, gives

(2.30)

Integrating over the area of the cross-section, eqn (2.30) can be arranged in matrix form as

(2.31)

Assuming suitable displacement functions for w, w0 and u (see eqns (2.12) and (2.13)) and proceeding as in Section 2.3, eqn (2.31) can be reduced to the form

{δq}^T{δp}={δq}^T[[KL]+[KGA]]{δq}

(2.32)

[KL] is as defined in Section 2.3 and [KGA] is referred to as the geometric stiffness matrix for incremental analysis. The derivation of [KGA] from the second integral in eqn (2.31), assuming w and w0 to be cubic polynomials, is complex. The derivation of [KGA]

can be simplified however by assuming linear polynomials for w and w0, defined by the nodal values of w and w0 only (Roberts and Azizian, 1983). All terms in the second integral of eqn (2.31) then become constants and integration is simply a matter of multiplying by the length of the element. This procedure, as well as simplifying the derivation considerably, is advantageous for convergence of finite element solutions.

After transformation to global coordinates and assembly of elements, the incremental equations for the whole structure take the form (see eqn (2.22))

(2.33) Solutions can be obtained by incrementing either loads or displacements. The geometric stiffness matrix has to be reformed prior to each increment to account for the current geometry or state of stress and the total forces and displacements accumulated from the increments { } and { }. Stresses at any stage can be determined from eqn (2.27) and the accumulated displacements.

The accuracy of incremental solutions will depend upon the size of each increment.

Also, if the simple linearisation techniques described are not employed, it may prove necessary to use a large number of elements to model each member, beam or column, of the frame.

2.4.2 Iterative Analysis

Derivation of the secant stiffness matrix for use in iterative methods of analysis can be based on eqn (2.3). Assuming the nonlinear expression for the axial strain, as defined by eqn (2.27), and substituting in eqn (2.3) gives (Roberts, 1970)

(2.34)

Integrating over the area of the cross-section, eqn (2.34) can be arranged in matrix form as

(2.35)

Following the previously described finite element procedures, eqn (2.35) can be reduced to a form similar to eqn (2.32) with the last two terms contributing a constant column

vector on the right hand side. As in the previous section, eqn (2.35) indicates full interaction between axial and bending displacements.

However, the major nonlinear influence in the behaviour of frames is the influence of axial forces on the flexure of members. Assuming, therefore, that the axial force t is simply EAux, as given by a linear elastic analysis, and assuming w0=0 for simplicity, eqn (2.35) reduces to the form (Chajes, 1974)

(2.36) Equation (2.36) can be simplified further by linearising w when deriving the geometric stiffness matrix, as discussed in Section 2.4.1.

Assuming suitable displacement functions for w and u, eqn (2.36) can be reduced to the form

{δq}^T{p}={δq}^T[[KL]+[tKGB]]{q}

(2.37) [KL] is as defined by eqn (2.19) and [tKGB] is the geometric stiffness matrix for iterative analysis. Assuming w to be a cubic polynomial

(2.38)

After transformation to global coordinates and assembly of elements, the secant stiffness equations for the complete structure take the form

(2.39) Solutions of eqn (2.39) can be obtained iteratively. For the first cycle [ ] is assumed zero and a linear elastic solution is obtained. The member axial forces given by the first cycle are then used to determine [ ] for the second cycle, and so on until the assumed and calculated member axial forces are consistent. Solutions of this type usually converge rapidly since the member axial forces do not vary much from those given by the first linear elastic analysis.

2.4.3 Iterative Analysis Using Stability Functions

An alternative way of expressing the influence of axial forces on the flexural behaviour of members is in terms of so-called ‘stability func-

FIG. 2.3. Member of a frame.

tions’ (Livesley and Chandler, 1956; Horne and Merchant, 1965). The differential equation governing the flexure of the member shown in Fig. 2.3 is

m=EIwxx=ƒA.x−mA−tA(w−wA)

(2.40) Solutions of eqn (2.40) subject to various boundary conditions lead to a secant stiffness matrix [KSC] of the form

(2.41)

In eqn (2.41) s, c and r are defined as follows (eqns (2.42)–(2.44)):

(2.42) For compressive axial forces

(2.43) For tensile axial forces

(2.44) In eqn (2.41), the terms relating axial forces and displacements are as for linear elastic analysis and terms which represent the interaction between flexural and axial displacements (see Section 2.4.2) are omitted.

After transformation to global coordinates and assembly of elements, the secant stiffness equations for the complete structure take the form

(2.45) Solutions of eqn (2.45) can be obtained iteratively as described in Section 2.4.2, noting that when member axial forces are assumed zero, .