NON-LINEAR ANALYSIS

The preceding section has dealt with the general problem of the FE discretisation process for an arbitrary linear system. Thus, linearity was assumed to apply at the three levels of statics (equilibrium equations written in the initial, undeformed geometry, as displacements are assumed to be small), kinematics (linearised strain–displacement compatibility equations, as strains are also taken to be small) and constitutive relations (Hooke’s law or its gener-alised version deemed to be applicable). Such linearisation assumptions lead directly to a mathematical model possessing the following desirable features: uniqueness of solution, use of superposition, and, most important perhaps, the ready availability of efficient programs for the solving of linear systems of equations based on well-established mathematical tools and algorithms. Although nature tends to be distinctly non-linear – often highly so – it turns out that a large number of structural-engineering problems may be tackled on the basis of linearly elastic concepts, at least for purposes of achieving an adequate level of performance under ordinary ‘working’ or ‘serviceability’ conditions. On the other hand, the understand-ing of ultimate-load conditions or, indeed, the more rational and/or economic design of structures necessitates the consideration of non-linear effects: this is especially true in the case of structural-concrete problems. Unlike linear analyses, non-linear systems cannot be solved directly but rely on various iterative or ‘search’ techniques; all of these, however, are based invariably on repeated solutions of linear systems until convergence is achieved.

Owing to the possibility of non-uniqueness, care must be exercised – often by appeal to physical reasoning – in order to ensure that the converged solution attained is actually the correct one.

The analysis of a non-linear structural system, which has been discretised in accordance with the stiffness formulation, still proceeds through the solution of the set of Equations 4.22, but now the stiffness matrix is a function of the load/displacement level. For conve-nience, this will be denoted by the statement [K] = [K(d)]. In what follows, only the briefest of outlines on the main iterative procedures for non-linear problems will be presented, with priority of choice given eventually to incremental methods. Discussion will be restricted to

‘softening’ structures, where this term is now used to denote systems for which the f − d path is ‘convex’, in the sense that the stiffness decreases with increasing f, as in the case of structural concrete where a steady degradation of stiffness occurs as the load is augmented;

‘hardening’ or ‘concave’ systems (i.e., exhibiting a steady increase of stiffness with load-ing) – which may require different iteration strategies – need not be considered for present purposes.

4.2.1 Direct iteration method

It is useful to begin with a concise description of the main numerical devices for attain-ing the solution at a given load level in a sattain-ingle series of iterations, without regard to the previous f − d path, that is non-incrementally; subsequently, it will become evident that the incremental techniques are based essentially on identical principles. Perhaps the most basic

solution type is that of ‘direct iteration’ (also known as ‘functional iteration’ or ‘succes-sive substitution/approximation’). In this method, succes‘succes-sive solutions are performed, each iteration making use of the previous solution for the unknown(s) d to predict the improved, current value of [K(d)]

dn+1 =[K d( )n ]−1f; n = 0 1 2 3 … , , , , (4.63) The initial guess is usually taken to be do = 0, and the process is deemed to have converged when dn + 1 – dn → 0. It will be found convenient to illustrate this method (and, also, subse-quent ones) graphically by reference to the 1-D case (i.e., f, d constitute a single-DOF sys-tem), although, clearly, the same type of iterative behaviour extends to multi-DOF systems.

Figure 4.2 shows the implementation of direct iteration in the search for the solution cor-responding to the load level f in the given non-linear response f − d, with the initial guess taken as do = 0. It can be seen that the unknowns are the displacements, the secant slope being used in each iteration. This ‘secant modulus’ or ‘variable stiffness’ approach tends to be expensive since [K] must be revised and a new set of linear equations solved for each iteration. Furthermore, symmetry in the matrix of coefficients need not necessarily result when direct iteration is employed (Zienkiewicz 1977), and this means that the more efficient algorithms, based on the fact that [K] for linear problems is symmetrical, may not always be applicable. Another drawback of the scheme is that its convergence is not guaranteed and cannot be predicted a priori. In addition, as the number of DOF increases, coupling of stiffness terms might lead to instability of the iterative technique (Owen and Hinton 1980).

4.2.2 Newton–Raphson method

A more sophisticated process of iteration is the well-known Newton–Raphson (NR) method. This can be outlined as follows. Unless convergence has occurred, [K]d = f will not be satisfied at any stage of the iteration, and hence a system of residual forces Δf can be assumed to exist, so that

Δf = f – [ ] K d (4.64)

that is Δf may be viewed as a measure of the system’s current departure from the required state of equilibrium. Now, a better approximation exists at

d₃ d₂ d₁ d_o f

d Figure 4.2 Direct iteration method.

dn+1 = dn + Δdn; n =0 1 2 3, , , ,… (4.65) where the NR approximation for the increment or correction Δdn may be written as

Δdn =[K dt( )n ]⁻¹Δf d( )n ; n = 0 1 2 3, , , ,… (4.66) and where the subscript in [K] indicates that the latter is the tangential stiffness matrix.

With increasing number of iterations n, convergence is achieved as Δf and/or Δdn → 0. The process is shown schematically in Figure 4.3 for a single-DOF system, with initial guess d_o (= 0 here) leading to Δfo which, in turn, yields the correction so that d₁ = do + Δdo becomes an improved approximation, and so on. In this figure, the slopes of the f − d characteristic at the locations corresponding to the various d_n are the 1-D counterparts of the tangential stiffness matrix [K_t] used in multiple-DOF problems, as can readily be seen by invoking the well-known argument whereby Taylor’s series are curtailed beyond the first derivative (Cook 1981). (For a version of the NR technique in which the matrix of coefficients link-ing Δf and Δd is not symmetric, see Owen and Hinton 1980; [Kt], on the other hand, is always symmetric, and hence conforms to those equation-solving algorithms that make use of this property.) In general, the NR or ‘tangential stiffness’ method converges more rapidly and exhibits superior stability than the direct-iteration scheme. Again, however, there is no guarantee of convergence, especially if the initial guess is not close to the actual solution and/or combinations of ‘convex’ and ‘concave’ characteristics are encountered throughout the region of iteration. As for the direct-iteration process, the NR technique is demanding computationally, since each iteration requires the assembly of the updated matrix [K_t] and the concomitant linear-equation solving.

4.2.3 Modified

NR method instead of tackling a new system of equations for each iteration, the following approximation could be made:

[ ( )]K dt n ⊕ [K dt( )o] (4.67)

∆f₂ ∆f₁ f

d d_o d₁ d₂ d₃

∆d_o ∆d₁ ∆d₂

∆f_o = ∆f₁(d₁)

Figure 4.3 Newton–Raphson method.

so that

Δdn =[Kt( )do ]⁻¹Δf d( )n ; n = 0 1 2 3 … , , , , (4.68) throughout the entire search process. This algorithm is known as the ‘initial/constant stiff-ness’ method, and also as the ‘modified Newton–Raphson’ approach; its schematic illustra-tion is depicted in Figure 4.4 for the 1-D case (with d_o ≠ 0). The tangential stiffness matrix corresponding to the initial guess is assembled only once and, on reduction or factorisation of the set of equations and the storing of the result, solutions required in subsequent itera-tive cycles can be obtained at a much reduced computational effort. This significant sav-ing in computsav-ing cost per iteration, however, is countered by a lower rate of convergence when compared with the formal NR algorithm. Once again, although convergence is usu-ally achieved, this cannot be guaranteed for all cases, and sometimes divergence may be encountered in situations where the more rapidly converging NR technique is successful.

The relative economics and convergence rates of the initial and tangential stiffness methods depend on the degree and type of non-linearity of the system considered. The optimum algorithm is usually obtained by combining both methods so that [K] is updated to [K_t] only occasionally during iterations.

4.2.4 Generalised Newton–Raphson method

As stressed repeatedly in the preceding paragraphs, none of the previous methods, in which the unknowns were the total displacements d, converge in all cases. Only incremental cedures, where the unknowns are the changes Δd due to increments in loading Δf, can pro-vide some assurance on convergence. Furthermore, such methods enable a full study of the load deformation behaviour of a structure to be made; besides its obvious usefulness, the complete knowledge of the f − d characteristic followed at the structural level becomes mandatory when the solution is dependent, not only on the current displacements, but also on the previous loading history. Evidently, with sufficiently small increments, convergence may be ensured and the local linearisation at each iterative step becomes fully justified. Then

∆f₂

∆f₁ f

d

d_o d₁ d₂ d₃

∆d_o ∆d₁∆d₂

∆f_o

Figure 4.4 Modified NR method.

the NR method suggests itself as a natural iterative technique (its incremental version being termed the ‘generalised Newton–Raphson’ method), with the initial value now always taken as d_o = 0. A possible general algorithm might be

[K dt( )]n Δdn = Δfn + Δfn−1; n = 0 1 2 3, , , ,… (4.69) where Δfn represents the increment in the actual load, while Δfn−1 stands for the residual out-of-balance forces from the previous load step. This is often referred to as the ‘incremental with one-step NR correction’, and is illustrated for the 1-D case in Figure 4.5 by means of the dashed lines. It is evident that, despite the fact that the single residual-force corrections result in effective load increments (Δ Δfo; f1 +Δfo;Δf2 +Δf₁; etc.), a small (usually cumula-tive) drift from the true solution path occurs. On the other hand, when Δf = 0 in Equation 4.69, the algorithm becomes in essence the matrix counterpart of Euler’s numerical method for the solution of a differential equation; as can be seen by the path indicated by the dot-ted lines in Figure 4.5, the cumulative drift now becomes larger in this purely incremental algorithm without corrections. At the other extreme, by setting Δf = 0 in Equation 4.69, the formal NR method (in its incremental form) is recovered, namely sufficient iterations are performed for each load increment in order to converge to the actual solution before the next external-load increment is applied. Obviously, any degree of transition between the extremes Δf = 0 and Δf = 0 could be specified: for example, many small external-load increments with few iterations in each, or fewer but larger external-load steps coupled with a substantial number of corrective iterations for each of them. For overall economy, any such gradations may be combined with the constant-stiffness iteration algorithm.

4.2.5 Concluding remarks

Reliable algorithms for the non-linear analysis of concrete structures by means of the FEM should employ incremental techniques. On the basis of the preceding outline, a summary of the three most widely used versions associated with the incremental NR method (INRM) is contained in Figure 4.6. (It should be noted that all plots refer to a given load step or incre-ment.) Whether on their own, in combination, or slightly amended, they will be found to constitute the backbone of the iterative-search process in the present work.

∆f₂

∆f₁

∆f_o

∆f₂

∆f₃

∆f1

∆d₁ ∆d₂ ∆d₃ f

Figure 4.5 Incremental with one-step NR correction.

4.3 NON-LINEAR FINITE ELEMENT MODEL FOR