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Chapter 10 Second-order Direct and Advanced Analysis of Structures
10.3 Methods of analysis
10.3.2 Formulation for Nonlinear Numerical Methods
Every nonlinear numerical method has its own merits and limitations. None of them is remarkably superior to others in all cases. Their selection for a particular problem depends heavily on the type and constraint of the problem and the objective of study. For example, to determine the displacement of a structure under specified applied loads as required in most practical design, a load-control scheme should be chosen. If a prescribed displacement is imposed, a displacement-control scheme should be adopted.
However, these two methods may not achieve convergence in tracing the snap-through curve or the snap-back curve. To select an appropriate nonlinear numerical method, the user should therefore have a general understanding on the characteristics of these methods. The properties and formulations of some commonly used schemes are briefly described in this section. In Section 10.3.4, a comparison among the schemes is made.
In general, the incremental-iterative equilibrium equation of a system, which is not necessarily controlled by the load, can be written as,
F
F
K T
u ki
u
k
i
(Eqn 10.26)
in which
F and
u are respectively the out-of-balance forces and the corresponding displacement increments in the system;
F and
u are respectively the reference load vector and the resulting displacements; and ik is a control parameter to be determined according to various imposed constrained conditions. The superscript k refers to the number of load cycle while the subscript i represents the number of equilibrium iteration within a load cycle. By selecting a suitable numerical scheme for a particular problem considered, the above incremental-iterative equation can used to trace the nonlinear load-deformation curve of the structure. If the selected numerical scheme is successful, the load limit or load-carrying capacity of the structure can be determined from the curve. Furthermore, the structural response for the post-buckling range can also be obtained.In software NIDA, to use the nonlinear numerical methods, a nonlinear analysis case must be set up first by clicking the <Analysis> <Set Analysis Cases …> in top tool bar and the following template is popped,
To use various numerical methods, click <Add>, followed by <Nonlinear Analysis>
and the following template is popped up.
The choices of the numerical methods include Newton-Raphson method, single displacement control method and arc length method + minimum residual displacement method. To select one of the numerical methods, go to the <Numerical Method>
selection.
10.3.2.1 The Pure Incremental Method
The pure incremental method for nonlinear analysis is simple and is the earliest nonlinear solution method. Its basic procedure is to divide the total load into a number of small load increments. In each load step, the stiffness of a structure is determined first from the last known structure geometry and the loading state. It is then used to predict the next displacement increment. The sign of the determinant of the updated stiffness matrix will govern the direction of subsequent load step. The linearized displacement increment is calculated by solving the tangent stiffness matrix and the load increment.
Once the displacement increment is obtained, the coordinates of structure are updated and then the process is repeated until the desired load level is reached.
In general, this approach is capable of handling both the snap-through and the snap-back problems because it does not require any iteration and thus does not have divergence problem. However, as no equilibrium check or iteration is carried out, unavoidable drift-off error is accumulated in each increment and the error after a number of load steps may make the solution greatly deviated from the true equilibrium path. This drift-off error cannot be estimated and thus the accuracy of the resulting load-deflection curve cannot be assessed. The method to minimize this error is to employ a smaller load step of which the magnitude is, unfortunately, quite difficult to assess. Indeed, there is no guideline suggested for each load step. More importantly, the pure incremental method usually over-estimates the ultimate capacity or the limit load of a structure. This is unsafe and undesirable in practical design. Nevertheless, this simple method is still widely used for nonlinear analysis, especially in commercial packages for nonlinear analyses.
Drift-off Error in Displacement
Drift-off Error in Force
FFFF
Linearized Path
True Equilibrium Path
Load, F
Displacement, u
Figure 11 Pure Incremental Method with Constant Load Increments
Figure 10.11 Pure incremental method with constant load increment
10.3.2.2 The Newton-Raphson Method
Only the Newton-Raphson method gives the response of a structure at the input load in terms of buckling strength and therefore it should be use when the engineers want to check whether or not a structure is adequate when under a set of factored design loads. In t method, iteration is activated to obtain the equilibrium condition between the applied forces and the internal structural resistance within a load step. Unlike the pure incremental method in which no equilibrium check is performed, the unbalanced force is dissipated via the iterative procedure and can therefore be eliminated by this method.
Being free from the drift-off error, the solution is more accurate but the computational time is increased when compared with the pure incremental method.
Conventional Newton-Raphson Method
Modified Newton-Raphson Method Figure 10.12 The Newton Raphson method
10.3.2.3 The Displacement Control Method
Unlike the load control methods previously described, a constraint equation for displacement is imposed in this approach. This method simultanesouly possesses the capacity of traversing the limit point without destroying the symmetrical property of the tangent stiffness method. A single degree of freedom is chosen to be the steering displacement degree of freedom for control of the advance of the solution for equilibrium path, and the magnitude for each increment must be decided.
Figure 10.13 The displacement control method
The constant displacement method does not exhibit any difficulty in passing the snap-through limit point but fails to converge in snap-back problems. Thus, it is usually used in conjunction with other solution schemes in order to solve general nonlinear problems.
10.3.2.4 The Arc-Length Method
The basic concept of the spherical arc-length method is to constrain the load increment so that the dot product of displacement along the iteration path remains constant in the 2-dimensional plane of load versus deformation.
The procedure of the spherical arc-length method is illustrated below. Owing to its accuracy, reliability and satisfactory rate of convergence, it is probably the most popular method for nonlinear analysis and it was noted to be robust and stable for pre- and post-buckling analysis.
Figure 10.14 The arc-length method
10.3.2.5 The Minimum Residual Displacement Method
The basic idea of this method originally proposed by the author of this book (Chan, 1988) is to minimize the norm of residual displacement in each iteration.
The graphical representation of the procedure is demonstrated below. From Figure 10.15, it can be seen that this constraint condition enforces the iteration path to follow a path normal to the load-deformation curve. It adopts the shortest path to arrive at the solution path by error minimization and thus is considered to be an optimum solution. In addition, the procedure is much simpler to use than the arc-length method.
Generally speaking, owing to its efficiency and effectiveness in tracing the equilibrium path, the minimum residual displacement method is usually chosen to perform the iterative procedure and combined with the part for load size determination in the first iteration by the arc-length method.
Figure 10.15 The minimum residual displacement method