Stiffness of structures depends on its geometry and material. For small displacements, the changes in geometry are small and the linear analysis is adequate to predict the displacement. However for the large displacement problems, changes in geometry alter the stiffness of the structure significantly and for accurate determination of displacement, the effect of geometric nonlinearities should be taken into account in analysis. In the case of plate bending problem, stress due to membrane action is normally neglected in linear analysis. However, when the plate undergoes large bending deformation, the membrane tensile stress induced in the plate stiffens it and produces less bending deformation. This stiffening action in plate due to large displacement effect is shown in Figure-5.3. An exactly opposite effect can also be observed for some cases where in the stiffness of the structure decrease with the increase in load. Figure-5.4 shows the example of softening action due to effect of large displacement. The compressive stress induced in the cylindrical shell due to pressure reduces the stiffness of the structure and at high pressure load this reduction is significant and geometric nonlinear analysis will predict more displacement for the same load compared to that of linear analysis.
Several topics related to geometric nonlinear analysis are discussed in this chapter. The terminology associated with the load-deflection curves and some important locations in the load deflection curves are discussed. The finite element formulations for total and updated lagrangian concepts of geometric nonlinear problem are explained. The procedure for evaluating the large displacement and geometric stiffness matrices are explained for a simple 2D-truss element. The concept of linear and nonlinear stability analysis and the procedure for evaluating the buckling load are discussed. The role of imperfection in the pre and post buckling range is also covered in this chapter.
5.7.1 Load Deflection Curves
The overall nonlinear behaviour of many structures can be characterized by a load - deflection curve. A typical load-deflection (f–d) curve is shown in Figure-5.5. The terminology associated with the f–d curves are discussed below:
Equilibrium path
A smooth curve in the f–d chart is called a path and if this path represents a static equilibrium configuration it is called an equilibrium path. Each point in equilibrium path is equilibrium point.
Chapter 5: Structural Nonlinearity and Finite Element Analysis
Fig. 5.3 Central deflection of clamped square plate under uniform pressure (Stiffening behaviour)
Fig. 5.4 Central deflection of cylindrical shell under uniform pressure (Softening behaviour)
Fig. 5.5 Primary and secondary equilibrium paths
Reference state
The reference state is the origin of the f–d curve as it is the state from which the loads and deflection starts and also measured.
Primary and secondary path
The path that originates from the reference state is termed as the primary or fundamental path. It continues from the reference state until a state called critical point is
Chapter 5: Structural Nonlinearity and Finite Element Analysis
reached. The paths that are not primary but links with it at a critical point is called secondary path.
Various types of f-d curves are shown in Figure.5.6. Some important locations in these curves are discussed below.
Critical Point
The critical point is a location in equilibrium path at which the structure becomes physically uncontrollable or marginally controllable. The relation between the load and deflection is not unique at critical points. Two types of critical points are discussed here:
• Parallel to defection axis and at the location which is tangential to the to the equilibrium path horizontally is called the Limit point (L)
• The crossing point of two or more equilibrium paths is called the Bifurcation point (B)
Turning Point (T)
Parallel to the load axis and at the location which is tangential to the equilibrium path vertically i s called as turning points. As such points have potential to affect the performance of certain solution methods; they are only of some computational significance and are not critical points .
Failure Points (F)
Some points at which a path suddenly stops or break due to because of physical failure are called failure points. The funct i onal equil i bri um of t he s t ruct ure woul d be regai ned if the failure is local in nature even after moving to another equilibrium path dynamically. On the other hand, it would be detrimental if the failure is global in nature and the functional equilibrium of the structure is never regained.
Different possible types of equilibrium paths are given in Figure.5.6. Symbols F and L denotes the failure and limit points respectively. Composites and glassy crystals exhibit linear responses as in Figure.5.6 (a). A stiffening response shown in Figure.5.6 (b) is typical of structures under tensile loadings. The stiffness effect comes from the geometry transformations in response to the applied loads. Many common structures exhibit response as in figure.5.6(c) wherein immediately after softening a linear response is followed. Curved
structures like shallow arches exhibit snap through response as shown in Figure.5.6 (d) where in after the second limit the softening response is combined with hardening behaviour. These responses are unstable and have negative stiffness’s between the limit points. Typical structures like trusses, domed shaped members, exhibit snap-back responses shown in Figure.5.6 (e) where in the response curve reverses back in itself with resurfacing of turning points. The state between the two turning points may be in equilibrium and stable which can be of physical possibility.
In all the responses discussed so far, the response behaviour itself is unique in nature. In Figure.5.6 (f) and (g), interesting points called as bifurcation points exist which are popularly known as buckling points. These points provide more than one response path and proceeds towards a path based on least energy over the others. Typically thin structures under compressive loads exhibit such bifurcation points. Many combinations of bifurcation and turning points may occur under compressive load as shown in Figure.5.6 (g). Thin cylindrical shells under axial compression exhibit such a complicate response.