3.2 Application to a Snapping-through Plate

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Prediction and modeling of local instabilities is an important issue in engineering problems.

These phenomena are characterized by a local or temporal decrease of the load-carrying

capacity. This means that the load cannot be used as a time parameter since it is not

monotonically increasing. In general, powerful techniques such as the Riks method (Riks

1987) (Crisfield 1983), which use the path length as the time parameter, have to be followed.

However, in some applications, such as the one discussed in this section, other more simple

time parameters can be selected.

Consider the bent plate in Figure 3.4 loaded by a force in the center. As the force

increases, the plane bends until it snaps through. The snapping is an instability accompanied

by a complete loss of force-carrying capacity. Therefore, if the force is increased with time

(or pseudo-time), equilibrium is lost at the onset of instability. The time increments are

decreased, but the Newton–Raphson procedure fails to find a solution. This problem can

be solved by taking the displacement u of the loading point in the direction of the force

as a parameter since it is monotonically increasing with time. Figure 3.5 shows the forcedisplacement

curve for the loading point. Before the onset of instability, marked by the

force maximum, the force steadily increases. During snapping-through, the force crosses the

zero-axis (unstable equilibrium, characterized by a negative force-displacement slope) while

GEOMETRIC NONLINEAR EFFECTS 149

R

R

F

F

R/20

ν = 0.3

Figure 3.4 Bent plate

−2

−1

0

0

0.5 1

1

1.5 2

2

2.5 3

3

3.5 4

4

5

6

10 u

R (−)

104 F

ER2 (−)

Figure 3.5 Force-displacement curve for the bent plate

decreasing steadily, reaches a minimum and increases again until a new stable configuration

is found (stable equilibrium, characterized by a positive force-displacement slope). Notice

that at times an upward force must be exerted to keep the plate in its position. In the new

stable configuration, the force is zero. Increasing the force again now leads to a monotonic

force-displacement curve. This is an example of a strongly nonlinear behavior. It also shows

how a diligent choice of the loading parameter can lead to convergent solutions even in the

presence of instabilities. Other applications of the instability theory are treated in (Mang et

al. 2001) and (Kim et al. 2003).

150 GEOMETRIC NONLINEAR EFFECTS

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