1.3.2 Double cantilever elastic beam under point load

Product: ABAQUS/Standard  

This example concerns the response of an elastic beam, built-in at both ends, subject to a suddenly applied load at its midspan (see Figure 1.3.2–1). The central part of the beam undergoes displacements several times its thickness, so the solution quickly becomes dominated by membrane effects that significantly stiffen its response. The purpose of the example is to illustrate the effect of time step choice on solution accuracy, to compare direct and automatic time stepping, and to verify that the standard Newton and quasi-Newton solution techniques provide the same results in a relatively nonlinear case.

A number of factors are involved in controlling solution accuracy in a nonlinear dynamic problem. First, the geometry must be modeled with finite elements, which involves a discretization error. In this example the beam is modeled with five elements of type B23 (cubic interpolation beam for planar motion). Since a 10 element model gives almost the same response, we assume that this model is reasonably accurate. Second, the time step must be chosen. This source of error is studied in this example by comparing results based on different time steps and different tolerances on the automatic time stepping scheme. Third, convergence of the nonlinear solution within each time step must be controlled. This aspect of solution control is common to all nonlinear problems.

The quasi-Newton solution technique can be less expensive in terms of computer time than the standard Newton technique because it avoids the complete recalculation of the Jacobian. Each newly computed Jacobian is based on the current Jacobian. This savings becomes significant in large models, in cases when the Jacobian is expected to vary smoothly over time. This example is too small for the quasi-Newton method to show significant savings in computer time, but it demonstrates that, with correctly chosen tolerances, the quasi-Newton method solves the nonlinear system with no loss in accuracy.

Problem description

Results and discussion

Input files

Tables

Table 1.3.2–1 Energy balance at end of run—analyses with fixed time increments.

Time incrementKinetic energyStrain energyExternal workNumerical energy loss
N-mlb-inN-mlb-inN-mlb-in
25 s5.5649.219.1016924.862200.8%
50 s5.5949.516.9515023.162052.7%
100 s6.2355.213.5612021.921949.7%

Table 1.3.2–2 Energy balance at end of run—analyses with automatic time increments.

Half-stepKinetic energyStrain energyExternal workNumerical energy loss
toleranceN-mlb-inN-mlb-inN-mlb-in
44.5 N (10 lb)4.8042.520.2317925.20223 0.7%
222 N (50 lb)5.6149.618.6516524.642181.6%
4448 N (1000 lb)4.7742.215.4913721.931947.6%


Figures

Figure 1.3.2–1 Double cantilever elastic beam.

Figure 1.3.2–2 Fixed time step results for an elastic beam under point load.

Figure 1.3.2–3 Automatic time step results for an elastic beam under point load.

Figure 1.3.2–4 Peak half-step residuals for the elastic beam under point load.