Contact Constraints as Boundary Conditions in FEA Problem: The rectangular cross-section beam of Example 8-2 is supported in sliding bearings and loaded in bending by a transverse force at a fixed location in x as shown in Figure 8-11a. Construct an FEA model using contact constraints for the

Question

Contact Constraints as Boundary Conditions in FEA

Problem: The rectangular cross-section beam of Example 8-2 is supported in sliding bearings and loaded in bending by a transverse force at a fixed location in [latex]x[/latex] as shown in Figure 8-11a. Construct an FEA model using contact constraints for the boundary conditions and compare its prediction of deflection with the results from Example 8-2.

Given: The material is steel. The beam dimensions are shown in Figure 8-11a. Consider two loading cases, a load of [latex]F[/latex] = 250 lb as in Example 8-2 and a larger load of [latex]F[/latex] = 1000 lb.

Assumptions: The supporting bearings are considered to be much stiffer than the beam. Beam weight is negligible compared to the applied load. The bearings have 0.001-inch clearance around the beam cross section to allow sliding.

Accepted Answer

1 Figure 8-15 shows an FEA mesh with contact-constraints (boundary conditions) applied at [latex]A[/latex] and [latex]B[/latex]. These constraints are applied to the bearings, not to the beam. They do not allow the beam to intrude upon the bearing geometry but do allow nodes to pull away from the bearing surface. Thus they can support a compressive load between beam and bearing but not a tensile load. As the load is applied to the beam and it deflects, the nodal contact will vary along the bearing length, just as happens with the real system. The beam is fixed against movement in the [latex]x[/latex] direction by a boundary constraint at point [latex]C[/latex] on the neutral axis. By placing this [latex]x[/latex]-constraint at the beam center, both ends of the beam are allowed to slide along the bearings in the [latex]x[/latex] direction as in the real system. The bearing geometry in the FEA model is given rounded edges to avoid creating infinite stresses that would result from point contact at the ends of the bearings as the beam deflects.

2 Figure 8-16a shows the stresses and deflection that result when a 250-lb transverse load is applied in the center of this beam. Note the similarity to the simply-supported case with the same load as shown in Figure 8-14. The beam contacts the bearings at their bottom inner corners (points [latex]A[/latex] and [latex]B[/latex]), where stress concentration can be seen. The upper surface of the beam does not contact the top surface of the bearings at points [latex]A^{\prime}[/latex] and [latex]B^{\prime}[/latex]. The clearance there is 0.000517 in, similar to that shown in Figure 8-12. The maximum deflection calculated with this model at this load is –0.00099 in, the same as that of the simply supported FEA model in Example 8-2.

3 Figure 8-16b shows the stresses and deflection that result when a 1000-lb transverse load is applied in the center of the beam. The increased load causes the beam to deflect sufficiently to cause the top of the beam to contact the upper bearing surfaces at points [latex]A^{\prime}[/latex] and [latex]B^{\prime}[/latex] where stress concentration can be seen. The beam is now behaving more like the fixed-fixed model of Example 8-2. Note that the deflection with a 1000 lb load is only 3.18x as large as that with a 250 lb load. If the simply supported model were valid in both cases, then one would expect the deflection to be 4x larger, in direct proportion to the increased load. The change in boundary conditions as a function of load makes it a nonlinear system.

4 Figure 8-17 shows a plot of the beam deflection as a function of the fraction of maximum load applied in each case, which can be thought of as resulting from a slowly increasing application of each load from zero to its maximum value. For the 250-lb load case, the increase in (negative) deflection with increasing load is linear. For the 1000-lb load case, the deflection increases linearly until the beam ends hit the upper bearings, at which point (labeled [latex]A[/latex]), the slope abruptly changes because the beam is suddenly stiffer. The curve becomes nonlinear beyond point [latex]A[/latex] due to the changing points of contact as it deflects. This shows the nonlinear response of the beam to a load that “takes up” the bearing clearance, and changes the boundary conditions.*

* This phenomenon is often used to obtain a desired result such as with variable rate valve springs used in automotive valve trains. Some of the spring coils are wound closer together than others (see Figure 14-2a on p. 789) so that as the spring deflects in use, the closely-spaced coils touch and become solid. This changes the spring stiffness in mid-deflection and helps reduce spring vibration. Truck suspension systems use a similar concept by adding stiffer (helper) springs that are engaged only when suspension travel exceeds a certain level, as happens when overloading the vehicle or in hitting a large bump. When the gap is taken up, the suspension stiffness suddenly increases to limit deflection.

Question 8.3: Contact Constraints as Boundary Conditions in FEA Problem: T...

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