Why do I get no stresses on the top or bottom side of a member loaded with temperature (heating on the top side) if the member has no elastic foundation? Or more specifically, why does the upward curved member (due to heating on the top side) have tension stress on the bottom side if the member has elastic foundation? There must be compression stress on the bottom side.


The topic can be easily illustrated on a single-span beam. For this, three structural systems are described below. These models are documented in the attached file.

System 1

Statically determined system (no foundation), dT = 80 ° on the top side

The member is curved upwards, but is free of stress in itself.

System 2a

Like System 1, but with an additional member elastic foundation. The member elastic foundation is entered without a possible failure (nonlinearity).

If you would display the stresses sigma_x of the member for System 2a, you obtain compression on the top side of the member and tension on the bottom side of the member (see Figure 01).

Due to the curvature of the member and the existing member elastic foundation, the contact force p-z occurs, which should prevent the member curvature upwards (see Figure 02).

These contact forces p-z (Figure 02) are caused by the member curvature due to the temperature and the applied member elastic foundation. The illustrated contact forces can be replaced by the member load opposed to the curvature. This is shown in System 2b in the example file.

System 2b

The member elastic foundation is removed and a variable member load is entered in the Z-direction.

When comparing the results (for example, deformations u-z) on both System 2a and System 2b, you obtain the results with the same value (see Figure 03). 

Moreover, you can also display the stresses sigma_x for both System 2a and System 2b. These have also the same value (see Figure 04).

System 3 should only document the stresses due to the temperature difference on a statically determined system (without foundation).

The results documented in the "single-span beam" example can also be transferred to the surfaces with elastic foundations.


temperature stresses tension surfaces beam temperature load foundation


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