Plane stress software

There are also no through thickness shear stresses. We could load the Plane Stress model in Figure 1 with a biaxial load and calculate sigma x and sigma y. There is no sigma z. We can also calculate the corresponding in-plane strains e-x and e-y. If we apply a shear load we can find shear stress sigma xy and shear strain e-xy.

What is interesting is that we can also calculate the through thickness strain e-z. This is not zero, as the model is free to thin down in z as it stretches in x and y. There is no out of plane constraint to prevent this. In some solvers we can recover the through thickness strain e-z and use it to calculate the change in thickness If the model is in bi-axial compression then the z section will get thicker.

This is usually considered a secondary strain and may not be available for output — but it is there and can be calculated manually if needed! Figure 2. Thin walled Aluminium Tang transferring load into composite structure. Figure 2 shows an aluminum lug component. The lug protrudes from a composite sheet layup which has plies positioned and bonded over the tangs or legs and lower body section. The tangs transfer the load applied to the lug into the composite structure.

In practice the plies would be stepped to allow a smooth shear transfer through the bond into the composite. These balance the applied lug load. The key assumption here is that through thickness stresses are zero and the in-plane stresses are constant through thickness in the component. This means the local detail of the shear load transfer from composite to tang is poorly modeled.

However the focus of this analysis is to check sizing of lug and tang cross section clear of the composite, using in-plane stresses. The thickness of the component is small compared to other dimensions. This value is input as the actual thickness in the Plane Stress element definition. The areas of interest are around the lug and the shoulder radiuses.

In the real world the stress state at the stress concentrations would be 3D and through thickness sigma z stresses and shear stresses would balance locally. However it is very reasonable here to assume the in-plane stresses dominate. This is the same assumption implicit in most traditional stress concentration Kt calculations found in handbooks.

Figure 3. One of the convenient features of the Plane Stress analysis is that it is a strictly 2D analysis, so only 3 Degrees of Freedom DOF have to be constrained in-plane translations x, y and rotation about z axis. This lends itself to the minimum constraint method with balanced load, described in a previous article. In a 2D case this degenerates to a method. This allows the reaction load in the tangs to be applied directly as diffused balancing loads. It would be difficult to simulate this boundary condition via constraints to ground.

The through thickness e-z strain and hence thinning of the tangs could be calculated as a secondary effect. Figure 4 shows the essence of the Plane Strain method. Again, 2D planar Elements are used, but with subtly different assumptions.

The in-plane stresses x, y and xy are developed as before. This works well in representing thick structures such as shown. The presence of this much material tends to stabilize the component and prevent it straining in z. This also means that constant through thickness z stresses are developed in the structure. This stress-strain material relationship is defined in 2D Plane Strain elements used in this type of analysis. The assumption is that the stress state at this cut section will be duplicated at any xy plane cut z station through the component.

The component is assumed to be prismatic having a constant cross section down its length. In practice we use this method where the stress state is varying slowly from plane to plane in a deep component. There should be enough material through depth to stabilize and eliminate the through thickness strain. This is the same principle used on fracture toughness specimens shown in Fig. A failure under plane strain conditions is shown for the center section of the thick specimen.

The failure at the free edges and the thin section is a different mode, more like a plane stress state. A plane strain FEA model would by definition be a good representation of the centerline thick specimen behavior, but not of the free edges or the thin specimen.

Here the section is constant and deep enough so that we can assume the stresses are also constant with depth. The 2D plane strain analysis mesh is shown sectioned into the 3D component in Fig. A very fine 2D plane strain mesh can be used, which will run very quickly compared to a full 3D model.

The constraint method is used as before. The loading needs to be considered carefully. It is useful to pick a section, such as the single tang and estimate the nominal or average stress in this section for the full component. This can be used as a sanity check in the plane strain analysis. Incorrect loading is probably the main cause of error in this method. The results of the analysis are shown in Fig. The stress quantities used will depend on the solver used. Some solvers ignore the z direction stresses as secondary and recover the in-plane stresses.

The principal stresses and von Mises stresses then relate to a 2D in-plane stress state. If the z direction stress is recovered then it should be clearly identified, so that the 2D in-plane stress state in the x-y plane can be identified.

What exactly does the z direction stress represent? It is the stress developed due to the enforcement of zero z direction strain. The stress acts as if the free end faces of the prismatic section were fixed. At the central plane of a deep section component these will be the complementary stresses needed to hold the zero z direction strain state.

In reality as we move toward the free surface faces, the z-stress drops to zero and becomes a plane stress distribution as seen in the thick fracture mechanics specimen. In many cases, such as a pressurized cylinder, the end faces are capped and will in fact develop an axial stress due to axial forces. This will be a different stress from the induced axial stress in the plane strain analysis.

A hand calculation will be needed to calculate the axial stresses, or possible a supplementary axisymmetric model for pressure vessels.

For comparison, a half symmetry full 3D analysis of the deep tang component was done and the results are shown in Fig. The nominal stress across the upper single tang leg is identical in both cases—remember this is the basis of any sanity check. The local shoulder stresses are lower by a small percentage in the full model. This is for three reasons. First, the relatively coarse 3D tet mesh is inferior to the very fine 2D plane strain local mesh. A convergence check on the 3D model has not been carried out.

In this case, the effect is negligible as the fillets are away from the shoulder regions. In many components, however, there will be local fillets, and run out details. It takes advantage of the strength of Python script language and sets up the analysis model parametrically. SO-Consolidation is aimed at the evaluation of one-dimensional consolidation tests.

In order to obtain coefficient of consolidation and secondary compression index, each loading step can be analyzed by two methods, logarithm and square root of time. All software and resources in the CESDb. All downloadable or viewable content available on CESDb. You agree that you bear sole responsibility for your own decisions to download or use any of the software listed.

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