Strand7: Web notes: Nonlinear material

Running material nonlinear problems

Beam

Beam elements can consider both the nonlinear axial stress vs axial strain behaviour and nonlinear bending moment vs curvature behaviour.

Cutoff Bar

Cutoff bars can be used to model simple elasto-plastic axial behaviour with either "Ductile" or "Brittle" cutoff. If the ductile option is chosen and the axial force in the beam reaches the set limits, the axial force is held constant but the axial stiffness is reduced to zero (i.e. the beam yields).

Stress vs Strain Curve

The plate and brick elements have a general material nonlinear capability that includes the ability to define an arbitrary stress stain curve via a stress strain table. The stress strain table is defined in one of two ways:

A series of points with stress and strain values. The solver calculates the values of stress and strain between the input points using a linear interpolation.
A simplified alternative representation of a stress strain curve is the Ramberg-Osgood equation. In this approach the stress strain curve for various metals has been reduced to the definition of four parameters (Eo, E0.7, E0.85 and n). The values of these parameters for common materials are available in publications such as MIL-HDBK 5.

When defining the stress strain curve it is advisable to keep the curve as smooth as possible. If the curve has many step changes in gradient then convergence will be difficult to achieve. Also the performance is best when the stress strain curve has a positive gradient. In practice many curves reach a yield point after which the curve has a negative gradient. This is illustrated in the following figure.

Typical Stress Strain Curve

When representing a curve of this type the solution will converge much more readily if the stress strain curve is simplified as shown in the following figure.

Idealised Stress Strain Curve

Curves with negative gradients like that in above figure can be used in Strand7 when the nonlinear elastic option is selected in the property sets. For the ElastoPlastic option, the gradient cannot be negative.

In the general plate/shell case, bending actions may be present and thus the magnitude of stress can vary through the thickness of the element. In order to consider the effect of this stress variation on the modulus of the material at the different planes throughout the thickness, the element is considered to be constructed of a number of layers. The stress is monitored at each layer and the appropriate modulus, from the stress strain curve, is used for each layer. This approach allows some parts of the plate element to be yielded whilst other parts remain unyielded. The number of layers is defined in the plate properties input.

The yield criterion is actually checked at each of the Gauss points on the plate for each layer and the modulus calculated at each of these points. The number of Gauss points depends on the plate type. The Quad4 element has 4 Gauss points whilst the Quad8 element has 9.

For brick elements strain is monitored at the Gauss points and the modulus at each point adjusted using the stress strain table.

Failure criterion - The plate and brick properties allow the user to select between various yield criteria such as Von Mises, Tresca and Max Stress. The strains calculated from these yield criteria are used to calculate the modulus from the stress strain table. Different criteria are required for different materials.

Most stress strain tables are defined in the positive stress strain quadrant only and the same curve is assumed to apply to behaviour in tension and compression. This type of stress strain curve is shown in the figure above.

If the tension and compression behaviour of the material is different, a stress strain curve can be defined for both positive and negative strains for use with the Max Stress criterion in nonlinear elastic material. For elasto-plastic material, only the positive side is used.

Stress Strain curve with different behaviour for tension and compression.

Example - A typical material nonlinear analysis is shown in below.

Simply Supported I Beam

In this analysis a steel I-beam is subjected to a large overload and a plastic hinge is formed in the centre of the beam. This problem was run using the material and geometric nonlinear solvers. As the collapse load is approached the load must be applied very gradually (1% steps) to ensure convergence. Collapse occurs when the material completely yields through the depth of the beam and the stiffness of the beam is no longer sufficient to resist the applied load. Complete collapse is signified by a singular stiffness matrix or very large displacements. The collapse load predicted by Strand7 for this model agrees very closely with the traditional manual methods of analysis for a plastic hinge. The deflection of the beam as a function of the applied load is shown in below. Note the nonlinear shape of the curve.

Beam Deflection vs Applied Load