# Use Inplane Effect

Compressive and tensile loads change the capacity of a structure to resist bending. Compressive loads decrease resistance to bending. This phenomenon is called stress softening. On the other hand, tensile forces increase bending stiffness. This phenomenon is called stress stiffening.

To consider the effects of in-plane loading to the stiffness of the model select Use Inplane Effect in the Static dialog box.

By activating Inplane Effect, the stiffness properties become a function of both the static loads and the deformed shape. A geometric stiffness matrix K_{G} (also known as initial stress, differential
stiffness matrix, or stability coefficient matrix) is added to the conventional
structural stiffness matrix.

The displacements are computed with respect to the original geometry of the structure, and the change in geometry is reflected only in the geometric stiffness matrix. It is also assumed that the magnitude and direction of the loads remain fixed and their points of application move with the structure.

Since the geometric stiffness matrix depends on the displacements, the linear
static analysis is performed in two stages. In the first stage, the displacements
{u_{i}} are computed using the conventional stiffness matrix [K]. In the second
stage, the geometric stiffness matrix [K_{G}(u_{i})] is established based on the computed displacements, {u_{i}}, and added to the conventional stiffness matrix [K]
to solve for the new displacements, {u_{i+1}}. The system of equations
for linear static stress analysis in the presence of in-plane effects can be written as:

( [K] + [K_{G}(u_{i}) ]){u_{i+1}} = {F}

The geometric stiffness matrix K_{G} is built from the same
shape functions used to form the conventional stiffness matrix. It is symmetric,
but unlike the conventional stiffness matrix, it does not contain terms with elastic
moduli. It depends on the element geometry, displacement field, and the state of
stress. The geometric stiffness matrix K_{G} is in general indefinite, and hence cannot
be inverted.

Ideally, the displacements {u_{i+1}} could be used to compute the new geometric
stiffness matrix [K_{G}(u_{i+1})] and hence compute yet another set of solutions,
{u_{i+2}}, and so on. The iterations can be carried out until successive solutions do
not differ by more than the specified tolerance. In Simulation, the in-plane effects are considered
by performing one iteration only.

An accurate solution for considering the effect of loads on the stiffness (capacity of resisting loads) requires the use of geometrically nonlinear analysis.

If the applied in-plane (compressive) load is in the vicinity of the buckling load, the iterations may diverge, indicating instability. Such problems warrant the use of buckling analysis. In buckling analysis, the overall structural stiffness matrix, which is composed of the normal and geometric stiffness matrix, becomes singular with respect to the buckling modes.