### Understanding Geometric Nonlinearity - 1 by Kotur Raghavan

**Preamble**

Any structure deforms when
subjected to loads. The basic question is whether the deformations affect the
structural behavior to any practically significant extent. If they do, we know
that the small deformation theories are inadequate and it is necessary to
invoke large deformation formulation to obtain acceptable results. In other
words, one has to invoke the “Geometric
Nonlinearity” option while using FEA packages to obtain good results. The
general thumb-rule is that when the deformations are comparable to the
dimensions of the structure, the effects of geometric nonlinearity become
noticeable.

In the small deformation theory
or linear elasticity, the basic assumption is that the displacements in orthogonal
directions are uncoupled. In Cartesian geometry, the displacement fields X- and
Y- directions are independent of one another. In the cylindrical coordinate
system, the displacement fields in the radial and tangential directions are
assumed to be independent of one another. In the present article, a simple
example of a simply supported bar is chosen to demonstrate the effect of the
coupling of displacement fields.

**Case Study Problem and
Modeling Options**

The problem for the study is
shown in the figure below.

Fig. 1

Only the left half of the beam is considered for analysis. Two-dimensional plane stress elements are chosen. At the plane of symmetry, the axial displacements are constrained. At the supported end, two different sets of constraints are applied. They are described in the figure below.

Fig. 2

The constraint sets are referred
to as C1 and C2. The black dots denote the nodes where the transverse
displacements are constrained. For both the models, linear as well as nonlinear
analyses have been carried out. The four models will be referred to as LC1,
LC2, NLC1, and NLC2.

**Results of Analysis**

The computed values of the
maximum deflections for the four cases are summarized in the table below.

Table 1

There is no difference between
the responses for the two linear cases. For the NLC1 case, we observe a small amount
of nonlinearity effect. The nonlinearity effect is noticeable when the
deflection is comparable to the beam thickness. For the NLC2 case, however, the
effect of geometric nonlinearity is more pronounced. This is due to the effect
of coupling of displacements in orthogonal directions which is inherent in
formulations applicable to large deformations. This aspect is discussed in the
next section.

**Analysis**

In the figure below, the line AB
corresponds to the supported edge before deformation. Due to the applied load and
the resultant deformation, it will take a new position that corresponds to the
broken line A1-B1.

Fig. 5

Due to requirements of geometric compatibility,
points A and B will have non-zero displacements in the vertical (-) direction.
This requirement does not arise in a linear analysis. In the case of nonlinear
analysis, the constraint set C1 permits the points A and B to take the new
positions A1 and B1.

However, for the nonlinear
analysis involving the constraints C2, points A and B do not have the freedom
to occupy the positions as indicated. Instead, points A and B are restrained to
remain along lines PP and QQ respectively. This factor adds additional stiffness.
This additional stiffness is dependent on the extent of deformation and hence can
be considered to be the “geometric nonlinear” effect. From the figure, it can
be noticed that this effect starts manifesting at a deflection-to-thickness
ratio of 0.5.

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