Strain displacement relationship

First-order shear deformation theory calculations are based on a Mindlin assumption which states that sections perpendicular to the reference plane remain planar. This assumption allows one to write:

where:

• uo is the reference plane displacement of point (x, y) in the x direction

• vo is the reference plane displacement of point (x, y) in the y direction

• wo is the reference plane transverse displacement of point (x, y)

The figures below shows the laminate deformed shape in the XZ and YZ planes, respectively.

The strains are then calculated by differentiating the displacement functions:

These equations can be rewritten in the form:

where εx0, εy0 and γxy0 are called the mid plane strains and:

are called the curvatures. Positive curvatures are shown in the figure below.

Transverse shear deformations are given by:

Note:

In CLT, both γxz and γyz are neglected.

Look up more details

First order shear deformation theory

Constitutive equations

Off-axis relationships

Shell stress resultants

Laminate stiffness matrices

Transverse shear stiffness matrix

Shear deformation theory references

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