### 5.2.2. Interlaminar Normal Stresses

A cylindrical coordinate system is used for describing an arbitrary doubly curved shell [ 9 ]. Curved shell geometry, illustrated in Figure : Doubly Curved FE Geometry, is described by the coordinates and it is subdivided into angular segments with the apex angles and and the constant curvature radii of the centerline and .

Figure : Doubly Curved FE Geometry

The radial equilibrium equation becomes

| (5–6)

where:

Each segment is embedded between four cross-sections which are assumed to remain straight and perpendicular to the midplane. Studies on singly curved plates ([ 10 ] and [ 13 ]) show that the shear terms have only small effect and are thus neglected here. Equation 5–6 reduces to

| (5–7)

where only direct stresses appear and the material law reduces to

| (5–8)

where are components of the 3-D stiffness matrix expressed in reference coordinates which are parallel to the principal direction. The evaluation is described in Reference Coordinates. The term indicates free layer strains due to spatially constant changes of temperature and moisture content .

| (5–9)

Direct strains in Equation 5–8 can be expressed through the displacements u, v, and w. The kinematic relations in the modified coordinate system are:

| (5–10)

The in-plane deformations u and v are expressed by the laminate deformations and . The through-the-thickness coordinate is replaced with the radial coordinate and the curvature radii of the midplane:

| (5–11)

The direct strains become

| (5–12)

The combination of the material law of Equation 5–8 with the kinematic relations of Equation 5–10 leads to the direct stresses expressed by the deformations:

| (5–13)

Equation 5–13 is combined with the radial equilibrium Equation 5–7 and the differential equation of the through-the-thickness displacement is:

| (5–14)

where:

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