Ga 6 A

in orthotropic materials, with one direction parallel, and one at right angles to the fibers. Multilayer plates, in which layers of fabric or of roving are laid up parallel or perpendicular to each other, are also orthotropic. If the same number of strands or yarns is found in each principal direction (balanced construction), the strength and elastic properties are the same in those directions but not at intermediate angles; if the number of strands or yarns is different in the two principal directions (unbalanced construction), the strength and elastic properties are different in those directions as well as at all intermediate angles.

In the foregoing discussion the direction perpendicular to the plane of the plate has been neglected because the plate is assumed to be thin and the stresses are assumed to be applied in the plane of the plate rather than perpendicular to it. This assumption, which considerably simplifies the theory, carries through all of the following discussion. It is true, of course, that properties perpendicular to the plane of the plate are undoubtedly different than in the plane of the plate, and in thick plates this difference has to be taken into account, particularly when stresses are not planar.

For isotropic materials, such as mat-reinforced construction, if E is the modulus of elasticity in any reference direction, the modulus at any angle to this direction is the same, and the ratio E^/E is therefore unity. Poisson's ratio v is similarly a constant in all directions, and the shearing modulus G = E/l (1 + v). If v, for example, is 0.3, G/E = 0.385 at all angles. These relationships are shown in Fig. 2.33.

The following familiar relationships between direct stress <7 and strain, t', and shearing stress x and strain y hold:

A transverse strain (contraction or dilation) £ris caused by <7equal to

For orthotropic materials, such as fabric and roving-reinforced construction, £Land ET are the elastic moduli in the longitudinal (L) and transverse (T) directions, GLT is the shearing modulus associated with these directions, vLr is the Poisson's ratio giving the transverse strain caused by a strett in the longitudinal direction, and vLT is Poisson's ratio giving the longitudinal strain caused by a stress in the transverse direction. The modulus at any intermediate angle is and if Ci is a stress applied in the 1 -direction at an angle a with a longitudinal direction (Fig. 2.34, top), the stress CT] causes a strain e = <j/E y= T/G

in which Ey may be found from

This relationship is plotted as Ei/El in Fig. 2.34, in which 0° corresponds to the longitudinal direction and 90° to the transverse direction.

A transverse strain e2 is caused by <Ti

Unlike isotropic materials, stress CTj, when applied at any angle except 0° and 90°, causes shear distortion and the shear strain yl2 is found from yu = m:Oi/EL (2-95)

m, = sin 2 a { vLT+ | - 1 | -cos' a (1 + 2 vtr+ | -1) } (2-96)

A shearing stress Jt12 applied in the 1-2 directions causes a shear strain

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