Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 38

A beam of rectangular cross section of length 2L, width 2W and depth h is considered within a Cartesian coordinate system such that the x₁-axis is in the axial direction and the x₂-axis is in the in-plane transverse direction whereas the x₃-axis is in the through-thickness direction, as shown in Figure 2.2. The origin is selected to lie at the mid-point of the upper surface of the beam.

Figure 2.2 Schematic diagram of part of a rectangular orthotropic plate of length 2L and depth h, and coordinate system. The x₂-axis and u₂ displacement are directed out of the plane of the page, and the width is denoted by 2W.

The modelling assumes that the beam is in equilibrium such that the equilibrium equations (2.120)–(2.122) are satisfied where the stress tensor is symmetric as in (2.123). The displacement components are denoted by ui. The infinitesimal strains are then given by the relations (2.143). The stress-strain relations are assumed to be of the orthotropic form (2.196), namely,

(2.209)

The beam is assumed to be in a state of orthogonal bending combined with uniform through-thickness loading such that

(2.210)

and

(2.211)

where the strain parameters ε¯A, ε¯T, ε^A and ε^T, the through-thickness stress σt and the temperature difference ΔT are assumed for the moment to be known. From (2.143), (2.209) and (2.211), the in-plane strains and shear stress are given by

(2.212)

The expression for ε22 in (2.209) is first solved for the stress component σ22 so that

(2.213)

It then follows from (2.209) that

(2.214)

where

(2.215)

In addition, from (2.209)

(2.216)

where

(2.217)