ABSTRACT
In Chapter 5, we saw that when the material behaviour is linear-elastic, the stress at any point on a cross-section subjected to bending about the z-axis given by:
σ ε θ
= = − = −E yE x
My I
d d
(6.1)
and the curvature is calculated as:
κ θ
( ) ( ) ( )
x x
v x
x
M x EI
= = =
d d
d d
2 (6.2)
In addition, we saw that when the curvature can be expressed as a continuous function of x (i.e. the distance along the beam), the deflection of the beam can be obtained by twice integrating Equation 6.2:
v x x x x M x EI
x x( ) ( ) ( )
= =∫∫ ∫∫κ d d d d (6.3) where, for each integration, a constant of integration must be introduced and evaluated from the support conditions or boundary conditions of the problem. For nonlinear material behaviour, the deflection can still be obtained by double integration of the curvature provided the variation of curvature can be expressed as a function of x. This will be illustrated in Chapter 15.
