ABSTRACT

Consider a prismatic straight-line member, having cross sections of arbitrary geometry, 1made from a homogeneous, isotropic, linearly elastic material. We choose the x axis to be

the axis of the member, that is, the locus of the centroids of its cross sections. The member is initially at a reference stress-free, stain-free state (undeformed state) of mechanical and

othermal equilibrium at a uniform temperature T . Subsequently, the member reaches a second state (deformed state) of mechanical, but not necessarily thermal, equilibrium due to the application on it of one or more of the following loads (see Fig. 9.1):

1. A distribution of body forces throughout its volume as well as a distribution of traction on its lateral surfaces which are equivalent to specified transverse forces and bending

1 1moments whose vector is normal to the axis of the member (M = 0 and m = 0). The forces 2 1 3 1 2 2 3could be distributed p (x ) and p (x ) and concentrated P (n = 1, 2, ..., n ) and P (n = 1,

3 2 1 3 1 22, ..., n ). The moments could be distributed m (x ) and m (x ) and concentrated M (m = (m)

2 3 31, 2, ..., m ) and M (m = 1, 2, ..., m ). The line of action of the transverse forces lies in a (m)

plane which contains the shear centers of the cross sections of the member (see Section 9.7). 2 32. A specified change of temperature which is a linear function of x and x and moreover

2 3 c 2vanishes at the centroid (x = x = 0) of the cross section of the member [)T = 0, *T … 0, 3 *T … 0 (see Section 8.11)]. Notice that if in the second state of mechanical equilibrium the

temperature of the surface of the member varies with the space coordinates, the temperature inside the member will be non-uniform. Thus, heat will flow into or out of the member and consequently, the member will not be in a state of thermal equilibrium.