## Internal Strain Energy

ii The strain energy density of an arbitrary material is defined as, Fig. 13.1

The complementary strain energy density is defined

The strain energy itself is equal to e e

 U d= i U0dQ (13.6) JQ U * d=f i U^dQ (13.7) JQ

14 To obtain a general form of the internal strain energy, we first define a stress-strain relationship accounting for both initial strains and stresses a = D:(e - £o) + (13.8)

where D is the constitutive matrix (Hooke's Law); e is the strain vector due to the displacements u; e0 is the initial strain vector; a0 is the initial stress vector; and a is the stress vector.

15 The initial strains and stresses are the result of conditions such as heating or cooling of a system or the presence of pore pressures in a system.

16 The strain energy U for a linear elastic system is obtained by substituting a = D:e with Eq. 13.4 and 13.8

where Q is the volume of the system.

17 Considering uniaxial stresses, in the absence of initial strains and stresses, and for linear elastic systems, Eq. 13.10 reduces to

18 When this relation is applied to various one dimensional structural elements it leads to Axial Members:

Flexural Members:

Mzy EIz di = dAdx

Vo EIz

0 0