Basic Equations: Differential Form - Mechanics of Structures: Variational and Computational Methods - page 7

σ + σ + σ + 2 σ + 2 σ + 2 σ = σ + σ + σ + σ γ +

11 11 22 22 33 33 12 12 13 13 23 23 xx xx yy yy zz zz xy xy

yz . This is the same expression obtained using σ T .
To produce positive work by positively defined stresses and strains, it is important to
adopt consistent sign conventions. Stresses and strains are defined to be positive as indicated
in Fig. 1.6, i.e., components on the positive face of an element are positive when they
are acting along the positive direction of the coordinates. In addition, stress and strain
components are defined to be positive when their components on the negative face are
acting in the negative direction of the axis. An element's face with its outward normal
along the positive direction of a coordinate axis is defined to be a positive face. A face with
its normal in the opposite direction is said to be a negative face.

σ γ + σ γ

xz xz yz

It is essential that the reader is able to visualize matrix multiplication. It is helpful to be
familiar with a matrix multiplication scheme to aid in organizing practical multiplication
calculations. To illustrate one such procedure, consider a column vector x and a row
matrix y . The product yx can be obtained in the form

Similarly, AB = C can be written as

b 11 b 12

B =↓

b 21 b 22

c 11 = a 11 b 11 + a 12 b 21

−→

c 12 = a 11 b 12 + a 12 b 22

a 11 a 12 c 11 c 12

c 21 = a 21 b 11 + a 22 b 21

A =

a 21 a 22 c 21 c 22

c 22 = a 21 b 12 + a 22 b 22

= C

This scheme is suitable for organizing the matrix calculations that so frequently occur in
structural analysis. For example, to find ABCD use either

D
C CD
B BCD
A ABCD

B C D
A AB ABC ABCD

or

1.2 Deformation Relationships

1.2.1 Kinematical Equations

We begin by formulating the relationships between displacements and strains in a solid.
The three components, u x ,u y ,u z or u , , of the displacement vector at a point in a solid

are mutually orthogonal in a Cartesian coordinate system and they are taken to be positive
in the direction of the positive coordinate axes.

v , w