Structural Analysis Methods II: Structural Systems - Mechanics of Structures: Variational and Computational Methods - page 273

The stiffness matrix for a bar element with the uniaxial stiffness property EA / = k is given

by (Chapter 4, Example 4.1)

N xa
N xb

i 1 i k i

− 1 u xa − k u xa

EA

= = (5.84)

− 11 u xb − k k u xb

k i

p i v i =

which is written in terms of the local coordinates of a truss member. Equation (5.28), k i =

k i T i , permits the stiffness matrix to be expressed in terms of the global coordinates.
Thus,

T iT

T aa T aa i

T aa T aa

−

EA

k i T i

k i T iT

= = (5.85)

T aa T aa T aa T aa

−

with

cos 2 i

α − cos α sin α

T aa T aa = (5.86)

sin 2 − cos α sin α α

The element stiffness matrix represented in global coordinates is of size 4 4. It is convenient

to employ notation for the element stiffness matrix which indicates that Eq. (5.85) can be
partitioned into 2

×

× 2 matrices:

k i jj

k i jk

k i

= (5.87)

k i kj k i kk

where k i jj , k i jk , k i kj , and k i kk are 2 × 2 matrices and j , k are the initial and final nodes of

element i .
With these relationships for the element, we can proceed to use the displacement method
to solve plane truss problems.

EXAMPLE 5.3 Three-Bar Truss
The solution for the displacements and forces in the three-bar truss of Fig. 5.17 will illustrate
the use of the displacement method.

FIGURE 5.17
Three-bar truss.