Advanced Methods of Structural Analysis

388 11 Matrix Stiffness Method

3m 3m

4m

1

2

3

4

6 5

Fig. 11.21 Construction of the internal stiffness matrix for truss

Stiffness matrices for each member are shown below.

k 1 D

EA

l 1

Œ1D

EA

3

Œ1I k 2 D

EA

4

Œ1I k 3 D

EA

6

Œ1I

k 4 Dk 5 D

EA

5

Œ1I k 6 D

EA

p

62 C 42

Œ1D

EA

p

52

Œ1

The internal stiffness matrix of the truss becomes

kQDEA

2

6

6

(^66)
6
6
6
4
1=30000 0
0 1=4 0 0 0 0
001=600 0
0001=50 0
00001=5 0
000001=
p
52
3
7
7
(^77)
7
7
7
5
For bending elements, we can use the TablesA.3–A.6. If a uniform fixed-pinned
beam is subjected to angular displacementeof the fixed support, then a bending
moment at this support equalsMD^3 EIle, so the stiffness matrix of such element in
local coordinates is
kfpD
EI
l
Œ3 : (11.6)
If a fixed-fixed uniform beam is subjected to unit angular displacementse 1 and
e 2 of the fixed ends, then the following bending moments arise at the both supports:
M 1 D
EI
l
.4e 1 C2e 2 /;
M 2 D
EI
l
.2e 1 C4e 2 /:
These formulas may be presented in matrix form (11.3), i.e.,
M 1
M 2

D
EI
l

42
24

e 1
e 2

(11.7)
where the vector of internal forces isESD
M 1 M 2
̆T
, vector of angular displace-
ments at the end of elementEe D
e 1 e 2
̆T
and the stiffness matrix in local