Advanced Methods of Structural Analysis

384 11 Matrix Stiffness Method

F 1 =16kN

F 2 =10kN

3m 3m

4m

a

B

A D

C

a

2

(^13)
4
Z-P
b
S-e
5
2
1 3
4
c
d P 1
S 2
S 1
Joint B
P 3
P 2
S 2
S 5 S 3
Joint C
S 3
S 4
Joint D P 4
Fig. 11.18 (a–c) Design diagram of truss and corresponding Z-P and S-e diagrams;
(d) Construction of static matrixAfor truss
These equations in the ordered form can be rewritten as follows
P 1 D 1 S 1 C 0 S 2 C 0 S 3 C 0 S 4 C 0 S 5 ;
P 2 D 0 S 1 C 1 S 2 0:6S 3 C 0 S 4 C0:6S 5 ;
P 3 D 0 S 1 C 0 S 2 C0:8S 3 C 0 S 4 C0:8S 5 ;
P 2 D 0 S 1 C 0 S 2 C0:6S 3 C 1 S 4 C 0 S 5 :
These equations allows presenting the vector of possible joint forcesPE.41/D
P 1 P 2 P 3 P 4
̆T
in terms of the vector of unknown internal forcesS.51/ D
S 1 S 2 S 3 S 4 S 5
̆T
using the static matrixA.45/
2 6 6 6 6 6
P 1
P 2
P 3
P 4
3 7 7 7 7 7
D
2
6
6
4
10 0 0 0
01 0:600:6
0 0 0:8 0 0:8
00 0:6 1 0
3
7
7
5
„ ƒ‚ ...
A.45/

2 6 6 6 6 6 6 6 6
S 1
S 2
S 3
S 4
S 5
3 7 7 7 7 7 7 7 7
Each row of this matrix presents the corresponding equilibrium equation; the entries
of the matrix are coefficient atSi. We can see that static matrix describes the struc-
ture. First, this structure is statically indeterminate, since the number of rowsmD 4
while the number of columnsnD 5. Moreover, the matrixAdescribes the ways
of connections of different members. For example, second row shows that members
2, 3, and 5 are connected together, while the last row shows that members 3 and 4
are connected. At last, this matrix describes the inclination of different members.
Therefore, the static matrixAis very informative matrix.