Advanced Methods of Structural Analysis

7.3 Analysis of Statically Indeterminate Structures 231

while the multiplication of the summary unit bending moment diagramMN†by a

bending moment diagramMP^0 due to applied load in the primary structure

MN†MP^0

EI

D

1

1 EI



5

6



18  57 C 13  32 C 4 38:25

13 C 18

2





1

2 EI



4

6

.2 10 C6/ 32 D

3;455:25

EI

:

Therefore, the coefficients and free termsof equations (a) are computed correctly.
4.Primary unknowns.Canonical equations for primary unknownsX 1 andX 2
becomes

666:67X 1 C275X 2 2;294D0;

275X 1 C170:67X 2 1;161:25D0: (d)

All coefficients and free terms contain factor 1/EI, which can be cancelled. It

means that primary unknowns of the force method depend only onrelativestiff-

nesses of the elements.

The solution of these two equations leads to

X 1 D1:8915kN;X 2 D3:7562kN:

5.Internal force diagrams.The bending moment diagram of the structure will be
readily obtained using the expression

MDMN 1 X 1 CMN 2 X 2 CMP^0 : (e)

Location of the specified points 1–8 is shown in Fig.7.11b. Corresponding calcula-

tion is presented in Table7.4. The sign of bending moments is chosen arbitrarily for

summation purposes only.

Ta b l e 7. 4 Calculation of bending moments at the specified points

Points MN 1 MN 1 X 1 MN 2 MN 2 X 2 MP^0 M

1  10 18.915 8.0 30.049 C57.0 C8.036

2 ^10 18.915 5.5 20.659 C38.25 1.324

3  10 18.915 3.0 11.268 C32.0 C1.817

4 0.0 0.0 3.0 11.268 C32.0 11.268

5 0.0 0.0 0.0 0.0 0.0 0.0

6  10 18.915 0.0 0.0 C32.0 C13.085

7 6.0 11.349 0.0 0.0 0.0 11.349

8 0.0 0.0 0.0 0.0 0.0 0.0

signs of

bending moments

+ −

+

−

The resulting bending moment diagram and corresponding elastic curve are pre-

sented in Fig.7.11g.