Advanced Methods of Structural Analysis

228 7 The Force Method

version 2 by introducing a hinge at the jointC. Therefore, the primary unknown

in version 1 is reactions of support; primary unknown in the version 2 is bending

moment at jointC, so the dimensions of theı 11 is m=kN for version 1 and rad=kN m

for version 2.

The unit bending moment diagrams foranyprimary system aresimilar.This

is a general property for a statically indeterminate structure with first degree of

redundancy.

After bending moment diagram is constructed, the kinematical verification must

be performed. This procedure involves multiplication of the final and unit bending

moment diagram in any primary system. It is obvious that multiplication of the final

and unit bending moment diagram for each version will be equal to zero. However,

meaning of these multiplications will be different. For exampleMMNvers 1 D 0

means that horizontal displacement at pointBis zero;MMNvers 2 D 0 means that

mutual angle of rotation of two sections at jointCis zero.

Detailed analysis of statically indeterminate frame using canonical equations of

the force method is presented in Example 7.2; this design diagram will be analyzed

more detail at a later time (settlements of supports, change of temperature, other

methods of analysis).

Example 7.2.A frame is clamped at pointAand rolled at pointsBandCas

presented in Fig.7.11a. The frame is loaded by forcePD 8 kN and uniformly dis-

tributed load,qDkN=m. The relative flexural stiffness of each element is shown in

a circle. Construct the bending moment, shear, and normal force diagrams. Deter-

mine the reactions of supports.

Solution.1.Primary system and primary unknowns.The structure has five un-

known reactions. The degree of indeterminacy isnD 5  3 D 2 .Oneversionof

the primary system with primary unknownsX 1 andX 2 (vertical and horizontal

reactions at pointsBandC) is presented in Fig.7.11b.

Canonical equations of the force method are

ı 11 X 1 Cı 12 X 2 C1PD0;

ı 21 X 1 Cı 22 X 2 C2PD0: (a)

These equations show that for the adoptedprimary system the vertical displace-
ment at supportBand horizontal displacement at supportC in the primary
system caused by both primary unknowns and the given loads are zero.
2.The unit displacements and free terms of canonical equations.Figure7.11c–e
present the bending moment diagramsMN 1 ,MN 2 in the unit states and diagram
MP^0 caused by applied load in the primary system; also these diagram show the
unit and loaded displacementsıikandiP.
The graph multiplication method leads following results:

ı 11 D

MN 1 MN 1

EI

D

1

1 EI

 10  5  10 C

1

2 EI



1

2

 10  10 

2

3

 10 D

666:67

EI

m

kN



;