Advanced Methods of Structural Analysis

7.2 Canonical Equations of Force Method 217

Ta b l e 7. 2 Two versions of primary system and corresponding solution using principle of super-
position
Ve r s i o n 1 Ve r s i o n 2
Primary unknown Vertical reactionXDRA Support momentXDMB
Primary system

B

X= RA

q

A

B

q

A

X= MB

Compatibility
condition

yAD 0 BD 0

Displacement due to
given loadqin
primary system

q

A

yAq

B

q

A

θBq

Displacement due to
primary
unknownX
in primary
system X= RA

A

yAX B

X= MB

θBX

Solution of
compatibility
equation

yADyAqyAXD 0

yAqD

ql^4

8 EI

;yAXD

Xl^3

3 EI

XDRAD

3

8

ql

BDBqBXD 0

BqD

ql^3

24 EI

;BXD

Ml

3 EI

XDMBD

ql^2

8

Statically
determinate
structure and
bending moment
diagramMfor
entire structure

B

ql

8

3

q

A

8

ql^2

M

B

q

A

8

ql^2

8

ql^2

M

7.2 Canonical Equations of Force Method................................

Canonical equations of force method offer a unified procedure for analysis of stat-
ically indeterminate structures of different types. The word “canonical” indicates
that these equations are presented in standard, or in an orderly fashion form. Very
important is that canonical equations of the force method may be presented in a
matrix form. Thus, this set of equations is a first bridge between classical analytical
methods and numerical ones.

7.2.1 The Concept of Unit Displacements.........................

Analysis of any statically indeterminate structure by the force method begins with
determination of degree of statical indeterminacy. Primary system is obtained by