Advanced Methods of Structural Analysis

220 7 The Force Method

ıikD

XZ MNiMNk

EI

ds; iPD

XZ MNiMP^0

EI

ds: (7.5)

Accordingly to expression6.21, these formulas may be presented in conventional

forms. Properties of unit coefficients are as follows:

1.Main displacements are strictly positive (ıii>0).
2.Secondary displacementsıik,i¤kmay be positive, negative, or zero.
3.Secondary displacements satisfy the reciprocal displacement theorem

ıikDıki: (7.5a)

It means that unit displacements symmetrically placed with respect to principal di-

agonal of canonical equations are equal.

The unit of displacementsıikpresents the ratio of unit for displacement accord-

ingtoindexiand units for force according to indexk.

Construction of internal force diagrams.Solution of (7.4) is the primary un-

knownsXi,i D1;:::;n. After that the primary system may be loaded by de-

termined primary unknowns and given load. Internal forces may be computed as

for usual statically determinate structure. However, the following way allows once

again an effective use of the bending moment diagrams in primary system. The final

bending moment diagramMPmay be constructed by formula

MPDMN 1 X 1 CMN 2 X 2 CCMNnXnCMP^0 : (7.6)

Thus in order to compute the ordinates of the resulting bending moment diagram,

it is necessary to multiply each unit bending moment diagramsMNkby correspond-

ing primary unknownXkand summing up with bending moment diagram due to

applied load in the primary systemMP^0. This formula expresses the superposition

principle. Advantage of formula (7.6) is that it may be effectively presented in tab-

ulated form.

Shear forces may be calculated on the basis of bending moment diagram using

Schwedler theorem and axial forces may becalculated on the basis of shear force di-

agram by considering equilibrium of joints of the structure. Finally, having internal

force diagrams, all reactions are easy to determine.

Procedure for analysis The following procedure provides analysis of statically in-

determinate beams and frames using the canonical equations of the force method:

1.Provide the kinematical analysis and define the degree of statically indetermi-

nacynof a structure.

2.Choose the primary system and replace the eliminated redundant constraints by

corresponding primary unknownsXi,iD1;:::;n.

3.Formulate the canonical equations of the force method.

4.Apply the successive unit forcesX 1 D1; X 2 D1;:::; Xn D 1 to primary

system and for each unit primary unknownconstruct corresponding bending

moment diagramsMN 1 ;MN 2 ;:::;MNn.