Advanced Methods of Structural Analysis

7.2 Canonical Equations of Force Method 221

5.Calculate the unit coefficientsıik.
6.Construct the bending moment diagramMP^0 due to applied load in primary
system and calculate the load termsiPof (7.4).
7.Solve the system of equations with respect to primary unknownsX 1 ,X 2 ;:::;Xn.
8.Construct the bending moment diagrams by (7.6), next compute the shear and
construct corresponding shear force diagram, and lastly compute axial forces
and construct the corresponding axial force diagram.
9.Having internal force diagrams, calculate the reactions of supports. Other way is
consider primary system subjected to determined primary unknowns and given
load and provide computation of all internal forces by definition; this way is less
effective.
10.Provide the static control for all structure (or any its part).
11.Provide the kinematical control (7.2) for displacements of an entire structure in
direction of primary unknowns.
Intermediate checking of computationThese verifications are recommended to be
performedbeforesolving canonical equations for determining primary unknowns
Xi, i.e., on the steps 5 and 6 of the algorithm above. For control of unit displace-
ments and free terms, it is necessary to constructsummaryunit bending moment
diagramMN†DMN 1 CMN 2 CCMNn. The following types of controls are sug-
gested as follows:
(a)Row verification of unit displacements: Multiply summary unit bending
moment diagramMN†on a primary bending moment diagramMNi:

MN†MNi

EI

D

XZ 

MN 1 CMN 2 CCMNn



MNi

ds

EI

Dıi1Cıi2CCıin:

The result of this multiplication equals to the sum of unit displacements of the

i-equation.

(b)Total verification of unit displacements: Multiply the summary unit bending

moment diagramMN†on itself.

MN†MN†

EI

Dı„ 11 CCƒ‚ ı1n...

1 st equation

Cı„ 21 CCƒ‚ ı2n...

2 nd equation

CCı„n1CCƒ‚ ınn...

nth equation

D

Xn

i;kD 1

ıik:

The result of this multiplication equals tothe sum of all unit displacements of canon-

ical equations. It is quite sufficient to perform only total control, however, if errors

occur, then the row control should be performed for tracking the wrong coefficient.

(c)Verification of the load displacements: Multiply summary unit bending moment

diagramMN†on a bending moment diagramMP^0.