Advanced Methods of Structural Analysis

300 8 The Displacement Method

8.5.2 Errors of Fabrication.........................................

In cases of fabrication errors, analysis of the frame can be effectively performed
by the displacement method in canonical form. Construction of the primary system
and calculation of unit reactions should be performed as usual. Canonical equations
should be presented in form (8.4), but the free termsRiPshould be replaced byRie.
These free terms present the reactions of the introduced constraints caused by the
errors of fabrication.

Example 8.4.A design diagram of the frame is presented in Fig.8.12a. MemberAB
has been fabricatedD0:8cm too long. The moment of inertia of all cross sections
is ID9:9895 10 ^5 m^4 , and the modulus of elasticity isE D 2  108 kN=m^2
EID19;979kN m^2



. Calculate the angle of rotation of jointBand construct the
bending moment diagram.

Δ

A B

C

l = 5m

h = 2.8m

1

ab

r 11

Z 1 =1

M 1

l

4 EI

l

2 EI

h

3 EI

R1e

Δ

Meo

Δ

3 EI

h^2

A

C

B

cd

13.05

26.1

M(kN⋅m)

e

Fig. 8.12 (a,b) Errors of fabrication: Design diagram and primary system. (c,d) Bending moment
diagrams for unit and loading states. (e) Resulting bending moment diagram

Solution.The primary system of the displacement method is shown in Fig.8.12b.
The primary unknown is the angle of rotation of included constraint 1. The
canonical equation of the displacement method isr 11 Z 1 CR 1 eD 0. The primary
unknown becomesZ 1 DR 1 e=r 11.