Advanced Methods of Structural Analysis

170 6 Deflections of Elastic Structures

6.3.3 Summary

1.Maxwell–Mohr integral presents the fundamental and power method for calcu-
lation of arbitrary displacements of any elastic structure. Displacements may be
the result of any types of loads and change of temperature.
2.In order to calculate any displacement, itis necessary to consider two states of
a structure, i.e., given and unit states. Unit state presents the same structure, but
loaded by unit generalized force corresponding to the required displacement.
3.According to the type of structure, which terms of (6.11) should be taken into
account can be decided. For both states, given and unit, it is necessary to set up
expressions for corresponding internal forces and calculate the required displace-
ment by the Maxwell–Mohr integral.

6.4 Displacement Due to Settlement of Supports

and Errors of Fabrication

A settlement of supports often occurs in engineering practice. If the settlement of

support happens in direction of reaction of this support, then in case of statically

determinate structures such influence leads to a new position of structure without

deformation of its separate members; it means that internal stresses are not induced.

Computation of displacement of any point ofstatically determinate structures due

to settlement of supports is considered below.

Let us consider a portal frame; supportBsettles onas shown in Fig.6.15a. The

new position of the frame members after settlement of supportBis shown by dotted

line. It is necessary to calculate the linear displacementkof the pointk. Unit state

presents the same frame subjected to unit forceX, which acts in the direction of the

required displacementk(Fig.6.15b). This unit forceXproduces the reactionR

at the supportB. Assume that direction of this reaction coincides with settlement

of support.

Δ

AB

k

Δk

abc

AB

Δ

R

k

X=1

AB

Δ

Δ 1

Δ 2

Δ 3

K

l

h

Fig. 6.15 (a) Settlementof supportB;(b) unit state; (c) displacements at pointK

Effective method for solution of this type of problem is the principle of virtual

displacements X

ıWactD0: (6.16)