Special cases of uncracked sections and calculation of displacements - Concrete Structures: Stresses and Deformations - page 89

3.8.1 Unit load theory
The most e

nd the displacement at a coordinate j is the
unit load theory, based on the principle of virtual work. 3 For this purpose, a
fi

ff ective method to fi

ctitious virtual system of forces in equilibrium is related to the actual dis-
placements and strains in the structure. The virtual system of forces is com-
posed of a single force F j

1 and the corresponding reactions at the supports,
where the displacements in actual structure are known to be zero. When shear
deformations are ignored the displacement at any coordinate j on a plane
frame is given by:

=

D j = ε O N u j d l + ψ M u j d l (3.32)

where ψ

in any cross-section of the frame; N u j and M u j are the axial normal force and
bending moment at any section due to unit virtual force at coordinate j . The
cross-section is assumed to have a principal axis in the plane of the frame and
the reference point O is arbitrarily chosen on this principal axis. The axial
force N u j acts at O and M u j is a bending moment about an axis through O. The
integral in Equation (3.32) is to be performed over the length of all members
of the frame.
The principle of virtual work relates the deformations of the actual struc-
ture to any virtual system of forces in equilibrium. Thus, in a statically
indeterminate structure, the unit virtual load may be applied on a released
statically determinate structure obtained by removal of redundants. This
results in an important simpli

ε O and ψ are the axial strain at a reference point O and the curvature

cation of the calculation of N u j and M u j and in
the evaluation of the integrals in Equation (3.32). For example, consider the
transverse de

fi

ection at a section of a continuous beam of several spans. The
unit virtual load may be applied at the section considered on a released
structure composed of simple beams. Thus, M u j will be zero for all spans
except one while N u j is zero everywhere.
Only the second integral in Equation (3.32) needs to be evaluated and the
value of the integral is zero for all spans except one.

fl

3.8.2 Method of elastic weights
The rotation and the de

ection in a beam may be calculated respectively as
the shearing force and the bending moment in a conjugate beam subjected to a
transverse load of intensity numerically equal to the curvature

fl

ψ for the

actual beam. This load is referred to as elastic load (Fig. 3.6).
The method of elastic weights is applicable for continuous beams. The
conjugate beam is of the same length as the actual beam, but the conditions
of the supports are changed, 4 whereas for a simple beam, the conjugate and
actual beam are the same (Fig. 3.6(a) and (b) ).