Module
3
Analysis of Statically
Indeterminate
Structures by the
Displacement Method
Version 2 CE IIT, Kharagpur

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Lesson
15
The Slope-Deflection
Method: Beams
(Continued)
Version 2 CE IIT, Kharagpur

Instructional Objectives
After reading this chapter the student will be able to
1. Derive slope-deflection equations for the case beam with yielding supports.
2. Estimate the reactions induced in the beam due to support settlements.
3. Analyse the beam undergoing support settlements and subjected to external
loads.
4. Write joint equilibrium equations in terms of moments.
5. Relate moments to joint rotations and support settlements.
15.1 Introduction
In the last lesson, slope-deflection equations were derived without considering
the rotation of the beam axis. In this lesson, slope-deflection equations are
derived considering the rotation of beam axis. In statically indeterminate
structures, the beam axis rotates due to support yielding and this would in turn
induce reactions and stresses in the structure. Hence, in this case the beam end
moments are related to rotations, applied loads and beam axes rotation. After
deriving the slope-deflection equation in section 15.2, few problems are solved to
illustrate the procedure.
Consider a beam
AB
as shown in Fig.15.1.The support
B
is at a higher
elevation compared to
A
by an amount
Δ
. Hence, the member axis has rotated
by an amount
ψ
from the original direction as shown in the figure. Let
L
be the
span of the beam and flexural rigidity of the beam
EI
, is assumed to be constant
for the beam. The chord has rotated in the counterclockwise direction with
respect to its original direction. The counterclockwise moment and rotations are
assumed to be positive. As stated earlier, the slopes and rotations are derived by
superposing the end moments developed due to
(1) Externally applied moments on beams.
(2) Displacements
A
θ
,
B
θ
and
Δ
(settlement)
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