Module
12
Yield Line Analysis for
Slabs
Version 2 CE IIT, Kharagpur

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Lesson
31
Nodal Forces and Two-way
Slabs
Version 2 CE IIT, Kharagpur

Instructional Objectives:
At the end of this lesson, the student should be able to:
•
derive the expression for determining the work done by bending and
twisting moments when the yield lines are at angles with the directions of
reinforcing bars,
•
state the need for considering the nodal forces and to estimate their values
when one yield line meets another yield line or a free edge,
•
to select the possible yield pattern of a two-way slab supported at four sides
either by simple supports or fixed supports,
•
to finalise the yield patterns and to evaluate the collapse loads of two-way
slabs, either simply supported or clamped at four sides,
•
apply the theory in solving numerical problems of slabs to finalise the yield
pattern and to determine the collapse load employing (i) the method of
segmental equilibrium and (ii) the method of virtual work.
12.31.1
Introduction
Lesson 30 introduces the yield line analysis, which is an upper bound
method of analysis for slabs. The different rules for predicting the yield lines are
stated. The two methods i.e., (i) method of segmental equilibrium and (ii) method
of virtual work are explained. Applications of both the methods are illustrated
through numerical problems of one-way slabs – either simply supported or
continuous.
This lesson presents the derivations of the expressions for determining
bending and torsional moments when yield lines are at angles with the directions
of reinforcement. The need for the nodal forces and their determinations are
explained when yield line meets another yield line or the free edge. Thereafter,
different possible yield patterns of two-way slabs are explained. Numerical
illustrative problems of two-way slabs with or without nodal forces are illustrated.
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12.31.2
Work Done by Yield Line Moments
Normally, the reinforcing bars are placed in two mutually perpendicular
directions parallel to the sides of rectangular and square slabs. However, the
yield lines may be at an angle with the direction of reinforcing bars as shown in
Fig. 12.31.1, in which the yield line AB of length
L
has bending moment
M
b
and
twisting moment
M
t
per unit length of the yield line. The slab segment is
undergoing rigid body rotation whose components are
θ
x
and
θ
y
. The horizontal
and vertical projections of the yield line are having moment capacities of
M
x
and
M
y
per unit length, respectively. All moments and rotations are shown using the
right hand thumb rule. The following expression is derived for obtaining the
absolute values of the work done by
M
b
and
M
t
on the yield line AB.
With reference to Fig. 12.31.1, the total work done by the bending and
twisting moments
M
b
and
M
t
is,
W = M
b
L
(
θ
x
cos
θ
+
θ
y
sin
θ
) +
M
t
L
(-
θ
x
sin
θ
+
θ
y
cos
θ
)
(12.19)
Version 2 CE IIT, Kharagpur