2015-2016 40
Prof. Dr. Mustafa B. Dawood Dr. Bilal Ismaeel Al-Shraify
Chapter Two
Yield line
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Yield line analysis for slabs. Most concrete slabs are designed for moments found by the methods based essentially upon elastic theory. On the other hand, ...
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Chapter Two
Yield line
Yield line analysis for slabs
Most concrete slabs are designed for moments found by the methods based
essentially upon elastic theory. On the other hand, reinforcement for slabs is
calculated by strength methods. That account for the actual inelastic
behavior of member at the factored load stage. A corresponding
contradiction exists in the process by which beams and frames are analyzed
and designed, and the concept of limit, or plastic analysis of reinforced
concrete was introduced.
For slabs there is a good reason for interest in limit analysis. The
elasticity-based methods are restricted in important ways. But in the
practice, many slabs do not meet these restrictions, for example for round or
triangular slabs, slabs with large openings, slabs supported on two or three
edges only (as shown in fig. below), and slabs carrying concentrated loads.
Limit analysis provides a powerful tool for treating such problems.
Fig(2- ): Slabs supported at two edges or three.
For slabs which typically have tensile reinforcement ratio much below the
balanced value and consequently have large rotation capacity it can be
safely assumed that the necessary ductility is present. Practical methods for
the plastic analysis are thus possible and Yield line theory is one of these.
The plastic hinge was defined as a location along a member in a
continuance beam or frame at which upon over loading, there would be
large inelastic rotation at essentially a constant resisting moment. For slabs
The fixed-fixed slab shown
in Fig(2-3) is loaded
uniformly and assumed to
be equally reinforced for
(+m) and (−m). As the
load in increased the more
hingly stressed section at
support start yielding.
Rotations occur at the
support line hinges. The
load can be increased
further, until the moment at
mid span becomes equal to
the moment capacity there,
and a third yield forms.
The slab is now a
mechanism. Large
deflection occurs and
collapse takes place.
Elastic distribution of
moments shown in Fig (2- 2
b) the ratio of these
moments just before
collapse (Fig 2-2 c)
Fig(2-2) Fixed-end uniformly
loaded one-way slab
The terms positive yield line and negative yield line are used to distinguish
between those associated with tension at the bottom and tension at the top
of the slab respectively
Notation
Column
Simply supported
Either cautious of fixed end
Beam
+ve Y.L [Tension at bottom face]
- ve Y.L [Tension at Top face]
Point
Axes of rotation
Free edge four Column
Axes of rotation
Fixed End
Col.
B
A
C
Fixed End
S.S
S.S
Simply supported all sides
(a)
(c)
(b)
(d)
(f)
(g)
Fig.(2-4) Typical yield line patterns
Method of analysis for yield line
1 - Method of segment equilibrium
It requires consideration of the equilibrium of the individual slab
segments forming the collapse mechanism and leads to a set of
simultaneous equations permitting solution for the unknown
geometric parameters and for the relation between load capacity and
resisting moments.
2 - Method of virtual work
This method is based on equating the internal work done at the plastic
hinges with the external work done by the loads as the predefined
failure mechanism is given a small virtual displacement.
The yield line method of analysis for slabs is an upper bound
approach in the sense that the true collapse load will never be higher,
but may be lower, and then the load predicted. The solution has two
essential parts:
a- Establishing the correct failure pattern
b- Finding the geometric parameters that define the exact location and
orientation of the yield lines and solving for the relation between
applied load and resisting moments.
Internal work done by resisting moments:
The internal work done during the assigned virtual displacement is found by
summing the production of yield moment (݉ ) per unit length of hinge
times the plastic rotation (ߠ) at the respective yield lines. If the resisting
moment(݉ ) is constant along a yield line of length (ܮ) and if a rotation (ߠ)
is experienced, the internal work is :
For the entire system, the total internal work done is the sum of the
contributions from all yield lines.
Ex1: Determine the load capacity of the one-way uniformly loaded (ݓ)
simply supported slab shown in Fig. using the method of virtual work. The
resisting moment of slab in (m).
Solution
ଶ
ଶ
ଶ
For two-way slab for
Orthotropic moment capacities (݉
௫
௬
- Orthotropic
- Isotropic
For two-way slab orthotropic moment capacities (݉
௫
௬
) the internal
work is
௫
௫
௬
௬
Where ݉
௫
=ultimate moment per unit length in the direction of x-axis
Ex:- The slab shown is simply supported and it is orthotropic reinforced and
loaded by uniformly distributed load find the moment capacity of the slab.
Solution
௫
௫
௬
௬
௫
௫
௬
௬
,,,
or ∑ߜݓ = ܮߙݓ( 1 − 2 ߚ)ܮ ∗
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
ସ
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
Case 2 (ߜ ܽ ݐ ܾ & ܿ = 1 )
,
௫
௫
௬
௬
௫
௫
௬
௬
,,
ଶ
ଶ
ଶ
ଶ
ଶ
ଶ
Ex:- for the reinforced concrete isotropic slab shown in Fig. find the
relationship between the resistance moment (m) and uniform load (ݓ)
using yield line theory.
Sol:
ଶ
Ex:- using Y.L theory find the relationship between the moment of
resistance of the isotropic slab (m) and uniform load (ݓ) for the slab shown
in Fig.
Ex:-
ݔ
ݔ
ݕ
ݕ
ܣ
1
1
1
ݔ
ݔ
ݕ
ݕ
2
2
2
,
ଵ
ଶ
ଶ
ଶ
ଵ
ଶ
ଵ
ଶ
ଶ
ଶ
ଶ