Theory of Plates

1. 1.Introduction
2.Yield lines and Yield Line Theory
3.Basic Principles of Yield Line Theory
4.Virtual Work and Yield Line Theory
5.Advantages of Yield Line over Linear Elastic
Analysis
YIELD LINE THEORY OF
PLATES

2. Introduction
• The conventional principles of analysis of reinforced concrete slabs and beams
is not the same as for the design process.
• The design of beams and slabs considers the actual inelastic behaviour when
subjected to factored loads.
• Accordingly, it is desirable that the structural analysis of beams and slabs be
done taking into account the inelastic behaviour.
 For example, for reinforced concrete slabs and beams, the draft Eurocode
EC2 allows yield line analysis provided the ratio, x/d, is sufficiently low.
 On the other hand, BS8110 does not allow yield line unless results are
‘similar’ to those from a linear elastic method.
• Therefore, there are inconsistencies between the methods of analysis and the
methods of design, hence, there is need for adopting the inelastic analysis or
collapse limit state analysis for all structures.

3. Introduction (Cont’d….)
Assumptions made in the elastic analysis of slabs
and beams:
1. Slab panels are square or rectangular.
2. One-way slab panels must be supported
along two opposite sides only; the other two
edges remain unsupported.
3. Two-way slab panels must be supported
along two pairs of opposite sides, supports
remaining unyielding.
4. Applied loads must be uniformly distributed.
5. Slab panels must not have large openings.

4. Introduction (Cont’d….)

5. Introduction (Cont’d….)
The collapse limit state analysis has been
found to be a powerful and versatile method for
the following cases:
• Slabs of triangular, circular and other plan
forms
• Loads other than uniformly distributed
• Support conditions other than those specified
in Figures 1(a) to (d)

6. • Yield line analysis was pioneered
by K.W. Johansen in the 1940s
• Only applicable to ductile and
under-reinforced slabs
Yield lines and Yield Line Theory

7. Yield lines and Yield Line
Theory (Cont’d….)
Figure 1: Idealized moment-curvature relationship for ductile and under-
reinforced slabs

8. Yield lines and Yield Line
Theory (Cont’d….)
• Yield line is closely related to the plastic
collapse or limit analysis of steel frames
• This is an Upper Bound or Mechanism
approach.
• Yield lines, are lines of maximum yielding
moments in ductile and under-reinforced
slabs.

9. Yield lines and Yield Line
Theory (Cont’d….)
• Lines of maximum yielding moments of
the reinforcement of slab.
• The essence of Yield Line Theory is to
find out the locations of the appropriate
yield lines.
• Note: only under-reinforced bending
failure is considered ignoring the
following:
 Effects due to shear,

10. Basic Principles of Yield Line
Theory

11. Basic Principles of Yield
Line Theory (Cont’d….)
1.Establish the positions of the yield lines
and position them on a plan of the slab;
2.Show on a section of the slab, the
loading, deformation, slopes and
curvature distribution;
3.Illustrate the curvature in relation to its
position along the slab sections well as
the vertical displacement due to the load
application;
4.Illustrate the collapse mechanism;

12. Basic Principles of Yield
Line Theory (Cont’d….)
1. Establish the positions of the yield lines and position
them on a plan of the slab

13. Basic Principles of Yield
Line Theory (Cont’d….)
2. Show on a section of the slab, the loading, deformation,
slopes and curvature distribution

14. Basic Principles of Yield
Line Theory (Cont’d….)
3. Illustrate the curvature in relation to its position along the
slab sections well as the vertical displacement due to the
load application

15. Basic Principles of Yield
Line Theory (Cont’d….)
4. Illustrate the collapse mechanism

16. Basic Principles of Yield
Line Theory (Cont’d….)
5. Illustrate the bending moment diagram

17. Virtual Work and Yield Line
Theory
The conditions during collapse are such as to develop
enough plastic hinges to make the structure a
mechanism.
The collapse condition is represented by hogging and
sagging yield lines due to the corresponding to a
collapse load with a factor λc that needs to be
determined.
The amount of rotation () at the plastic hinges

18. Virtual Work and Yield Line
Theory (Cont’d….)
(a) Plan view of the slab, (b) hogging and sagging yield lines, (c)
plastic hinges

19. Basic Principles of Yield
Line Theory (Cont’d….)
At collapse, external work will be done, which is the total load on
the slab times the average displacement it moves through.
The internal work is the moment capacity of the plastic hinge (m’l2
and ml2) times the rotation it moves through.
The external work is balanced by the internal work.
The amount of rotation () is related to the corresponding
displacements caused by the rotation.
The analysis involves solving the following equation:
External Work = Internal Work

20. Advantages of Yield Line over
Linear Elastic Analysis

21. Advantages of Yield Line
Analysis (Cont’d….)
 Simpler to use (computer not
necessary);
 Linear elastic only tells you when first
yield occurs. Yield Line gives the
ultimate
 capacity of the slab - what it takes to
cause collapse;
 Helps understanding of ultimate

22. Disadvantages of Yield Line
Analysis
 Requires experience to know a likely
failure mechanism;
 Dangerous designs are possible
without checking or experience;
 Does not give an idea of slab
behaviour in service.