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Advanced Structural

Concrete Design

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Collapse Load Methods for SlabDesign

Yield Line Theory Hillerborgs Strip Method

Openings in Slabs

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Class Objectives

Explain the principles of plastic analysis.Discuss the differences between upper and

lower bound theorems and the limitations.

Determine the design moments using yield line

(upper bound solution) and Hillerborgs strip

(lower bound solution) methods

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Recommended Reading

James G. MacGregor, REINFORCED CONCRETE:

Mechanics and Design, 3rd Ed., Prentice-Hall,

1997, Ch. 15.

The next few slides are from the Reinforced Concrete Council

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Plastic Analysis

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Upper and Lower Bound Theorems

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Membrane Action

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Principles

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Limitations

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Example on Beam

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Yield Line Theory

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Development of cracks and conditionsat collapse

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Collapse Load Analysis for Slabs

For any slab the conditions of collapse are:

The moments and other forces satisfy equilibrium and

boundary conditions

The yield condition is satisfied everywhere in the slab

The yield criterion is satisfied along yield lines which

constitute a valid collapse mechanism

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Definitions

Yield Line - this is analogous to a plastic hinge in a beam

Yield condition the relevant yield condition requires the

bending moment normal to any section to be less than

or equal to the ultimate moment of resistance (yield

moment) of that section

Yield criterion the associated yield criterion is that thebending moment normal to an yield line is equal to the

yield moment of the section containing the yield line.

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Yield Line Method

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Notation

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Notations

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Yield Line Patterns

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Selection of Yield Line Patterns

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Moment-axis Notation

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Moment Vectors

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External Work

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Stepped Yield Criterion

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Orthogonal Slabs

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Isotropic Slabs

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Energy Dissipation in a yield line

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Energy Dissipation in YL - Example

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Why does the yield line not pass thruD?

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Another Collapse Mechanism

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Comments

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Energy Dissipation in a Rigid Region

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Energy Dissipation in RZ - Example

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Example

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EXAMPLEDetermine the ultimate uniformly distributed load, wu, per unit area of a simply

supported square slab of sidet. The slab is isotropically reinforced with ultimate

positive moment of resistance per unit width ofmu.

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Components of Internal work done

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Minimum Load Principle

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Example

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Analysis by Equilibrium Equations

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Refer to notes (YL-Equilibrium Eq Example.pdf) for

examples solved using Analysis by Equilibrium Equations.

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Fan Mechanism

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Refer to notes (Yield Line Examples.pdf) for more examples.

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Hillerborgs Strip Method

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Moments and Reinforcement

Parts of the slab that will crack

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Moments and Reinforcement

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Division into Strips

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Bending Moments for the 6 strips

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Hillerborgs Strip Method

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Examples

Refer to notes (Hillerborgs Strip Method.pdf) for more

examples.

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Example on Slab with Opening

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Draw Lines indicating the strips (strong

bands which contains concentration of

reinforcement)

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See notes to sketch BM diagrams