2 Introduction RC

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An Introduction to R/C structures

Materials Properties

Basic Theorems of Limit Analysis


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Concrete Uniaxial compression

Approximately linear elastic up to = 0.300.40fc Gradual decrease of slope up to = 0.750.90fc Above = 0.90fc there is a sharp decrease of the slope, up to the

peak stress fc (= compressive strength) Beyond the peak, there is a descending branch, until a crushing

failure occurs (at the ultimate strain cu 0.003)

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2D and 3D RC Structural Elements A.Y. 2009/2010


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Cyclic behaviour

If unloaded below = 0.50fc strains are essentially recoverable

In the range 0.500.75fc nonlinearities and plastic offsets begin toappear

Nonlinearities become pronounced at the peak stress and in thedescending branch

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Uniaxial tension

Generally, the stress-strain response is nearly linear up to the peakstress fct (= tensile strength)

After cracking, a quickly decaying response is observed: thisphenomenon is known as TENSION SOFTENING

The tensile behaviour of concrete can be measured by means ofdifferent tests:

direct tensile test

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indirect tensile tests (by splitting or in bending)


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Biaxial behaviour of concreteStrength

In biaxial stress conditions, the compressive strength of concreteincreases (with respect to uniaxial conditions)

by 25% for 2/1 = 0.50

by 16% for

2/

1 = 1.00 Under biaxial compression/tension, the tensile strength decreases

almost linearly as the compressive stress increases

Under biaxial tension, strength is unaffected by /

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Triaxial behaviour of concrete

Under triaxial compression, increases in strength and ductility can bevery significant

Depending on the confining stress level, concrete can act as:

quasi-brittle plastic softening

plastic-hardening

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Uniaxial behaviour of steel

Initial linear elastic region, up to the yield point

Yield plateau from the yield strain to the point (st), where strain-hardening begins

Strain-hardening region, from

st to the ultimate strain (

u) Decreasing branch from the ultimate strain to the fracture strain (f)

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Introduction to Limit Analysis

Obtaining a progressive-failure solution to a problem involvingreinforced concrete is usually a rather complex task...

Simple example: simply-supported beam subjected to a concentratedload at mid-span

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beam without stirrups

beam with stirrups


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A rational analytical solution to this problem involves considering:

cracking

bond slip

dowel action aggregate interlock

A possible way to include all the aforementioned aspects is to resort

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is not justified for everydays applications. Idealizations are necessary to obtain a reasonable approximate

solution for cases of practical interest.

It is often satisfactory to know the loads or load level at which

the structure will collapse the structure will deform excessively

This is the main objective of Limit Analysis.


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Applicability of Limit Analysis

Limit Analysis techniques are based on the statements of the two limittheorems by Drucker et al. (1952).

The theorems can be established directly for a structure satisfying thefollowing properties:

the material exhibits perfect or ideal plasticity (no work hardeningor work softening);

the yield surface of the material is convex, and the plastic strain

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rates can be derived from an associated flow rule;

changes in geometry of the structure at the collapse load areinsignificant; hence the equations of virtual work can be applied.

The collapse load of an idealized structure having the ideal propertieslisted above is called the limit load; it can be shown that collapseoccurs under constant load.


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Is reinforced concrete plastic? 12


Typically, concrete sections with low reinforcement ratios (such as inslabs), exhibit a plastic (= ductile) behaviour. Another way to reducethe brittleness of concrete is to provide an adequate confinement (=closely-spaced stirrups).


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Comparison between steel and R/C sections 13

rectangular

steel section


rectangularR/C section


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Theorems of Limit Analysis

The limit theorems are then as follows:

1. Lower-bound theorem: if an equilibrium distribution of stress

can be found, which balances the applied loads and is everywherebelow yield or at yield, the structure will not collapse or will just beat the point of collapse.

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2. Upper-bound theorem: the structure will collapse if there isany compatible pattern of plastic deformation, for which the rate atwhich the external forces do work exceeds the rate of internaldissipation.


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Consequences of the Limit Theorems

Theorem 1 reaffirms the important consideration that a plastic materialcan adjust itself to carry the applied load, if this is possible. It giveslower bounds of the collapse load, and hence it is on the safe side.

Theorem 2 states that if a failure path exists, the structure will not beable to carry the loads. It gives upper bounds of the collapse load, andhence it is on the unsafe side.

Since the two theorems give, respectively, a lower and an upper

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oun o e co apse oa , s poss e o rac e e rea va ue o

the collapse load close enough for practical engineering purposes.

These two theorems are implemented in current engineering practicethrough the two methods for evaluating the collapse load:

Static Approach

Kinematic Approach


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The Static Approach

The static method is based on the satisfaction of two (out of three)basic requirements of structural mechanics:

Equilibrium

Resistance (= yield limit in a plastic material) Consider the example of a beam over 4 supports:

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load q


elastic moments

The load q, correspondingto the moment distributionMd, is a lower bound ofthe collapse load. This isthe typical application inthe design of beams andother structures.


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The Kinematic Approach

The kinematic approach is based on the satisfaction of two (out ofthree) basic requirements of structural mechanics:

Equilibrium

Compliance (= formation of a plastic mechanism) In plasticity, compliance is typically enforced through the subdivisionof the structure into two zones:

an elastic (= rigid) zone, where the deformations are negligible;

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a plastic zone, where energy dissipation takes place. The plastic zone can have different names (= plastic hinge, yield

line...) depending on the structure under consideration.

Equilibrium is enforced through application of the principle of virtual

works.


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Worked-out examplePropped cantilever with point loads

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Consider a beam with one end built-in and the other simply-supported.


The beam is subjected to two concentrated loads, characterizedby the same intensity Q. The resisting moment Mp (= yieldmoment) of the beam is the same in hogging and sagging.


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Application of the Static Approach 19


RA = RB = Q Mp = QL/3 Qu 3Mp/L

Note that the moment distribution is not necessarily the elastic distribution.


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Application of the Kinematic Approach 20

plastic hinges


Equilibrium is enforced via the Principle of Virtual Works:

Q L/3 = Mp + Mp 2 Qu 9Mp/L


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Alternative moment distributionStatic Approach

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RA = Q + Mp/L, RB = Q Mp/L Mp = RBL/3

Qu 4Mp/L


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Alternative moment distributionKinematic Approach

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Q L/3 + Q 2L/3 = Mp + Mp 3

Qu 4Mp/L

Note that the value of the collapse load using the two approachesis exactly the same; hence, this value is the exact value of thecollapse load.

2 Introduction RC

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