Lec-5-Flexural Analysis and Design of Beamns

8/13/2019 Lec-5-Flexural Analysis and Design of Beamns

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Plain & Reinforced

Concrete-1CE-313

Lecture # 528th Feb 2006

Flexural Analysis andDesign of Beams

By Engr. Azhar


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Plain & Reinforced Concrete-1

Different Types of Cracks1. Pure Flexural Cracks

Flexural cracks start appearing at the section of maximumbending moment. These verticalcracks initiate from the

tension face and move towards N.A.


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Different Types of Cracks (contd)

2. Pure Shear/Web Shear Cracks

These inclined cracks appear at the N.A due to shear stressand propagate in both direction

45o

1 445on = f

3 2

1and 2are Major Principle Stresses


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Pure Shear/Web Shear Cracks


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Different Types of Cracks (contd)

3. Flexural Shear CracksIn the regions of high shear due to diagonal tension,the inclined cracks develop as an extension of flexural

cracks and are termed as Flexural Shear Cracks.

Flexural Shear Cracks.


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Tensile Strength of ConcreteThere are considerable experimental difficulties in determiningthe true tensile strength of concrete. In direct tension testfollowing are the difficulties:

1. When concrete is gripped by the machine it may be crusheddue the large stress concentration at the grip.

2. Concrete samples of different sizes and diameters show largevariation in results.

3. If there are some voids in sample the test may show verysmall strength.

4. If there is some initial misalignment in fixing the sample theresults are not accurate.


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Tensile Strength of Concrete (contd)Following are the few indirect methods through with tensilestrength of concrete is estimated.

A. Split cylinder Test

This test is performed by loading a standard 150mm x 300mmcylinder by a line load perpendicular to its longitudinal axis withcylinder placed horizontally on the testing machine platen.

The tensile strength can be defined as

ft= 2P / (

DL)WhereP = Total value of line load registered by machineD = Diameter of concrete cylinderL = Cylinder height


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Tensile Strength of Concrete (contd)

B. Double Punch Test

In this test a concrete cylinder is placed vertically between the loadingplatens of the machine and is compressed by two steel punches placed

parallel to top and bottom end surfaces. The sample splits across manyvertical diametrical planes radiating from central axis.

Tensile strength can be defined as

ft= Q / [ (1.2bH-a2)]

Q = Crushing Load

H

Q

Q

2b

2a


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Tensile Strength of Concrete (contd)

C. Modulus of Rupture Test

For many years, tensile strength has been measured in termof themodulus of rupture fr,the computed flexural tensile stress at which a

test beam of plain beam fractures.Because this nominal stress is computed on the assumption thatconcrete is an elastic material, and because this bending stress islocalized at the outermost surface, it is larger than the strength ofconcrete in uniform axial tension.

It is a measure of, but not identical with the real axial tensile strength.

Two point loading


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Tensile Strength of Concrete (contd)There are some relationships which relate fr with compressive strengthof concrete

fr= 0.69 fcfc and fr are in MPaACI code give a formula for fr

fr= 0.5 fc

For deflection control

fr= 0.625 fc


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Tensile Strength of Concrete (contd)By some testing methods few empirical formulae are developed for thetensile strength of concrete:

ft= 5/12 fcand

ft= 1/4 fc

Tensile strength compressive strength.

Tensile strength of concrete is generally 8 to 15% of compressivestrength.


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Transformed SectionBeam is a combination of concrete and steel so as a whole it is not ahomogeneous material. In transformed section the steel area is replacedby an equivalent concrete areain order to calculate the sectionproperties.

Width of the extended area is same as diameter of steel bar and itsdistance from compression face remains same.

d

Transformed Section


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Modular Ratio, n

The ratio of modulus of elasticity of steelto modulus of elasticity of concrete is

known as modular Ratio.

n = Es/ Ec

Normally the value of n is 8 to 10

It is unit-less quantity


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Un-cracked Transformed SectionWhen both steel and concrete are in elastic range and tensile stressat the tension face of concrete is less than tensile strength ofconcrete the section is un-cracked.Within the elastic range, perfect bond (no slippage) exists between

concrete and steel, so s= cfs/Es= fc/ Ecfs= (Es/ Ec) fc

fs= n fc

Using this relationship, stress in steel can be calculated ifstress in concrete and modular ratio are known.


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Un-cracked Transformed Section (contd)Consider a beam having steel area As . In order to obtain a transformed section,the area of steel (As) is replaced by an equivalent area of concrete so that equalforce is developed in both

Ps= Pc

fsAs= fcAceqnfcAs= fcAc eqnAs =Ac eq

nAs/2 nAs/2

b

h


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Un-cracked Transformed Section (contd)The equivalent area nAs/2 is shown on either side but steel inside thebeam is removed which creates a space that is filled by area of concrete,thus the equivalent area on either side becomes

nAs/2 As/2(n-1)As/ 2

Once the transformed sectionhas been formed the sectionalproperties (A, Location of N.A., I, Setc)are calculated in usual manner

(n-1)As/2(n-1)As/2

Transformed Section

Total Area of transformed section = b x h + (n-1)As= Ag+ (n-1)As


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Concluded

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Lec-5-Flexural Analysis and Design of Beamns