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**CONCRETE DESIGN****BEAMS AND SLABS(I) Working Stress Method **

(a) Modular Ratio (m): It is the ratio of modulus of elasticity of steel to that of concrete

Where, E

**(b) Equivalent Areas of Composite Section**

Consider an R.C.C. section shown in Fig below subjected to a compressive load P.

Let

A = Area of cross-section

A_{c} = Area of concrete

A_{st} = Area of steel

m = Modular ratio

S_{s} = Stress in steel

S_{c} = Stress in concrete

e_{s} = Strain in steel

e_{c} = Strain in concrete

P_{s} = Load carried by steel

P_{c} = Load carried by concrete

A_{eqc} = Equivalent area of section in terms of concrete

E_{s} = Yong’s modulus of elasticity of steel

E_{c} = Young’s modulus of elasticity of concrete

P=P_{s}+P_{c}

The bond between steel and concrete is assumed to be perfect so the strains in steel and the surrounding concrete will be equal

It means that stress in steel is m times the stress in concrete or load carried by steel is m times the load carried by concrete of equal area. Using Eqns. (i) and (ii)

The expression in the denominator

is called the equivalent area of the section in terms of concrete. It means that the area of steel Ast, can be replaced by an equivalent area of concrete equal to m.Ast

Therefore, the concept of modular ratio makes it possible to transform the composite section into an equivalent homogeneous section, made up of one material.

**(c) Assumptions in working stress method:**

-Plane Section before bending will remain plane after bending

-Bond between steel and concrete is perfect with in elastic limit of steel.

-The steel and concrete behaves as linear elastic material

-All tensile stresses are taken by reinforcement and none by concrete

-The stresses in steel and concrete are related by a factor known as “modular ratio

-The Stress-strain relationship of steel and concrete is a Straight line under working load

**(d) Effective Length**

- Simply supported beams and slabs (I
_{eff})

l_{eff} = minimum

Here, l_{0} = clear span

w = width of support

d = depth of beam or slab

- For continuous beam

(i) If width of support < (1/12) of clear span

l_{eff} = minimum

(ii) If width of support > (1/12) of clear span

(a) When one end fixed other end continuous or both end continuous. l_{ef} = l_{0}

(b) When one end continuous and other end simply supported.

l_{eff} = minimum (1 + w / 2,1 + d / 2)

**Cantilever**

**Frames**

l_{eff }= Centre to Centre distance

In the analysis of continuous frame centre to centre distance shall be used.

**(e) Control of Deflection **

(i) This is one of the most important check for limit state of serviceability.

(a) The final deflection due to all loads including the effect of temperature, creep and shrinkage and measured from as cast level of the support of floors, roofs and other horizontal members should not and other horizontal members should not normally exceed **span/250**

(b) The deflection including the effect of temperature, creep and shirnkage occuring after erection of partition and application of finishes should not normally exceed**span/350** or 20 mm which ever is less.

(ii) The vertical deflection limit may generally be assumed to be satisfied provided that span to depth ratio are not greater than the value obtained as below: (a) Basic span to effective depth ratio for span upto 10 m is

**Type of beams:span/ effective depth**

For Cantilever — 7

For simply supported — 20

For continuous — 26

(b) For span > 10 m effective depth

where 'A' is span to effective depth ratio for span upto 10m.

(c) Depending upon the tension reinforcement the value 'A' can be modified by multiplying a factor called modification factor (MF_{1}) effective depth =

**(d) **Depending** **upon area of compression reinforcement, value (A) can be further modified using a modification factor (MF_{2}) effective depth =

**(e) For flanged beam : **A reduction factor is used

(f) **Deflection check for two way slab:****Slenderness Limit**:

1. For simply supported or continuous beams.

where, l_{0} = Clear span

B = Width of the section and

d = Effective depth

2. For Cantilever beam

** (i) Minimum tensions reinforcement**

**(ii) Maximum tension reinforcement =** 0.04bD

**(iii) Maximum compression reinforcement **= 0.04 bD

where D = overall depth of the section

**(iv)** Where D > 750 mm, side face reinforcement is provided and it is equal to 0.1% of gross cross-section area (b × D). It is provided equally on both face.**(v) **Maximum spacing of side face reinforcement is 300 mm.

**(v)** Maximum size of reinforcement for slab/beam is 1/8 of total thickness of the member.

**(vi)** Nominal cover for different members

Beams — 30 mm

Slab — 20 to 30 mm

Column — 40 mm

Foundation — 50 mm

**(vii)** Moment and shear coefficiently beam/slabs

**One way slab:**

**Two way slab:****(i) ****(ii) **Slab is supported on all edges.

**(g)**** Permissible value of strength in concrete in tension for mild steel **

- ζ
_{bd}given in table is only for plain mild steel bar in tension. - ζ
_{bd}value should be increased by 60% for deformed bars both in LSM and WSM. - For bars in compression the value should be increased by 25%.

**(h) Permissible stresses in steel**

(Clauses B-2.2, B-2.2.1, B -2.3 an B-4.2)

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