Flanged Beam by Working Stress Method

In monolithic construction slabs and beams are cast together. If slab in such cases in compression zone they become effective, either partially or wholly, in adding significantly to the area of concrete in compression in the beam. However, if slabs are located in tension zone, concrete in the slab becomes effective in cracked section analysis.

A flanged beam will be designed as a rectangular beam even if cast monolithically when the bending moment is negative (Hogging moment). As in the case of support in a continuous beam. Away from the support, the slab will be in compression (Sagging moment). Therefore, in this region, it will be designed as a flanged beam.

Inverted Beams are for architectural requirement i.e. to provide high overhead clearance. Such beams are also designed as a rectangular beam because the slab is in tension zone and does not resist any compression.

${b}_{f}$ shown in the diagram above is the effective width of the flange. It is defined as the width of the flange with constant compressive stress equal to the peak actual flexural compressive stress which leads to the same longitudinal compressive force as due to the original stress distribution.

Effective Flange Width for T and L beams as per IS 456

     L Beam
${b}_{f}={b}_{w}+3{D}_{f}+\frac{{l}_{0}}{12}$
${b}_{f}<{b}_{w}+\frac{{l}_{1}}{2}$
     T Beam
${b}_{f}={b}_{w}+6{D}_{f}+\frac{{l}_{0}}{6}$
${b}_{f}<{b}_{w}+\frac{{l}_{1}+{l}_{2}}{2}$

     Isolated L Beam
${b}_{f}={b}_{w}+\frac{0.5{l}_{0}}{\frac{{l}_{0}}{b}+4}\le b$
     Isolated T Beam
${b}_{f}={b}_{w}+\frac{{l}_{0}}{\frac{{l}_{0}}{b}+4}\le b$
     ${b}_{w}=\text{Width of the web}$
     ${D}_{f}=\text{Thickness of the flange}$
     ${l}_{0}=\text{distance between points of zero moment in the beam}$
     ${l}_{0}=\text{effective span for simply supported beam and 0.7 times effective span for continuous beam}$

Analysis of Flanged Section

1. Neutral axis lies in the flange:
   $\frac{{b}_{f}{{D}_{f}}^2}{2}\ge m {A}_{st}(d-{D}_{f})$
   If flanged beam satifies above condition then flanged beam will be designed as singly reinforced rectangular beam of width ${b}_{f}$.


2. Neutral axis lies in the web: If the condition mentioned in 1 is not satisfied then the NA lies in the web.
   
   
   
   
        ${C}_{1}= \text{Compression carried by area }({b}_{f}x)$
        ${C}_{2}=\text{Compression carried by the negative (blue) area }[({b}_{f}-{b}_{w})(x-{D}_{f})]$
   
   NA can be find using the following relation,
   
   $\frac{{b}_{f}x^2}{2}-({b}_{f}-{b}_{w})\frac{(x-{D}_{f})^2}{2}=m {A}_{st}(d-x)$
   
   Determination of Stresses in concrete and steel
   
   By internal couple method:
   ${C}_{1} \times {Z}_{1}-{C}_{2} \times {Z}_{2} = M$
        ${C}_{1}=\frac{1}{2}{f}_{cbc} \times x \times {b}_{f}$
   
        ${Z}_{1} = d - \frac{x}{3}$
   
        ${C}_{2}=\frac{1}{2}{f}_{c}(x-{D}_{f})({b}_{f}-{b}_{w})=\frac{1}{2}[\frac{{f}_{cbc}(x-{D}_{f})}{x}](x-{D}_{f})({b}_{f}-{b}_{w})$
   
        ${Z}_{2}=d-{D}_{f}-\frac{x-{D}_{f}}{3}$
   
   ${f}_{cbc}$ can be find out using above relation and once ${f}_{cbc}$ is known ${f}_{st}$ can be find out from the following relation,
   
   ${f}_{st}=\frac {m \times {f}_{cbc}}{x}(d-x)$
   
   By flexural formula:
   $I=\frac{{{b}_{f}}{{x}^{2}}}{3}-({{b}_{f}}-{{b}_{w}})\frac{{{(x-{{D}_{f}})}^{3}}}{3}+m{{A}_{st}}{{(d-x)}^{2}}$
   
   ${{f}_{cbc}}=\frac{Mx}{I}$
   
   ${{f}_{st}}=m\frac{M(d-x)}{I}$
   



3. Moment of Resistance of the section
   
   If $n=\frac{x}{d} \text{(NA coefficient) < } {n}_{0} \text{(critcal NA coefficient)}$ i.e the section is under reinforced then,
   
   ${f}_{st}={\sigma}_{st}$, ${f}_{cbc}=\frac{{\sigma}_{st}\times x}{m \times (d-x)}$
   
   If $n \text{(NA coefficient) > } {n}_{0} \text{(critcal NA coefficient)}$ i.e the section is over reinforced then,
   
   ${f}_{cbc}={\sigma}_{cbc}$
   
        Moment of Resistance,
   $MOR = {C}_{1} \times {Z}_{1}-{C}_{2} \times {Z}_{2}$
   
   ${C}_{1},{C}_{2},{Z}_{1} \text{and } {Z}_{2}$ are same as explained above in "internal couple method for determination of stresses".

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Doubly RCC beam by Working Stress Method

When the bending moment to be borne by a beam becomes greater than the balanced moment of resistance, the section becomes over-reinforced section and IS 456 does not recommend an over-reinforced section due to its brittle nature of the failure.
If there is no restriction on the size of the beam, we can increase its size so that the beam becomes under-reinforced. But if the size is restricted due to some reason then either we can increase the concrete mix to increase the capacity of the section or we can provide compression reinforcement in compression zone to give additional strength to the concrete in compression and such beams are called doubly reinforced beam.

Advantages of compression reinforcement:

1. It permits smaller size beams which look aesthetic.
2. It reduces the long-term deflection and increases the ductility of the beam.
3. It can be used as anchor bars for positioning the shear reinforcement.
4. As the compression reinforcement increases the ductility of the beam, they are provided in the seismic zone to withstand repeated reversals produces.

where,
       ${A}_{sc}=\text{Area of Compression Steel}$
       ${A}_{st}=\text{Area of Tension Steel}$
       $m=\text{Modular Ratio of tension steel}$
       $m'=1.5m=\text{Modular Ratio of compression steel}$
       ${f}_{sc}=\text{Stress in compression steel} $
       ${f}_{st}=\text{Stress in tension stell}$


The value of the modular ratio of compression steel $(m')$ is higher than that of tension steel because of the long-term plastic deformation is known as creep. The creep deformation of concrete produces additional strain in compression steel and gradually raises the level of stress. To account for this increase in stress, the modular ratio of compression steel is increased.

Depth of Neutral Axis,

$\frac{b{{x}^{2}}}{2}+(m'-1){{A}_{sc}}(x-d')=m{{A}_{st}}(d-x)$

Stresses in Concrete and steel,

1. By flexure formula,
   Moment of Inertia about NA, $I=\frac{b{{x}^{3}}}{3}+(m'-1){{A}_{sc}}{{(x-d')}^{2}}+m{{A}_{st}}{{(d-x)}^{2}}$
   
   ${{f}_{cbc}}=\frac{Mx}{I}$
   
   $\frac{{{f}_{sc}}}{m'}=\frac{M(x-d')}{I}$
   
   $\frac{{{f}_{st}}}{m}=\frac{M(d-x)}{I}$


2. By Internal Couple Method,
   
   
   
   ${C}_{1}=\frac{1}{2}\times{{f}_{cbc}\times{xb}}$
   
   ${C}_{2}=(m'-1){A}_{sc}\times{{f}_{sc}}$
   
   ${f}_{sc}={f}_{cbc}\frac{x-d'}{x}$
   
   $M={C}_{1}(d-\frac{x}{3})+{C}_{2}(d-d')$
   Using above equations ${f}_{cbc}$ can be found out.
   
   $\frac{{f}_{sc}}{m'}={f}_{cbc}\frac{x-d'}{x}$
   
   $\frac{{f}_{st}}{m}={f}_{cbc}\frac{d-x}{x}$
   ${f}_{sc}$,${f}_{st}$ can be found out using above equations
   
   ${C}_{1}=\text{Compression carried by concrete}$
   ${C}_{2}=\text{Additional force carried by compression steel}$

Moment of Resistance of Doubly Reinforced Section

If neutral axis coefficient ${n}_{0}$ for singly reinforced balanced section is greater than actual neutral axis coefficient $n$

${n}_{0}> n$ then, ${f}_{st}={\sigma}_{st}$

${f}_{cbc}=\frac{{\sigma}_{st}}{m}(\frac{x}{d-x})$

$\text{MOR}={C}_{1}(d-\frac{x}{3})+{C}_{2}(d-d')$

${C}_{1}=\frac{1}{2}\times{{f}_{cbc}\times{xb}}$

${C}_{2}=(m'-1){A}_{sc}\times{{f}_{sc}}=(m'-1){A}_{sc}\times {f}_{cbc}\frac{x-d'}{x}$

If neutral axis coefficient ${n}_{0}$ for singly reinforced balanced section is smaller than actual neutral axis coefficient $n$

${n}_{0}< n$ then, ${f}_{cbc}={\sigma}_{cbc}$

$\text{MOR}={C}_{1}(d-\frac{x}{3})+{C}_{2}(d-d')$

${C}_{1}=\frac{1}{2}\times{{\sigma}_{cbc}\times{xb}}$

${C}_{2}=(m'-1){A}_{sc}\times {\sigma}_{cbc}\frac{x-d'}{x}$
Area of steel in tension and compression zone

To calculate or rather to design a doubly reinforced beam, the beam is divided in to two parts. First part will resemble a balanced singly reinforced section while the other will show only tension and compression reinforcement.

Area of Steel, ${A}_{st}={A}_{st1}+{A}_{st2}$

${A}_{st}=\frac{{R}_{w}bd^2}{{\sigma}_{st}({J}_{0}d)}+\frac{M-{R}_{w}bd^2}{{\sigma}_{st}(d-d')}$

$M-{{R}_{w}}b{{d}^{2}}=(m'-1){{A}_{sc}}\left( {{\sigma }_{cbc}}\frac{x-d'}{x} \right)(d-d')$
We can canculate the compression steel from above expression.

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RCC Beam

A beam is a structural member which has one dimension greater than the other two and placed in a horizontal plane. It's a flexural member which can take bending moment and shear force. The cross-section of a beam can either be rectangular or T-shaped or circular etc.

To resist the bending moment and shear force, an adequate section is required which is neither too bulky nor too small.

The geometry of the section of the beam depends on section modulus and section modulus itself depends on the moment of Inertia of that section.

IS 456 has given the criteria for the selection of the section of a beam which is based on the deflection of the beam.

As per IS 456:2000 the ratio of the span and effective depth should not be greater than 20, 7 and 26 for the simply supported beam, cantilever beam and continuous beam respectively. These values are valid for span up to 10 m above which these values must be modified by multiplying them with (10/span in meter).

Effective depth of a beam is the depth of the beam from the top fibre to the CG of the reinforcement provided in the tension zone. 

The difference between the overall depth and effective depth is called effective cover.

Effective cover = nominal cover + diameter of stirrups + half the diameter of main reinforcement steel bar.

And, the Nominal cover is provided so that the steel bars are fully embedded, and it is not exposed to exterior conditions like rain. It also helps in maintaining the required connection between concrete and steel bars.
IS 456:2000 has provided the values of Nominal Cover on the bases of exposure condition. Click here

Sections of beams

There are 3 types of sections on the bases of stress condition of concrete and that of steel.

1. Under Reinforced Section: When the permissible stress in steel is reached in a beam prior to that of concrete.
2. Balanced Section: When the permissible stress in steel and that of concrete reaches at the same time.
3. Over Reinforced Section: When the permissible stress in concrete reaches prior to that of steel.

IS 456:2000 does not recommend a over reinforced section because of its brittle failure nature. In case of under reinforced section the failure is ductile which gives sufficient warning in terms of excessive cracks and deflection to the inmates before failure. 

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Methods of Design

There are 3 methods to design a structural component.

1. Working Stress Method
2. Ultimate Load Method
3. Limit State Method 

Working Stress Method

This is a classical method used to design RCC structures. In this method, the material is assumed to behave elastically. The relationship between stress and load is linear and stresses within the material is not allowed to exceed the permissible stress.

               $\text{Permissible Stress}=\frac{\text{Strength of Material}}{\text{Factor of Safety}}$

        Permissible Stress in Tension steel: $0.55{{f}_{y}}={{\sigma }_{st}}$
        FOS for Steel = 1.8

        Permissible Stress in Concrete in bending: ${{\sigma }_{cbc}}=0.33{{f}_{ck}}$
        FOS for Concrete = 3

Assumptions in Working Stress Method

1. A section which is plane before bending remains plane after bending.
2. The bond between concrete and steel is perfect within the elastic limit of steel.
3. Tension is borne entirely by steel.
4. The Modulus of Elasticity of concrete is same for all stresses.
5. There are no initial stresses in steel when it is embedded in concrete.

Deficiencies in Working Stress Method

1. Due to the long-term effect of creep and shrinkage and stress concentration, it may not be possible to keep the stresses within permissible limit.
2. Actual margin of safety is not equal to the factor of safety used in WSM because the stress-strain curve is not linear up to collapse. 
3. WSM does not discriminate between different types of loads that act simultaneously.

Although WSM has deficiencies still it is used to design structures like Bridges, Water tanks, chimneys etc because of its simplified approach.

Ultimate Load Method

This method was introduced in the 1960s. In this method stress condition at the state of impending failure is analysed and non-linear stress-strain curve of steel and concrete are made use of.
A safety measure is introduced by an appropriate choice of load factor.

               $\text{Load Factor =}\frac{\text{Ultimate Load}}{\text{Working Load}}$

In this method distribution of stress resultants at ultimate load is taken as distribution at service load magnified by load factor. This is clearly an error because significant inelastic behaviour and redistribution of stress resultant take place as loading is increased from service loads to ultimate loads.

Limit State Method

There is uncertainty in loading, properties of material and dimensions of a member and to account for these uncertainties FOS and Load Factor was introduced in WSM and Ultimate load Method respectively. But there was no theoretical justification for use of FOS and load factors.

To overcome this, a reliability-based analysis was performed and factors of safety for both loading and material properties were established and these factors were called Partial factors of safety. Selection of partial factors of safety was done on the probabilistic basis.

This analysis was called Limit State Method. Limit state is a state in which the structure becomes unfit for use.

There are 2 types of Limit States:

1. Limit state of serviceability:  Satisfactory performance under service load. Such as discomfort caused by excessive deflection, crack width, vibration, leakage, loss of durability etc.
2. Limit state of collapse: Adequate margin of safety for normal overloads. These include limit state of strength, overturning, sliding, buckling, fatigue etc.

Assumptions in Limit State of Collapse

1. A section which is plane before bending remains plane after bending.
2. The maximum strain in concrete at the outermost compression fibre is taken as 0.0035 in bending.
3. The relationship between compressive stress distribution in concrete and strain in concrete may be assumed to be rectangular, trapezoidal, parabolic or any other shape which results in the prediction of strength in substantial agreement with the result of the test.
   
   
                                                  Source: IS 456
   
   
4. The Tensile strength of concrete is ignored.
5. Partial Factor of Safety of Steel is 1.15 and that for concrete is 1.5.
6. Maximum strain in Tension reinforcement in the section at failure shall not be less than
   
   

                  ${{\varepsilon }_{st}}=\frac{{{f}_{y}}}{1.15{{E}_{s}}}+0.002$
   Where,
                 ${{\varepsilon }_{st}}=$ Strain in Tension Steel
                 ${{f}_{y}}=$ Characterstic Strength of Steel
                 ${{E}_{s}}=$ Modulus of Elasticity of Steel

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RCC Basics

IS 456:2000 is the code for Plain and Reinforced Concrete. As per this code, the concrete is used for an RCC structure should have the grade of M20 or above.

The letter "M" here stands for mix and the number is the characteristic strength of standard cube of size 150 mm which when tested under compression load test after 28 days, not more than 5% of the cubes are expected to fail.

For example, if 100 standard cubes of M20 is tested after 28 days then at least 95 cubes should be able to resist more than 20 MPa before failure. If it is less than 95 then the test will be declared invalid and the concrete will be rejected.

But at a construction site, site engineers cannot wait for 28 days to test the compressive strength of the concrete used for construction. To know about the strength of the concrete at early stages they test the concrete cubes at 3 and 7 days. The 3-day strength of a concrete cube is nearly 40% of the 28-day strength and 7-day strength is nearly 65% of the 28-day strength.
This test is conducted on at least 3 cubes and the average of these 3 cubes is taken, provided that individual variation should not be more than $\pm $ 15 % average.

Characteristic compressive strength compliance requirement (IS 456 Cl 16.1 and 16.3)

Specified Grade	Mean of the group of 4 Non-overlapping consecutive test results in $N/m{{m}^{2}}$	Individual test results in $N/m{{m}^{2}}$
M 15	$\ge {{f}_{ck}}+0.825\times $ established standard deviation (rounded off to nearest $0.5 N/m{{m}^{2}}$) or ${{f}_{ck}}+3$ $N/m{{m}^{2}}$ , whichever is greater	${{f}_{ck}}-3$
M 20 or above	$\ge {{f}_{ck}}+0.825\times $ established standard deviation (rounded off to nearest  $0.5 N/m{{m}^{2}}$) or ${{f}_{ck}}+4$ $N/m{{m}^{2}}$ , whichever is greater	${{f}_{ck}}-4$

Characteristic Strength (${{f}_{ck}}$)

                                              ${{f}_{ck}}={{f}_{m}}-1.65\sigma $

Where, ${{f}_{m}}=$ Mean Strength
             $ \sigma=$ Standard Deviation
             $=\sqrt{\frac{\sum\limits_{{}}^{{}}{(f-{{f}_{m}})}}{m}}$ when test samples are $\ge$ 30
             $=\sqrt{\frac{\sum\limits_{{}}^{{}}{(f-{{f}_{m}})}}{m-1}}$ when test samples are $<$ 30
             $m=$ number of samples

Compressive Strength of Concrete in Structures

Strength of concrete is found to decrease with increase in the size of the specimen. However, beyond 450 mm size, there is no decrease in the compressive strength of concrete.
Thus, compressive strength of concrete in structure is taken as $0.67{{f}_{ck}}$

Flexural Strength of Concrete (Modulus of Rupture)
                                              ${{f}_{cr}}=0.7\sqrt{{{f}_{ck}}}$

Tensile Strength of Concrete

Tensile strength of plain concrete is obtained by the splitting test.

Splitting tensile strength,

                                              ${{f}_{ct}}=\frac{2P}{\pi dL}=0.6{{f}_{cr}}$

                                              Source:Research Gate

Stress Strain Curve of Concrete

                                      Source: IS 456

Maximum compressive stress occurs at a strain value of 0.002 i.e., 0.2%. The value of stress at 0.002 strain is called compressive strength of concrete.

Modulus of elasticity of concrete for all practical purpose is taken as secant modulus at a stress of around $0.33{{f}_{ck}}$.

Modulus of elasticity of concrete is primarily influenced by the elastic properties of aggregate and to a lesser exent by the condtions of curing, mix proportion and type of cememnt.

As per IS 456:2000 short term modulus of elasticity,
                                            ${{E}_{C}}=5000\sqrt{{{f}_{ck}}}$
Long-term Modulus of elasticity depends on Creep,
                                            ${{E}_{ce}}=\frac{{{E}_{C}}}{1+\theta }$
$\theta=$ Creep Coefficient

Age at loading	Creep Coefficient
7 days	2.2
28 days	1.6
1 year	1.1

Exposure Conditions

Exposure	Minimum Grade	Minimum Cement Content ${KG}/{{{m}^{3}}}\;$	Maximum freewater cement ratio	Nominal Cover, mm
Mild	M20	300	0.55	20
Moderate	M25	300	0.50	30
Severe	M30	320	0.45	45
Very Severe	M35	340	0.45	50
Extreme	M40	360	0.40	75

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