Learn : Analysis of singly reinforced beam Working stress method,Equivalent or Transformed Section,Strain Diagram,Stress Diagram,Neutral Axis,Stresses in Concrete and Steel , Dimensions of the Beam and Area of Steel,Percentage of Steel,Lever arm, moment of resistance

**Analysis of singly reinforced beam Working stress method**

Analysis of singly reinforced beam Working stress method : A singly reinforced beam section is shown in Fig. 2.3(a). To analyse this section, it is necessary to convert it into a transformed or equivalent section of concrete.

**Equivalent or Transformed Section**

As per the assumption (3), all the tensile stresses are taken by steel and none by concrete i.e., concrete in the tensile zone is cracked. So, the concrete area below the neutral axis is neglected and the effective area or the equivalent area of the section in terms of concrete is shown in Fig. 2.3(b). The equivalent area is equal to the area of concrete in the compression zone and an additional concrete area mA_{st }of conrete corresponding to steel area, A_{st}

**Strain Diagram**

As per the assumption (1) of elastic theory, the strain distribution is linear, with value zero at the neutral axis to maximum at the top and bottom fibre. The strain diagram for the given R.C.C. section is shown in Fig. 2.3(c).

### Stress Diagram

As per the assumption (4) of the elastic theory the stress-strain relationship is linear for concrete. So, the stress diagram is also a straight line with value zero at neutral axis and varying linearly with the distance as shown in Fig. 2.3(d).

Maximum permissible stress at the top most fibre in concrete =σ_{cbc}

Maximum permissible stress in steel =σ_{st}

Maximum stress in equivalent concrete area at the level of steel= \[\frac{\sigma _{st}}{m}\]

**Note:** 1. The suffix cbc in σ_{cbc} stands for permissible stress in *concrete in bending compression*.

- The suffix
*st*in σ_{st}stands for permissible stress in*steel in tension*.

** Neutral Axis (n)**

Neutral axis lies at the centre of gravity of the section. *It is defined as that axis at which the stresses are zero.* It divides the section into tension and compression zone. The position of the neutral axis depends upon the shape (dimensions) of the section and the amount of steel provided. The position of neutral axis of any rectangular section can be found by the following two methods :

** Stresses in Concrete and Steel are Known**

Let us consider the R.C.C. section shown in Fig. 2.4(a) the stress σ_{c} in concrete’s top most fibre and σ_{s} in steel reinforcement are known.

From stress diagram:

\[\frac{\sigma _{c}}{n}=\frac{\sigma _{s}/m}{d-n}\]

[Similar triangles]

\[\frac{m.\sigma _{c}}{\sigma _{s}}=\frac{n}{d-n}\]

If the stresses in concrete and steel are permissible then equation for n is written as

\[\frac{m.\sigma _{cbc}}{\sigma _{st}}=\frac{n}{d-n}\]

*This neutral axis, corresponding to permissible values of stresses of concrete and steel is called as critical neutral axis n _{c} .*

* n _{c} *=kd where k is the neutral axis depth factor.

\[\frac{m.\sigma _{cbc}}{\sigma _{st}}=\frac{kd}{d-kd}\]

On rearranging, we get

\[k=\frac{m.\sigma _{cbc}}{m.\sigma _{cbc}+\sigma _{st}}\]

Putting \[m=\frac{280}{3\sigma _{cbc}}\] in the above equation for k, we can see that k does not depend upon grade of concrete. It depends upon grade of steel only.

\[k=\frac{280/3}{280/3+\sigma _{st}}\]

#### ** Dimensions of the Beam and Area of Steel are Known **

The moment of the tensile and compressive area should be equal at the neutral axis. The neutral axis obtained by this method is called as *actual neutral axis*.

Moment of compressive area = Area in compression × Distance between c.g. of compressive area and neutral axis

\[=b.n.\frac{n}{2}=\frac{bn^{2}}{2}\]

Moment of tensile area = Equivalent tensile area × Distance of centroid of steel reinforcement from neutral axis

\[=m.A_{st}×(d-n)\]

Moment of compressive area = Moment of tensile area

\[\frac{bn^{2}}{2}=m.A_{st}(d-n) (iii)\]

It is a quadratic equation which will give two values of n. Out of these two values only one value (+ve) of n is possible.

**Percentage of Steel ***P*_{t}

*P*

_{t}The percentage of steel in R.C.C. sections means the area of steel (A_{st}) expressed as percentage of total area of concrete.

\[\therefore P_{t}=\frac{A_{st}}{bd}\times 100\]

\[By equation (iii), \frac{b.n^{2}}{2}=m.A_{st}(d-n)\]

On rearranging, we get \[ A_{st}=\frac{b.n^{2}}{2m(d-n)}\]

\[P_{1}=\frac{50n^{2}}{md(d-n)}\]

Putting n = kd \[ P_{1}=\frac{50k^{2}}{m(1-k)}\]

** Lever Arm**

Lever arm is the distance between the resultant compressive force and the resultant tensile force. It is denoted as a in the stress diagram. As the compressive area is triangular, the resultant compressive force (C) will act at \[\frac{n}{3}\] from the top compressive fibre. The resultant tensile force (T) will act the centroid of the steel reinforcement.

Lever arm = \[a=d-\frac{n}{3}\] , it is also expressed as a = jd where j is the lever arem depth factor.

\[jd=d-\frac{kd}{3}\]

\[j=1-\frac{k}{3}\]

** Moment of Resistance (M**_{r})

_{r})

*Moment of resistance is the resistance offered by the beam against external loads. *As there is no resultant force acting on the beam and the section is in equilibrium, the total compressive force is equal to the total tensile force. These two forces (equal and opposite separated by a distance) will form a couple (Fig. 2.5) and the moment of this couple is equal to the resisting moment or moment of resistance of the section.

Total compression = C=\[\left ( \frac{1}{2}\sigma _{cbc}\times n \right )b=\frac{1}{2}\sigma _{cbc}bn acting at \frac {n}{3}\] from top

Total tension = T=\[\sigma _{st}.A _{st}\] acting at centroid of steel reinforcement.

Moment of resistance = C . a or T . a

\[M_{r}=\frac {1}{2}\sigma _{cbc}b.n\left ( d-\frac{n}{3} \right ) ….(iv) \] \[ \left [ where a is the lever arm \therefore a=d-\frac{n}{3} \right ]\]

\[M _{r}=\sigma _{st}.A _{st}\left ( d-\frac{n}{3} \right ) …..(v)\]

Putting n = kd in the equation (iv),

\[M _{r}=\frac{1}{2}\sigma _{cbc}b.kd\left ( d-\frac{kd}{3} \right )\]

\[=\frac{1}{2}\sigma _{cbc}k.\left ( 1-\frac{k}{3} \right )b.d^{2}\]

\[M _{r}=\frac{1}{2}\sigma _{cbc}k.j.b.d^{2} \] \[ \left [ \because j=1-\frac{k}{3} \right ]\]

\[M _{r}=Rbd^{2} \] where R is called as resisting moment factor.

\[R=\frac{1}{2}\sigma _{cbc}kj\]

The factor *k, j* and *R* are constant for a given type of steel and concrete and do not depend upon the beam dimension. These are called as *design constants.* The value of *k, j, R* and *P _{t} * are given in Table 2.3.

**TABLE 2.3. Values of Design Constants**

Grade of Concrete |
σ_{cbc} |
Modular ratio m |
Mild Steel σ= 140 N/mm_{st} ^{2} |
Fe 415 σ=230 N/mm_{st}^{2} |
Fe 500 σ = 275 N/mm_{st}^{2} |
|||||||||

k |
j |
R |
P_{t} |
k |
j |
R |
P_{t} |
k |
j |
R |
P_{t} |
|||

M15 | 5 | 18.67 | 0.4 | 0.867 | 0.867 | 0.72 | 0.29 | 0.904 | 0.65 | 0.314 | 0.25 | 0.916 | 0.58 | 0.23 |

M20 | 7.0 | 13.33 | 0.4 | 0.867 | 1.214 | 1.0 | 0.29 | 0.904 | 0.914 | 0.44 | 0.25 | 0.916 | 0.81 | 0.32 |

M25 | 8.5 | 10.98 | 0.4 | 0.867 | 1.48 | 1.21 | 0.29 | 0.904 | 1.11 | 0.534 | 0.25 | 0.916 | 0.985 | 0.39 |

M30 | 10.0 | 9.33 | 0.4 | 0.867 | 1.73 | 1.43 | 0.29 | 0.904 | 1.306 | 0.628 | 0.25 | 0.916 | 1.16 | 0.46 |

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