Learn : Types of problem in T Beam : Procedure for Determining the Moment of Resistance of the Given Section in T beam,Find Stresses in Steel and Concrete in T beam,To design the section for given loads.Design of T beam

**TYPES OF PROBLEM IN T BEAM**

- To determine the moment of resistance of the given section in T beam.
- To Find Stresses in Steel and Concrete in T beam.
- To design of T beam.

** Procedure for Determining the Moment of Resistance of the Given Section in T beam**

Given : Dimensions of the section i.e., b_{w}, b_{f}, D_{f}, d.

Area of steel A_{st}

Material – grade of concrete and steel

(i) Determine σ_{cbc} and σ_{st} from the Tables 2.1 and 2.2 for given grades of concrete and steel Calculater modular ratio (m)

\[m=\frac{280}{3\sigma_{cbc}}\]

(ii) Determine critical neutral axis (n_{c})

\[\frac{m.\sigma_{cbc}}{\sigma_{st}}=\frac{n_{c}}{d-n_{c}}\]

(iii) Determine actual neutral axis (n) : To reduce the trial of calculations, it is better to assume n>D_{f}.

\[\therefore b_{f}.D_{f}\left ( n-\frac{D_{f}}{2} \right )=m.A_{st}(d-n)\]

If n comes out to be less than D_{f} on solving the above equation, then use following equation to calculate n.

\[\therefore b_{f}\times \frac{n^{2}}{2}=m.A_{st}(d-n)\]

**(iv) Compare n and n _{c}**

(i) If n<n_{c}, then under reinforced section

(ii) If n>n_{c}, then over reinforced section

(v) Determine moment of resistance using appropriate formula after determining the stress in following ways.

** (a) For under reinforced section,** σ_{st} is known and σ_{cbc} can be calculated as follows :

\[\therefore \frac{m.\sigma_{cbc}}{\sigma_{st}}=\frac{n}{d-n}\]

** (b) Determine **σ_{c} and ȳ

\[\sigma’_{c}=\sigma_{cbc}\left ( \frac{n-D_{f}}{n} \right )\]

\[\bar y=\left ( \frac{\sigma_{cbc}+2\sigma’_{c}}{\sigma_{cbc}+\sigma’_{c}} \right )\frac{D_{f}}{3}\]

(c) For over reinforced section, σ_{cbc} is known hence calculating σ** _{c}**.

\[\sigma’_{c}=\sigma_{cbc}\left ( \frac{n-D_{f}}{n} \right )\]

** Problem to Find Stresses in Steel and Concrete in T beam**

** Given :**

Dimension of the beam

Area of steel

Maximum B.M. or load on the beam

** Determine actual neutral axis**

** Write **σ_{cbc}**in terms of **σ_{c}

\[\sigma’_{c}=\sigma_{cbc}\left ( \frac{n-D_{f}}{n} \right )\]

** Find ȳ**

\[\bar y=\left ( \frac{\sigma_{cbc}+2\sigma’_{c}}{\sigma_{cbc}+\sigma’_{c}} \right )\times\frac{D_{f}}{3}\]

** Find moment of resistance (M _{r})**

\[M_{r}=b_{f}.D_{f}\left ( \frac{\sigma_{cbc}+\sigma’_{c}}{2} \right )(d-\bar y)\]

Equate moment of resistance to maximum bending moment and find σ_{cbc}

** Find **σ_{st}

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

**Design of T Beam**

In Design of T beam of problem, the dimensions of the beam and the area of steel is to be determined.

**Given :** Maximum bending moment or loading

Materials i.e., grade of concrete and steel span of the beam.

- Calculate design constants (m, k.j)
- Assume total depth of the beam as \[\frac{1}{12} to \frac{1}{15}\] of the span and calculate effective depth.
- Determine the maximum bending moment coming on the beam due to given loads.
- Determine area of steel required.

\[A_{st}=\frac{M}{\sigma_{st}jd}\]

- Check the trial section as follows :

(i) Determine actual neutral axis depth (n)

(ii) Write σ_{c} in terms of σ_{cbc} and find ȳ.

(iii) Write M_{c} in terms of σ_{cbc}.

(iv) Equating M and M_{r} and calculate σ_{cbc}and σ_{st}

(v) If the values of σ_{cbc} and σ_{st} are less than the permissible stresses, then design is OK, but if not, then revise the trial section and repeat steps from 2 to 5.

- Design for shear
- Check for development length.