Types of problem in T Beam | Working Stress Method

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

1. To determine the moment of resistance of the given section in T beam.
2. To Find Stresses in Steel and Concrete in T beam.
3. 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_cbcin 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.

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

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

5. 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_cbcand 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.

6. Design for shear
7. Check for development length.