Learn : Analysis of T beam working stress method : Neutral Axis is Within the Flange (n < Df), Neutral Axis Lies in the Web of the Beam (n >Df)
ANALYSIS OF T BEAM
Consider the section of a T-beam shown in Fig. 2.14 (a). The analysis of a T-beam comprises of following two cases :
(i) Neutral axis is within the flange.
(ii) Neutral axis is in the web.
Case 1 : Neutral Axis is Within the Flange (n < Df)
Equivalent or Transformed Section
The equivalent section of the T-beam in terms of concrete is shown in Fig. 2.13 (b).
The concrete below the neutral axis is assumed to be cracked and the area of steel is replaced by an equivalent concrete area which is equal to m.Ast. The compression area is rectangular in shape as n < Df. Thus, this flanged beam can be analyzed exactly as a rectangular beam having a width bf instead of b.
Neutral Axis
The neutral axis is determined in the following two ways :
Critical Neutral Axis :
The critical neutral axis is determined by the safe permissible stresses.
\[\frac{n_{c}}{d-n_{c}}=\frac{m.\sigma_{cbc}}{\sigma_{st}}\]
Actual Neutral Axis :
The actual neutral axis is determined by equating the moments of areas of compressive and tensile zones about neutral axis. Considering Fig. 2.13.
Moment of compressive area about neutral axis = \[b _{f}.n.\frac {n}{2}\]
Moment of tensile area about neutrl axis = \[m.A _{st}.(d-n)\]
\[\therefore b _{f}.\frac{n^{2}}{2}=m.A _{st}(d-n)\]
Lever Arm
The lever arm is calculated as under :
\[a=d-\frac{n}{3}\]
Moment of Resistance
The moment of resistance of the given T-beam section is determined by taking the moment of total compressive force about the centroid of steel reinforcement.
\[M _{r} = Total compressive force × Lever arm = C × a\]
\[M _{r}=\frac{1}{2}\sigma _{cbc}.b _{f}.n(d-\frac{n}{3})\]
Case II: Neutral Axis Lies in the Web of the Bam (n >Df)
Equivalent Section
The equivalent section of the T-beam when n>Df is shown in Fig. 2.15(b).
The compression zone of the given section consists of :
\[(i) b_{f}\times D_{f}\]
\[(ii) b_{w}\times (n-D_{f})\]
The tensile steel is replaced by an equivalent concrete area =m.Ast.
Neutral Axis
The actual neutral axis of the given T-beam section can be determined in following two ways :
When Compression in the Web is taken into Consideration :
Considering Fig. 2.14 (b) and taking moments of the area of compressive and tensile zones about the neutral axis.
Moment of compression area of flange = \[b_{f}.D_{f}\left ( n-\frac{D_{f}}{2} \right )\]
Moment of compression area of web = \[b_{w}(n-D_{f})\left ( \frac{n-D_{f}}{2} \right )\]
Moment of tensile area = m.Ast(d-n)
\[\therefore b_{f}D_{f}\left ( n-\frac{D_{f}}{2} \right )+b_{w}(n-D_{f})\left ( \frac{n-D_{f}}{2} \right )=m.A_{st}(d-n)\]
The above equation can be solved for getting n.
When Compression in the Web is not taken into Consideration:
The contribution of the web area in compression is generally very small as compared to the flange area. So it may be neglected while analysing a T-beam. neglecting \[b_{w}(n-D_{f})\left ( \frac{n-D_{f}}{2} \right )\], we get
\[b_{f}.D_{f}\left ( \frac{n-D_{f}}{2} \right )=m.A_{st}(d-n)\]
MOMENT OF RESISTANCE (n > Df)
When compression in the Web is Taken into Consideration
The stress diagram for this case is shown in Fig. 2.14 (c).
Mr = M1 + M2
where M1 = Moment of resistance due to compressive force in flange
M2 = Moment of resistance due to compressive force in web
M1 = Total compressive force × Lever arm
\[=C_{1}\times a\]
\[C_{1}=\frac{\left ( \sigma _{cbc}+\sigma ‘_{c} \right )}{2}\times D_{f}\times b_{f}\]
where σc is the compressive stress in concrete at the junction of flange and web.
\[\sigma’_{c}=\sigma_{cbc}\left ( \frac{n-D_{f}}{n} \right )\] [From similar triangles]
\[a=d-\bar{y}\]
where ȳ is the distance of centre of gravity of the dark shaded portion of stress diagram from the top of the beam
\[\bar{y}=\left( \frac{\sigma_{cbc}+2\sigma’_{c}}{\sigma_{cbc}+\sigma’_{c}} \right )\times \frac{D_{f}}{3}\]
and \[M_{1}=\frac{1}{2}(\sigma_{cbc}+\sigma’_{c})b_{f}.D_{f}(d-\bar{y})\]
\[M_{2} = Total compressive force in web × Lever arm\]
\[= C_{2}\times a’\]
\[C_{2}=b_{w}(n-D_{f})\left ( \frac{\sigma’_{c}}{2} \right )\]
\[a’=\left ( d-D_{f}-\frac{(n-D_{f})}{3} \right )\]
\[\therefore M_{2}=b_{w}(n-D_{f})\left ( \frac{\sigma’_{c}}{2} \right )\left ( d-D_{f}\frac{(n-D_{f})}{3} \right )\]
\[M_{r}=M_{1}+M_{2}\]
\[M_{r}=b_{f}.D_{f}\left ( \frac{\sigma_{cbc}+\sigma’_{c}}{2} \right )(d-\bar y)+b_{w}(n-D_{f})\left ( \frac{\sigma’_{c}}{2} \right )\left ( d-D_{f}-\frac{(n-D_{f})}{3} \right )\]
Web Compression is not Taken into Consideration :
In this case M2 is neglected as it is very small as compared to M1.
\[\therefore M_{r}=M_{1}\]
\[=b_{f}.D_{f}\left ( \frac{\sigma_{cbc}+\sigma’_{c}}{2} \right )(d-\bar y)\]
where \[\sigma’_{c}=\sigma_{cbc}\frac{(n-D_{f})}{n}\]
\[\bar y=\left ( \frac{\sigma_{cbc}+2\sigma’_{c}}{\sigma_{cbc}+\sigma’c} \right )\times \frac {D_{f}}{3}\]
Note : While calculating the moment of resistance of the T-beam sections, first it is to be checked whether the section is under-reinforced or over reinforced by comparing n and nc and the values of stresses are determined appropriately.