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
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
${b}_{f}<{b}_{w}+\frac{{l}_{1}}{2}$
T Beam
${b}_{f}<{b}_{w}+\frac{{l}_{1}+{l}_{2}}{2}$
Isolated L Beam
Isolated T Beam
${b}_{w}=\text{Width of the web}$
${D}_{f}=\text{Thickness of the flange}$
${l}_{0}=\text{distance between points of zero moment in the beam}$
${l}_{0}=\text{effective span for simply supported beam and 0.7 times effective span for continuous beam}$
Analysis of Flanged Section
-
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 ${b}_{f}$. - 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:
$I=\frac{{{b}_{f}}{{x}^{2}}}{3}-({{b}_{f}}-{{b}_{w}})\frac{{{(x-{{D}_{f}})}^{3}}}{3}+m{{A}_{st}}{{(d-x)}^{2}}$
${{f}_{cbc}}=\frac{Mx}{I}$
${{f}_{st}}=m\frac{M(d-x)}{I}$
-
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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