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**Design stress-strain curve at the ultimate state**

Design value of strength For concrete

where, y_{mc} = Partial factor of safety for concrete = 1.5

f_{d} = design value of strength

For steel

f_{d} = f_{y} / 1.15 = 0.87f_{y}

**Singly Reinforced Beam**

- Limiting depth of the neutral axis (x
_{u}, lim)

Here d = effective depth of the beam

- Actual depth of neutral axis (X
_{u}) - Lever arm = d – 0.42 X
_{u} - Ultimate moment of resistance

**Some special cases**

**When X**_{u}< X_{u}_{,lim}

It is an under-reinforced section

M_{u}= 0.36 f_{ck}bX_{u}(d - 0.42X_{u})

or M_{u}= 0.87f_{y}A_{st}(d - 0.42X_{u})**When X**_{u}= X_{u}_{,lim}

It is a balanced section

M_{u}= 0.36f_{ck}bX_{u,lim}(d - 0.42X_{u,lim})

or M_{u}= 0.87f_{y}A_{st}(d - 0.42X_{u,lim})**When X**_{u}> X_{u},_{lim}

It is over reinforced section. In this case, keep X_{u}limited to X_{u},lim and moment of resistance of the section shall be limited to limiting moment of resistance, (M_{u},lim)

**Doubly Reinforced Section**

- Limiting depth of neutral axis.
- For actual depth of neutral axis (X
_{u})

- Ultimate moment of resistance

M_{u}= 0.36f_{ck}bX_{u}(d - 0.42X_{u}) + (f_{sc}- 0.45f_{ek})A_{sc}(d - d_{c})

where f_{SC} = stress in compression steel and it is calculated by strain at the location of compression steel (f_{SC})

- Effective width of flange Discussed in WSM
- Limiting depth of neutral axis

**Singly reinforced T-Beam**

**Case-1:** When NA is in the flange area

i.e., X_{u} < D_{f}

**(i) for Xu**

**(ii) Ultimate moment of resistance**

Mu = 0.36f_{ck}b_{f}X_{u}(d - 0.42X_{u})

or M_{u} = 0.87 f_{y}A_{st}(d - 0.42X_{u})

**Case-2:** When NA is in the web area (X_{u} > D_{f})

Case (a) when Xu > Df and

i.e., depth of flange in less than the depth of the rectangular portion of the stress diagram.

**For actual depth of neutral, a is**

0.36f_{ek}b_{w}x_{u}+ 0.45f_{ek}(b_{f}- b_{w})D_{f}= 0.87f_{y}A_{st}**Ultimate moment of resistance**

**Special Case (2):** When Xu > Df and

i.e., the depth of the flange is more than the depth of the rectangular portion of the stress diagram.

**As per IS 456 : 2000**

(b_{f} – b_{w}) Df portion of the flange is converted into (b_{f} – b_{w})y_{f} section for which stress is taken constantly throughout the section is 0.45 f_{ck}.

As per IS 456 : 2000

y_{f} = 0.15X_{u} + 0.65D_{f} < D_{f}

For actual depth of neutral axis

0.36f_{ek}b_{w}X_{u} + 0.45f_{ek}(b_{f} - b_{w}) y_{f} = 0.87f_{y}A_{st1} + 0.87f_{y}A_{st2}

or 0.36f_{ck}b_{w}X_{u} + 0.45f_{ck}(b_{f} - b_{w})y_{f} = 0.87f_{y}A_{st}

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