Types of problem in doubly reinforced beams working stress method: Determination of moment of resistance of the given section,Determination of actual stresses in concrete and steel,Design of the section.

## Types of problem in doubly reinforced beams working stress method

- Determination of moment of resistance of the given section.
- Determination of actual stresses in concrete and steel.
- Design of the section.

** Determination of Moment of Resistance**

** Given :**

(i) Dimension of the beam section (b and d)

(ii) Area of tensile steel A_{st} and area of compressive steel A_{sc}

(iii) Permissible stress in concrete {σ_{cbc}) and permissible stress in steel (σ_{st})

#### **Procedure :**

- Calculate \[m=\frac{280}{3\sigma_{cbc}}\]
- Calculate critical neutral axis (n
_{c})

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

- Calculate actual neutral axis depth (n
_{c})

\[\frac{b.n^{2}}{2}+(1.5m-1)A_{sc}(n-d_{c})=m.A_{st}(d-n)\]

- Compare n and n
_{c}

(a) If n>n_{c} the section is under reinforced (fully stressed)

Maximum tensile stress developed in steel = σ_{st} Maximum compressive stress developed in concrete

\[\sigma_{cbc}(where \sigma’_{cbc})is less than \sigma_{cbc}\]

\[\frac{\sigma’_{cbc}}{\sigma_{st}/m}=\frac{n}{d-n}\] (from stress diagram)

\[\therefore\sigma’_{cbc}=\frac{\sigma _{st}}{m}\left ( \frac{n}{d-n} \right )\]

The stress in concrete at the level of compression steel (σ_{c}) can be obtained as

\[\sigma’_{c}=\frac{\sigma _{cbc}}{n}(n-d’)\]

The moment of resistance of the (under reinforced) doubly reinforced section is calculated as :

\[M_{r}=\frac{1}{2}\sigma_{cbc}.bn\left ( d-\frac{n}{3} \right )+(1.5m-1)\sigma’_{c}.A_{sc}(d-d’)\]

(b) If n>n_{c}, then section is over reinforced and max compressive stress in concrete is σ_{cbc}. The moment of resistance is calculated as

\[\therefore M_{r}=\frac{1}{2}\sigma_{cbc}.b.n\left ( d-\frac{n}{3} \right )+(1.5m-1)\sigma’_{c}.A_{sc}(d-d’)\]

where \[\sigma’_{c}=\frac{\sigma_{cbc}}{n}(n-d’)\]

** Determination of Stress in Steel and Concrete **

** Given :**

(i) Dimensions of beam (b and d)

(ii) Area of tensile and compressive reinforcement i.e., A_{st} and A_{sc}

(iii) Material used grade of concrete and steel.

(iv) Maximum bending moment or loading on the section.

#### **Procedure :**

- Determine actual neutral axis of the section

\[\frac{bn^{2}}{2}+(1.5m-1)A_{sc}.(n-d’)=m.A_{st}(d-n)\]

- Determine the value of σ
_{c }in terms of σ_{cbc}

\[\sigma’_{c}=\frac{\sigma_{cbc}}{n}(n-d’)\]

- Determine the maximum bending moment (M) on the section due to loads and equate it to moment of resistance of the section (M
_{r})

\[M=M_{r}=\frac{1}{2}\sigma_{cbc}b.n.\left ( d-\frac{n}{3} \right )+(1.5m-1)\sigma’_{c}.A_{sc}(d-d’)\]

Putting the value of σ_{c} in this equation

\[M=\frac{1}{2}\sigma_{cbc}b.n\left ( d-\frac{n}{3} \right )+(1.5m-1)A_{sc}\frac{\sigma _{cbc}}{n}(n-d’)(d-d’)\]

In the above equation, only σ_{c} as unknown and it can be calculated easily.

- Knowing σ
_{cbc}the stress in tensile steel (σ_{st}) and stress in compressions steel (σ_{sc}) are calculated as under

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

and \[\sigma’_{st}=\frac{\sigma_{cbc}}{n}(n-d’)\]

\[\sigma_{sc}=m_{c}.\sigma’_{c}\]

** Design of the Section**

** Given :**

(i) Span of the beam (*l*) and its dimensions (b and d).

(ii) Loading on the beam.

(iii) Material used-grade of concretes and steel i.e., σ_{cbc} and σ_{st}.

**Procedure :**

- Determine maximum bending moment (M) coming on the section due to loads (including self weight of the beam).
- Calculate the design constants k, j and R for the given materials.
- Calculate M
_{1}=Rbd^{2}. - Calculate the area of tensile reinforcement (A
_{st1}) corresponding to M_{1}.

\[A_{st}=\frac{M_{1}}{\sigma_{st}jd}\]

- Calculate M
_{2}

M_{2} =M-M_{1}

- Calculate the additional area of the tensile reinforcement (A
_{st2}) needed to resist M_{2}.

\[M _{2}=\sigma _{st}.A _{st _{2}}(d-d’)\]

\[\therefore A_{st^{2}}=\frac{M_{2}}{\sigma_{st}(d-d’)}\]

- Determine total area of tensile steel (A
_{st})

\[A _{st} =A _{st _{1}}+A _{st _{2}}\]

Selecting suitable diameter of the bar, provide A_{st}.

- Determine the area of compressive steel (A
_{sc}) by equating the moment area of compressive steel (A_{sc}) to the moment of the area of additional tensile steel (A_{st2}) about neutral axis.

Moment of the area of compressive steel (A_{sc}) about neutral axis.

\[= (m _{c}-1) A _{sc} (n-d’) = (1.5m-1) A _{sc} (n-d’)\]

Moment of the additional area of tensile steel (A_{st2}) about neutral axis

\[= m.A _{st _{2}}(d-n)\]

Equating them and calculating A_{SC}.

\[1.5(m-1)A _{sc} (n-d’)=m.A _{st _{2}}(d-n)\]

\[A_{sc}=\frac{m.A_{st_{2}}(d-n)}{1.5(m-1)(n-d_{c})}\]

Calculate the number of bars required for providing A_{sc}.

- Design for shear and bond is same as that of singly reinforced beam.
- Draw a neat sketch and give summary of design.

**Note :** Design consideration for a doubly reinforcement beam as per IS 456:2000.

(i) Maximum Compression Reinforcement (A_{sc}): The maximum compression reinforcement in a beam cannot be more than 0.04 bD. (4%) of the gross cross-sectional area).