# Working Stress Design of Reinforced Concrete

## Working Stress Design

Working Stress Design is called **Alternate Design Method** by NSCP (*National Structural Code of the Philippines*) and ACI (*American Concrete Institute, ACI*).

**Code Reference**

*NSCP 2010, Section 424:*Alternate Design Method

*ACI 318M-99, Appendix A:*Alternate Design Method

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## Notation

*f*= allowable compressive stress of concrete

_{c}*f*= allowable tesnile stress of steel reinforcement

_{s}*f'*= specified compressive strength of concrete

_{c}*f*= specified yield strength of steel reinforcement

_{y}*E*= modulus of elasticity of concrete

_{c}*E*= modulus of elasticity of steel

_{s}*n*= modular ratio

*M*= design moment

*d*= distance from extreme concrete fiber to centroid of steel reinforcement

*kd*= distance from the neutral axis to the extreme fiber of concrete

*jd*= distance between compressive force

*C*and tensile force

*T*

ρ = ratio of the area of steel to the effective area of concrete

β

*= ratio of long side to short side of concentrated load or reaction area*

_{c}*A*= area of steel reinforcement

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## Design Principle

The design concept of WSD is based on Elastic Theory in which the stress-strain diagram is within the proportional limit and will obey Hooke's Law.

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## Allowable Stresses

(NSCP 2010, 424.4.1 / ACI 318M, A.3.1)

**Concrete**

1. | Flexure | |

Extreme fiber stress in compression | $0.45f'_c$ | |

2. | Shear | |

Beams and one-way slabs and footings: | ||

Shear carried by concrete, $v_c$ | $0.09\sqrt{f'_c}$ | |

Maximum shear carried by concrete plus shear reinforcement, $v_c$ | $0.38\sqrt{f'_c}$ | |

Joists: | ||

Shear carried by concrete, $v_c$ | $0.09\sqrt{f'_c}$ | |

Two-way slabs and footings: | ||

Shear carried by concrete, $v_c$ but not greater than $\frac{1}{6}\sqrt{f'_c}$ | $\frac{1}{12}(1 + 2/\beta_c)\sqrt{f'_c}$ | |

3. | Bearing on loaded area | $0.3f'_c$ |

**Steel Reinforcement**

1. | Grade 275 or Grade 350 reinforcement | 140 MPa |

2. | Grade 420 reinforcement or greater and welded wire fabric (plain or deformed) | 170 MPa |

3. | For flexural reinforcement, 10 mm or less in diameter, in one-way slabs of not more than 4 m span but not greater than 200 MPa | 0.50f_{y} |

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## Modulus of Elasticity

**Modulus of Elasticity of Concrete**(NSCP 408.6.1)

For weight of concrete,

*w*between 1500 and 2500 kg/m

_{c}^{3}

$E_c = {w_c}^{1.5}0.043\sqrt{f'_c}$

For normal weight concrete

$E_c = 4700\sqrt{f'_c}$

**Modulus of Elasticity of Steel Reinforcement** (NSCP 408.6.2)

For nonprestressed reinforcement

$E_s = 200\,000 ~ \text{MPa}$

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## Modular Ratio

**Modular Ratio for Beams with Compression Steel** (NSCP 424.6.5)

In doubly reinforced flexural members, an effective modular ratio of $2n$ shall be used to transform compression reinforcement for stress computations.

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## Assumptions in WSD

- Plane section remains plane before and after bending.
- Concrete stress varies from zero at the neutral axis to a maximum at the extreme fiber.
- Concrete do not carry tensile stress. All tensile stress will be carried by steel reinforcement.
- The steel and concrete are perfectly bond so that no slippage between the two will occur.

Moment of area:

$Q_{\text{above NA}} = Q_{\text{below NA}}$

$\frac{1}{2}bx^2 = nA_s(d - x)$

Tensile and Compressive Forces

$T = f_s A_s$

$C = \frac{1}{2}f_c bx$

$C = T$

Moment in the section

$y = d - \frac{1}{3}x$

$M = Cy$

$M = Ty$

Bending stresses

$I_{NA} = \dfrac{bx^3}{3} + nA_s(d - x)^2$

$f_c = \dfrac{Mx}{I_{NA}}$

$\dfrac{f_s}{n} = \dfrac{M(d - x)}{I_{NA}}$