[RC] Chapter 2

Beams in Working Stress Design

Rectangular cross-section

Most commonly shaped reinforced concrete beam

Single-reinforced Beam

It is a preliminary type of beam and the reinforcement is provided near the tension face.

Mc and Ms

Resisting Moments

Formula of Mc

Mc = 1/2 fc k j b d2

Formula of Ms

Ms = As fs j d

Strain of concrete

ec = fc / Ec

Strain of steel

es = fs / Es

Modular ratio

n = Es / Ec

Stress ratio

r = fs / fc

k for design problem

k = n / \[n + (fs/fc)\]

Steel ratio

p = As / (b d)

k for investigation problem

k = sqrt \[ (np)^2 + 2np \] - np

Condition for k

0\.3 < = k < = 0.45

Formula for j

j = 1 - k/3

Condition for j

0\.85 < = j < = 0.90

Number of bars

nb = As / Ab

Spacing of bars

b = 2 (side cover) + nb db + Sb (nb -1)

1. Sb > = 25 mm
2. Sb > = db
3. Sb > = 1 - 1/3” max size of coarse agg

NSCP Code Specs for Sb

\[w\] Max moment for Simply supported beam

M = wL^2 / 8

\[w\] Midspan deflection for Simply supported beam

delta = 5wL^4 / 384EI

\[p\] Max moment for simply supported beam

M = PL / 4

\[p\] Midspan deflection for simply supported beam

delta = PL^3 / 48EI

\[w\] Max moment for fully restrained beam

M = wL^2 / 12

\[w\] Midspan deflection for fully restrained beam

delta = wL^4 / 384EI

\[p\] Max moment for fully restrained beam

M = PL / 8

\[p\] Midspan deflection for fully restrained beam

delta = PL^3 / 192EI

\[w\] Max moment for cantilever beam

M = wL^2 / 2

\[w\] Midspan deflection for cantilever beam

delta = wL^4 / 8EI

\[p\] Max moment for cantilever beam

M = PL

\[p\] End deflection for cantilever beam

delta = PL^3 / 3EI

Bond Stress

It is primarily the result of shear interlock between the reinforcing element and the enveloping concrete caused by various loads.

Bond Stress

This stress acts as acts as the outer interface of reinforcement to the surrounding concrete when various forces pull out the steel reinforcements from the hardened concrete.

Bond Stress

This stress helps in keeping the bond between the steel reinforcement and concrete together.

Actual/Applied Bond Stress

mact = V / \[(nb•pi•db)(j)(d)\]

For top bars in tension, the maximum allowable bond stress formula is:

mall = 10.14 sqrt(fc’) / db

Other than top bars, the maximum allowable bond stress formula is:

mall = 7.18 sqrt(fc’) / db

The shear failure effect is usually

Sudden or brittle

Actual shear stress

vact = V / bd

Maximum allowable shear stress

vall = 0.09 sqrt(fc’)

Spacing of stirrups reinforcement

S = Av fs / v’ b

Minimum area of steel to resist shear

Avmin = bS / 3 fy

Formula for Smax

Smax = d/2