Question

For a beam of rectangular section of width b and depth d, the maximum bending stress in the cross section for a moment m is

Answer 1

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For a rectangular beam with cross section having width b and depth d and loaded as shown in the diagram , Choose the ratio of maximum shear stress to maximum bending stress

A) b/2a

B) d/4a

C) b/4a

D) d/2a

The ratio of maximum shear stress to maximum bending stress is B)d/4a.

Formula for stress of a beam for rectangular section is

$\tau = \frac{3}{2} \times \frac{ v_{max}}{a}$

and formula for bending stress is

$\sigma = \frac{m}{z}$

Where

$v_{max} = maximum \: sheer \: force$

z = section module.

m = maximum bending moment.

assume maximum shear force as p

so,

$v_{max} = p$

and hence the maximum shear stress will be

$\tau = \frac{3}{2} \times \frac{p}{bd}$

____consider the above as (A)

assume bending moment as pa

so,

$\sigma = \frac{pa}{\frac{ {bd}^{2} }{6} } = \frac{6pa}{ {bd}^{2} }$

____consider the above as (B)

And from both (A) and (B)

the ratio of maximum shear stress to the maximum bending stress is

$\frac{ \tau}{ \sigma} = \frac{ \frac{3}{2} \times \frac{p}{bd} }{ \frac{6pa}{ {bd}^{2} } }$

and therefore

$\frac{ \tau}{ \sigma} = \frac{d}{4a}$

The ratio of maximum shear stress to maximum bending stress is d/4a.