A beam of rectangular cross-section carries a total load of 20 kN which is uniformly distributed over the total span of 3.6 m. If the depth of the beam is twice its width and maximum stress in tension or compression is not to exceed 7 MPa, find out the dimension of the beam (refer Figure 6.10)

Question

A beam of rectangular cross-section carries a total load of 20 kN which is uniformly distributed over the total span of 3.6 m. If the depth of the beam is twice its width and maximum stress in tension or compression is not to exceed 7 MPa, find out the dimension of the beam (refer Figure 6.10).

Accepted Answer

From the overall free-body diagram of the beam shown in Figure 6.10(b), let us take moment about point B. It gives [latex] R_{ A }=10 kN [/latex]. Now if we take a section at a distance x from the left end and consider the equilibrium of its left side as shown in Figure 6.10(c), we can write

[latex] M_x=R_{ A } x-w \frac{x^2}{2} [/latex] (1)

Putting [latex] R_{ A }=10 kN [/latex] and w = 20/3.6 kN/m, we get

[latex] M_x=10 x-\frac{20}{3.6} \frac{x^2}{2} [/latex]

It is evident from the load-distribution pattern that shear force will vary along the length and will be zero at midpoint (refer to Example 5.1 in the previous chapter). So, maximum bending moment will occur there. Putting x = 3.6/2 = 1.8 m in Eq. (1), we obtain

[latex] M_x=M_{\max }=10 imes 1.8-\frac{20}{3.6} \cdot \frac{(1.8)^2}{2}=9 kN m [/latex]

Now bending stress from Eq. (6.5) is

[latex] \sigma_x=\frac{M y}{I} [/latex] (6.5)

[latex] \sigma=\frac{M y}{I} [/latex]

Hence, the maximum bending stress at the bottom-most fibre will be tensile and maximum bending stress at topmost fibre will be compressive. The magnitude for both is given by [latex] M_{\max } / Z [/latex] where Z is the section modulus. Equating this with allowable stress 7 MPa and remembering Z = bh³/6 for a rectangular cross-section having width b and depth h, we get

[latex] \frac{M_{\max }}{b h^2 / 6}=7 imes 10^6 [/latex]

or [latex] \frac{9 imes 10^6 imes 6}{b h^2}=7 imes 10^6 [/latex]

or [latex] \frac{9 imes 10^6 imes 6}{b imes(2 b)^2}=7 imes 10^6 \quad( ext { as } h=2 b) [/latex]

or b = 124.5 mm and h = 249 mm. So, depth is 249 mm and width is 124.5 mm.

Question 6.1: A beam of rectangular cross-section carries a total load of ...

Related Answered Questions

An I-section of a beam has the following dimensions: flanges, 10 cm × 1 cm and web, 12 cm × 1 cm (Figure 6.20). If this section is subjected to a bending moment of 5 kN m and a shearing force of 5 kN, find out maximum tensile stress and maximum shearing stress induced in it. ...

Verified Answer:

A simply supported beam 100 × 200 mm carries a central concentrated load W. The permissible stress in bending and shear are 15 N/mm² and 1.2 N/mm², respectively. Determine the safe load, W, if the span of the beam is 3.0 m. ...

Verified Answer:

A circular log of timber has diameter d. Find the ratio of breadth to depth of the rectangular beam that can be cut from a circular log for the strongest section in bending. ...

Verified Answer:

A cantilever beam of circular solid cross-section is fixed at one end and carries a concentrated load P at the free end. The diameter of the section is 200 mm at free end and increases uniformly to 400 mm at the fixed end over a length of 2 m. At what distance from the free end will the bending ...

Verified Answer:

A timber beam 15 cm wide and 20 cm deep carries a uniformly distributed load over a span of 4 m and is simply supported. If the permissible stresses are 30 N/m² longitudinally and 3 N/mm² in transverse shear, calculate the maximum load which can be carried by the timber beam. ...

Verified Answer:

A simply supported beam carries a concentrated load at its centre as shown in Figure 6.19. If the allowable working stress in bending is 70 kg/cm² and that in shear is 11 kg/cm², what is the safe value of P? ...

Verified Answer:

A simply supported beam made of rolled steel joist (I-section: 450 mm × 200 mm) has a span of 5 m and it carries a concentrated load W. The flanges are strengthened by two 300 mm × 20 mm plates, one riveted to each flange, over the entire length of the flanges. The second moment of area of the ...

Verified Answer:

Verified Answer:

Verified Answer:

Three beams have same length, allowable stress and bending moment (Figure 6.11). Find the ratios of weights of circular and rectangular beam with respect to square beam. ...

Verified Answer: