A timber beam AB of Length L and rectangular cross section carries a uniformly distributed load w and is supported as shown. (a) Show that the ratio τm/σm of the maximum values of the shearing and normal stresses in the beam is equal to 2h/L, where h and L are, respectively, the depth and the length of the beam. (b) Determine the depth h and the width b of the beam, knowing that L = 5 m, w = 8 kN/m, τm = 1.08 MPa, and σm = 12 MPa.
Fig. P6.20
(a)
To show that: The ratio
Answer to Problem 20P
The ratio
Explanation of Solution
Given information:
The length of the beam AB is L.
The depth of the beam is h.
Calculation:
Calculate the area of the cross section as shown below.
Here, b is the width of the beam and h is the depth of the beam.
Calculate the section modulus of the cross section as shown below.
Due to the symmetry of the beam reaction at supports C and D are Equal.
Calculate the shear force as shown below.
Shear force at A,
Shear force at A right,
Shear force at C,
Shear force at D,
Shear force at B,
Calculate the bending moment as shown below.
BM at A,
BM at C,
BM at D,
BM at B,
Sketch the shear force and bending moment diagram as shown in Figure 1.
Calculate the maximum shear stress as shown below.
Substitute
Calculate the maximum normal stress as shown below.
Substitute
Calculate the ratio
Therefore, the ratio
(b)
The depth and width of the beam.
Answer to Problem 20P
The depth of the beam is
The width of the beam is
Explanation of Solution
Given information:
The length (L) of the beam is
The load is
The maximum shear stress is
The maximum normal stress is
Calculation:
Refer to part (a).
Calculate the depth of the beam as shown below.
Substitute
Hence, the depth of the beam is
Calculate the width of the beam as shown below.
Substitute
Therefore, the width of the beam is
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Chapter 6 Solutions
Mechanics of Materials, 7th Edition
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