EBK MECHANICS OF MATERIALS
EBK MECHANICS OF MATERIALS
7th Edition
ISBN: 8220100257063
Author: BEER
Publisher: YUZU
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Textbook Question
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Chapter 6.2, Problem 20P

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.

Chapter 6.2, Problem 20P, A timber beam AB of Length L and rectangular cross section carries a uniformly distributed load w

Fig. P6.20

(a)

Expert Solution
Check Mark
To determine

To show that: The ratio τmσm of the maximum values of the shearing and normal stresses in the beam is equal to 2hL.

Answer to Problem 20P

The ratio τmσm of the maximum values of the shearing and normal stresses in the beam is equal to 2hL is proved.

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.

A=bh

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.

S=16bh2

Due to the symmetry of the beam reaction at supports C and D are Equal.

RC=RD=wL2

Calculate the shear force as shown below.

Shear force at A, VA=0.

Shear force at A right, VA,right=wL4

Shear force at C, VC=wL2wL4=wL4

Shear force at D, VD=3wL4+wL2=wL4

Shear force at B, VB=wL+wL2+wL2=0

Calculate the bending moment as shown below.

BM at A, MA=0

BM at C, MC=wL4×L8=wL232

BM at D, MD=3wL4×3L8+wL2×L2=9wL232+wL24=(9+832)wL2=wL232

BM at B, MB=wL×L2+wL2×3L4+wL2×L4=wL22+3wL28+wL28=(4+3+18)wL2=0

Sketch the shear force and bending moment diagram as shown in Figure 1.

EBK MECHANICS OF MATERIALS, Chapter 6.2, Problem 20P

Calculate the maximum shear stress as shown below.

τm=32VmaxA

Substitute wL4 for Vmax and bh for A.

τm=32(wL4)bh=3wL8bh (1)

Calculate the maximum normal stress as shown below.

σm=MmaxS

Substitute wL232 for Mmax and 16bh2 for S.

σm=wL23216bh2=wL232×6bh2=3wL216bh2

Calculate the ratio τmσm of the maximum values of the shearing and normal stresses as shown below.

τmσm=3wL8bh3wL216bh2=3wL8bh×16bh23wL2=2hL

Therefore, the ratio τmσm of the maximum values of the shearing and normal stresses in the beam is equal to 2hL is proved.

(b)

Expert Solution
Check Mark
To determine

The depth and width of the beam.

Answer to Problem 20P

The depth of the beam is 225mm_.

The width of the beam is 61.7mm_.

Explanation of Solution

Given information:

The length (L) of the beam is 5m.

The load is 8kN/m.

The maximum shear stress is τm=1.08MPa.

The maximum normal stress is σm=12MPa.

Calculation:

Refer to part (a).

τmσm=2hL (2)

Calculate the depth of the beam as shown below.

Substitute 5m for L, 1.08MPa for τm, and 12MPa for σm in Equation (2).

1.0812=2×h52h=0.45h=225×103m×1,000mm1mh=225mm

Hence, the depth of the beam is 225mm_.

Calculate the width of the beam as shown below.

Substitute 8kN/m for w, 1.08MPa for τm, 5m for L and 225mm for h in Equation (1).

1.08MPa×103kN/m21MPa=3×8kN/m×5m8b×225mm×1m1,000mm1944b=120b=0.0617m×1,000mm1mb=61.7mm

Therefore, the width of the beam is 61.7mm_.

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Chapter 6 Solutions

EBK MECHANICS OF MATERIALS

Ch. 6.2 - 6.9 through 6.12 For beam and loading shown,...Ch. 6.2 - 6.9 through 6.12 For beam and loading shown,...Ch. 6.2 - 6.13 and 6.14 For a beam having the cross section...Ch. 6.2 - 6.13 and 6.14 For a beam having the cross section...Ch. 6.2 - For a timber beam having the cross section shown,...Ch. 6.2 - Two steel plates of 12 220-mm rectangular cross...Ch. 6.2 - Two W8 31 rolled sections may be welded at A and...Ch. 6.2 - For the beam and. loading shown, determine the...Ch. 6.2 - Fig. P6.19 6.19 A timber beam AB of length L and...Ch. 6.2 - A timber beam AB of Length L and rectangular cross...Ch. 6.2 - 6.21 and 6.22 For the beam and loading shown,...Ch. 6.2 - 6.21 and 6.22 For the beam and loading shown,...Ch. 6.2 - 6.23 and 6.24 For the beam and loading shown,...Ch. 6.2 - 6.23 and 6.24 For the beam and loading shown,...Ch. 6.2 - 6.25 through 6.28 A beam having the cross section...Ch. 6.2 - 6.25 through 6.28 A beam having the cross section...Ch. 6.2 - Prob. 27PCh. 6.2 - 6.25 through 6.28 A beam having the cross section...Ch. 6.5 - The built-up timber beam shown is subjected to a...Ch. 6.5 - The built-up beam shown is made by gluing together...Ch. 6.5 - The built-up beam was made by gluing together...Ch. 6.5 - Several wooden planks are glued together to form...Ch. 6.5 - The built-up wooden beam shown is subjected to a...Ch. 6.5 - Knowing that a W360 122 rolled-steel beam is...Ch. 6.5 - 6.35 and 6.36 An extruded aluminum beam has the...Ch. 6.5 - 6.35 and 6.36 An extruded aluminum beam has the...Ch. 6.5 - Knowing that a given vertical shear V causes a...Ch. 6.5 - The vertical shear is 1200 lb in a beam having the...Ch. 6.5 - The vertical shear is 1200 lb in a beam having the...Ch. 6.5 - 6.40 and 6.47 The extruded aluminum beam has a...Ch. 6.5 - Prob. 41PCh. 6.5 - Prob. 42PCh. 6.5 - Three planks are connected as shown by bolts of...Ch. 6.5 - A beam consists of three planks connected as shown...Ch. 6.5 - A beam consists of five planks of 1.5 6-in. cross...Ch. 6.5 - Four L102 102 9.5 steel angle shapes and a 12 ...Ch. 6.5 - A plate of 14-in. thickness is corrugated as shown...Ch. 6.5 - Prob. 48PCh. 6.5 - An extruded beam has the cross section shown and a...Ch. 6.5 - Prob. 50PCh. 6.5 - The design of a beam calls for connecting two...Ch. 6.5 - The cross section of an extruded beam is a hollow...Ch. 6.5 - Prob. 53PCh. 6.5 - Prob. 54PCh. 6.5 - Prob. 55PCh. 6.5 - 6.56 and 6.57 A composite beam is made by...Ch. 6.5 - 6.56 and 6.57 A composite beam is made by...Ch. 6.5 - Prob. 58PCh. 6.5 - Prob. 59PCh. 6.5 - Prob. 60PCh. 6.6 - 6.61 through 6.64 Determine the location of the...Ch. 6.6 - 6.61 through 6.64 Determine the location of the...Ch. 6.6 - 6.61 through 6.64 Determine the location of the...Ch. 6.6 - Prob. 64PCh. 6.6 - 6.65 through 6.68 An extruded beam has the cross...Ch. 6.6 - 6.65 through 6.68 An extruded beam has the cross...Ch. 6.6 - 6.65 through 6.68 An extruded beam has the cross...Ch. 6.6 - 6.65 through 6.68 An extruded beam has the cross...Ch. 6.6 - 6.69 through 6.74 Determine the location of the...Ch. 6.6 - Prob. 70PCh. 6.6 - Prob. 71PCh. 6.6 - Prob. 72PCh. 6.6 - Prob. 73PCh. 6.6 - Prob. 74PCh. 6.6 - Prob. 75PCh. 6.6 - 6.75 and 6.76 A thin-walled beam has the cross...Ch. 6.6 - 6.77 and 6.78 A thin-walled beam of uniform...Ch. 6.6 - Prob. 78PCh. 6.6 - Prob. 79PCh. 6.6 - Prob. 80PCh. 6.6 - Prob. 81PCh. 6.6 - Prob. 82PCh. 6.6 - Prob. 83PCh. 6.6 - Prob. 84PCh. 6.6 - Prob. 85PCh. 6.6 - Solve Prob. 6.85, assuming that the thickness of...Ch. 6.6 - Prob. 87PCh. 6.6 - Prob. 88PCh. 6 - Three boards are nailed together to form the beam...Ch. 6 - For the beam and loading shown, consider section...Ch. 6 - For the wide-flange beam with the loading shown,...Ch. 6 - For the beam and loading shown, consider section...Ch. 6 - The built-up timber beam is subjected to a 1500-lb...Ch. 6 - Knowing that a given vertical shear V causes a...Ch. 6 - Three planks are connected as shown by bolts of...Ch. 6 - Three 1 18-in. steel plates are bolted to four L6...Ch. 6 - The composite beam shown is made by welding C200 ...Ch. 6 - Prob. 98RPCh. 6 - A thin-walled beam of uniform thickness has the...Ch. 6 - Determine the location of the shear center O of a...
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