Mechanics of Materials, 7th Edition
Mechanics of Materials, 7th Edition
7th Edition
ISBN: 9780073398235
Author: Ferdinand P. Beer, E. Russell Johnston Jr., John T. DeWolf, David F. Mazurek
Publisher: McGraw-Hill Education
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Chapter 4.7, Problem 120P

a)

To determine

Show that the maximum compressive stresses are in the ratio 4:5:7:9.

a)

Expert Solution
Check Mark

Explanation of Solution

Given information:

The load act on the point of the bars is P.

Calculation:

At the point A:

Show the cross-sectional diagram of the square bar as in Figure 1.

Mechanics of Materials, 7th Edition, Chapter 4.7, Problem 120P , additional homework tip  1

Here, a is sides of the square bar.

Refer to Figure 1.

The maximum compressive stress of the square bar (σA)s is as follows:

(σA)s=PAsPecAIs=PAs(1+AsecAIs) (1)

Here, e is the eccentricity of the load and cA is the distance between the centroid from extreme fibre.

The cross-sectional area of the square bar (As) is a2.

The eccentricity of the load (e) is a2.

The distance between the centroid from extreme fibre (cA) is a2.

The moment of inertia (Is) of the section of the square bar is a412.

Calculate the maximum compressive stress of the square bar (σA)s:

Substitute a2 for As, a2 for e, a2 for cA, and a412 for Is in Equation (1).

(σA)s=PAs(1+a2×a2×a2a412)=PAs12a4P4a4As=PAs3PAs=4PAs

Show the cross-sectional diagram of the circular bar as in Figure 2.

Mechanics of Materials, 7th Edition, Chapter 4.7, Problem 120P , additional homework tip  2

Here, c is radius of the circular bar.

Refer to Figure 2.

The maximum compressive stress of the circular bar (σA)c is as follows:

(σA)c=PAcPecAIc=PAc(1+AcecAIc) (2)

The cross-sectional area of the circular bar (Ac) is as follow:

Ac=πc2=a2

The eccentricity of the load (e) is aπ.

The distance between the centroid from extreme fibre (cA) is aπ.

The moment of inertia (Ic) of the section of the circular bar is as follows:

Ic=π4c4=π4×(a4π)=π4×a4π2=a44π2

Calculate the maximum compressive stress of the circular bar (σA)c:

Substitute a2 for Ac, aπ for e, aπ for cA, and a44π2 for Ic in Equation (2).

(σA)c=PAc(1+a2×aπ×aπa44π2)=PAc4π2a4Pa4π2Ac=PAc4PAc=5PAc

Show the cross-sectional diagram of the diamond shape bar as in Figure 3.

Mechanics of Materials, 7th Edition, Chapter 4.7, Problem 120P , additional homework tip  3

Here, a is side of the diamond shape bar.

Refer to Figure 3.

The maximum compressive stress of the diamond shape bar (σA)d is as follows:

(σA)d=PAdPecAId=PAd(1+AdecAId) (3)

The cross-sectional area of the diamond shape bar (Ad) is as follow:

Ad=a2

The eccentricity of the load (e) is 2a2.

The distance between the centroid from extreme fibre (cA) is 2a2.

The moment of inertia (Id) of the section of the diamond shape bar is as follows:

Id=a412

Calculate the maximum compressive stress of the diamond shape bar (σA)d:

Substitute a2 for Ad, 2a2 for e, 2a2 for cA, and a412 for Id in Equation (3).

(σA)d=PAd(1+a2×2a2×2a2a412)=PAd12×2a4P4a4Ad=PAd6PAd=7PAd

Show the cross-sectional diagram of the triangular bar as in Figure 4.

Mechanics of Materials, 7th Edition, Chapter 4.7, Problem 120P , additional homework tip  4

Here, s is side of the triangular bar.

Refer to Figure 4.

The maximum compressive stress of the triangular bar (σA)t is as follows:

(σA)t=PAtPecAIt=PAt(1+AtecAIt) (4)

The cross-sectional area of the triangular bar (At) is as follow:

At=12×s×3s2=3s24

The distance between the centroid from extreme fibre (cA) is as follows:

cA=23×3s2=3s3×33=3s33=s3

The eccentricity of the load (e) is s3.

The moment of inertia (It) of the section of the triangular bar is as follows:

It=136×s×(3s2)3=s36×33s38=3s496

Calculate the maximum compressive stress of the triangular bar (σA)t:

Substitute 3s24 for At, s3 for e, s3 for cA, and 3s496 for It in Equation (4).

(σA)t=PAt(1+3s24×s3×s33s496)=PAt96s4P3×4s4At=PAt8PAt=9PAt

Calculate the maximum compressive stresses are in the ratio:

A=(σA)s:(σA)c:(σA)d:(σA)t

Substitute (4PAs) for (σA)s, (5PAc) for (σA)c, (7PAd) for (σA)d, and (9PAt) for (σA)t.

A=4PAs:5PAc:7PAd:9PAt=4As:5Ac:7Ad:9At

The four bars shown have the same cross-sectional area.

A=4:5:7:9

Hence the maximum compressive stresses are in the ratio 4:5:7:9 is proved.

b)

To determine

Show that the maximum tensile stresses are in the ratio 2:3:5:3.

b)

Expert Solution
Check Mark

Explanation of Solution

Given information:

The load act on the point of the bars is P.

Calculation:

At the point B:

Refer to Figure 1.

The maximum tensile stress of the square bar (σB)s is as follows:

(σB)s=PAs+PecBIs=PAs(AsecBIs1) (5)

Here, the e is the eccentricity of the load and cB is the distance between the centroid from extreme fibre.

The cross-sectional area of the square bar (As) is a2.

The eccentricity of the load (e) is a2.

The distance between the centroid from extreme fibre (cB) is a2.

The moment of inertia (Is) of the section of the square bar is a412.

Calculate the maximum tensile stress of the square bar (σB)s:

Substitute a2 for As, a2 for e, a2 for cB, and a412 for Is in Equation (5).

(σB)s=PAs(a2×a2×a2a4121)=12a4P4a4AsPAs=3PAsPAs=2PAs

Refer to Figure 2.

The maximum tensile stress of the circular bar (σB)c is as follows:

(σB)c=PAc+PecBIc=PAc(AcecBIc1) (6)

The cross-sectional area of the circular bar (Ac) is as follow:

Ac=πc2=a2

The eccentricity of the load (e) is aπ.

The distance between the centroid from extreme fibre (cB) is aπ.

The moment of inertia (Ic) of the section of the circular bar is as follows:

Ic=π4c4=π4×(a4π)=π4×a4π2=a44π2

Calculate the maximum tensile stress of the circular bar (σB)c:

Substitute a2 for Ac, aπ for e, aπ for cB, and a44π2 for Ic in Equation (6).

(σB)c=PAc(a2×aπ×aπa44π21)=4π2a4Pa4π2AcPAc=4PAcPAc=3PAc

Refer to Figure 3.

The maximum tensile stress of the diamond shape bar (σB)d is as follows:

(σB)d=PAd+PecBId=PAd(AdecBId1) (7)

The cross-sectional area of the diamond shape bar (Ad) is as follow:

Ad=a2

The eccentricity of the load (e) is 2a2.

The distance between the centroid from extreme fibre (cB) is 2a2.

The moment of inertia (Id) of the section of the diamond shape bar is as follows:

Id=a412

Calculate the maximum tensile stress of the diamond shape bar (σB)d:

Substitute a2 for Ad, 2a2 for e, 2a2 for cB, and a412 for Id in Equation (7).

(σB)d=(a2×2a2×2a2a4121)=12×2a4P4a4AdPAd=6PAdPAd=5PAd

Refer to Figure 4.

The maximum tensile stress of the triangular bar (σB)t is as follows:

(σB)t=PAt+PecBIt=PAt(AtecBIt1) (8)

The cross-sectional area of the triangular bar (At) is as follows:

At=12×s×3s2=3s24

The distance between the centroid from extreme fibre (cB) is as follows:

cB=13×3s2=3s3×2×33=3s3×23=s23

The eccentricity of the load (e) is s3.

The moment of inertia (It) of the section of the triangular bar is as follows:

It=136×s×(3s2)3=s36×33s38=3s496

Calculate the maximum tensile stress of the triangular bar (σB)t:

Substitute 3s24 for At, s3 for e, s23 for cB, and 3s496 for It in Equation (8).

(σB)t=PAt(3s24×s3×s233s4961)=96s4P3×4×2s4AtPAt=4PAtPAt=3PAt

Calculate the maximum tensile stresses are in the ratio:

A=(σA)s:(σA)c:(σA)d:(σA)t

Substitute (2PAs) for (σA)s, (3PAc) for (σA)c, (5PAd) for (σA)d, and (3PAt) for (σA)t.

B=2PAs:3PAc:5PAd:3PAt=2As:3Ac:5Ad:3At

The four bars shown have the same cross-sectional area.

B=2:3:5:3

Hence the maximum tensile stresses are in the ratio 2:3:5:3 is proved.

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Chapter 4 Solutions

Mechanics of Materials, 7th Edition

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