EBK MECHANICS OF MATERIALS
EBK MECHANICS OF MATERIALS
7th Edition
ISBN: 8220100257063
Author: BEER
Publisher: YUZU
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Chapter 5.1, Problem 2P

5.1 through 5.6 For the beam and loading shown, (a) draw the shear and bending-moment diagrams, (b) determine the equations of the shear and bending-moment curves.

Fig. P5.2

Chapter 5.1, Problem 2P, 5.1 through 5.6 For the beam and loading shown, (a) draw the shear and bending-moment diagrams, (b)

(a)

Expert Solution
Check Mark
To determine

To draw: The shear and bending-moment diagrams.

Explanation of Solution

Determine the reactions of the beam.

Show the free-body diagram of the entire beam as in Figure 1.

EBK MECHANICS OF MATERIALS, Chapter 5.1, Problem 2P , additional homework tip  1

Determine the vertical reaction at point C by taking moment about point A.

MA=0Cy(L)P(a)=0Cy=PaL

Determine the vertical reaction at point A by resolving the vertical component of forces.

Fy=0AyP+Cy=0

Substitute wL2 for By.

AyP+PaL=0Ay=PPaL=P(LaL)=PbL

Determine the horizontal direction at point A by resolving the horizontal component of forces.

Fx=0Ax=0

Show the free-body diagram of the section 1-1 and 2-2 as in Figure 2.

EBK MECHANICS OF MATERIALS, Chapter 5.1, Problem 2P , additional homework tip  2

Section 1-1:

Determine the shear force at the section by resolving the vertical component of forces.

Fy=0PbLV=0V=PbL

Determine the moment at the section by taking moment about the section.

Mx=0PbL(x)+M=0M=PbxL

Section 2-2:

Determine the shear force at the section by resolving the vertical component of forces.

Fy=0PbLV=0V=PbL

Determine the moment at the section by taking moment about the section.

Mx=0PbL(x)+M=0M=PbxL

Show the free-body diagram of the section 3-3 and 4-4 as in Figure 3.

EBK MECHANICS OF MATERIALS, Chapter 5.1, Problem 2P , additional homework tip  3

Section 3-3:

Determine the shear force at the section by resolving the vertical component of forces.

Fy=0PbLVP=0V=PbLP=P(bL1)

V=P(bLL)=P(LbL)=PaL

Determine the moment at the section by taking moment about the section.

Mx=0PbL(x)+M+P(xa)=0M=PbxLP(xa)=PbxLPx+Pa

When the section 3-3 is at point B, x=a.

MB=Pb(a)LP(a)+Pa=PabL

Section 4-4:

Determine the shear force at the section by resolving the vertical component of forces.

Fy=0PbLVP=0V=PbLP=P(bL1)

V=P(bLL)=P(LbL)=PaL

Determine the moment at the section by taking moment about the section.

Mx=0PbL(x)+M+P(xa)=0M=PbxLP(xa)=PbxLPx+Pa=PbxPLx+PLaL

M=Px(bL)+PLaL=Pxa+PLaL=Pa(Lx)L

Shear force and bending moment values:

Show the calculated shear force and bending moment values as in Table 1.

Location (x)Shear force (V)Bending Moment (M)
1-1PbL0
2-2PbLPabL
3-3PaLPabL
4-4PaL0

Plot the shear force and bending moment diagrams as in Figure 4.

EBK MECHANICS OF MATERIALS, Chapter 5.1, Problem 2P , additional homework tip  4

(b)

Expert Solution
Check Mark
To determine

The equations of the shear and bending-moment curves.

Answer to Problem 2P

The equation of shear force and bending-moment curves is:

For section AB;

V=PbL;M=PbxL_.

For section BC;

V=PaL;M=Pa(Lx)L_.

Explanation of Solution

Determine the reactions of the beam.

Show the free-body diagram of the entire beam as in Figure 5.

EBK MECHANICS OF MATERIALS, Chapter 5.1, Problem 2P , additional homework tip  5

Determine the vertical reaction at point C by taking moment about point A.

MA=0Cy(L)P(a)=0Cy=PaL

Determine the vertical reaction at point A by resolving the vertical component of forces.

Fy=0AyP+Cy=0

Substitute wL2 for By.

AyP+PaL=0Ay=PPaL=P(LaL)=PbL

Determine the horizontal direction at point A by resolving the horizontal component of forces.

Fx=0Ax=0

Show the free-body diagram of the section 1-1 and 2-2 as in Figure 6.

EBK MECHANICS OF MATERIALS, Chapter 5.1, Problem 2P , additional homework tip  6

Section 1-1:

Determine the shear force at the section by resolving the vertical component of forces.

Fy=0PbLV=0V=PbL

Determine the moment at the section by taking moment about the section.

Mx=0PbL(x)+M=0M=PbxL

Section 2-2:

Determine the shear force at the section by resolving the vertical component of forces.

Fy=0PbLV=0V=PbL

Determine the moment at the section by taking moment about the section.

Mx=0PbL(x)+M=0M=PbxL

Show the free-body diagram of the section 3-3 and 4-4 as in Figure 7.

EBK MECHANICS OF MATERIALS, Chapter 5.1, Problem 2P , additional homework tip  7

Section 3-3:

Determine the shear force at the section by resolving the vertical component of forces.

Fy=0PbLVP=0V=PbLP=P(bL1)

V=P(bLL)=P(LbL)=PaL

Determine the moment at the section by taking moment about the section.

Mx=0PbL(x)+M+P(xa)=0M=PbxLP(xa)=PbxLPx+Pa

When the section 3-3 is at point B, x=a.

MB=Pb(a)LP(a)+Pa=PabL

Section 4-4:

Determine the shear force at the section by resolving the vertical component of forces.

Fy=0PbLVP=0V=PbLP=P(bL1)

V=P(bLL)=P(LbL)=PaL

Determine the moment at the section by taking moment about the section.

Mx=0PbL(x)+M+P(xa)=0M=PbxLP(xa)=PbxLPx+Pa=PbxPLx+PLaL

M=Px(bL)+PLaL=Pxa+PLaL=Pa(Lx)L

Therefore, the equation of shear force and bending-moment curves is:

For section AB;

V=PbL;M=PbxL_.

For section BC;

V=PaL;M=Pa(Lx)L_.

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Chapter 5 Solutions

EBK MECHANICS OF MATERIALS

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Understanding Shear Force and Bending Moment Diagrams; Author: The Efficient Engineer;https://www.youtube.com/watch?v=C-FEVzI8oe8;License: Standard YouTube License, CC-BY
Bending Stress; Author: moodlemech;https://www.youtube.com/watch?v=9QIqewkE6xM;License: Standard Youtube License