Draw the influence lines for the vertical reactions at supports A and C . Draw the influence lines for the shear and bending moment at point B .

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Answer 1

Question

Chapter 8, Problem 1P

To determine

Draw the influence lines for the vertical reactions at supports A and C.

Draw the influence lines for the shear and bending moment at point B.

Expert Solution & Answer

Explanation of Solution

Calculation:

Apply a 1 k unit moving load at a distance of x from left end A.

Sketch the free body diagram of beam as shown in Figure 1.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 1

Refer Figure 1.

Find the equation of support reaction $(Ay)$ at A using equilibrium equation:

Take moment about point C.

Consider moment equilibrium at point C.

Consider clockwise moment as positive and anticlockwise moment as negative.

Sum of moment at point C is zero.

$ΣMC=0Ay(28)−1(28−x)=0Ay(28)−28+x=028Ay=28−xAy=1−x28$ (1)

Find the equation of support reaction $(Cy)$ at C using equilibrium equation:

Apply vertical equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$Ay+Cy=1$

Substitute $1−x28$ for $Ay$ .

$1−x28+Cy=1Cy=1−1+x28Cy=x28$ (2)

Consider Equation (1).

Find the value of influence line ordinate of reaction $Ay$ at support A.

Substitute 0 for x in Equation (1).

$Ay=1−028=1 k/k$

Similarly calculate the influence line ordinate of reaction $Ay$ for different value of x and summarize the result in Table 1.

x $(ft)$	$Ay (k/k)$
0	1
14	0.5
28	0

Draw the influence line diagram for the vertical reactions at support A using Table 1 as shown in Figure 2.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 2

Consider Equation (2).

Find the influence line ordinate of reaction $Cy$ at support C.

Substitute 28 for x in Equation (2).

$Cy=2828=1$

Similarly calculate the influence line ordinate of reaction $Cy$ for different value of x and summarize the result in Table 2.

x $(ft)$	$Cy (k/k)$
0	0
14	0.5
28	1

Draw the influence line diagram for the vertical reactions at support C using Table 2 as shown in Figure 3.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 3

Find the equation of shear force at B of portion AB $(0≤x<14 ft)$ .

Sketch the free body diagram of the section AB as shown in Figure 4.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 4

Refer Figure 4.

Apply equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$ΣFy=0$

$Ay−SB−1=0SB=Ay−1$

Substitute $1−x28$ for $Ay$ .

$SB=1−x28−1=−x28$

Find the equation of shear force at B of portion BC $(14 ft<x≤28 ft)$ .

Sketch the free body diagram of the section BC as shown in Figure 5.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 5

Refer Figure 5.

Apply equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$ΣFy=0$

$SB−1+Cy=0SB=1−Cy$

Substitute $x28$ for $Cy$ .

$SB=1−x28$

Thus, the equations of the influence line for $SB$ are,

$SB=−x28 0≤x<14 ft$ (3)

$SB=1−x28 14 ft<x≤28 ft$ (4)

Find the value of influence line ordinate of shear force at various points of x using the Equations (3) and (4) and summarize the value as in Table 3.

x $(ft)$	$SB (k/k)$
0	1
$14−$	$−12$
$14+$	$12$
28	0

Draw the influence lines for the shear force at point B using Table 3 as shown in Figure 6.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 6

Refer Figure 4.

Consider clockwise moment as positive and anticlockwise moment as negative.

Find the equation of moment at B of portion AB $(0≤x<14 ft)$ .

$MB=Ay(14)−(1)(14−x)$

Substitute $1−x28$ for $Ay$ .

$MB=(1−x28)(14)−(1)(14−x)=14−x2−14+x=x2$

Refer Figure 5.

Consider clockwise moment as negative and anticlockwise moment as positive.

Find the equation of moment at B of portion BC $(14 ft<x≤28 ft)$ .

$MB=Cy(14)−(1)[14−(28−x)]=14Cy−14+(28−x)=14Cy+14−x$

Substitute $x28$ for $Cy$ .

$MB=14(x28)+14−x=x2+14−x=14−x2$

Thus, the equations of the influence line for $MB$ are,

$MB=x2 0≤x<14 ft$ (5)

$MB=14−x2 14 ft<x≤28 ft$ (6)

Find the value of influence line ordinate of moment at various points of x using the Equations (5) and (6) and summarize the value as in Table 4.

x $(ft)$	$MB$ $(k-ft/k)$
0	0
14	$7.0$
28	0

Draw the influence lines for the moment at point B using Table 4 as shown in Figure 7.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 7

Therefore, the influence lines for the vertical reactions at supports A and C and the influence lines for the shear and bending moment at point B are drawn.

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Answer 2

Question

Chapter 8, Problem 1P

To determine

Draw the influence lines for the vertical reactions at supports A and C.

Draw the influence lines for the shear and bending moment at point B.

Expert Solution & Answer

Explanation of Solution

Calculation:

Apply a 1 k unit moving load at a distance of x from left end A.

Sketch the free body diagram of beam as shown in Figure 1.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 1

Refer Figure 1.

Find the equation of support reaction $(Ay)$ at A using equilibrium equation:

Take moment about point C.

Consider moment equilibrium at point C.

Consider clockwise moment as positive and anticlockwise moment as negative.

Sum of moment at point C is zero.

$ΣMC=0Ay(28)−1(28−x)=0Ay(28)−28+x=028Ay=28−xAy=1−x28$ (1)

Find the equation of support reaction $(Cy)$ at C using equilibrium equation:

Apply vertical equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$Ay+Cy=1$

Substitute $1−x28$ for $Ay$ .

$1−x28+Cy=1Cy=1−1+x28Cy=x28$ (2)

Consider Equation (1).

Find the value of influence line ordinate of reaction $Ay$ at support A.

Substitute 0 for x in Equation (1).

$Ay=1−028=1 k/k$

Similarly calculate the influence line ordinate of reaction $Ay$ for different value of x and summarize the result in Table 1.

x $(ft)$	$Ay (k/k)$
0	1
14	0.5
28	0

Draw the influence line diagram for the vertical reactions at support A using Table 1 as shown in Figure 2.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 2

Consider Equation (2).

Find the influence line ordinate of reaction $Cy$ at support C.

Substitute 28 for x in Equation (2).

$Cy=2828=1$

Similarly calculate the influence line ordinate of reaction $Cy$ for different value of x and summarize the result in Table 2.

x $(ft)$	$Cy (k/k)$
0	0
14	0.5
28	1

Draw the influence line diagram for the vertical reactions at support C using Table 2 as shown in Figure 3.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 3

Find the equation of shear force at B of portion AB $(0≤x<14 ft)$ .

Sketch the free body diagram of the section AB as shown in Figure 4.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 4

Refer Figure 4.

Apply equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$ΣFy=0$

$Ay−SB−1=0SB=Ay−1$

Substitute $1−x28$ for $Ay$ .

$SB=1−x28−1=−x28$

Find the equation of shear force at B of portion BC $(14 ft<x≤28 ft)$ .

Sketch the free body diagram of the section BC as shown in Figure 5.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 5

Refer Figure 5.

Apply equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$ΣFy=0$

$SB−1+Cy=0SB=1−Cy$

Substitute $x28$ for $Cy$ .

$SB=1−x28$

Thus, the equations of the influence line for $SB$ are,

$SB=−x28 0≤x<14 ft$ (3)

$SB=1−x28 14 ft<x≤28 ft$ (4)

Find the value of influence line ordinate of shear force at various points of x using the Equations (3) and (4) and summarize the value as in Table 3.

x $(ft)$	$SB (k/k)$
0	1
$14−$	$−12$
$14+$	$12$
28	0

Draw the influence lines for the shear force at point B using Table 3 as shown in Figure 6.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 6

Refer Figure 4.

Consider clockwise moment as positive and anticlockwise moment as negative.

Find the equation of moment at B of portion AB $(0≤x<14 ft)$ .

$MB=Ay(14)−(1)(14−x)$

Substitute $1−x28$ for $Ay$ .

$MB=(1−x28)(14)−(1)(14−x)=14−x2−14+x=x2$

Refer Figure 5.

Consider clockwise moment as negative and anticlockwise moment as positive.

Find the equation of moment at B of portion BC $(14 ft<x≤28 ft)$ .

$MB=Cy(14)−(1)[14−(28−x)]=14Cy−14+(28−x)=14Cy+14−x$

Substitute $x28$ for $Cy$ .

$MB=14(x28)+14−x=x2+14−x=14−x2$

Thus, the equations of the influence line for $MB$ are,

$MB=x2 0≤x<14 ft$ (5)

$MB=14−x2 14 ft<x≤28 ft$ (6)

Find the value of influence line ordinate of moment at various points of x using the Equations (5) and (6) and summarize the value as in Table 4.

x $(ft)$	$MB$ $(k-ft/k)$
0	0
14	$7.0$
28	0

Draw the influence lines for the moment at point B using Table 4 as shown in Figure 7.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 7

Therefore, the influence lines for the vertical reactions at supports A and C and the influence lines for the shear and bending moment at point B are drawn.

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 3

Question

Chapter 8, Problem 1P

To determine

Draw the influence lines for the vertical reactions at supports A and C.

Draw the influence lines for the shear and bending moment at point B.

Expert Solution & Answer

Explanation of Solution

Calculation:

Apply a 1 k unit moving load at a distance of x from left end A.

Sketch the free body diagram of beam as shown in Figure 1.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 1

Refer Figure 1.

Find the equation of support reaction $(Ay)$ at A using equilibrium equation:

Take moment about point C.

Consider moment equilibrium at point C.

Consider clockwise moment as positive and anticlockwise moment as negative.

Sum of moment at point C is zero.

$ΣMC=0Ay(28)−1(28−x)=0Ay(28)−28+x=028Ay=28−xAy=1−x28$ (1)

Find the equation of support reaction $(Cy)$ at C using equilibrium equation:

Apply vertical equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$Ay+Cy=1$

Substitute $1−x28$ for $Ay$ .

$1−x28+Cy=1Cy=1−1+x28Cy=x28$ (2)

Consider Equation (1).

Find the value of influence line ordinate of reaction $Ay$ at support A.

Substitute 0 for x in Equation (1).

$Ay=1−028=1 k/k$

Similarly calculate the influence line ordinate of reaction $Ay$ for different value of x and summarize the result in Table 1.

x $(ft)$	$Ay (k/k)$
0	1
14	0.5
28	0

Draw the influence line diagram for the vertical reactions at support A using Table 1 as shown in Figure 2.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 2

Consider Equation (2).

Find the influence line ordinate of reaction $Cy$ at support C.

Substitute 28 for x in Equation (2).

$Cy=2828=1$

Similarly calculate the influence line ordinate of reaction $Cy$ for different value of x and summarize the result in Table 2.

x $(ft)$	$Cy (k/k)$
0	0
14	0.5
28	1

Draw the influence line diagram for the vertical reactions at support C using Table 2 as shown in Figure 3.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 3

Find the equation of shear force at B of portion AB $(0≤x<14 ft)$ .

Sketch the free body diagram of the section AB as shown in Figure 4.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 4

Refer Figure 4.

Apply equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$ΣFy=0$

$Ay−SB−1=0SB=Ay−1$

Substitute $1−x28$ for $Ay$ .

$SB=1−x28−1=−x28$

Find the equation of shear force at B of portion BC $(14 ft<x≤28 ft)$ .

Sketch the free body diagram of the section BC as shown in Figure 5.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 5

Refer Figure 5.

Apply equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$ΣFy=0$

$SB−1+Cy=0SB=1−Cy$

Substitute $x28$ for $Cy$ .

$SB=1−x28$

Thus, the equations of the influence line for $SB$ are,

$SB=−x28 0≤x<14 ft$ (3)

$SB=1−x28 14 ft<x≤28 ft$ (4)

Find the value of influence line ordinate of shear force at various points of x using the Equations (3) and (4) and summarize the value as in Table 3.

x $(ft)$	$SB (k/k)$
0	1
$14−$	$−12$
$14+$	$12$
28	0

Draw the influence lines for the shear force at point B using Table 3 as shown in Figure 6.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 6

Refer Figure 4.

Consider clockwise moment as positive and anticlockwise moment as negative.

Find the equation of moment at B of portion AB $(0≤x<14 ft)$ .

$MB=Ay(14)−(1)(14−x)$

Substitute $1−x28$ for $Ay$ .

$MB=(1−x28)(14)−(1)(14−x)=14−x2−14+x=x2$

Refer Figure 5.

Consider clockwise moment as negative and anticlockwise moment as positive.

Find the equation of moment at B of portion BC $(14 ft<x≤28 ft)$ .

$MB=Cy(14)−(1)[14−(28−x)]=14Cy−14+(28−x)=14Cy+14−x$

Substitute $x28$ for $Cy$ .

$MB=14(x28)+14−x=x2+14−x=14−x2$

Thus, the equations of the influence line for $MB$ are,

$MB=x2 0≤x<14 ft$ (5)

$MB=14−x2 14 ft<x≤28 ft$ (6)

Find the value of influence line ordinate of moment at various points of x using the Equations (5) and (6) and summarize the value as in Table 4.

x $(ft)$	$MB$ $(k-ft/k)$
0	0
14	$7.0$
28	0

Draw the influence lines for the moment at point B using Table 4 as shown in Figure 7.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 7

Therefore, the influence lines for the vertical reactions at supports A and C and the influence lines for the shear and bending moment at point B are drawn.

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 4

Question

Chapter 8, Problem 1P

To determine

Draw the influence lines for the vertical reactions at supports A and C.

Draw the influence lines for the shear and bending moment at point B.

Expert Solution & Answer

Explanation of Solution

Calculation:

Apply a 1 k unit moving load at a distance of x from left end A.

Sketch the free body diagram of beam as shown in Figure 1.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 1

Refer Figure 1.

Find the equation of support reaction $(Ay)$ at A using equilibrium equation:

Take moment about point C.

Consider moment equilibrium at point C.

Consider clockwise moment as positive and anticlockwise moment as negative.

Sum of moment at point C is zero.

$ΣMC=0Ay(28)−1(28−x)=0Ay(28)−28+x=028Ay=28−xAy=1−x28$ (1)

Find the equation of support reaction $(Cy)$ at C using equilibrium equation:

Apply vertical equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$Ay+Cy=1$

Substitute $1−x28$ for $Ay$ .

$1−x28+Cy=1Cy=1−1+x28Cy=x28$ (2)

Consider Equation (1).

Find the value of influence line ordinate of reaction $Ay$ at support A.

Substitute 0 for x in Equation (1).

$Ay=1−028=1 k/k$

Similarly calculate the influence line ordinate of reaction $Ay$ for different value of x and summarize the result in Table 1.

x $(ft)$	$Ay (k/k)$
0	1
14	0.5
28	0

Draw the influence line diagram for the vertical reactions at support A using Table 1 as shown in Figure 2.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 2

Consider Equation (2).

Find the influence line ordinate of reaction $Cy$ at support C.

Substitute 28 for x in Equation (2).

$Cy=2828=1$

Similarly calculate the influence line ordinate of reaction $Cy$ for different value of x and summarize the result in Table 2.

x $(ft)$	$Cy (k/k)$
0	0
14	0.5
28	1

Draw the influence line diagram for the vertical reactions at support C using Table 2 as shown in Figure 3.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 3

Find the equation of shear force at B of portion AB $(0≤x<14 ft)$ .

Sketch the free body diagram of the section AB as shown in Figure 4.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 4

Refer Figure 4.

Apply equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$ΣFy=0$

$Ay−SB−1=0SB=Ay−1$

Substitute $1−x28$ for $Ay$ .

$SB=1−x28−1=−x28$

Find the equation of shear force at B of portion BC $(14 ft<x≤28 ft)$ .

Sketch the free body diagram of the section BC as shown in Figure 5.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 5

Refer Figure 5.

Apply equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$ΣFy=0$

$SB−1+Cy=0SB=1−Cy$

Substitute $x28$ for $Cy$ .

$SB=1−x28$

Thus, the equations of the influence line for $SB$ are,

$SB=−x28 0≤x<14 ft$ (3)

$SB=1−x28 14 ft<x≤28 ft$ (4)

Find the value of influence line ordinate of shear force at various points of x using the Equations (3) and (4) and summarize the value as in Table 3.

x $(ft)$	$SB (k/k)$
0	1
$14−$	$−12$
$14+$	$12$
28	0

Draw the influence lines for the shear force at point B using Table 3 as shown in Figure 6.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 6

Refer Figure 4.

Consider clockwise moment as positive and anticlockwise moment as negative.

Find the equation of moment at B of portion AB $(0≤x<14 ft)$ .

$MB=Ay(14)−(1)(14−x)$

Substitute $1−x28$ for $Ay$ .

$MB=(1−x28)(14)−(1)(14−x)=14−x2−14+x=x2$

Refer Figure 5.

Consider clockwise moment as negative and anticlockwise moment as positive.

Find the equation of moment at B of portion BC $(14 ft<x≤28 ft)$ .

$MB=Cy(14)−(1)[14−(28−x)]=14Cy−14+(28−x)=14Cy+14−x$

Substitute $x28$ for $Cy$ .

$MB=14(x28)+14−x=x2+14−x=14−x2$

Thus, the equations of the influence line for $MB$ are,

$MB=x2 0≤x<14 ft$ (5)

$MB=14−x2 14 ft<x≤28 ft$ (6)

Find the value of influence line ordinate of moment at various points of x using the Equations (5) and (6) and summarize the value as in Table 4.

x $(ft)$	$MB$ $(k-ft/k)$
0	0
14	$7.0$
28	0

Draw the influence lines for the moment at point B using Table 4 as shown in Figure 7.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 7

Therefore, the influence lines for the vertical reactions at supports A and C and the influence lines for the shear and bending moment at point B are drawn.

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 5

Chapter 8, Problem 1P

To determine

Draw the influence lines for the vertical reactions at supports A and C.

Draw the influence lines for the shear and bending moment at point B.

Expert Solution & Answer

Explanation of Solution

Calculation:

Apply a 1 k unit moving load at a distance of x from left end A.

Sketch the free body diagram of beam as shown in Figure 1.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 1

Refer Figure 1.

Find the equation of support reaction $(Ay)$ at A using equilibrium equation:

Take moment about point C.

Consider moment equilibrium at point C.

Consider clockwise moment as positive and anticlockwise moment as negative.

Sum of moment at point C is zero.

$ΣMC=0Ay(28)−1(28−x)=0Ay(28)−28+x=028Ay=28−xAy=1−x28$ (1)

Find the equation of support reaction $(Cy)$ at C using equilibrium equation:

Apply vertical equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$Ay+Cy=1$

Substitute $1−x28$ for $Ay$ .

$1−x28+Cy=1Cy=1−1+x28Cy=x28$ (2)

Consider Equation (1).

Find the value of influence line ordinate of reaction $Ay$ at support A.

Substitute 0 for x in Equation (1).

$Ay=1−028=1 k/k$

Similarly calculate the influence line ordinate of reaction $Ay$ for different value of x and summarize the result in Table 1.

x $(ft)$	$Ay (k/k)$
0	1
14	0.5
28	0

Draw the influence line diagram for the vertical reactions at support A using Table 1 as shown in Figure 2.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 2

Consider Equation (2).

Find the influence line ordinate of reaction $Cy$ at support C.

Substitute 28 for x in Equation (2).

$Cy=2828=1$

Similarly calculate the influence line ordinate of reaction $Cy$ for different value of x and summarize the result in Table 2.

x $(ft)$	$Cy (k/k)$
0	0
14	0.5
28	1

Draw the influence line diagram for the vertical reactions at support C using Table 2 as shown in Figure 3.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 3

Find the equation of shear force at B of portion AB $(0≤x<14 ft)$ .

Sketch the free body diagram of the section AB as shown in Figure 4.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 4

Refer Figure 4.

Apply equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$ΣFy=0$

$Ay−SB−1=0SB=Ay−1$

Substitute $1−x28$ for $Ay$ .

$SB=1−x28−1=−x28$

Find the equation of shear force at B of portion BC $(14 ft<x≤28 ft)$ .

Sketch the free body diagram of the section BC as shown in Figure 5.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 5

Refer Figure 5.

Apply equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$ΣFy=0$

$SB−1+Cy=0SB=1−Cy$

Substitute $x28$ for $Cy$ .

$SB=1−x28$

Thus, the equations of the influence line for $SB$ are,

$SB=−x28 0≤x<14 ft$ (3)

$SB=1−x28 14 ft<x≤28 ft$ (4)

Find the value of influence line ordinate of shear force at various points of x using the Equations (3) and (4) and summarize the value as in Table 3.

x $(ft)$	$SB (k/k)$
0	1
$14−$	$−12$
$14+$	$12$
28	0

Draw the influence lines for the shear force at point B using Table 3 as shown in Figure 6.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 6

Refer Figure 4.

Consider clockwise moment as positive and anticlockwise moment as negative.

Find the equation of moment at B of portion AB $(0≤x<14 ft)$ .

$MB=Ay(14)−(1)(14−x)$

Substitute $1−x28$ for $Ay$ .

$MB=(1−x28)(14)−(1)(14−x)=14−x2−14+x=x2$

Refer Figure 5.

Consider clockwise moment as negative and anticlockwise moment as positive.

Find the equation of moment at B of portion BC $(14 ft<x≤28 ft)$ .

$MB=Cy(14)−(1)[14−(28−x)]=14Cy−14+(28−x)=14Cy+14−x$

Substitute $x28$ for $Cy$ .

$MB=14(x28)+14−x=x2+14−x=14−x2$

Thus, the equations of the influence line for $MB$ are,

$MB=x2 0≤x<14 ft$ (5)

$MB=14−x2 14 ft<x≤28 ft$ (6)

Find the value of influence line ordinate of moment at various points of x using the Equations (5) and (6) and summarize the value as in Table 4.

x $(ft)$	$MB$ $(k-ft/k)$
0	0
14	$7.0$
28	0

Draw the influence lines for the moment at point B using Table 4 as shown in Figure 7.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 7

Therefore, the influence lines for the vertical reactions at supports A and C and the influence lines for the shear and bending moment at point B are drawn.

Answer 6

Chapter 8, Problem 1P

To determine

Draw the influence lines for the vertical reactions at supports A and C.

Draw the influence lines for the shear and bending moment at point B.

Expert Solution & Answer

Explanation of Solution

Calculation:

Apply a 1 k unit moving load at a distance of x from left end A.

Sketch the free body diagram of beam as shown in Figure 1.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 1

Refer Figure 1.

Find the equation of support reaction $(Ay)$ at A using equilibrium equation:

Take moment about point C.

Consider moment equilibrium at point C.

Consider clockwise moment as positive and anticlockwise moment as negative.

Sum of moment at point C is zero.

$ΣMC=0Ay(28)−1(28−x)=0Ay(28)−28+x=028Ay=28−xAy=1−x28$ (1)

Find the equation of support reaction $(Cy)$ at C using equilibrium equation:

Apply vertical equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$Ay+Cy=1$

Substitute $1−x28$ for $Ay$ .

$1−x28+Cy=1Cy=1−1+x28Cy=x28$ (2)

Consider Equation (1).

Find the value of influence line ordinate of reaction $Ay$ at support A.

Substitute 0 for x in Equation (1).

$Ay=1−028=1 k/k$

Similarly calculate the influence line ordinate of reaction $Ay$ for different value of x and summarize the result in Table 1.

x $(ft)$	$Ay (k/k)$
0	1
14	0.5
28	0

Draw the influence line diagram for the vertical reactions at support A using Table 1 as shown in Figure 2.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 2

Consider Equation (2).

Find the influence line ordinate of reaction $Cy$ at support C.

Substitute 28 for x in Equation (2).

$Cy=2828=1$

Similarly calculate the influence line ordinate of reaction $Cy$ for different value of x and summarize the result in Table 2.

x $(ft)$	$Cy (k/k)$
0	0
14	0.5
28	1

Draw the influence line diagram for the vertical reactions at support C using Table 2 as shown in Figure 3.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 3

Find the equation of shear force at B of portion AB $(0≤x<14 ft)$ .

Sketch the free body diagram of the section AB as shown in Figure 4.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 4

Refer Figure 4.

Apply equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$ΣFy=0$

$Ay−SB−1=0SB=Ay−1$

Substitute $1−x28$ for $Ay$ .

$SB=1−x28−1=−x28$

Find the equation of shear force at B of portion BC $(14 ft<x≤28 ft)$ .

Sketch the free body diagram of the section BC as shown in Figure 5.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 5

Refer Figure 5.

Apply equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$ΣFy=0$

$SB−1+Cy=0SB=1−Cy$

Substitute $x28$ for $Cy$ .

$SB=1−x28$

Thus, the equations of the influence line for $SB$ are,

$SB=−x28 0≤x<14 ft$ (3)

$SB=1−x28 14 ft<x≤28 ft$ (4)

Find the value of influence line ordinate of shear force at various points of x using the Equations (3) and (4) and summarize the value as in Table 3.

x $(ft)$	$SB (k/k)$
0	1
$14−$	$−12$
$14+$	$12$
28	0

Draw the influence lines for the shear force at point B using Table 3 as shown in Figure 6.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 6

Refer Figure 4.

Consider clockwise moment as positive and anticlockwise moment as negative.

Find the equation of moment at B of portion AB $(0≤x<14 ft)$ .

$MB=Ay(14)−(1)(14−x)$

Substitute $1−x28$ for $Ay$ .

$MB=(1−x28)(14)−(1)(14−x)=14−x2−14+x=x2$

Refer Figure 5.

Consider clockwise moment as negative and anticlockwise moment as positive.

Find the equation of moment at B of portion BC $(14 ft<x≤28 ft)$ .

$MB=Cy(14)−(1)[14−(28−x)]=14Cy−14+(28−x)=14Cy+14−x$

Substitute $x28$ for $Cy$ .

$MB=14(x28)+14−x=x2+14−x=14−x2$

Thus, the equations of the influence line for $MB$ are,

$MB=x2 0≤x<14 ft$ (5)

$MB=14−x2 14 ft<x≤28 ft$ (6)

Find the value of influence line ordinate of moment at various points of x using the Equations (5) and (6) and summarize the value as in Table 4.

x $(ft)$	$MB$ $(k-ft/k)$
0	0
14	$7.0$
28	0

Draw the influence lines for the moment at point B using Table 4 as shown in Figure 7.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 7

Therefore, the influence lines for the vertical reactions at supports A and C and the influence lines for the shear and bending moment at point B are drawn.

Answer 7

Chapter 8, Problem 1P

To determine

Draw the influence lines for the vertical reactions at supports A and C.

Draw the influence lines for the shear and bending moment at point B.

Answer 8

Expert Solution & Answer

Explanation of Solution

Calculation:

Apply a 1 k unit moving load at a distance of x from left end A.

Sketch the free body diagram of beam as shown in Figure 1.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 1

Refer Figure 1.

Find the equation of support reaction $(Ay)$ at A using equilibrium equation:

Take moment about point C.

Consider moment equilibrium at point C.

Consider clockwise moment as positive and anticlockwise moment as negative.

Sum of moment at point C is zero.

$ΣMC=0Ay(28)−1(28−x)=0Ay(28)−28+x=028Ay=28−xAy=1−x28$ (1)

Find the equation of support reaction $(Cy)$ at C using equilibrium equation:

Apply vertical equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$Ay+Cy=1$

Substitute $1−x28$ for $Ay$ .

$1−x28+Cy=1Cy=1−1+x28Cy=x28$ (2)

Consider Equation (1).

Find the value of influence line ordinate of reaction $Ay$ at support A.

Substitute 0 for x in Equation (1).

$Ay=1−028=1 k/k$

Similarly calculate the influence line ordinate of reaction $Ay$ for different value of x and summarize the result in Table 1.

x $(ft)$	$Ay (k/k)$
0	1
14	0.5
28	0

Draw the influence line diagram for the vertical reactions at support A using Table 1 as shown in Figure 2.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 2

Consider Equation (2).

Find the influence line ordinate of reaction $Cy$ at support C.

Substitute 28 for x in Equation (2).

$Cy=2828=1$

Similarly calculate the influence line ordinate of reaction $Cy$ for different value of x and summarize the result in Table 2.

x $(ft)$	$Cy (k/k)$
0	0
14	0.5
28	1

Draw the influence line diagram for the vertical reactions at support C using Table 2 as shown in Figure 3.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 3

Find the equation of shear force at B of portion AB $(0≤x<14 ft)$ .

Sketch the free body diagram of the section AB as shown in Figure 4.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 4

Refer Figure 4.

Apply equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$ΣFy=0$

$Ay−SB−1=0SB=Ay−1$

Substitute $1−x28$ for $Ay$ .

$SB=1−x28−1=−x28$

Find the equation of shear force at B of portion BC $(14 ft<x≤28 ft)$ .

Sketch the free body diagram of the section BC as shown in Figure 5.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 5

Refer Figure 5.

Apply equilibrium equation of forces.

Consider upward force as positive $(↑+)$ and downward force as negative $(↓−)$ .

$ΣFy=0$

$SB−1+Cy=0SB=1−Cy$

Substitute $x28$ for $Cy$ .

$SB=1−x28$

Thus, the equations of the influence line for $SB$ are,

$SB=−x28 0≤x<14 ft$ (3)

$SB=1−x28 14 ft<x≤28 ft$ (4)

Find the value of influence line ordinate of shear force at various points of x using the Equations (3) and (4) and summarize the value as in Table 3.

x $(ft)$	$SB (k/k)$
0	1
$14−$	$−12$
$14+$	$12$
28	0

Draw the influence lines for the shear force at point B using Table 3 as shown in Figure 6.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 6

Refer Figure 4.

Consider clockwise moment as positive and anticlockwise moment as negative.

Find the equation of moment at B of portion AB $(0≤x<14 ft)$ .

$MB=Ay(14)−(1)(14−x)$

Substitute $1−x28$ for $Ay$ .

$MB=(1−x28)(14)−(1)(14−x)=14−x2−14+x=x2$

Refer Figure 5.

Consider clockwise moment as negative and anticlockwise moment as positive.

Find the equation of moment at B of portion BC $(14 ft<x≤28 ft)$ .

$MB=Cy(14)−(1)[14−(28−x)]=14Cy−14+(28−x)=14Cy+14−x$

Substitute $x28$ for $Cy$ .

$MB=14(x28)+14−x=x2+14−x=14−x2$

Thus, the equations of the influence line for $MB$ are,

$MB=x2 0≤x<14 ft$ (5)

$MB=14−x2 14 ft<x≤28 ft$ (6)

Find the value of influence line ordinate of moment at various points of x using the Equations (5) and (6) and summarize the value as in Table 4.

x $(ft)$	$MB$ $(k-ft/k)$
0	0
14	$7.0$
28	0

Draw the influence lines for the moment at point B using Table 4 as shown in Figure 7.

Structural Analysis (MindTap Course List), Chapter 8, Problem 1P , additional homework tip 7

Therefore, the influence lines for the vertical reactions at supports A and C and the influence lines for the shear and bending moment at point B are drawn.

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

Ch. 8 - Prob. 1P Ch. 8 - Prob. 2P Ch. 8 - Prob. 3P Ch. 8 - Prob. 4P Ch. 8 - Prob. 5P Ch. 8 - Prob. 6P Ch. 8 - Prob. 7P Ch. 8 - Prob. 8P Ch. 8 - Prob. 9P Ch. 8 - Prob. 10P

Draw the influence lines for the vertical reactions at supports A and C . Draw the influence lines for the shear and bending moment at point B .

Concept explainers

Draw the influence lines for the vertical reactions at supports A and C.

Draw the influence lines for the shear and bending moment at point B.

Explanation of Solution

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Chapter 8 Solutions

Draw the influence lines for the vertical reactions at supports A and C . Draw the influence lines for the shear and bending moment at point B .

Concept explainers

Draw the influence lines for the vertical reactions at supports A and C. Draw the influence lines for the shear and bending moment at point B.

Explanation of Solution

Want to see more full solutions like this?

Chapter 8 Solutions

Draw the influence lines for the vertical reactions at supports A and C.

Draw the influence lines for the shear and bending moment at point B.