UCD FUND OF STRUCTURAL ANALYSIS 5E
UCD FUND OF STRUCTURAL ANALYSIS 5E
5th Edition
ISBN: 9781264843923
Author: Leet
Publisher: MCG
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Chapter 12, Problem 6P

(a)

To determine

Draw the influence lines for the reactions B,C,E,andG of the beam and moments at C and E.

Determine the reactions at B, C, D and E, moments at C and D.

(a)

Expert Solution
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Explanation of Solution

Given Information:

The uniform load (w) is 2 kips/ft.

Calculation:

Influence line for reaction at G.

Consider the portion AF (0x<80ft).

Apply a 1 kip unit moving load at a distance of x from A in the portion AF.

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

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  1

Refer Figure 1.

Find the equation support reaction (RG).

Take moment about point F from H.

Consider clockwise moment as negative and anticlockwise moment as positive

ΣMF=0RG(22)=0RG=0

Consider the portion FH (80ft<x114ft).

Apply a 1 kip unit moving load at a distance of x from A in the portion GH.

Sketch the free body diagram of beam as shown in Figure 2.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  2

Refer Figure 2.

Consider clockwise moment as negative and anticlockwise moment as positive.

Find the equation support reaction (RG).

Take moment about point F from H.

ΣMF=0RG(22)1(x80)=0RG=x8022

Thus, the equations of the influence line for RG are,

RG=0 0x80ft        (1)

RG=x8022 80ftx114ft        (2)

Find the value of influence line ordinate of RG for various points of x using the Equations (1) and (2) and summarize the value as in Table 1.

Pointsx (ft)RG
A00
B120
C420
D500
E720
F800
G1021
H1141.55

Draw the influence lines for RG using Table 1 as shown in Figure 3.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  3

Refer Figure 3.

Determine the reaction at G.

RG=12(1.55)(11480)(w)=12(1.55)(34)(2)=52.7k

Therefore, the reaction at G is 52.7k_

Influence line for reaction RE.

Consider the portion AD (0x50ft).

Apply a 1 kip unit moving load at a distance of x from A in the portion AD.

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

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  4

Refer Figure 4.

Find the equation support reaction (RE).

Take moment about point D from H.

Consider clockwise moment as negative and anticlockwise moment as positive

ΣMD=0RG(52)+RE(22)=0(0)(52)+RE(22)=0RE=0

Consider the portion DF (50ftx80ft).

Apply a 1 kip unit moving load at a distance of x from A in the portion DF.

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

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  5

Refer Figure 5.

Find the equation support reaction (RE).

Take moment about point F from H.

ΣMD=0RG(52)+RE(22)1(x50)=0(0)(52)+RE(22)=x50RE=x5022

Consider the portion FH (80ftx114ft).

Apply a 1 kip unit moving load at a distance of x from A in the portion FH.

Sketch the free body diagram of beam as shown in Figure 6.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  6

Refer Figure 6.

Find the equation support reaction (RE).

Take moment about point F from H.

ΣMD=0RG(52)+RE(22)1(x50)=0(x8022)(52)+RE(22)=x502.364x189.1+22RE=x50

22RE=x502.364x+189.122RE=139.11.364xRE=6.3230.062x

Thus, the equations of the influence line for RE are,

RE=0 0x50ft        (3)

RE=x5022 50ftx80ft        (4)

RE=6.3230.062x 80ftx114ft        (5)

Find the value of influence line ordinate of RE for various points of x using the Equations (1) and (2) and summarize the value as in Table 2.

Pointsx (ft)RE
A00
B120
C420
D500
E721
F801.364
G1020
H114‑0.744

Draw the influence lines for RE using Table 2 as shown in Figure 7.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  7

Refer Figure 7.

Determine the reaction at E.

RE=12(1.364)(10250)(w)+12(0.744)(114112)(w)=12(1.364)(10250)(2)+12(0.744)(114102)(2)=70.9288.928=62k

Therefore, the reaction at E is 62k_

Influence line for reaction RC.

Consider the portion AD (0x50ft).

Apply a 1 kip unit moving load at a distance of x from A in the portion AB.

Sketch the free body diagram of beam as shown in Figure 8.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  8

Refer Figure 8.

Find the equation support reaction (RC).

Take moment about point B from H.

ΣMB=0RG(70)+RE(60)1(x12)+RC(30)=0(0)(90)+(0)(60)x+12+30RC=030RC=x12RC=0.033x0.4

Consider the portion DF (50ftx80ft).

Apply a 1 kip unit moving load at a distance of x from A in the portion DF.

Sketch the free body diagram of beam as shown in Figure 9.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  9

Refer Figure 9.

Find the equation support reaction (RE).

Take moment about point B from H.

ΣMB=0RG(90)+RE(60)1(x12)+RC(30)=0(0)(90)+(x5022)(60)x+12+30RC=060x3,00022x+12+30RC=030RC=124.3641.727xRC=4.150.0576x

Consider the portion FH (80ftx114ft).

Apply a 1 kip unit moving load at a distance of x from A in the portion FH.

Sketch the free body diagram of beam as shown in Figure 10.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  10

Refer Figure 6.

Find the equation support reaction (RD).

Take moment about point B from H.

Consider clockwise moment as negative and anticlockwise moment as positive

ΣMB=0RG(90)+RE(60)1(x12)+RC(30)=0(x8022)(90)+(6.3230.062x)(60)x+12+30RC=04.1x327.27+379.383.72xx+12+30RC=00.62x+64.11+30RC=0RC=0.021x2.14

Thus, the equations of the influence line for RE are,

RC=0.033x0.4 0x50ft        (6)

RC=4.150.0576x 50ftx80ft        (7)

RC=0.021x2.14 80ftx114ft        (8)

Find the value of influence line ordinate of RC for various points of x using the Equations (6), (7) and (8) and summarize the value as in Table 3.

Pointsx (ft)RC
A0‑0.4
B120
C421
D501.27
E720
F80‑0.46
G1020
H1140.25

Draw the influence lines for RC using Table 3 as shown in Figure 11.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  11

Refer Figure 11.

Determine the reaction at C.

RC=w[12(0.4)(120)+12(1.27)(7212)+12(0.46)(10272)+12(0.25)(114102)]=2(2.4+38.16.9+1.5)=60.6k

Therefore, the reaction at C is 60.6k_

Influence line for reaction RB.

Consider the portion AD (0x50ft).

Refer Figure 8.

Consider upward force as positive and anticlockwise moment as negative.

Find the equation support reaction (RB).

Consider vertical equilibrium equation.

ΣFy=0RB+RC+RE+RG1=0RB+0.033x0.4+0+01=0RB=1.40.033x

Consider the portion DF (50ftx80ft).

Refer Figure 9.

Find the equation support reaction (RB).

Consider vertical equilibrium equation.

ΣFy=0RB+RC+RE+RG1=0RB+4.150.0576x+x5022 +01=0RB+4.150.0576x+0.0455x2.2731=0RB=0.012x0.877

Consider the portion FH (80ftx114ft).

Refer Figure 6.

Find the equation support reaction (RB).

Consider vertical equilibrium equation.

Consider upward force as positive and anticlockwise moment as negative.

ΣFy=0RB+RC+RE+RG1=0RB+0.021x2.14+6.3230.062x+x80221=0RB+0.021x2.14+6.3230.062x+0.0455x3.6361=0RB=0.450.0045x

Thus, the equations of the influence line for RB are,

RB=1.40.033x 0x50ft        (9)

RB=0.0121x0.877 50ftx80ft        (10)

RB=0.450.0045x 80ftx114ft        (11)

Find the value of influence line ordinate of RB for various points of x using the Equations (9), (10) and (11) and summarize the value as in Table 4.

Pointsx (ft)RB
A01.4
B121
C420
D50‑0.25
E720
F800.1
G1020
H1140.063

Draw the influence lines for RB using Table 4 as shown in Figure 12.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  12

Refer Figure 12.

Determine the reaction at B.

RB=w[12(1.4)(420)+12(0.27)(7242)+12(0.1)(10272)+12(0.063)(114102)]=2(29.44.05+1.50.378)=53k

Therefore, the reaction at B is 53k_.

Influence line for the moment at section C:

Consider portion AC (0x<42m)

Find the equation of moment at C for portion AC.

Apply a 1 kip in the portion AC from A.

Sketch the free body diagram of the section CH as shown in Figure 13.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  13

Find the equation of moment at C of portion AC.

RG(60)+RE(30)MC=0MC=(0)(60)+(0)(30)MC=0

Consider portion CD (42<x50m):

Find the equation of moment at C for portion CD.

Apply a 1 kip in the portion CD from A.

Sketch the free body diagram of the section AC as shown in Figure 14.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  14

Find the equation of moment at C of portion CD.

RB(30)+MC=0MC=(1.40.033x)(30)MC=42x

Consider portion DF (50x80m):

Find the equation of moment at C for portion DF.

Apply a 1 kip in the portion DF from A.

Sketch the free body diagram of the section AC as shown in Figure 15.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  15

Find the equation of moment at C of portion DF.

RB(30)+MC=0MC=(0.0121x0.877)(30)MC=0.363x26.31

Consider portion FH (80x114m):

Find the equation of moment at C for portion FH.

Apply a 1 kip in the portion FH from A.

Sketch the free body diagram of the section AC as shown in Figure 16.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  16

Find the equation of moment at C of portion FH.

RB(30)+MC=0MC=(0.450.0045x)(30)MC=13.50.135x

Thus, the equations of the influence line for MC are,

MC=0 0x42ft        (12)

MC=42x 42ftx50ft        (13)

MC=0.363x26.31 50ftx80ft        (14)

MC=13.50.135x 80ftx114ft        (15)

Find the value of influence line ordinate of RC for various points of x using the Equations (12), (13), (14) and (15) and summarize the value as in Table 5.

Pointsx (ft)MC
A00
B120
C420
D50‑8
E720
F80+2.73
G1020
H114‑1.89

Draw the influence lines for MC using Table 4 as shown in Figure 17.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  17

Refer Figure 17.

Determine the moment at C.

MC=w[12(8)(7242)+12(2.73)(10272)+12(1.89)(114102)]=2(120+40.9511.34)=180.8ft-k

Therefore, the moment at C is 180.8ft-k_.

Influence line for the moment at section E:

Consider portion AE (0x<72ft)

Find the equation of moment at D for portion AE.

Apply a 1 kip in the portion AE from A.

Sketch the free body diagram of the section EH as shown in Figure 18.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  18

Find the equation of moment at E of portion AE.

RG(30)+ME=0ME=0

Consider portion EF (72<x80ft)

Find the equation of moment at E for portion EF.

Apply a 1 kip in the portion DF from A.

Sketch the free body diagram of the section AC as shown in Figure 19.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  19

Find the equation of moment at E of portion EF.

RB(60)RC(30)+ME=0ME=(0.0121x0.877)(60)+(4.150.0576x)(30)ME=0.726x52.62+124.51.728xME=72x

Consider portion FH (80x114ft)

Find the equation of moment at E for portion FH.

Apply a 1 kip in the portion FH from A.

Sketch the free body diagram of the section AC as shown in Figure 20.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  20

Find the equation of moment at D of portion FH.

RB(60)RC(30)+ME=0ME=(0.45340.0045x)(60)+(0.021x2.14)(30)ME=27.2040.27x+0.63x64.2ME=0.36x37

Thus, the equations of the influence line for MC are,

ME=0 0x72ft        (16)

MC=72x 72ftx80ft        (17)

ME=0.36x37 80ftx114ft        (18)

Find the value of influence line ordinate of RC for various points of x using the Equations (16), (17), and (18) and summarize the value as in Table 6.

Pointsx (ft)ME
A00
B120
C420
D500
E720
F80‑8
G1020
H1144.04

Draw the influence lines for ME using Table 5 as shown in Figure 21.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 12, Problem 6P , additional homework tip  21

Refer Figure 21.

Determine the moment at E.

ME=w[12(8)(10272)+12(4.04)(114102)]=2(120+24.24)=191.52ft-k

Therefore, the moment at E is 191.52ft-k_.

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