Connect Access Card For Fundamentals Of Structural Analysis (one Semester Access) 5th Edition
Connect Access Card For Fundamentals Of Structural Analysis (one Semester Access) 5th Edition
5th Edition
ISBN: 9781259820960
Author: Leet, Kenneth
Publisher: McGraw-Hill Education
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Question
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Chapter 12, Problem 49P

(a)

To determine

Find the position of moving load to produce maximum moment.

Find the absolute maximum moment produced by the moving loads.

Find the maximum deflection produced by the load.

(a)

Expert Solution
Check Mark

Answer to Problem 49P

The position of moving load to produce maximum moment is 18ft_ when the maximum moment produced under wheel load 1.

The maximum moment is 405ft-k_.

The maximum deflection is 107,587,492k-in3EI_.

Explanation of Solution

Refer Table 2.3 “Live Load Impact Factor” in the text book.

Take the value of impact factor (I) for Cab-operated traveling crane support girders and their connections as 25%.

Find the increasing moving load using impact factor.

P=Pgiven+25%(Pgiven)=20+(25%)(20)=20+25100(20)=25k

Sketch the loading diagram as in Figure 1.

Connect Access Card For Fundamentals Of Structural Analysis (one Semester Access) 5th Edition, Chapter 12, Problem 49P , additional homework tip  1

Refer Figure 1.

Find the resultant force.

R=25+25=50k

Find the position of resultant using the equation.

x¯=82=4ft

Find the maximum moment.

The wheel loads are equal and so the moment produced under any one of the load.

Assume the maximum moment occurs under the wheel load 1. Therefore, the beam’s centerline divides the distance between the wheel load 1 and the resultant.

Draw the position of loading diagram as in Figure 2.

Connect Access Card For Fundamentals Of Structural Analysis (one Semester Access) 5th Edition, Chapter 12, Problem 49P , additional homework tip  2

Refer Figure 2

Find the position of resultant from center of the beam.

x¯=42=2ft

Find the reaction at A.

Consider moment at B.

ΣMB=0RA(40)50(20x¯)=0RA(40)50(202)=040RA=900RA=22.5k

Find the moment under wheel load 1.

Consider moment at wheel load 1.

Connect Access Card For Fundamentals Of Structural Analysis (one Semester Access) 5th Edition, Chapter 12, Problem 49P , additional homework tip  3

Find the moment under wheel load 2.

Consider moment at wheel load 2.

ΣM2=0M2=RA(20+6)25(8)=22.5(26)200=385ft-k

Therefore, the position of moving load to produce maximum moment is 18ft_ when the maximum moment produced under wheel load 1.

Therefore, the maximum moment is 405ft-k_ produced under wheel load 1

Draw the moment diagram using moment values at load 1 and 2 as in Figure 3.

Connect Access Card For Fundamentals Of Structural Analysis (one Semester Access) 5th Edition, Chapter 12, Problem 49P , additional homework tip  4

Assume maximum deflection occurs at M.

Draw the MEI diagram using Figure 3 as in Figure 4.

Connect Access Card For Fundamentals Of Structural Analysis (one Semester Access) 5th Edition, Chapter 12, Problem 49P , additional homework tip  5

Refer Figure 4.

Find distance y using similar triangle.

y8xM=4053858y8xM=2.5y=202.5xM

The maximum deflection occurs between point 1 and 2.

Find the deflection ΔDA.

ΔDA=1EI[12(14)(385)(23×14)+(8)(385)(14+4)+12(8)(405385)(14+23×8)+12(405)(18)(22+183)]=1EI(25,153.33+55,440+1,546.67+102,060)=184,200EI

Find the slope at A.

θA=184,200EIL=184,20040(EI)=4,605EI

Find the slope θMA

θMA=1EI[12(18)(405)+(xm)(385)+(20+y2)(xM)]=1EI[3,645+385xM+(20+202.5xM2)(xM)]=1EI[3,645+385xM+(20+1.25xM)(xM)]=1EI[3,645+385xM+20xM+1.25xM2]=1EI(1.25xM2+405xM+3,645)

Consider slope at M from A is equal to slope at A.

Find the point of maximum deflection (xM) from point 2.

θMA=θA1.25xM2+405xM+3,645=4,605EI1.25xM2+405xM960=0        (1)

Solve Equation (1),

xM=2.3533ft2.4ftfrom right of wheel load 1

Find the maximum deflection.

Δmax=ΔAM=1EI[12(405)(18)(23×18)+(xM)(405)(18+xM2)12(208xM)(xM)(18+23×xM)]=1EI[43,740+7,290xM+202.5xM212(2.5xM)(xM)(18+23xM)]=1EI[43,740+7,290(2.4)+202.5(2.4)212(2.5×2.4)(2.4)(18+23×2.4)]=1EI[43,740+17,496+1,166.4141.12]=62,261.28k-ft3EI=62,261.28×(12)3k-in3EI107,587,492k-in3EI

Therefore, the maximum deflection is 107,587,492k-in3EI_.

(b)

To determine

Find the maximum moment and maximum deflection when the moving load placed symmetrically.

Compare the deflections of both part.

(b)

Expert Solution
Check Mark

Answer to Problem 49P

The maximum moment is 400ft-k_

The maximum deflection at center of the beam is 108,749,000k-in3EI_

Explanation of Solution

Find the maximum moment.

Place the moving load symmetrically.

Draw the position of loading diagram as in Figure 5.

Connect Access Card For Fundamentals Of Structural Analysis (one Semester Access) 5th Edition, Chapter 12, Problem 49P , additional homework tip  6

Refer Figure 5.

Find the reaction at A and B.

The loading are symmetrical.

RA=RB=25+252=25k

Find the moment under wheel load 1.

Consider moment at wheel load 1 and 2.

ΣM1=0M1=M2=RA(204)=25(16)=400ft-k

Therefore, the maximum moment is 400ft-k_.

Draw the moment diagram using the calculated values as in Figure 6.

Connect Access Card For Fundamentals Of Structural Analysis (one Semester Access) 5th Edition, Chapter 12, Problem 49P , additional homework tip  7

Refer Figure 6.

The maximum deflection occurs at center for symmetrical loading of simply supported beam.

Find the maximum deflection.

Δmax=ΔAM=1EI[12(16)(400)(23×16)+(400)(4)(16+2)]=1EI[34,133.33+28,800]=62,933.333k-ft3EI=62,933.333×(12)3k-in3EI108,749,000k-in3EI

Therefore, the maximum deflection at center of the beam is 108,749,000k-in3EI_.

Comparison of deflection for both part (a) and (b):

Maximum deflection occurs when loads are centered symmetrically on the beam span.

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