Fundamentals of Structural Analysis
Fundamentals of Structural Analysis
5th Edition
ISBN: 9780073398006
Author: Kenneth M. Leet Emeritus, Chia-Ming Uang, Joel Lanning
Publisher: McGraw-Hill Education
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Chapter 8, Problem 38P
To determine

Find the vertical deflection and horizontal deflection at B.

Find the vertical deflection and horizontal deflection at the mid span of the member CD.

Expert Solution & Answer
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Answer to Problem 38P

The vertical deflection at B is 8.6in._.

The horizontal deflection at B is 10.61in._.

The vertical deflection at mid span of the member CD is 20.30in._.

The horizontal deflection at mid span of the member CD is 10.61in._.

Explanation of Solution

Given information:

The moment of inertia of all members is I=180in.4.

The area of the column is 6in.2 and the area of the girder is 10in.2.

The value of Young’s modulus is 29,000kips/in.2.

Apply the sign conventions for calculating reactions, forces and moments using the three equations of equilibrium as shown below.

  • For summation of forces along x-direction is equal to zero (Fx=0), consider the forces acting towards right side as positive (+) and the forces acting towards left side as negative ().
  • For summation of forces along y-direction is equal to zero (Fy=0), consider the upward force as positive (+) and the downward force as negative ().
  • For summation of moment about a point is equal to zero (Matapoint=0), consider the clockwise moment as negative and the counter clockwise moment as positive.

Calculation:

Find the vertical deflection at B (δBV):

Find the moment MP of the P-system:

For span BC,

MB=0

MC=10(5)=50kip-ft

Consider the point E as midspan of span CD.

For span CED,

MD=0

ME=2.4(10)(5)=120kip-ft

MC=2.4(20)(10)=480kip-ft

For span AC,

MC=2.4(20)(10)+10(5)=430kip-ft

Matthepointwhere6kipsacts=2.4(20)(10)+10(5)=430kip-ft

MA=2.4(20)(10)+10(5)6(6)=466kip-ft

Find the force FP of the P-system:

FC=2.4(20)10=58kips

FA=2.4(20)10=58kips

Sketch the MP diagram and FP diagram for the structure as shown in Figure 1.

Fundamentals of Structural Analysis, Chapter 8, Problem 38P , additional homework tip  1

Apply 1 kips vertically at point B for vertical deflection (δBV).

Find the moment MQ of the Q-system:

For span BC,

MB=0

MC=1(5)=5kip-ft

For span AC,

MC=1(5)=5kip-ft

MA=1(5)=5kip-ft

Find the force FQ of the Q-system:

FC=1kips

FA=1kips

Sketch the MP diagram and FP diagram for δBV as shown in Figure 2.

Fundamentals of Structural Analysis, Chapter 8, Problem 38P , additional homework tip  2

Find the vertical deflection at B (δBV) using Figure 1 and Figure 2 as follows:

ΣQδBV=Σ1EI(CM1M3L)+ΣFQFPLAE(1kips)(δBV)=[12EI(13×50×5)×51EI(430×5)×61EI(12((430+466)×5))×6+58(12)AE]δBV=26,131.67EI+696AEδBV=26,131.67(1,728)(29,000)(180)+696(12)(6)(29,000)

δBV=8.6in.8.6in.

Therefore, the vertical deflection at B is 8.6in._.

Apply 1 kips horizontally at point B for horizontal deflection (δBH).

Sketch the MP diagram and FP diagram for δBH as shown in Figure 3.

Fundamentals of Structural Analysis, Chapter 8, Problem 38P , additional homework tip  3

Find the moment MQ of the Q-system:

For span AC,

MC=0

MA=1(12)=12kip-ft

Find the force FQ of the Q-system:

FB=1kips

FC=1kips

Find the horizontal deflection at B (δBH) using Figure 1 and Figure 3 as follows:

ΣQδBH=Σ1EI(CM1M3L)(1kips)(δBH)=[1EI(12×6×430)×6+1EI(16×6×(2(430)+466))×6+1EI(16(12(430+2(466))))×6]δBH=32,040EIδBH=32,040(1,728)(29,000)(180)

δBH=10.61in..

Therefore, the horizontal deflection at B is 10.61in._.

Apply 1 kips vertically at point E for vertical deflection (δEV).

Find the moment MQ of the Q-system:

For span CED,

MD=0

ME=0

MC=1(10)=10kip-ft

For span AC,

MC=1(10)=10kip-ft

MA=1(10)=10kip-ft

Find the force FQ of the Q-system:

FC=1kips

FA=1kips

Sketch the MP diagram and FP diagram for δEV as shown in Figure 4.

Fundamentals of Structural Analysis, Chapter 8, Problem 38P , additional homework tip  4

Find the vertical deflection at E (δEV) using Figure 1 and Figure 4 as follows:

ΣQδEV=MQMPdxEI+Σ1EI(CM1M3L)+ΣFQFPLAE(1kips)(δEV)=[12EI010(10+x)(1.2x2+48x480)dx+1EI(10×430)×6+1EI(12(10(430+466)))×6+58(12)AE](1kips)(δEV)=[12EI010(12x2480x+4,8001.2x3+48x2480x)dx+25,800EI+26,880EI+696AE]δEV=[12EI[12(10)33480(10)22+4,800(10)1.2(10)44+48(10)33480(10)22]+25,800EI+26,880EI+696AE]δEV=61,180EI+696AE=61,180(1,728)(29,000)(180)+696(12)(6)(29,000)=20.30in.

Therefore, the vertical deflection at mid span of the member CD is 20.30in._.

Apply 1 kips horizontally at point E for horizontal deflection (δEH).

Sketch the MP diagram and FP diagram for δEH as shown in Figure 5.

Fundamentals of Structural Analysis, Chapter 8, Problem 38P , additional homework tip  5

Find the moment MQ of the Q-system:

For span AC,

MC=0

MA=1(12)=12kip-ft

Find the force FQ of the Q-system:

FC=1kips

FE=1kips

The moment MQ for horizontal deflection at E and horizontal deflection at H are same and no axial loads are developed in the member BCED due to applied loading.

Therefore, δBH=δEH=10.61in..

Thus, the horizontal deflection at mid span of the member CD is 10.61in._.

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