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 7, Problem 40P
To determine

Sketch the slope and deflection diagrams of the beam.

Expert Solution & Answer
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Explanation of Solution

Given information:

The flexural rigidity of segment ABCD is 2I and DEF is I.

Calculation:

Apply the sign conventions for calculating reactions 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 positive and the counter clockwise moment as negative.

Use conjugate beam method.

Sketch the Free Body Diagram of the beam as shown in Figure 1.

Fundamentals of Structural Analysis, Chapter 7, Problem 40P , additional homework tip  1

Refer to Figure 1.

Consider span DEF.

Use equilibrium equations:

Summation of moments about D is equal to 0.

MDDEF=0RF×1610×8=0RF=5 kips()

Summation of forces along y-direction is equal to 0.

+Fy=0RDR+105=0RDR=5 kips()

The vertical reaction at the left of the hinge at D is RDL=5 kips().

Consider span ABCD.

Use equilibrium equations:

Summation of moments about A is equal to 0.

MAABCD=05×20RC×8+25×4=0RC=0

Summation of forces along y-direction is equal to 0.

+Fy=0RA+525=0RA=20kips()

Calculate the moment at each point as shown below.

MA=0MB=20×4=80 kipftMC=20×825×4=60 kipft

MD=20×2025×16=0ME=20×2825×24=40 kipftMF=20×3625×32+10×8=0

Sketch the MEI diagram as shown in Figure 2.

Fundamentals of Structural Analysis, Chapter 7, Problem 40P , additional homework tip  2

In the real beam A is the hinged support, F and C are the roller support, and D is the internal hinge but in conjugate-beam method the end conditions has to be change.

Change the hinged support at A as hinged support, the roller at F as roller support, the roller at C as internal hinge, and the internal hinge at D as roller support.

Sketch the conjugate beam of the loading from M/EI diagram as shown in Figure 3.

Fundamentals of Structural Analysis, Chapter 7, Problem 40P , additional homework tip  3

Refer to Figure 3.

Consider span ABC.

Use equilibrium equations:

Summation of moments about A is equal to 0.

MC=0RA×8+12×4×40EI×(13×4+4)+12×4×10EI×(23×4)+4×30EI×42=0RA=90EI()

Summation of forces along y-direction is equal to 0.

+Fy=0RCL+12×4×40EI+12×4×10EI+4×30EI90EI=0RCL=130EI()

Consider span CDEF.

Use equilibrium equations:

Summation of moments about F is equal to 0.

MF=0(130EI×28+12×12×30EI×(23×12+16)+RD×1612×8×40EI×(13×8+8)12×8×40EI×(23×8))=0RD=337.5EI()

Summation of forces along y-direction is equal to 0.

+Fy=0RF337.5EI+130EI+12×12×30EI12×16×40EI=0RF=347.5EI

Calculate the shear (slope) of conjugate beam at each point as shown below.

VA=90EIVB=90EI+12×4×40EI=10EIVC=90EI+12×4×40EI+12×4×10EI+4×30EI=130EI

V@ left of D=90EI+12×4×40EI+12×16×40EI             =310EIVD=90EI+12×4×40EI+12×16×40EI337.5EI    =27.5EI

VE=90EI+12×4×40EI+12×16×40EI337.5EI12×8×40EI   =187.5EIV@ left of F=90EI+12×4×40EI+12×16×40EI337.5EI12×16×40EI            =347.5EIVE=90EI+12×4×40EI+12×16×40EI337.5EI12×16×40EI+347.5EI   =0

Calculate the moment (deflection) of conjugate beam at each point as shown below.

MA=0MB=90EI×4+12×4×40EI×(13×4)=253.3EIMC=90EI×8+12×4×40EI×(13×4+4)+12×4×10EI×(23×4)+4×30EI×42=0

MD=90EI×20+12×4×40EI×(13×4+16)+12×16×40EI×(23×16)=3000EIME=(90EI×28+12×4×40EI×(13×4+24)+12×16×40EI×(23×16+8)337.5EI×812×8×40EI×(13×8))=2353.3EIMF=(90EI×36+12×4×40EI×(13×4+32)+12×16×40EI×(23×16+16)337.5EI×1612×8×40EI×(13×8+8)12×8×40EI×(23×8))=0

Sketch the slope (shear) and deflection (moment) of the beam as shown in Figure 4.

Fundamentals of Structural Analysis, Chapter 7, Problem 40P , additional homework tip  4

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