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 20P
To determine

Find the slope and the vertical deflection of the cantilever beam at the points B and C.

Find the minimum value of moment of inertia, if the deflection of point C is 0.4 in.

Expert Solution & Answer
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Answer to Problem 20P

The slope at point C is θC=720EI_.

The deflection at point C is δC=PL33EI+wL48EI_.

The slope at point B is θB=576EI_.

The deflection at point B is δB=1,998EI_.

The required minimum value of I is 6,531.8in.4_.

Explanation of Solution

Given information:

The value of E is 4,000kips/in.2 and length of the beam is 12ft.

Deflection at point C is 0.4 in.

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 deflection at C of the cantilever beam as follows:

For Q system, remove all external loads and consider a dummy load of 1 kips at the point C of the beam.

Sketch the P-system and Q-system for deflection at C as shown in Figure 1.

Fundamentals of Structural Analysis, Chapter 8, Problem 20P , additional homework tip  1

From P-system,

The moment at a distance of x from C is,

MP=Px+wxx2=Px+wx22

From Q-system,

MQ=1(x)=x

Find the deflection at point C (δC) as follows:

(1kips)(δC)=0LMQMPdxEIδC=0L(x)(Px+wx22)dxEI=0L(Px2+wx32)dxEI=1EI[Px33+wx48]0L

δC=1EI(PL33+wL48)=PL33EI+wL48EI

Therefore, the deflection at point C is δC=PL33EI+wL48EI_.

Find the minimum value of I, if the value of δC is 0.4in.

Substitute 6 kips for P, 1kip/ft for w, 0.4in. for δC, and 12 ft for L.

0.4in.=(6kips)(12ft)33(4,000kips/in.2)I+(1kip/ft)(12ft)48(4,000kips/in.2)I0.4in.=(6kips)(12ft×12in.1ft)33(4,000kips/in.2)I+(1kip/ft×1ft12in.)(12ft×12in.1ft)48(4,000kips/in.2)I0.4=1,492.992I+1,119.744IImin=6,531.8in.4

Therefore, the required minimum value of I is 6,531.8in.4_.

Find the slope at C of the cantilever beam as follows:

For Q system, remove all external loads and consider a dummy moment of 1kipft at the point C of the beam.

Sketch the P-system and Q-system for slope at C as shown in Figure 2.

Fundamentals of Structural Analysis, Chapter 8, Problem 20P , additional homework tip  2

From P-system,

The moment at a distance of x from C is,

MP=Px+wxx2=Px+wx22

From Q-system,

MQ=1

Find the slope at point C (θC) as follows:

θC=0LMQMPdxEI=0L(1)(Px+wx22)dxEI=0L(Px+wx22)dxEI=1EI[Px22+wx36]0L

θC=PL22EI+wL36EI

Substitute 6 kips for P, 1kip/ft for w, 0.4in. for δC, and 12 ft for L.

θC=(6)(12)22EI+(1)(12)36EI=432EI+288EI=720EI

Therefore, the slope at point C is θC=720EI_.

Find the deflection at B of the cantilever beam as follows:

For Q system, remove all external loads and consider a dummy load of 1 kips at the point C of the beam.

Sketch the P-system and Q-system for deflection at B as shown in Figure 3.

Fundamentals of Structural Analysis, Chapter 8, Problem 20P , additional homework tip  3

From P-system,

The moment at a distance of x from B is,

MP=w(6+x)(6+x2)+P(6+x)=w2(36+12x+x2)+6P+Px=18w+6wx+wx22+6P+Px

From Q-system,

MQ=1(x)=x

Find the deflection at point B (δB) as follows:

(1kips)(δB)=0LMQMPdxEIδB=06(x)(18w+6wx+wx22+6P+Px)dxEI=06(18wx+6wx2+wx32+6Px+Px2)dxEI=1EI[18wx22+6wx33+wx48+6Px22+Px33]06

δB=1EI(18w(6)22+6w(6)33+w(6)48+6P(6)22+P(6)33)=1EI(324w+432w+162w+108P+72P)=1EI(324(1)+432(1)+162(1)+108(6)+72(6))=1,998EI

Therefore, the deflection at point B is δB=1,998EI_.

Find the slope at B of the cantilever beam as follows:

For Q system, remove all external loads and consider a dummy moment of 1kipft at the point B of the beam.

Sketch the Q-system for slope at B as shown in Figure 4.

Fundamentals of Structural Analysis, Chapter 8, Problem 20P , additional homework tip  4

From P-system,

The moment at a distance of x from B is,

MP=w(6+x)(6+x2)+P(6+x)=w2(36+12x+x2)+6P+Px=18w+6wx+wx22+6P+Px

From Q-system,

MQ=1

Find the slope at point B (θB) as follows:

θB=0LMQMPdxEI=06(1)(18w+6wx+wx22+6P+Px)dxEI=06(18w+6wx+wx22+6P+Px)dxEI=1EI[18wx+6wx22+wx36+6Px+Px22]06

θB=1EI[18w(6)+6w(6)22+w(6)36+6P(6)+P(6)22]=1EI(108w+108w+36w+36P+18P)=1EI(108(1)+108(1)+36(1)+36(6)+18(6))=576EI

Therefore, the slope at point B is θB=576EI_.

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