EBK MECHANICS OF MATERIALS
EBK MECHANICS OF MATERIALS
7th Edition
ISBN: 8220100257063
Author: BEER
Publisher: YUZU
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Chapter 9, Problem 157RP

For the loading shown, determine (a) the equation of the elastic curve for the cantilever beam AB, (b) the deflection at the free end, (c) the slope at the free end.

Chapter 9, Problem 157RP, For the loading shown, determine (a) the equation of the elastic curve for the cantilever beam AB,

Fig. P9.157

(a)

Expert Solution
Check Mark
To determine

The equation of elastic curve (y) for the cantilever beam AB.

Answer to Problem 157RP

The equation of the elastic curve for the cantilever beam AB is w0EIL(L3x26L2x312+x5120)_.

Explanation of Solution

Calculation:

Show the free body diagram of the beam as in Figure 1.

EBK MECHANICS OF MATERIALS, Chapter 9, Problem 157RP , additional homework tip  1

Calculate the reaction (RA) at point A by resolving the vertical component of force.

Fy=0RAw0L2=0RA=w0L2

Calculate the moment MA at point A as below:

MA=0MAw0L2×2L3=0MA=w0L23MA=w0L23

Show the free body diagram of beam section AE as in Figure 2.

EBK MECHANICS OF MATERIALS, Chapter 9, Problem 157RP , additional homework tip  2

Calculate the moment (M) by taking moment about the point E.

ME=0w0L23w0Lx2+w0x22L×(x3)+M=0M=w0L23+w0Lx2w0x22L×(x3)M=w0L23+w0Lx2w0x36LM=w0L23+w0Lx2w0x36L (1).

Calculate the equation of the elastic curve (y) by integrating the Equation (1).

EId2ydx2=w0L23+w0Lx2w0x36LEIdydx=w0L23x+w0L2x22w06Lx44+C1EIdydx=w0L23x+w0Lx24w0x424L+C1 (2).

Substitute 0 for x and 0 for dydx in Equation (2).

EI(0)=w0L23(0)+w0L(0)24w0(0)424L+C10=0+00+C1C1=0

Integrate equation (2) and Substitute 0 for C1.

EIdydx=w0L23x+w0Lx24w0x424L+(0)EIdydx=w0L23x22+w0L4x33w024Lx55+C2EIy=w0L2x26+w0Lx312w0x5120L+C2 (3)

Substitute 0 for x and 0 for y in Equation (3).

EI(0)=w0L2(0)26+w0L(0)312w0(0)5120L+C20=0+00+C2C2=0

Integrate Equation (3) and Substitute 0 for C2.

EIy=w0L2x26+w0Lx312w0x5120L+0

Multiply the Equation by (LL).

EIy=w0L2x26LL+w0Lx312LLw0x5120LLLEIy=w0L3x26L+w0L2x312Lw0x5120Ly=w0EIL(L3x26L2x312+x5120)

Thus, the equation of the elastic curve for the cantilever beam AB is w0EIL(L3x26L2x312+x5120)_.

(b)

Expert Solution
Check Mark
To determine

The deflection (yB) the free end at point B.

Answer to Problem 157RP

The deflection (yB) the free end at point B is 1112w0L4EI(downward)_.

Explanation of Solution

Calculation:

Calculate the deflection (yB) using the Equation (3).

Substitute L for x.

EIy=w0L2(L)26+w0L(L)312w0(L)5120LEIy=w0L46+w0L412w0(L)5120LyB=w0L4EI(16112+1120)yB=11120w0L4EI

Thus, the deflection (yB) the free end at point B is 1112w0L4EI(downward)_.

(c)

Expert Solution
Check Mark
To determine

The slope (θB) the free end at point B.

Answer to Problem 157RP

The slope (θB) the free end at point B is 18w0L3EI_.

Explanation of Solution

Calculation:

Calculate the slope (θB) using the Equation (2).

Substitute L for x and 0 for C1

EIdydx=w0L23(L)+w0L(L)24w0(L)424L+0EIdydx=w0L33+w0L34w0L324dydx=w0L3EI(1314+124)dydx=18w0L3EI

Thus, the slope (θB) the free end at point B is 18w0L3EI_.

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Chapter 9 Solutions

EBK MECHANICS OF MATERIALS

Ch. 9.2 - For the beam and loading shown, (a) express the...Ch. 9.2 - (a) Determine the location and magnitude of the...Ch. 9.2 - For the beam and loading shown, determine the...Ch. 9.2 - Knowing that beam AE is a W360 101 rolled shape...Ch. 9.2 - For the beam and loading shown, knowing that a = 2...Ch. 9.2 - Knowing that beam AE is an S200 27.4 rolled shape...Ch. 9.2 - For the beam and loading shown, determine (a) the...Ch. 9.2 - For the beam and loading shown, determine (a) the...Ch. 9.2 - 9.19 through 9.22 For the beam and loading shown,...Ch. 9.2 - 9.19 through 9.22 For the beam and loading shown,...Ch. 9.2 - 9.19 through 9.22 For the beam and loading shown,...Ch. 9.2 - 9.19 through 9.22 For the beam and loading shown,...Ch. 9.2 - For the beam shown, determine the reaction at the...Ch. 9.2 - For the beam shown, determine the reaction at the...Ch. 9.2 - 9.25 through 9.28 Determine the reaction at the...Ch. 9.2 - 9.25 through 9.28 Determine the reaction at the...Ch. 9.2 - Prob. 27PCh. 9.2 - 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