Page 605.Consider the Figure9.9 (a)of Concept Problem 9.1,whichis a prismatic cantilever beam with a single load P applied at point A.First, solve theConcept Problem9.1as stated.b.Now let’s add a downward uniformly distributed load of w per unit length and solve the problem.c.With the results of b,assume now thatthe load P points upward. Under thiscase, what iswas a functionofP in order tohavezero displacement at point A?What is the slope at Points Aand B?d.Can you think of a different structure that would have the samedeflection profileas inc, with theappropriate boundary conditions. Explain in detail but concisely. A(a)B[x = L,8=0][x = Ly=0]BFig. 9.9 (a) Cantilever beam with endload. (b) Free-body diagram of section AC.(c) Deformed shape and boundaryconditions.XConcept Application 9.1The cantilever beam AB is of uniform cross section and carries a load Pat its free end A (Fig. 9.9a). Determine the equation of the elastic curveand the deflection and slope at A.Using the free-body diagram of the portion AC of the beam(Fig. 9.9b), where C is located at a distance x from end A,M =-Px(1)Substituting for M into Eq. (9.4) and multiplying both members by theconstant El givesIntegrating in x,EIwhich we carry back into Eq. (2):dydxdyEI =-Px²+C₁dxNow observe the fixed end B where x = L and 0 = dy/dx = 0 (Fig. 9.9c).Substituting these values into Eq. (2) and solving for C₁ givesC₁ =PL²EIYA =Integrating both members of Eq. (3).=-Pxy=PL²3EIEl y =-Px²+ PL²x + C₂But at B, x = L, y = 0. Substituting into Eq. (4),0=-PL³ ++PL² + C₂C₂ = -PL²³-Px²+ PL²P6EICarrying the value of C₂ back into Eq. (4), the equation of the elasticcurve isEl y=-Px²+PL²x - PL²(-x+ 3L²x - 21³)andThe deflection and slope at A are obtained by letting x = 0) in Eqs. (3)and (5).0₁ =(2)dxA(3)PL²2EI(4)(5)

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Answered: Page 605.Consider the Figure9.9 (a)of…

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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Page 605.Consider the Figure9.9 (a)of Concept Problem 9.1,whichis a prismatic cantilever beam with a single load P applied at point A.

First, solve theConcept Problem9.1as stated.b.Now let’s add a downward uniformly distributed load of w per unit length and solve the problem.

c.With the results of b,assume now thatthe load P points upward. Under thiscase, what iswas a functionofP in order tohavezero displacement at point A?What is the slope at Points Aand B?

d.Can you think of a different structure that would have the samedeflection profileas inc, with theappropriate boundary conditions. Explain in detail but concisely.

A
(a)
B
[x = L,8=0]
[x = Ly=0]
B
Fig. 9.9 (a) Cantilever beam with end
load. (b) Free-body diagram of section AC.
(c) Deformed shape and boundary
conditions.
X
Concept Application 9.1
The cantilever beam AB is of uniform cross section and carries a load P
at its free end A (Fig. 9.9a). Determine the equation of the elastic curve
and the deflection and slope at A.
Using the free-body diagram of the portion AC of the beam
(Fig. 9.9b), where C is located at a distance x from end A,
M =-Px
(1)
Substituting for M into Eq. (9.4) and multiplying both members by the
constant El gives
Integrating in x,
EI
which we carry back into Eq. (2):
dy
dx
dy
EI =-Px²+C₁
dx
Now observe the fixed end B where x = L and 0 = dy/dx = 0 (Fig. 9.9c).
Substituting these values into Eq. (2) and solving for C₁ gives
C₁ =PL²
EI
YA =
Integrating both members of Eq. (3).
=-Px
y=
PL²
3EI
El y =-Px²+ PL²x + C₂
But at B, x = L, y = 0. Substituting into Eq. (4),
0=-PL³ ++PL² + C₂
C₂ = -PL²³
-Px²+ PL²
P
6EI
Carrying the value of C₂ back into Eq. (4), the equation of the elastic
curve is
El y=-Px²+PL²x - PL²
(-x+ 3L²x - 21³)
and
The deflection and slope at A are obtained by letting x = 0) in Eqs. (3)
and (5).
0₁ =
(2)
dx
A
(3)
PL²
2EI
(4)
(5)

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