Determine the equations of the elastic curve for the beam using the and coordinates.. Determine the equations of the elastic curve usingthe coordinates and , specify the slope and deflec
Trang 1For
(1)
(2)For
(3)
(4)Boundary conditions:
atFrom Eq (2)
Due to symmetry:
atFrom Eq (3)
*8–1. Determine the equations of the elastic curve for the
beam using the and coordinates Specify the slope at
A and the maximum deflection EI is constant.
x2
x1
P P
L
x2
x1
Trang 2dx1 =
P2EI(x1 + a
Pa3
2 + C1 = Pa
3
PaL2
8–2. The bar is supported by a roller constraint at B, which
allows vertical displacement but resists axial load and
moment If the bar is subjected to the loading shown,
determine the slope at A and the deflection at C EI is
Trang 4*8–4. Determine the equations of the elastic curve using
the coordinates and , specify the slope and deflection
w
x1
C
Trang 5For
(3)(4)Boundary conditions:
From Eqs (1) and (3),
From Eqs (2) and (4),
Trang 62 7 2
For
(1)(2)For
(3)(4)Boundary conditions:
8–5. Determine the equations of the elastic curve using
the coordinates and , and specify the slope and
deflection at point B EI is constant.
w
x1
C
Trang 7Elastic curve and slope:
8–6. Determine the maximum deflection between the
supports A and B EI is constant Use the method of
Trang 82 7 4
Continuity conditions:
at
From Eqs (1) and (3)
Substitute C1into Eq (5)
(6)
The negative sign indicates downward displacement
(7)occurs when
Trang 9(2)Boundary conditions:
5woL3192EI
8–7. Determine the elastic curve for the simply supported
beam using the x coordinate Also, determine
the slope at A and the maximum deflection of the beam.
EI is constant.
0 … x … L>2
L x
w0
Trang 102 7 6
Support Reactions and Elastic Curve: As shown on FBD(a).
Moment Function: As shown on FBD(c) and (c).
Slope and Elastic Curve:
For ,
(1)(2)
For ,
(3)(4)
Boundary Conditions:
Continuity Conditions:
The Slope: Substituting into Eq [1],
Ans.
into Eqs [2] and [4], respectively
*8–8. Determine the equations of the elastic curve using
the coordinates and , and specify the slope at C and
displacement at B EI is constant.
x2
x1
B A
a
x1
x3
x2a
w C
Trang 11Support Reactions and Elastic Curve: As shown on FBD(a).
Moment Function: As shown on FBD(b) and (c).
Slope and Elastic Curve:
For ,
(1)(2)
For
(3)(4)
Boundary Conditions:
Continuity Conditions:
Ans.
into Eqs [2] and [4], respectively,
8–9. Determine the equations of the elastic curve using
the coordinates and , and specify the slope at B and
deflection at C EI is constant.
x3
x1
B A
a
x1
x3
x2a
w C
Trang 122 7 8
Using the diagram and the elastic curve shown in Fig a and b, respectively,
Theorem 1 and 2 give
in2d(500 in4)
= 2700 k#ft3EI
8–10. Determine the slope at B and the maximum
displacement of the beam Use the moment-area theorems
Trang 13The real beam and conjugate beam are shown in Fig b and c, respectively.
=
270 (122) k#in2c29(103)k
in2d (500 in4)
= 0.00268 rad
uB = V¿B = – 270 k
#ft2EI
-V¿B- 1
2a90 kEI#ftb(6 ft) = 0+ c a Fy = 0;
8–11. Solve Prob 8–10 using the conjugate-beam method
Trang 142 8 0
Using the diagram and the elastic curve shown in Fig a and b, respectively,
Theorem 1 and 2 give
*8–12. Determine the slope and displacement at C EI is
constant Use the moment-area theorems
Trang 15The real beam and conjugate beam are shown in Fig a and b, respectively.
uC = V¿C = -3937.5 k#ft2
3937.5 k#ft2EI
+ c a Fy = 0; -V¿C - 1
2a225 kEI#ftb(15 ft) - 2250 kEI#ft
B¿y = 2250 k#ft2
EI+ a MA= 0; B¿y(30 ft) - c12 a225 kEI#ftb(30 ft) d (20 ft)
8–13. Solve Prob 8–12 using the conjugate-beam method
Trang 162 8 2
Using the diagram and the elastic curve shown in Fig a and b,
respectively, Theorem 1 and 2 give
Then
Here, it is required that
Choose the position root,
uA>B = 1
2a4EIPLb(L) + 12a - EIPab(a + L)
M
EI
8–14. Determine the value of a so that the slope at A is
equal to zero EI is constant Use the moment-area theorems.
P
B C
Trang 17The real beam and conjugate beam are shown in Fig a and b,
respectively Referring to Fig d,
a
It is required that , Referring to Fig c,
Choose the position root,
Pa2
2EI = 0V'A = uA = 0
D¿y =
PL2
16EI
-PaL3EI+ a MB = 0; D'y(L) + c12aPaEIb(L) d a23Lb - c12a4EIPLb(L) d a12b = 0
8–15. Solve Prob 8–14 using the conjugate-beam method
P
B C
Trang 182 8 4
Using the diagram and the elastic curve shown in Fig a and b,
respectively, Theorem 2 gives
It is required that
Ans.
a = L3
2tD>B
= PL3
96EI
-PaL248EI
TC>B = c12a4EIPLb aL2b d c13aL2b d + c12a– 2EIPab aL2b d c13aL2b d
= PL3
16EI
-PaL26EI
tD>B= c12a4EIPLb(L) d aL2b + c12a– EIPab(L) d aL3b
M
EI
*8–16. Determine the value of a so that the displacement
at C is equal to zero EI is constant Use the moment-area
theorems
P
B C
Trang 19The real beam and conjugate beam are shown in Fig a and b, respectively.
PL332EI +
PaL212EI = 0
–c16EIPL2 - PaL
6EId aL2b = 0–c12a2EIPab aL2b d c13aL2b d+ a MC= 0; c12a4EIPLb aL2b d c13aL2b d
8–17. Solve Prob 8–16 using the conjugate-beam method
P
B C
Trang 202 8 6
Using the diagram and the elastic curve shown in Fig a and b, respectively,
Theorem 1 and 2 give
Pa34EI c
uC = uC>D= Pa2
4EI
uC>D =
1
2a2EIPab(a) = 4EIPa2
tC >D = c12a2EIPab(a) d aa + 23ab = 12EI5Pa3
tB >D = c12a2EIPab(a) d a23ab = 6EIPa3
M
EI
8–18. Determine the slope and the displacement at C EI
is constant Use the moment-area theorems
B
P
Trang 21The real beam and conjugate beam are shown in Fig a and b, respectively.
Trang 222 8 8
Using the diagram and the elastic curve shown in Fig a and b,
respectively, Theorem 1 and 2 give
= 0.00171 rad
uC= uA + uC>A = 6 kN#m2
18 kN#m2EI
uC>A = 1
2a12 kNEI#mb(6 m) + 12a - 12 kNEI#mb(9 m)
M
EI
*8–20. Determine the slope and the displacement at the
the moment-area theorems
E = 200 GPa, I = 70(106) mm4
B D
Trang 23The real beam and conjugate beam are shown in Fig a and b, respectively.
M¿C + c12a12 kNEI#m2b(3 m) d c 23(3 m)d+ a MC= 0;
8–21. Solve Prob 8–20 using the conjugate-beam method
B D
Trang 24M
EI
8–22. At what distance a should the bearing supports at A
and B be placed so that the displacement at the center of
the shaft is equal to the deflection at its ends? The bearings
exert only vertical reactions on the shaft EI is constant Use
a
L
P P
a
Trang 25The real beam and conjugate beam are shown in Fig a and b,
respectively Referring to Fig c,
Pa2
2EI(L – 2a) +
Pa33EI =
Pa8EI (L – 2a)
B¿y (L - 2a) - cPaEI (L – 2a)d a L – 2a2 b = 0+ a MA = 0;
8–23. Solve Prob 8–22 using the conjugate-beam method
a
L
P P
a
Trang 262 9 2
Using the diagram and elastic curve shown in Fig a and b, respectively,
Theorem 1 and 2 give
*8–24. Determine the displacement at C and the slope
at B EI is constant Use the moment-area theorems.
Trang 27The real beam and conjugate beam are shown in Fig a and b, respectively.
B¿y = uB = 18 kN
#m2
EI+ a MA = 0; B¿y (3 m) - a12 kNEI#m b(3 m)(1.5 m) = 0
8–25. Solve Prob 8–24 using the conjugate-beam method
Trang 282 9 4
Using the diagram and elastic curve shown in Fig a and b, respectively,
Theorem 1 and 2 give
8–26. Determine the displacement at C and the slope at B.
EI is constant Use the moment-area theorems.
P P
Trang 298–27. Determine the displacement at C and the slope at B.
EI is constant Use the conjugate-beam method.
P P
-M'C = 0+ c12a2EIPab(a) d aa3b - 7Pa4EI2(2a)+ a MC= 0; c12a2EIPa b(a) d a43 ab + c aPaEIb(a) d aa2b
Trang 30tC/A = c12a2EIPab(2a) d(2a) + c12a - FaEIb(2a) d c 13(2a) + ad
tB/A = c12a2EIPab(2a) d(a) + c12a - FaEIb(2a) d c 13(2a)d = 2EIPa3 - 2Fa
3
3EI
M
EI
*8–28. Determine the force F at the end of the beam C so
that the displacement at C is zero EI is constant Use the
Trang 31The real beam and conjugate beam are shown in Fig a and b, respectively Referring
+a MC = 0; c12aFaEIb(a) d c23 (a)d - a4EIPa2 - 2Fa
8–29. Determine the force F at the end of the beam C so
that the displacement at C is zero EI is constant Use the
Trang 322 9 8
Using the diagram and elastic curve shown in Fig a and b,
Theorem 1 and 2 give
3Pa34EI T
+ uB = - Pa2
12EI +
Pa22EI =
5Pa212EI
tC >A = c12a -PaEIb(2a) d(a) = - PaEI3
tB >A = c12a -PaEIb(a) d c13 (a)d = - Pa6EI3
8–30. Determine the slope at B and the displacement at C.
EI is constant Use the moment-area theorems.
A C
B
Trang 33The real beam and conjugate beam are shown in Fig c and d, respectively.
uB = B¿y = 5Pa2
12EI+ a MA = 0; c12aPaEIb(a) d aa + 23 ab - B¿y(2a) = 0
8–31. Determine the slope at B and the displacement at C.
EI is constant Use the conjugate-beam method.
A C
B
Trang 343 0 0
Using the diagram and the elastic curve shown in Fig a and b, respectively,
Theorem 1 and 2 give
*8–32. Determine the maximum displacement and the slope
at A EI is constant Use the moment-area theorems.
A
C B
L
2
L
2
Trang 358–33. Determine the maximum displacement at B and the
slope at A EI is constant Use the conjugate-beam method.
A
C B
L
2
L
¢max = ¢D = M¿D = -
0.00802M0L2EI
2EIb aL2b d aL3b = 0
Trang 363 0 2
Using the diagram and the elastic curve shown in Fig a and b, respectively,
Theorem 1 and 2 give
8–34. Determine the slope and displacement at C EI is
constant Use the moment-area theorems
a a
P
B
M0 Pa
Trang 37The real beam and conjugate beam are shown in Fig a and b, respectively.
2aPaEIb(a) - a2PaEI b(a) - 12aPaEIb(a) = 0
8–35. Determine the slope and displacement at C EI is
constant Use the conjugate-beam method
a a
P
B
M0 Pa
Trang 383 0 4
Using the diagram and the elastic curve shown in Fig a and b, respectively,
Theorem 1 and 2 give
*8–36. Determine the displacement at C Assume A is a
fixed support, B is a pin, and D is a roller EI is constant.
Use the moment-area theorems
Trang 398–37. Determine the displacement at C Assume A is a
fixed support, B is a pin, and D is a roller EI is constant.
Use the conjugate-beam method
The real beam and conjugate beam are shown in Fig a and b, respectively.
uD = Dy¿ = 75 kN
#m2EI+ c12a37.5 kNEI #mb(3 m) d(2 m) - D¿y (6 m) = 0+ a MB = 0; c12a37.5 kNEI #mb(6 m) d(3 m)
Trang 403 0 6
Using the diagram and elastic curve shown in Fig a and b, respectively,
Theorem 1 and 2 give
8–38. Determine the displacement at D and the slope at
D Assume A is a fixed support, B is a pin, and C is a roller.
Use the moment-area theorems
6 k
Trang 41The real beam and conjugate beam are shown in Fig a and b, respectively.
8–39. Determine the displacement at D and the slope at
D Assume A is a fixed support, B is a pin, and C is a roller.
Use the conjugate-beam method
6 k