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Problem 7.93
Determine the support reactions for the loading shown.
Solution
The distributed loads are replaced by the resultant forces
Using the FBD, the equilibrium equations are
Problem 7.94
In Fig. P7.88, replace the pin and roller supports with a built-in support at
C
, and determine the support
reactions.
Solution
The resultant force for the distributed load is
Using the FBD shown at the right, the equilibrium equations are
Problem 7.95
In Fig. P7.89, replace the pin and roller supports with a built-in support at
A
, and determine the support
reactions.
Problem 7.96
In Fig. P7.90, reposition the roller support to the midspan of the beam, and determine the support reactions.
2.20 ft/ .6kip=ft/D60 kip:(1)
The position for the resultant force on the right-hand portion of the beam
is
d2D20 ft C.2=3/20 ft D33:33 ft
. Using the FBD, the equilibrium
Problem 7.97
In Fig. P7.91 on p. 481, replace the pin and roller supports with a built-in support at
A
, and determine the
support reactions.
Problem 7.98
Determine the support reactions for the cantilever beam. Express your answers in
terms of parameters such as w1,w2,a, and L.
Solution
Using the FBD, the equilibrium equations are
XFxD0WAxD0; (5)
2.
2.
Problem 7.99
Determine the support reactions for the simply supported beam. Express your
answers in terms of parameters such as w0,a, and L.
Solution
The locations of these resultants relative to point Aare
Using the FBD, the equilibrium equations are
Remark A partial check of the accuracy of these answers is to consider the following cases:
Problem 7.100
A blade of the main rotor of a hovering helicopter is subjected to the
y
direction
distributed forces shown, where values of the distributed force are known at
points
A
,
B
, and
C
. Determine the constants
a
,
b
, and
c
so that the quadratic
polynomial
wDaCbx Ccx2
describes this loading. Using this polynomial,
determine the total force produced by this distribution and the
x
position of its
line of action.
The distributed forces on the main rotor of the helicopter described in
Prob. 7.100 may be approximated by using a linear distribution where values of
the distributed force are known at points
A
and
C
. Determine the constants
a
and
b
so that the linear polynomial
wDaCbx
describes this loading. Using
integration with this polynomial, determine the total force produced by this
distribution and the xposition of its line of action.
Problem 7.102
The wing of a jet is modeled as a two-dimensional structure that is supported by
a pin and torsional spring at point
A
. The distributed load on the bottom of the
wing is due to the lift forces, and the
5kN
,
3kN
, and
4kN
forces are due to the
weights of the fuel, engine, and wing, respectively. When there are no forces
applied to the wing, the wing is horizontal and the torsional spring produces no
moment. Due to the forces shown, the tip of the wing (point
E
) deflects upward
by
8cm
. Assuming the wing is rigid, determine the stiffness
kt
of the torsional
spring. Report your answer using degrees.
Problem 7.103
Consider a straight uniform member with length
L
and weight
W
. Two force
systems for representing the weight of this member are shown. In system 1, the
distributed force is uniform with value w0DW=L.
(a)
Use Eq. (4.16) on p. 248 to determine
Ay
and
By
in terms of
W
so that
the two force systems are equivalent.
(b)
Are the results for Part (a) in agreement with the load lumping scheme
described for trusses in Example 6.2 on p. 372?
(c)
Without calculations, but perhaps by use of an appropriate sketch, show
that the results of Part (a) do not apply for a member that is not straight
and/or has nonuniform weight distribution. Hint: Redraw Fig. P7.103
using a member that is not straight or using a straight member with
nonuniform weight distribution. Then argue that the results of Part (a) do
not constitute an equivalent force system for this situation.
Problem 7.104
A uniform curved beam with circular shape and weight
W
has a built-in support
at
A
. Determine the support reactions. Express your answers in terms of
parameters such as Wand R.
Problem 7.105
A uniform curved beam with circular shape and weight
W
is supported by a
frictionless bearing at
A
and a roller at
B
. Determine the support reactions.
Express your answers in terms of parameters such as Wand R.
Problem 7.106
A cube of material with edge lengths
d
and specific weight
2
is suspended by a
cable and is submerged to a depth
d
in a fluid having specific weight
. Determine
the force
T
in the cable. Express your answer in terms of parameters such as
d
,
,
etc.
Problem 7.107
In Fig. P7.107(a), the concrete wall of a building has a small water-filled gap
between it and the adjacent soil. In Fig. P7.107(b), a concrete wall is used to
retain water. If the depth
d
of water is the same for both walls, which wall will be
subjected to the larger forces due to water pressure? Explain.
Note: Concept problems are about explanations, not computations.
Problem 7.108
Water in a channel is retained by a gate with
5in:
width (into the plane of the
figure). The gate is supported by a pin at
B
and a vertical cable at
A
, and the
contact between the gate and the bottom of the channel at
A
is frictionless. The
vertical wall
BC
is fixed in position. If the gate’s
50 lb
weight is uniformly
distributed, determine the cable force Trequired to begin opening the gate.
Solution
The FBD for the gate is shown at the right. When the gate begins to open,
AyD0. The water pressures at Aand Bare
Summing moments about point Bprovides the equilibrium equation
Problem 7.109
Water in a channel is retained by a gate with
5in:
width (into the plane of the
figure). The gate is supported by a pin at
B
and a vertical cable at
A
, and the
contact between the gate and the bottom of the channel at
A
is frictionless. The
vertical wall
BC
is fixed in position. If the gate’s
50 lb
weight is uniformly
distributed, determine the cable force Trequired to begin opening the gate.
Solution
The weight of the water in the FBD is
Problem 7.110
Water in a channel is retained by a gate with
10 cm
width (into the plane of the figure).
The gate is supported by a pin at
B
and by frictionless contact with the bottom of
the channel at
A
. The gate is outfitted with a prewound torsional spring at
B
with
stiffness
ktD75 Nm=rad
. The vertical wall
BC
is fixed in position. If the gate has
negligible weight, determine the amount
0
the spring must be prewound so that the
gate will begin to open when dD50 cm.
Solution
The FBD for the gate is shown at the right where
MB
is the moment due to
the torsional spring. When the gate begins to open,
AxD0
. The density
of water is
D103kg=m3
. When
dD50 cm
, the water pressures at
A
and
Using the composite shapes shown in the FBD, and noting that the width
of the gate (into the plane of the figure) is
10 cm
, the forces due to the fluid
pressures are
Summing moments about point Bprovides the equilibrium equation
Problem 7.111
Water in a channel is retained by a gate with
10 cm
width (into the plane of the
figure). The gate is supported by a pin at
B
and by frictionless contact with the
bottom of the channel at
A
. The gate is outfitted with a prewound torsional spring at
B
with stiffness
ktD75 Nm=rad
. The vertical wall
BC
is fixed in position. If the
gate has negligible weight, determine the amount
0
the spring must be prewound
so that the gate will begin to open when dD50 cm.
2.pApB/ .0:3 m/ .0:1 m/D44:15 N:(5)
Summing moments about point Bprovides the equilibrium equation
