978-0073380292 Chapter 7 Part 7

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.

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 Nm=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 Nm=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.pApB/ .0:3 m/ .0:1 m/D44:15 N:(5)

Summing moments about point Bprovides the equilibrium equation