978-0073380292 Chapter 7 Part 6

Problem 7.80

(a)

Use integration (Eqs. (7.41) and (7.43) on p. 469) with the appropriate expression(s) for the dis-

tributed load

w

to determine the

x

position of the line of action for the resultant force produced by

the distributed load.

(b) Determine the support reactions using the results of Part (a).

(c) Determine the support reactions using composite shapes to represent the distributed load.

Loading (b) in Prob. 7.74 on p. 479.

Solution

Part (a) From the solution to Problem 7.74

Equation (7.41) may be applied by inspection to obtain the total force applied to the

beam as

Problem 7.81

(a)

Use integration (Eqs. (7.41) and (7.43) on p. 469) with the appropriate expression(s) for the dis-

tributed load

w

to determine the

x

position of the line of action for the resultant force produced by

the distributed load.

(b) Determine the support reactions using the results of Part (a).

(c) Determine the support reactions using composite shapes to represent the distributed load.

Loading (c) in Prob. 7.74 on p. 479.

Part (a) From the solution to Problem 7.74

Problem 7.82

(a)

Use integration (Eqs. (7.41) and (7.43) on p. 469) with the appropriate expression(s) for the dis-

tributed load

w

to determine the

x

position of the line of action for the resultant force produced by

the distributed load.

(b) Determine the support reactions using the results of Part (a).

(c) Determine the support reactions using composite shapes to represent the distributed load.

Loading (b) in Prob. 7.75 on p. 479.

Solution

Applying Eq. (7.43), the line of action of Fis positioned at

Problem 7.83

(a)

Use integration (Eqs. (7.41) and (7.43) on p. 469) with the appropriate expression(s) for the dis-

tributed load

w

to determine the

x

position of the line of action for the resultant force produced by

the distributed load.

(b) Determine the support reactions using the results of Part (a).

(c) Determine the support reactions using composite shapes to represent the distributed load.

Loading (c) in Prob. 7.75 on p. 479.

Solution

Part (a) From the solution to Problem 7.75

Statics 2e 1073

Part (c)

The FBD is shown at the right, where the distributed loading is repre-

sented using composite shapes. From the solution to Problem 7.75 (or by noting

Problem 7.84

Consider modeling the weights from books 8–11 from Example 7.11 on p. 475,

using the linear force distribution

wDaCbx

, as shown here. The constants

a

and

b

can be determined by requiring the two force systems shown to be

equivalent. That is,

4in:

Z

0

w dx D

11

X

iD8

Wiand

4in:

Z

0

xw dx D

11

X

iD8

MBi ;

where MBi is the moment about point B(the origin) of weight Wi.

(a)

Using the weights and geometry of books 8–11 given in Table 1 of

Example 7.11, evaluate the above expressions and solve for

a

and

b

to

show that wD35

8

lb

in. 15

16

lb

in.2x.

(b)

Evaluate the distributed force from Part (a) at

xD0

and

xD4in:

, and

compare these to the values used in Example 7.11, namely,

4lb=in:

and

1lb=in:, respectively.

(c)

Discuss why the distributed force from Part (a) is better than that used in

Example 7.11.

(d)

Using the distributed force from Part (a), determine the support reactions

for the bookshelf and compare to those found in Example 7.11.

0 4.375 lb=in. 4 lb=in.

4in:0.6250 lb=in. 1 lb=in.

Part (c)

The distributed force given by Eq. (7) and the distributed force used in Example 7.11 both satisfy

Eq. (1). However, the distributed force used in Example 7.11 does not satisfy Eq. (4), thus it is not an

2.4:375 lb=in. 0:625 lb=in./ .4in:/D7:50 lb;(11)

Using the right-hand FBD, the equilibrium equations are

Problem 7.84 Example 7.11

A 17:96 lb 17:9 lb

Problem 7.85

(a)

For the distributed loading shown, develop an expression (or multiple

expressions if needed) for the distributed force

w

as a function of position

x.

(b)

Use integration (Eqs. (7.41) and (7.43) on p. 469) with the results of

Part (a) to determine the total force produced by the distributed load and

the xposition of its line of action.

(c) Determine the support reactions using the results of Part (b).

(d)

Determine the support reactions using composite shapes for the dis-

tributed load.

2.6 ft/.4 kip=ft/D12 kip:(11)

The equilibrium equations are

Problem 7.86

A beam is loaded by a distributed force that begins at the left-hand end as an

800 N=m

uniform load with

dw=dx D0

and decreases to zero at the right-hand

end.

(a)

Determine the constants

a

,

b

, and

c

so that the quadratic polynomial

wDaCbx Ccx2describes this loading.

(b)

Determine the constants

d

and

f

so that the trigonometric function

wDdcos.f x/ describes this loading.

Problem 7.87

(a)

Use integration (Eqs. (7.41) and (7.43) on p. 469) with the results of

Part (a) of Prob. 7.86 to determine the total force produced by the dis-

tributed load, the

x

position of its line of action, and the support reactions.

(b)

Use integration (Eqs. (7.41) and (7.43) on p. 469) with the results of

Part (b) of Prob. 7.86 to determine the total force produced by the dis-

tributed load, the

x

position of its line of action, and the support reactions.

(c)

Compare the results of Parts (a) and (b), and discuss if these should be

the same or if differences are expected.