SOLUTION Horizontal plate:

In document Mechanics of Materials 7th Edition Beer Johnson Chapter 6 (Page 118-130)

0.5 in.

0.5 in.

4.5 in.

4.5 in.

0.5 in.

PROBLEM 6.98

The design of a beam requires welding four horizontal plates to a vertical 0.5 5-in. plate as shown. For a vertical shear V, determine the dimension h for which the shear flow through the welded surface is maximum.

SOLUTION Horizontal plate:

3 2

2

1 (4.5)(0.5) (4.5)(0.5) 12

0.046875 2.25

Ih h

h

 

 

Vertical plate: 1 3 4

(0.5)(5) 5.2083 in

v 12

I  

Whole section: I4IhIv9h25.39583 in4 For one horizontal plate, Q(4.5)(0.5)h2.25 inh 3

2

2.25 9 5.39583

VQ Vh

qIh

 To maximize q, set dq 0.

dh

2 2

2 2

(9 5.39583) 18

2.25 0

(9 5.39583)

h h

V h

  

 0.774 h in. 

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PROBLEM 6.99

A thin-walled beam of uniform thickness has the cross section shown. Determine the dimension b for which the shear center O of the cross section is located at the point indicated.

SOLUTION action of the forces in BDOGK pass through point O.

E

PROBLEM 6.100

Determine the location of the shear center O of a thin-walled beam of uniform thickness having the cross section shown.

SOLUTION

(0.75)(1.5) 0.5625 in

3 3

(2)(0.28125) (2)(0.5625) 1.6875 in

AB 3.375 3 (3.375)(3) 0.083333

(2)(0.083333)(3 sin 60 )

e  0.433 e in. 

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PROBLEM 6.C1

A timber beam is to be designed to support a distributed load and up to two concentrated loads as shown. One of the dimensions of its uniform rectangular cross section has been specified and the other is to be determined so that the maximum normal stress and the maximum shearing stress in the beam will not exceed given allowable values all and all. Measuring x from end A and using either SI or U.S. customary units, write a computer program to calculate for successive cross sections, from x to x L0  and using given increments  the shear, the bending moment, and x, the smallest value of the unknown dimension that satisfies in that section (1) the allowable normal stress requirement and (2) the allowable shearing stress requirement. Use this program to solve Prob. 5.65, assuming all 12 MPa and all 825 kPa and using

0.1 m.

 x

SOLUTION

See solution of Prob. 5.C2 for the determination of R R V x and A, B, ( ), M x ( ) We recall that

1 2

where STPA, STPB, STP1, STP2, STP3, and STP4 are step functions defined in Problem 5.C2.

(1) To satisfy the allowable normal stress requirement: If unknown dimension is h:

min | |/ all.

If unknown dimension is t:

min | |/ all.

(2) To satisfy the allowable shearing stress requirement:

We use Equation (6.10), Page 378: max 3 | | 3 | |

2 2

  VV

A th

If unknown dimension is h:

all

3| |

 2

h h V t If unknown dimension is t:

all

3

 2 

t t M h

PROBLEM 6.C1 (Continued)

Program Outputs Problem 5.65

2.40 kN 3.00 kN

A B

R R

 X

m

V kN

kN  m M HSIG mm

HTAU mm

0.00 2.40 0.000 0.00 109.09

0.10 2.40 0.240 54.77 109.09

0.20 2.40 0.480 77.46 109.09

0.30 2.40 0.720 94.87 109.09

0.40 2.40 0.960 109.54 109.09

0.50 2.40 1.200 122.47 109.09

0.60 2.40 1.440 134.16 109.09

0.70 2.40 1.680 144.91 109.09

0.80 0.60 1.920 154.92 27.27

0.90 0.60 1.980 157.32 27.27

1.00 0.60 2.040 159.69 27.27

1.10 0.60 2.100 162.02 27.27

1.20 0.60 2.160 164.32 27.27

1.30 0.60 2.220 166.58 27.27

1.40 0.60 2.280 168.82 27.27

1.50 0.60 2.340 171.03 27.27

1.60 –3.00 2.400 173.21 136.36 

1.70 –3.00 2.100 162.02 136.36 

1.80 –3.00 1.800 150.00 136.36 

1.90 –3.00 1.500 136.93 136.36 

2.00 –3.00 1.200 122.47 136.36 

2.10 –3.00 0.900 106.07 136.36 

2.20 –3.00 0.600 86.60 136.36 

2.30 –3.00 0.300 61.24 136.36 

2.40 0.00 0.000 0.05 0.00 

PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.

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PROBLEM 6.C2

A cantilever timber beam AB of length L and of uniform rectangular section shown supports a concentrated load P at its free end and a uniformly distributed load w along its entire length. Write a computer program to determine the length L and the width b of the beam for which both the maximum normal stress and the maximum shearing stress in the beam reach their largest allowable values.

Assuming all1.8 ksi and all120 psi,use this program to determine the dimensions L and b when (a)P1000 lband w 0, (b) 0P and w12.5 lb/in., (c) 500 lbP and w12.5 lb/in.

SOLUTION

Both the maximum shear and the maximum bending moment occur at A. We have

1 2

To satisfy the allowable normal stress requirement:

all 1 2 3

To satisfy the allowable shearing stress requirement:

We use Equation (6.10), Page 378.

all 2

PROBLEM 6.C2 (Continued)

Program Outputs

For P1000 lb, w0.0 lb/in. For P0 lb, w12.5 lb/in.

Increment 0.0010 in. Increment 0.0010 in.

37.5 in.,Lb1.250 in. 70.3 in.,Lb1.172 in.

For P500 lb, w12.5 lb/in.

Increment 0.0010 in.

59.8 in.,Lb1.396 in.

PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.

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PROBLEM 6.C3

A beam having the cross section shown is subjected to a vertical shear V. Write a computer program that, for loads and dimensions expressed in either SI or U.S. customary units, can be used to calculate the shearing stress along the line between any two adjacent rectangular areas forming the cross section. Use this program to solve (a) Prob. 6.10, (b) Prob. 6.12, (c) Prob. 6.22.

SOLUTION

1. Enter V and the number n of rectangles.

2. For 1i to n, enter the dimensions b and .i h i 3. Determine the area Aib hi i of each rectangle.

4. Determine the elevation of the centroid of each rectangle:

1 and the elevation y of the centroid of the entire section:

i i i

i i

yA y   A

 

   

 5. Determine the centroidal moment of inertia of the entire section:

3 2

The shearing stress on the surface between the rectangles i and i is 1

i i

PROBLEM 6.C3 (Continued)

Program Outputs

Problem 6.10

Shearing force 10 kN

75.000 mm

y above base

6 4

39.580 10 mm

I  

Between Elements 1 and 2:

 418.39 kPA

Between Elements 2 and 3:

 919.78 kPA (a)

Between Elements 3 and 4:

 765.03 kPA (b)

Between Elements 4 and 5:

 418.39 kPA

Problem 6.12

Shearing force 10 kips

2.000 in.

y

14.58 in4

I Between Elements 1 and 2:

 2.400 ksi

Between Elements 2 and 3:

 3.171 ksi (a)

Between Elements 3 and 4:

 2.400 ksi (b)

PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.

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1017

PROBLEM 6.C3 (Continued)

Program Outputs (Continued) Problem 6.22

Shearing force 90 kN

65.000 mm

y

6 4

58.133 10 mm

I  

Between Elements 1 and 2:

23.222 MPA (b)

Between Elements 2 and 3:

 30.963 MPA (a)

y1 y

x

y2

x2 x1 xn

PROBLEM 6.C4

A plate of uniform thickness t is bent as shown into a shape with a vertical plane of symmetry and is then used as a beam.

Write a computer program that, for loads and dimensions expressed in either SI or U.S. customary units, can be used to determine the distribution of shearing stresses caused by a vertical shear V. Use this program (a) to solve Prob. 6.47, (b) to find the shearing stress at a Point E for the shape and load of Prob. 6.50, assuming a thickness t14 in.

SOLUTION

For each element on the right-hand side, we compute (for i to n): 1 Length of element Li  (xixi1)2(yiyi1)2

Area of element AitLi

where 1

4 in.

t

Distance from x axis to centroid of element 1 1

( )

2

yiyiyi Distance from x axis to centroid of section:

i i

i

y A yA Note that yn  and that 0 xn1yn1 0.

Moment of inertia of section about centroidal axis:

2 2

1

2 1 ( ) ( )

i 12 i i i

I  A  yyyy  Computation of Q at Point P where stress is desired:

( )

i i

Q A yy where sum extends to the areas located between one end of section and Point P.

Shearing stress at P:

VQ

  It

PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.

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1019

PROBLEM 6.C4 (Continued)

Program Outputs Part (a):

0.5333 in4

I

max 2.02 ksi

  

1.800 ksiB  

Part (b):

22.27 in4

I

194.0 psiE  

x2 x1

y1

t2 t1

yn tn

e

y2

O

V y

x

PROBLEM 6.C5

The cross section of an extruded beam is symmetric with respect to the x axis and consists of several straight segments as shown. Write a computer program that, for loads and dimensions expressed in either SI or U.S. customary units, can be used to determine (a) the location of the shear center O, (b) the distribution of shearing stresses caused by a vertical force applied at O. Use this program to solve Prob. 6.70.

SOLUTION

In document Mechanics of Materials 7th Edition Beer Johnson Chapter 6 (Page 118-130)