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

*I**h* *h*

*h*

Vertical plate: 1 ^{3} ^{4}

(0.5)(5) 5.2083 in

*v* 12

*I*

Whole section: *I*4*I** _{h}*

*I*

*9*

_{v}*h*

^{2}5.39583 in

^{4}For one horizontal plate,

*Q*(4.5)(0.5)

*h*2.25 in

*h*

^{3}

2

2.25 9 5.39583

*VQ* *Vh*

*q* *I* *h*

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.

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

Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
**on a website, in whole or part. **

**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.

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

Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
**on a website, in whole or part. **

**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 L*0 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

*V* *V*

*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. **

Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
**on a website, in whole or part. **

**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) 0*P* and *w*12.5 lb/in., (c) 500 lb*P* 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.,*L* *b*1.250 in. 70.3 in.,*L* *b*1.172 in.

For *P*500 lb, *w*12.5 lb/in.

Increment 0.0010 in.

59.8 in.,*L* *b*1.396 in.

**on a website, in whole or part. **

**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 1*i to n, enter the dimensions b and .*_{i}*h ** _{i}*
3. Determine the area

*A*

**

_{i}*b h*

*of each rectangle.*

_{i i}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*

*y* *A 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) *

**on a website, in whole or part. **

**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)

*y*_{1}
*y*

*x*

*y*_{2}

*x*_{2}
*x*_{1}
*x*_{n}

**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 *t*^{1}_{4} in.

**SOLUTION **

For each element on the right-hand side, we compute (for *i to n): *1
Length of element *L** _{i}* (

*x*

**

_{i}*x*

_{i}_{}

_{1})

^{2}(

*y*

**

_{i}*y*

_{i}_{}

_{1})

^{2}

Area of element *A** _{i}*

*tL*

_{i}where 1

4 in.

*t*

Distance from x axis to centroid of element 1 _{1}

( )

2 ^{}

*y** _{i}*

*y*

**

_{i}*y*

*Distance from x axis to centroid of section:*

_{i}###

*i i*

###

*i*

*y* *A y* *A*
Note that *y** _{n}* and that 0

*x*

_{n}_{}

_{1}

*y*

_{n}_{}

_{1} 0.

Moment of inertia of section about centroidal axis:

2 2

1

2 1 ( ) ( )

*i* 12 *i* *i* *i*

*I* *A* *y* *y*_{} *y* *y*
Computation of Q at Point P where stress is desired:

( )

*i* *i*

*Q* *A y* *y* where sum extends to the areas located between one end of section and Point P.

Shearing stress at P:

*VQ*

*It*

**on a website, in whole or part. **

**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}*

*x*_{2}
*x*_{1}

*y*_{1}

*t*_{2} *t*_{1}

*y*_{n}*t*_{n}

*e*

*y*_{2}

*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 **