Home Work #8
HW 8-Problem #1
The beam is subjected to a uniform load w. Determine the placement a of the supports so
that the shear stress in the beam is as small as possible. What is this stress?
HW 8-Problem #1
For 0<x<a
wx  v  0
Fy  0
wl
For a<x<(L-a)
Fy  0
wx  v 
For (L-a)<x<L
Fy  0
wx  v  2 (
w(
l
 a )  wa
wl
v
x
 wx
2
)0
v  wl  wx

b/2
b / 2

d /2
ydyd  
bd
8
0
a  l/4
w (l / 2  l / 4 )b ( d / 8 )
2
 
V
2
Q 
2
v   wx
0
2
wl
w
1
12
3
bd b

3 wl
8 bd
2
I 
1
12
bd
2
HW 8-Problem #2
The fence board weaves between the three smooth fixed posts. If the
posts remain along the same line, determine the maximum bending stress
in the board. The board has a width of 150mm and a thickness of 12mm.
E=12Gpa. Assume the displacement of each end of the board relative to
its center is 75mm.
HW 8-Problem #2
For 0<x<L/2
EIy  M ( x )  px / 2
''
EIy 
'
P
4
EIy 
P
12
Y=0 at x=0; C2=0
x=L, dy/dx=0; C1=-PL2/16
x  C1
2
x  C1 x  C 2
3
1
y( x) 
(
P
3
EI 12
P
At x=1.2m; y= -0.075 m
P
x 
 0 . 075 
16
(1 . 2 ) 
3
12
9
1
12
P=67.5 N
My
I
P
2
( 2 . 4 ) (1 . 2 )
16
(12 * 10 )(
 max  
2
L x)
 11 . 25 MPa
)( 0 . 15 )( 0 . 012 )
3
HW 8-Problem #3
Determine the equation of the deflection of the beam and specify the
slope at B and deflection at C. EI is constant. (please ignore x1, x2 and
x3).
HW 8-Problem #3
For 0<x<a
For a<x<2a
F V  0
 M o  Fx  M  0 M  M o  Fx  wax  1 . 5 wa 2
F  w( x  a)  v  0
v  w(2a  x)
 M o  Fx  w ( w  a )(
x9
) M  0
2
M   2 wa  2 wax  0 . 5 wx
2
For 0<x<a
EIy  M
For x=0 y=0; C2=0
''
2
EIy   1 . 5 wa x  0 . 5 wax  C 1
1
2 2
3
EIy   0 . 75 wa x  wax  C 1 x  C 2
6
'
2
For x=0 dy/dx=0; C1=0
EIy   0 . 75 wa x 
2
2
1
6
wax
3
2
HW 8-Problem #3
For a<x<2a
EIy  M
''
EIy  
'
EIy  
w
6
w
x  awx
3
x 
4
24
1
3
EIy   0 . 75 wa x 
2
1
 2 wa x  C 3
2
awx  wa x  C 3 x  C 4
3
For x=a EIy '   1 . 5 wa 2 x  0 . 5 wax
2
2
3

6
C3 
At point B x=2a
At point C x=a
wa
w

2
wax
2
2
x  awx
3
6
w
x 
4
24
1
C4  
6
 
dy

dx
y
7 wa
6 EI
7 wa
4
12 EI
3
 2 wa x  C 3
2
awx  wa x  C 3 x  C 4
3
3
3
2
wa
24
4
2
2
HW 8-Problem #4
The beam is supported by a pin at A, a spring having a stiffness k at B,
and a roller at C. Use method of superposition to determine the force the
spring exerts on the beam. EI is constant.
HW 8-Problem #4