Problem 4-a
a
P
A
45o
B
O
D
45o
C
The semicircular rod ABCD is
maintained in equilibrium by
the small wheel at D and the
rollers at B and C. Knowing
that a = 45o, determine the
reactions at B , C , and D.
Problem 4-a
a
P
A
45
B
Solving Problems on Your Own
O
o
D
45
o
C
The semicircular rod ABCD is
maintained in equilibrium by
the small wheel at D and the
rollers at B and C. Knowing
that a = 45o, determine the
reactions at B , C , and D.
1. Draw a free-body diagram of the body. This diagram shows
the body and all the forces acting on it.
Problem 4-a
a
P
A
45
B
Solving Problems on Your Own
O
o
D
45
o
C
The semicircular rod ABCD is
maintained in equilibrium by
the small wheel at D and the
rollers at B and C. Knowing
that a = 45o, determine the
reactions at B , C , and D.
2. Write equilibrium equations and solve for the unknowns.
For two-dimensional structure the three equations might be:
SFx = 0 SFy = 0
SMO = 0
where O is an arbitrary point in the plane of the structure
or
SFx = 0 SMA = 0 SMB = 0
where point B is such that line AB is not parallel to the y axis
or
SMA = 0 SMB = 0 SMC = 0
where the points A, B , and C do not lie in a straight line.
Problem 4-a Solution
a
P
A
45
B
O
o
D
45
o
Draw a free-body diagram
of the body.
C
P sina
P
A
P cosa
O
45o
B
D
45o
C
R
B/ 2
B
D
B/ 2
C/ 2
C
C/ 2
Problem 4-a Solution
P sina
P
A
P cosa
O
45o
B
Write three equilibrium
equations and solve
for the unknowns.
D
45o
C
R
B/ 2
B
D
C/ 2
C
B/ 2
C/ 2
+ S MO = 0: (P sina) R _ D (R) = 0
+ S Fx = 0:
P cosa + B/ 2
+ S Fy = 0:
_
_
D = P sina
C/ 2=0
(1)
(2)
P sina + B/ 2 + C / 2 _ P sina = 0
_
2P sina + B/ 2 + C / 2 = 0
(3)
P sina
P
A
P cosa
O
45o
C
R
B/ 2
(2) + (3)
D
45o
B
B
Problem 4-a Solution
D
B/ 2
C/ 2
C
C/ 2
P(cosa _ 2sina) + 2 B/ 2 = 0
2
B=
(2sina _ cosa) P
2
(2) _ (3)
(4)
P(cosa + 2sina) _ 2 C/ 2 = 0
2
C=
(2sina + cosa) P
2
(5)
P sina
P
A
P cosa
O
45o
B
B
EQ. (4) :
EQ. (5) :
EQ. (1) :
D
45o
C
R
B/ 2
B/ 2
Problem 4-a Solution
D
For a = 45o
C/ 2
C
sina = cosa = 1/ 2
C/ 2
2 2
B=
(
2
2
2 2
C=
(
2
2
D = P/ 2
_
_
1
1
)P= P;
2
2
1
3
)P= P;
2
2
1
B= P
2
C= 3 P
2
D = P/ 2
45o
45o