Transcript Math for Truss Calculation

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Math for Truss Calculation
Trusses
Trusses are structures made up of beams joined together at
their endpoints.
Solve for the force in each member of the truss in this example
to find whether the members are in tension or compression.
Start be replacing the supports with reaction forces:
500
N
500
N
B
2m
45
A
C
2m
A
X
AY
CY
Trusses
This truss consists of:
•2 unknown member forces and 1 unknown external force at joint B
•2 unknown member forces and 2 unknown reaction forces at joint A
•2 unknown member forces and 1 unknown reaction force at joint C
Draw the free-body diagrams for each pin:
500
N
500 N
A
A
X
X
AY
CY
AY
CY
Trusses
These three separate free-body diagrams assume that:
•Member AB is in tension
•Member BC is in compression
•Member AC is in tension
If these assumptions are incorrect, then our solution will show
a negative quantity for force.
500 N
500 N
FAB FBC
FAB
AX
AX
AY
CY
FBC
FAC FAC
AY
CY
Trusses
To really “see” that
•Member AB is in tension
•Member BC is in
compression
•Member AC is in tension
We must look at the
free-body diagrams of
the beams, which
show the effects of the
pins on the beams.
Trusses
Before we can solve for the forces, we must break FBC into its
x and y components:
FBCY
FBC
45
FBCX
FBCX = FBC * cos (45)
FBCY = FBC * sin (45)
Trusses
+
FX = 0 ;
The following is the free-body
diagram of joint B. The force
FBC has been replaced with
its x and y components:
500N – FBCcos(45) = 0
FBCcos(45) = 500N
cos(45)
cos(45)
FBC = 707.1 N (C)
FBC sin(45)
+FY = 0 ;
FBCsin(45) – FAB = 0
FBC cos(45)
500 N
FAB
707.1*sin(45) – FAB = 0
FAB = 707.1*sin(45)
FAB = 500 N (T)
Trusses
+
FX = 0 ;
The (T) indicates tension
500N – FBCcos(45) = 0
and the (C) indicates
FBCcos(45) = 500N
compression. Both
solutions were positive
cos(45)
cos(45)
therefore our initially
FBC = 707.1 N (C)
assumptions were correct.
(If the solution had been a +F = 0 ;
Y
negative number, then we
FBCsin(45) – FAB = 0
would simply reverse our
assumption from tension to
707.1*sin(45) – FAB = 0
compression or vice versa)
F = 707.1*sin(45)
AB
FAB = 500 N (T)
Trusses
+
FX = 0 ;
The following is the freebody diagram of joint C.
The force FBC has been
replaced with its x and y
components:
– FAC+ FBC cos (45) = 0
– FAC+ 707.1 cos (45) = 0
FAC = 707.1 cos(45)
FAC = 500 N (T)
+FY = 0 ;
FBC
45
CY – FBC *sin(45) = 0
FAC
CY – 707.1 *sin(45) = 0
CY
CY = 500 N
Trusses
+
FX = 0 ;
The following is the
free-body diagram of
joint A:
FAC - Ax = 0
500 N - Ax = 0
AX = 500 N
+FY = 0 ;
FAB
FAB - AY = 0
AX
FAC
AY
500 N - AY = 0
AY = 500 N
Trusses
Solved!!!
500 N
500 N
B
2m
45
A
C
2m
500 N
500 N (T)
500 N
500 N