G
Truss A
C
D
E
F
P (pin)
A
B
R (roller)
a. Simple planar double truss bridge, 2kT total external forces applied to bridge (assume forces centered
on bridge). All bridge members are L units long. Solve for all internal member forces and external
reaction forces.
Truss A: Fill out the table with your coefficients (left side of equation) and constants (algebraically “move” the constants to
the right side of the equations first!)
Eqn
1x
1y
2x
2y
3x
3y
4x
4y
5x
5y
A
B
C
D
E
F
G
R1x
R1y
R2x
R2y
Constant
1. How many unknowns are there for this problem?
2. You have 10 independent equations using the variables for this problem (5 node x 2 Fx/Fy equations per node). Are there
enough equations to find a consistent, independent solution for the system of equations?
3. If you have “extra” equations, what do you do with them? Also, note that there is a blank line at the bottom of the table.
Can you look at the problem and develop one more equation that relates the some or all of the vectors?
4. Use the table above to create an augmented matrix using the table above in your graphing
calculator. Note that the dimensions of an augmented matrix are always N x N+1. Basically you
have an N x N coefficient matrix augmented with the N x 1 constant matrix.
5. Use your calculator’s matrix functions to solve this problem (find the reduced row echelon
form of the augmented matrix). Write your solutions in the table to the right.
6. What do the solutions mathematically represent in terms of the vectors we used to create the
free body diagrams for this problem?
7. What is the physical meaning of the solutions? I.e., how does this help you design the bridge?
Table for problem 5 - 10
Solution T or C?
A
B
C
D
E
F
G
R1
R2
8. Do your answers make sense (meet your expectations)? Is there any symmetry as we expected?
9. How do you interpret a negative answer?
Table for problem 11
Solution T or C?
10. Write a T or a C in the third column to indicate if the member is under tension or
compression.
11. Make a simple change to the problem…double the load on your truss. How does this change
effect the table above, your augmented matrix on your TI, and your answers? Write your revised
answers in the table to the right.
12. Now try solving the problem using symmetry WITHOUT using matrices (we’ll do this
together in class using the next page).
A
B
C
D
E
F
G
R1
R2
13. Look at the next two truss examples, Truss B and Truss C. Is there an easy way to modify your results above to take the
minor changes into account? Could you do the same for trusses D and E? Do it! Then compare the results to the original truss
and make some generalizations (write some conclusions on the comparative truss diagram pages).
Using Symmetry for Truss A – can we solve the problem with less work?
What is the minimum number of node equations you need to solve the problem? Do it without using
matrices!!
G
Truss A
(using symmetry)
C
D
E
F
P (pin)
A
B
R (roller)
a. Simple planar double truss bridge, 2kT total external forces applied to bridge (assume forces centered
on bridge). All bridge members are L units long. Solve for all internal member forces and external
reaction forces.
G
Truss B
C
D
E
F
A
B
P (pin)
R (roller)
b. Simple planar double truss bridge, 2kT total external forces applied to bridge (assume = 0.6kT
downward, = = 0.2kT downward forces on bridge). All bridge members are L units long. Solve for
all internal member forces and external reaction forces.
Hint: Re-draw only those free body diagrams that changed from the original. Which equations must
change as well?
G
Truss C
C
D
E
F
P (pin)
A
B
R (roller)
c. Simple planar double truss bridge, 2kT total external forces applied to bridge (assume = 0.2kT
downward, = 0.7kT and = 0.1kT downward forces on bridge). All bridge members are L units long.
Solve for all internal member forces and external reaction forces.
Hint: Re-draw only those free body diagrams that changed from the original. Which equations must
change as well?
Eqn
1x
1y
2x
2y
3x
3y
4x
4y
5x
5y
A
B
C
D
E
F
G
R1x
R1y
R2x
R2y
Solve for all internal member forces using matrices and write solutions in table
to the write and on the comparative solutions diagrams.
Eqn
1x
1y
2x
2y
3x
3y
4x
4y
5x
5y
A
B
C
D
E
F
G
R1x
R1y
R2x
Constant
Solution T or C?
A
B
C
D
E
F
G
R1
R2
R2y
Solve for all internal member forces using matrices and write solutions in table
to the write and on the comparative solutions diagrams.
Constant
Solution T or C?
A
B
C
D
E
F
G
R1
R2
G
Truss D
C
D
A
E
F
B
P (pin)
R (roller)
d. Simple planar double truss bridge, 2kT total external forces applied to bridge (assume forces centered
on bridge). Horizontal bridge members are L units long, others are 2L units long. Solve for all internal
member forces and external reaction forces.
Hint: How have the free body diagrams have changed from the original? Is there an easy way to update
the table?
Truss E
C
P (pin)
D
A
G
E
F
B
R (roller)
e. Simple planar double truss bridge, 2kT total external forces applied to bridge (assume forces centered
on bridge). Horizontal bridge members are L units long; others are 2/3 L units long. Solve for all internal
member forces and external reaction forces.
Hint: How have the free body diagrams have changed from the original? Is there an easy way to update
the table?
Eqn
1x
1y
2x
2y
3x
3y
4x
4y
5x
5y
A
B
C
D
E
F
G
R1x
R1y
R2x
R2y
Solve for all internal member forces using matrices and write solutions in table
to the write and on the comparative solutions diagrams.
Eqn
1x
1y
2x
2y
3x
3y
4x
4y
5x
5y
A
B
C
D
E
F
G
R1x
R1y
R2x
Constant
Solution T or C?
A
B
C
D
E
F
G
R1
R2
R2y
Solve for all internal member forces using matrices and write solutions in table
to the write and on the comparative solutions diagrams.
Constant
Solution T or C?
A
B
C
D
E
F
G
R1
R2
Solution Comparisons for Trusses A, B, and C
G
Truss A
C
D
E
F
A
B
P (pin)
R (roller)
G
Truss B
C
D
E
F
A
B
P (pin)
R (roller)
G
Truss C
C
D
E
F
A
B
P (pin)
R (roller)
Conclusions:
Solution Comparisons for Trusses A, D, and E
G
Truss A
C
D
E
F
A
B
P (pin)
R (roller)
G
Truss D
C
D
A
E
F
B
P (pin)
R (roller)
Truss E
C
P (pin)
Conclusions:
D
A
G
E
F
B
R (roller)