Project Statement:
The goal of this project was to design a truss that spans two piers A and B for a minimum
cost with the given constraints and loading.
Figure 1: Design Picture
Introduction:
A truss is
a structural system, made up of two force members, that is generally
economical and lightweight compared to the loads that it can hold. A truss must consist
of only straight force members joined at their ends to form a rigid frame. It must also
have join
ts that can be represented as pin connections and the centerline of each straight
member is assumed to intersect the center axis of the pin. Finally, a truss must carry all
external forces, and no moment, only at the joints. As a result of the requiremen
ts of a
truss, all members in the structure behave as two

force me
mbers.
The basic shape behind a truss is the triangle. The triangle is the simplest rigid structure,
meaning internally stable, that can be created with two

force members. A truss can e
asily
be analyzed to determine the force that each member is under due to a certain loading.
With the determined loading, the concurrent stress and desired area can be found to
adhere to a certain safety requirement. A two

force member can
be
under eithe
r tensile
or compressive forces. To find the tensile or compressive stresses in these members, the
simple equation
A
P
, where P is the applied load and A is the cross sectional area,
can be used to find the resulting stress in the mem
ber. Buckling can also occur only
when the member is in compression. The stress due to buckling is given by
A
P
cr
cr
where P
cr
is the critical load and A is the
cross sectional area. The critical load is
determined
by
2
2
L
EI
P
cr
, where E is the Modulus of Elasticity, I is the second moment
of area, and L is the length of the given member. The second moment of area can be
determined by
4
12
1
b
I
for a square cross section and
4
64
d
I
for a rectangul
ar cross
section. The area of the cross sections can be adjusted to meet a certain safety
requirement based on
allow
U
S
F
.
.
.
This truss design project called for truss that would cross a 36 ft long span. The supports
can be at the road level
or down up to 8 ft below the road level. The height below the
road level at the center is up to 12 ft. There is a choice between three met
als; aluminum,
brass, and steel (See table 5 in Appendix B for detailed material list).
The cross sections
of membe
rs can be either solid or hollow and square or round. The outside dimension of
cross sections must be in the increment of ½ inch and the thickness of hollow members
must be in the increment of ¼ inch. Each joint that is used costs $40.00. Members of the
truss must not yield if the member force is tensile or compressive and must not buckle if
the member force is compressive. All members must meet a factor of safety of at least 5.
The bridge must be symmetric about the mid span and it is assumed that the
re is a pin at
A and a roller at B, as shown in the above picture.
The trusses must be designed to hold the weight of the roadbed, 1,500 lb/ft, plus the
weight of two trucks that cross in the opposite direction, each weighing 72,000 lb. The
standardized
loads for the truss w
ere
determined and
are
shown in Figure 2.
Figure 2: Truss loads
This project requires a truss that will support the given loads with a factor of safety of at
least 5 for the lowes
t possible cost.
Approach:
Cost was dependent on weight and the number of joints, so to minimize the cost, the
weight and number of joints needed to be minimized. This was the goal of the initial
baseline design. Several truss bridge designs were researc
hed, and the design which
involved all equilateral t
riangles was selected because
an equilateral triangle is a very
rigid structure
. Also, for the first design, the cross sections were circular.
The first truss design (See Figure 1
and Table 1
in appendix
B
) adhered to all
design requirements; it met the factor of safety level of 5 and held all applied loads. The
design however, was really expensive with a cost of $2677.80. The high cost was a result
of four long compressive members that required a large
diameter to avoid buckling.
Some design changes include adding more nodes and shorter elements. This design also
went high above the road surface; this was changed for the next design by placing the
truss under the road surface and making it shorter. A
final design change was made by
varying the type of material used in separate members, depending on the required load
that the member must hold. For example, zero force members were made of aluminum
and hollow whi
le members with a high force were
made of
solid steel. After the initial
results, we found that our first design had some good aspects but there are design changes
to be made if the cost is to be minimized.
The second truss design (see Figure 2 and Table 2 in appendix
B
)
adhered to all
design re
quirements; it met the factor of safety level of 5 and held all applied loads. The
design however, was still quite expensive with a cost of $2211.88. The high cost was a
result of three long compressive members that require a large diameter to avoid buck
ling.
One design change made after this design was to change the height of the bridge to better
minimize the cost.
As seen in figure 2, t
here were several members whose factor of safety
was over 8. This occurred because the required diameter for the eleme
nt was just higher
than 3 in, so we had to round up to 3.5 in. This design went under the road surface, and
was changed again for the third design by placing the truss above the road surface and
using more of a triangular shape for the overall bridge inst
ead of the rectangular shape in
this design. This reduced the number of members used in the design and, consequently,
reduced the cost. A final design change can be made by varying the type of material used
in separate members, depending on the required lo
ad that the member must hold.
This
design used
aluminum for zero force members and steel for everything else.
The third truss design
(see Figure 3 and Table 3
in appendix B
)
adhered to all
design requirements; it met the factor of safety level of 5 and he
ld all applied loads. The
design however, is the most expensive with a cost of $3576.54. The high cost is a result
of long compressive members that require a large diameter to avoid buckling. Almost all
forces in this design are in compression and there
fore need to be made thick to avoid
buckling. One design change that was made was to
change the height of the bridge to
b
etter minimize the cost. As seen in figure 3, there were a few members whose factor of
safety was
over 8. This occurred because the re
quired diameter for the element was just
higher than 1/2 in. increment so we had to round up to the next available size. This design
went under the road surface; this could possibly be changed by placing the truss above
the road surface
, which was done for
the final design.
A final design change was
made by
varying the
cross sections of
separate members, depending on the required load that the
member must hold. This design uses steel for all members.
The final design was the combination of the design chan
ges made from the first
three designs.
To minimize cost, the truss was made much shorter than in previous
designs; only 4 feet tall. It was also determined that aluminum was the least expensive
material to be used. This was determined by minimizing the
ratio
E
c
for compressive
buckling forces and
c
for tensile forces. The derivations of these equations are shown
in Table 6 of A
ppendix
B
.
The final design took different aspects from the first iterations
to impr
ove each and make the le
ast expensive possible bridge.
The X design in the center
of the structure was taken from the second design (Figure 4), as it made the bridge
symmetric about the center as required. Also, triangular outer sections were taken from
de
sign 1 (Figure 3) to reduce the number of joints used.
Iteration Summaries
Figure 3: Design 1 Plot
Table 1: Iteration #1 Summary
Iteration #1
Number of Joints = 7 x $40 = Joint cost = $ 280
Total cost = Total material cost + Joint cost = $ 2677.80
Total Weight = 2349.444 lb
Figure 4:
Design 2 Plot
Table 2: Iteration #2 Summary
Iteration #2
Number of Joints = 8 x $40 = Joint cost = $ 320
Total cost = Total material cost + Joint cost = $ 2211.80
Total Weight = 1859.851
Figure 5: Desi
gn
3
Plot
Table 3: Iteration #3 Summary
Iteration #3
Number of Joints = 8 x $40 = Joint cost = $ 320
Total cost = Total material cost + Joint cost = $ 3576.54
Total Weight = 3190.86
Final Results
Figure 6: Final Design Plot
Table 4: Final Desig
n Iteration Summary
Final Iteration
Number of Joints = 6 x $40 = Joint cost = $ 240
Total cost = Total material cost + Joint cost = $ 1117.35
Total Weight = 1381.58319lb
The final truss design
, shown in figure 6,
adhered to all design requirements; it
me
t the factor of safety level of 5 and held all applied loads. The design is also the least
expensive with a cost of $1117.35. The lower cost is a result of making a shorter truss
and thus making shorter members. This requires less material to be used.
There were
also very few compressive forces in this design so it required very few thick members.
This design also implemented hollow cross sections of the members so the factor of
safety for all members came very close to the requirement of 5 and there w
as less wasted
material.
Verification of the Ansys results was done using the Method of Joints. This
design was statically indeterminate
because there were more forces than there were
equations. Using the method of joints, there are only 2(nodes)
–
3 eq
uations. This design
had 10 member forces and only 9 equations. The Method of Joints was used to find the
forces on nodes 2 and 4 , and the forces at the other nodes were verified by using the
forces from Ansys to ensure that the static equations were equa
l to zero (see appendix C
for
force
calculations
)
.
Table
5: Force Comparison
ELEMENT #
ANSYS FORCES (lbs)
CALCULATED FORCES (lbs)
1
81000
81000
2
79894
79894
3
81000
81000.09486
4

85381

85381.49682
5

82106

82106
6

85381

85381.49682
7
26631
26
631
8
1165.6
1165.6
9
1165.6
1165.6
10
26631
26631
Table
6: Stress Comparison
ELEMENT #
ANSYS STRESSES (psi)
CALCULATED STRESSES(psi)
1
10125
10125
2
11836
11836.14815
3
10125
10125
4

3595

3594.989474
5

4105.3

4105.3
6

3595

3594.989474
7
11836
11836
8
4662.5
4662.4
9
4662.5
4662.4
10
11836
11836
The final design used aluminum for all members because it was found to be the
cheapest material to use. Another design change that was implemented on this design
was the use of square cros
s sections. It was found that square cross sections used the
least amount of mater
ial to avoid buckling because the area moment of inertia was found
to be larger for the hollow square cross section than for the circular cross section
. This
design took as
pects from the other designs to create the most economical and strongest
possible truss.
Conclusions
Three test designs were submitted during the design process of the truss. As these trusses
were designed, it was determined what aspects made the stron
gest and least expensive
bridge. It was found that the cost of the final design was minimized at a height of only 4
feet. The final design used the least amount of material possible and the least amount of
joints, since each joint costs $40.00. It was a
lso determined that aluminum was the
cheapest material to use and square cross sections created a stronger member while using
the least amount of material. The final design also implemented hollow members. With
hollow members, it was possible to adhere t
o the safety requirement of 5 while using the
least amount of material. The final truss design came to a total cost of $1117.35. It used
6 joints and came to a total weight of 1381.58319 lbs. This was the least expensive of all
the designs created
and m
et all the requirements of the project.
Appendix A: An
s
y
s Details of the final Design
Figure 1: Ansys Element Plot
Table 1: Nodal Coordinates
NODE X Y
Z
1 0.00000000000 0.00000000000 0.00000000000
2 144.000000000 0.00000000000 0.00000000000
3 288.000000000 0.00000000000 0.00000000000
4 432.000000000 0.00000000000 0.00000000000
5
288.000000000 48.0000000000 0.00000000000
6 144.000000000 48.0000000000 0.00000000000
Table 2: Element Connectivity
ELEM MAT TYP REL ESY SEC NODES
1
1 1 2 0 1 1 2
2 1 1 3 0 1 2 3
3 1 1
2 0 1 3 4
4 1 1 4 0 1 5 4
5 1 1 5
0 1 6 5
6 1 1 4 0 1 1 6
7 1 1 6 0
1 6 2
8 1 1 7 0 1 6 3
9 1 1 7 0 1 2
5
10 1 1 6 0 1 5 3
Table 3: Element Stresses and Forces for Final Design
STAT
CURRENT
CURRENT
ELEM
AXFORCE
(lbs)
AXSTRESS
(psi)
1
81000.
10125.
2
79894.
11836.
3
81000.
10125.
4

85381.

3595.0
5

82106.

4105.3
6

85381.

3595.0
7
26631.
11836.
8
1165.6
4662.5
9
1165.6
4662.5
10
26631.
11836
.
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