Structural Analysis of Bridge Gusset Plates:
Steel vs. Composite
by
Stephen Ganz
An Engineering Report Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
In Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
August, 2012
i
© Copyright 2012
by
Stephen Ganz
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
TERMINOLOGY / LIST OF SYMBOLS / ACRONYMNS .......................................... vii
ACKNOWLEDGMENT ................................................................................................ viii
ABSTRACT ..................................................................................................................... ix
1. Introduction.................................................................................................................. 1
2. Methodology ................................................................................................................ 2
3. Bridge Design .............................................................................................................. 4
4. Loading ........................................................................................................................ 5
4.1
Dead Load .......................................................................................................... 5
4.2
Live Load ........................................................................................................... 5
4.3
Total Load (W) ................................................................................................... 5
5. Materials ...................................................................................................................... 6
5.1
Carbon Steel ....................................................................................................... 6
5.2
HexPly 8552 IM7 ............................................................................................... 7
6. FEA Models ............................................................................................................... 11
6.1
6.2
Plate Geometry ................................................................................................. 16
6.1.1
Bottom End Plates ................................................................................ 16
6.1.2
Mid Span Plates ................................................................................... 17
6.1.3
Upper End Plates .................................................................................. 18
Truss Geometry ................................................................................................ 18
7. Results........................................................................................................................ 19
7.1
Carbon Steel Plates .......................................................................................... 19
7.2
HexPly 8552 IM7 Plates .................................................................................. 21
7.2.1
HexPly 8552 IM7 [0 90]S .................................................................... 22
7.2.2
HexPly 8552 IM7 [0 45 90]S ............................................................... 24
iii
7.2.3
HexPly 8552 IM7 [0 15 30 45 60 75 90]S ........................................... 26
7.3
Factors of Safety .............................................................................................. 29
7.4
Deflections ....................................................................................................... 30
8. Conclusions................................................................................................................ 31
9. References.................................................................................................................. 32
10. Appendices ................................................................................................................ 34
iv
LIST OF TABLES
Table 1: Carbon Steel Properties ..................................................................................... 6
Table 2: HexPly 8552 IM7 Properties ............................................................................. 7
Table 3: Composite Layup Arrangements ....................................................................... 7
Table 4: Factors of Safety ................................................................................................ 30
Table 5: Deflections ......................................................................................................... 30
v
LIST OF FIGURES
Figure 1: Side and Plan views of the Bridge ................................................................... 4
Figure 2: Bridge Free Body Diagram .............................................................................. 5
Figure 3: Abaqus Material Editor for A36 Carbon Steel ................................................. 6
Figure 4: Abaqus Material Editor for HexPly 8552 IM7 ................................................ 8
Figure 5: Abaqus Fail Stress for HexPly 8552 IM7 ........................................................ 8
Figure 6: Abaqus Section Editor for HexPly 8552 IM7 [0 90]S ..................................... 9
Figure 7: Abaqus Section Editor for HexPly 8552 IM7 [0 45 90]S ................................ 10
Figure 8: Abaqus Section Editor for HexPly 8552 IM7 [0 15 30 45 60 75 90]S ............ 10
Figure 9: FEA Bridge showing loads and boundary conditions ...................................... 11
Figure 10: Iso view of a joint with shell thicknesses rendered ........................................ 11
Figure 11: Abaqus model showing all loads and constraints .......................................... 12
Figure 12: Abaqus model with z-constraint (U3) suppressed for clarity......................... 12
Figures 13a, b: Loading applied to the lower mid-span plates ........................................ 13
Figures 14a, b, c, d: Bottom end plate boundary conditions ........................................... 14
Figures 15a, b, c: Side view of bridge and close up showing tie constraints .................. 15
Figure 16: Bottom End Plate Detail Drawing (Plates A and L) ...................................... 16
Figures 17a, b: Mid Span Plate Detail Drawings (Plates B, D, E, F, G, H, I and J)........ 17
Figure 18: Upper End Plate Detail Drawings (Plates C and K) ...................................... 18
Figure 19: Abaqus FEA Von Mises Stress results for A36 Carbon Steel Model ............ 19
Figure 20: Abaqus FEA Deflection results for A36 Carbon Steel Model ....................... 20
Figure 21: Abaqus FEA Tsai-Wu results for HexPly [0 90]S Model.............................. 22
Figure 22: Abaqus FEA Deflection results for HexPly [0 90]S Model ........................... 23
Figure 23: Abaqus FEA Tsai-Wu results for HexPly [0 45 90]S Model......................... 24
Figure 24: Abaqus FEA Deflection results for HexPly [0 45 90]S Model ...................... 25
Figure 25: Abaqus FEA Tsai-Wu results for HexPly [0 15 30 45 60 75 90]S Model..... 27
Figure 26: Abaqus FEA Deflection results for HexPly [0 15 30 45 60 75 90]S Model .. 28
Figure 27: Tsai-Wu Equation .......................................................................................... 29
vi
TERMINOLOGY / LIST OF SYMBOLS / ACRONYMNS
FEA – Finite Element Analysis
FBD – Free Body Diagram
2D – 2 Dimensions
DOT – Department of Transportation
NTSB – National Transportation Safety Board
FHWA – Federal Highway Administration
E – Modulus of Elasticity (Msi)
G – Modulus of Rigidity (Msi)
ν – Poisson’s Ratio
ρ – Density (lbf/in3)
tp – Ply Thickness (in)
YS – Yield Strength (ksi)
UTS – Ultimate Tensile Strength (ksi)
σ1t – Tensile strength in the 1 (longitudinal) direction (ksi)
σ1c – Compressive strength in the 1 (longitudinal) direction (ksi)
σ2t – Tensile strength in the 2 (transverse) direction (ksi)
σ2c – Tensile strength in the 2 (transverse) direction (ksi)
τ12f – Shear Strength (ksi)
[Orientation
number of plies]S
– Laminate Layup which is characterized by ply orientation,
number of plies and symmetry about the mid-plane (S, if applicable).
Abaqus – Computer Software used to perform modeling and FEA
Isotropic – Same properties in all directions
Orthotropic – Different properties in different directions
TSAIW – An abbreviation for Tsai-Wu Abaqus uses
vii
ACKNOWLEDGMENT
I would like to thank Professor Ernesto Gutierrez-Miravete for his guidance throughout
master’s project. I would also like to thank Professor David Hufner for his guidance on
finite element analysis of composite materials using Abaqus. And I would also like to
thank all the other teachers and professors from Washingtonville High School, SUNY
Binghamton and RPI for sharing their knowledge.
viii
ABSTRACT
A structural comparison of gusset plates for a single span Warren Truss bridge was
performed using Abaqus/CAE for different materials. The two materials chosen are A36
carbon structural steel and HexPly brand 8552 IM7 prepreg composite. Four bridge
models were analyzed; one consisting of steel plates and three models with composite
plates of varying ply orientations. Performance of the two materials was evaluated
based on failure margin and deflection. The goal here is to compare the performance
differences in metallic and composite material to determine their acceptability for use in
bridge construction.
ix
1. Introduction
Bridges play a vital role in transportation networks the world over. Spanning up to
several thousand feet in length and towering up to several hundred feet high, these manmade engineering marvels allow safe, convenient passage for people and their cargo.
Although declining in popularity in new construction, many bridges still in service are
based on truss designs to efficiently transmit load back to their foundations. Warren
Truss bridges with verticals have appeared as early as the mid-1880’s and remained
popular in highway construction throughout the 20th century [1]. Analysis of their gusset
plates becomes ever more critical as these structures are nearing the end of their design
life.
Gusset plates are integral to a truss-based bridge because they serve as the attachment
point for the truss members.
Gusset plates have become the focal point of much
research since the collapse of the I-35W Bridge in Minneappolis, MN in 2007, in which
the National Transportation Safety Board (NTSB) reports that the probable cause is due
to inadequate plate design [2]. This prompted the Federal Highway Administration
(FHWA) to advise an immediate re-inspection of steel truss bridges across the United
States [3].
For this project, a structural comparison of bridge gusset plates of different materials
was performed. Loading is assumed to be in-plane (2 dimensional). The gusset plates
were modeled and analyzed in Finite Element Analysis (FEA) software, Abaqus/CAE.
Performance criterion was based on failure margin and deflection.
1
2. Methodology
In order to perform a comparative structural analysis for steel and composite gusset
plates, a Warren Truss bridge was constructed to state and federal guidelines. Loads
were determined based on the bridge’s design and load carrying requirements as shown
in Appendix A. The computer generated model used in FEA represents the vertical
section of either side of the bridge consisting of plates, trusses, loads and boundary
conditions. Bridge components are created using shell elements. The trusses are
connected to the plates with tie constraints to simulate a weld. The trusses have a simple,
robust geometry with a coarse mesh. Assigning these a coarse mesh is acceptable
because this project is not concerned with any analysis of the trusses themselves and
because their mesh refinement has no effect on the results for the plates. Other more
advanced analyses simulate the entire bridge in much greater detail such as in [2], but
this was part of an investigation into the failure of the I-35W Bridge. The model shown
in [2] is a global-local model where the joints of interest are two highly detailed solid
models with extremely dense mesh refinement embedded into a bridge model
constructed of beam elements, an analysis such as this is beyond the capabilities of the
resourses which I have access to and for the purposes this project, this high level of
complexity is not warranted.
The approach taken for this master’s project is similar to the analsysis shown in [4] in
which only the vertical truss side of a bridge was modeled. One small difference is [4]
uses beam elements to represent the trusses instead of shell elements. Gusset plates for
this mater’s project are sized based on dimensions shown in [4] which are typical for a
railroad bridge and are assumed to be a close enough approximation for the gusset plates
of a highway bridge. Since the goal is to compare material performance through
structural analysis of identical items, the size of the gusset plates is inconsequential.
What does matter is that the gusset plates are the same size from one model to another
(steel and composite models).
Loading will come from dead load (bridge weight) and live load (vehicles, snow) based
on building requirements in accordance with DOT rules and regulations [5, 6]. Detailed
2
calculations of these loads are provided in Appendix A. Transverse forces (wind) will
be ignored since these plates are not significantly loaded in the transverse direction, the
lateral members in the plane of the top and bottom chords resist wind loads [1]. This also
permits the use of shell elements for 2D analysis. Failure of these joints is more
commonly associated with tensile and buckling failure as shown in [2]. This will
provide enough information to make a structural comparison.
A mesh study summarized in Appendix B ensured accuracy of the steel and composite
bridge models. Stresses and deflections resulting from the final runs were compared to
draw conclusions about the different materials.
There are four total models being analyzed, one with steel plates and three with
composite plates of different layups. This was done to determine if increasing the
number of different orientations within the same material thickness improves its
performance by possibly increasing isotropic behavior. For consistency, the plates in all
models are 2 inches thick and the trusses in all models are made from A36 steel, because
this project is only concerned with the performance of the plates.
Once results from FEA are obtained, factors of safety are calculated based on failure
criterion. Therefore, the factors of safety for the steel model are calculated to be
Ultimate Tensile Strength (UTS) divided by the maximum Von-Mises stress. Factors of
safety for the composite models are calculated to the inverse of the maximum Tsai-Wu
value (1/Tsai-Wu). A qualitative comparison of deflections is also provided.
3
3. Bridge Design
The bridge chosen for this project is a single span Warren Truss with verticals. This
highway bridge does not actually exist, but has been constructed using state and federal
guidelines. Figure 1 below details the overall dimensions of the bridge which is assumed
to be 120' long and 20' high with angular members set at 45°. All of the vertical beams in
the Warren Truss and horizontal joists are assumed to have a cross section of 64in2 [6]
and are made from carbon structural steel that is 0.282 lb/in3 in density [7]. The clear
minimum width of the roadway for a bridge maintained on a state highway in
Connecticut is 28’ [5] and assumed to be 1’ thick. On either side of the bridge there is a
5’ wide sidwalk that is 6” thick [5]. Therefore, total bridge width is 38’. The two sides
of the bridge are connected by seven floor joists at the bottom, to support the roadway
and five joists connect the bridge at the top.
20’
6x 20’
120’
38’
Figure 1: Side and Plan views of the Bridge
4
4. Loading
In the case of bridges, loading comes from three major components: dead load
(structure), dynamic load such as wind and live load (vehicles and snow). For the
purposes of this project only dead load and live load will be considered.
4.1 Dead Load
Dead load is comprised of the weight of the Warren Truss section, sidewalks, asphalt,
roadway and floor joists. Based on the dimensions listed in Section 3 and calculations
detailed in Appendix A dead load has been determined to be 297,201 lbs.
4.2 Live Load
Live load is comprised of weight of passing vehicles and snow. Appendix A details the
calculations for live load and has been determined to be 279,435 lbs.
4.3 Total Load (W)
The combination of Dead Load and Live Load results in a Total Load (W) of 576,636
lbs. For the purposes of this project the load is assumed to be distributed evenly, so onefifth of the total load is applied at the joints as shown below. Figure 2 below shows how
the bridge is assumed to be loaded. The load W/5 is applied to each plate as shown as a
surface traction equal to 1153 psi.
C
E
G
I
K
A
B
D
F
H
J
L
R1
W
5
W
5
W
5
W
5
W
5
R2
Figure 2: Bridge Free Body Diagram
5
5. Materials
5.1 Carbon Steel
Carbon steel was chosen as the baseline material for the gusset plates because of its
widespread use in structural applications. It’s relatively low cost, ease of machining and
weld-ability makes it a popular choice as a building material. However, it requires
regular maintenance and inspection since it is subject to corrosion and must be protected
from the elements with paint. Table 1 lists the material properties and Figure 3 shows
how they were defined in Abaqus/CAE. The gusset plates in all models are 2 inches
thick. This material is used for the plates in the “steel” model and the trusses for all
models.
Table 1: Carbon Steel Properties
Property [7, 8]
E (Msi)
G (Msi)
ν
ρ (lb/in3)
YS (ksi)
UTS (ksi)
Figure 3: Abaqus Material Editor for A36 Carbon Steel
6
Value
30.0
11.5
0.292
0.282
36 min
58-80
5.2 HexPly 8552 IM7
The material chosen for the composite plates is the same as the composite of choice as
determined in [6] and Table 2 lists the mechanical properties of HexPly 8552 IM7.
Three different layups were chosen to determine if there is a significant performance
advantage that comes with varying fiber orientation (i.e. increasingly isotropic behavior)
and this will help hone in on the best performing composite layup. The plies were kept
symmetric about the mid-plane and the number of plies (or thickness of each orientation)
was kept equal per layer. This was done because results are the same in layers of the
same orientation at the same distance from the mid-plane and therefore reduces the
amount of time to evaluate results. Table 3 below shows the number of plies and
thickness for each orientation and verifies that total thickness was held to 2 inches for all
composite plates. Figures 4 and 5 show how the elastic and failure properties were
entered in Abaqus/CAE.
Table 2: HexPly 8552 IM7 Properties
Property
E1 (Msi)
E2 (Msi)
E3 (Msi)
ν12
ν13
ν23
G12 (Msi)
G13 (Msi)
G23 (Msi)
σ1t (ksi)
σ1c (ksi)
σ2t (ksi)
σ2c (ksi)
τ12f (ksi)
ρ (lb/in3)
tp (in)
Value
23.8
1.7
1.7
0.32
0.32
0.0229
0.75
0.75
0.831
395
-245
16.1
-32.3
17.4
0.047
0.006
Table 3: Composite Layup Arrangements
Layups
Thickness
Total Thickness
[083,9083]S
0.50 per oreintation
2.0
[056,4556,9056]S
0.333 per orientation
2.0
[024,1524,3024,4524,6024,7524,9024]S
0.143 per orientation
2.0
7
Figure 4: Abaqus Material Editor for HexPly 8552 IM7. The required elastic
values from Table 2 are entered here.
Figure 5: Abaqus Fail Stress for HexPly 8552 IM7. The user defines the
longitudinal and transverse tensile and compressive strengths as well as
shear strength and the cross-prod term coeff (f*) which is necessary for the
F12 term in the Tsai-Wu equation [9, 10]. The stress limit is not required,
Abaqus will use f* instead [10].
8
Figures 6, 7 and 8 show how the layups were defined in Abaqus. Abaqus requires a
thickness for each orientation rather than number of plies times a ply thickness. The
thickness or each orientation was kept so that the total thickness of the composite was 2
inches (the same as steel). The default number of integration points (3) was used.
Figure 6: Abaqus Section Editor for HexPly 8552 IM7 [0 90]S
9
Figure 7: Abaqus Section Editor for HexPly 8552 IM7 [0 45 90]S
Figure 8: Abaqus Section Editor for HexPly 8552 IM7 [0 15 30 45 60 75 90]S
10
6. FEA Models
As shown in Figure 9, the models used for analysis are a complete vertical side section
of a Warren Truss bridge consisting of plates, trusses, loads and boundary conditions.
Loads were applied to partitioned sections of the plate’s surface as surface tractions
which are defined as force per unit area (psi); refer to Figure 13 to see how these loads
were defined.
This reduces the chance of unusually high stress concentrations
commonly associated with point loads which can adversely affect the accuracy of the
model.
C
G
E
I
y (U2)
K
i
A
Pinned End
D
B
W
5
F
W
5
x (U1)
J
H
W
5
W
5
L
W
5
Roller End
Figure 9: FEA Bridge showing loads and boundary conditions. Results were recorded
only for plates which are circled.
Plates and trusses were constructed with shell elements and meshed with hex elements
following the guidance provided in [10] for creating composite sections using shell
elements. An iso-view of the plates and trusses with thicknesses rendered is shown in
Figure 10. Shell elements are appropriate for a 2D analysis and the use of hex elements
provides more accurate results than triangular elements.
Figure 10: Iso view of a joint with
shell thicknesses rendered
11
The Free Body Diagram (FBD) for the bridge is shown below. The entire bridge was
constrained in the out-of-plane (U3, z-direction) to keep the analysis 2D, this is
illustrated in Figures 11 and 12. The load W/5 is applied to each plate as shown as a
surface traction equal to 1153 psi. This is calculated in Appendix A and summarized in
Section 4 and shown in Figure 13.
y (U2)
x (U1)
Figure 11: Abaqus model showing all loads and constraints.
y (U2)
x (U1)
Figure 12: Abaqus model with z-constraint (U3) suppressed for clarity.
12
Surface traction
Figures 13a, b: Loading applied
to the lower mid-span plates.
Loading
was
applied
as
a
surface traction to the center
square section of plates B, D, F,
H and J.
Surface traction load (psi)
calculated in Appendix A.
13
The two end plates were assigned boundary conditions as follows: the bottom edge of
plate A is constrained in the x and y directions (U2, vertical and U1, horizontal) and the
edge of plate L is constrained in the y direction (U2, vertical). Rotational degrees of
freedom were left unconstrained to simulate a simple support condition.
L
A
Figureds 14a, b, c, d: Bottom end plate boundary conditions.
14
To simulate a welded joint, tie contraints were assigned at the mating surfaces between
the plates and trusses as shown in Figure 15. The master and slave surfaces were
selected to be the truss and plate surfaces respectively.
Figures 15a, b, c: Side view of
bridge and close up showing tie
constraints.
15
6.1 Plate Geometry
For this bridge model there are 3 major types of plates: the bottom ends, the top ends and
the mid-span plates. Figures 16, 17 and 18 shown below are the detail drawings for all
the plates. It can be seen that all the plates are based off of the mid-span plate design,
where the top and bottom end plates are basically modified mid-span plates. And the
mid-span plates are based on the dimensions of plates shown in [4]. All of the plates
could have been kept the same for simplicity, but this was avoided to minimize the total
amount of elements in the models. Also shown in the detail drawings is the geometry for
the surface partitions used for tie constraints with the truss members and for the
application of surface traction loads.
All dimensions shown are in inches unless
otherwise specified and as previously mentioned, all plates are 2 inches thick.
6.1.1
Bottom End Plates
These are the plates at the bottom corners on each side of the bridge. They connect only
two trusses and are assigned boundary conditions (described in Section 6) along their
bottom edges. They are basically the same as the mid-span plates, but are cut in half
down their vertical centerline. This reduces the number of mesh elements and cuts down
on time to solve the model.
40
22
2x 20
2x 10
45°
5
10
45
Figure 16: Bottom End Plate Detail Drawing (Plates A and L)
16
6.1.2
Mid Span Plates
These are the plates that are used everywhere on the bridge except the ends. They are the
largest and also connect the most truss members. There are two sub-types of these
plates, both are identical in external geometry, the only difference is the number of
partitioned surfaces to overlap with a corresponding number of trusses. Both plates are
symmetrical about the vertical centerline.
5
40
22
5x 20
10
45°
10
5x 10
10
90
Figures 17a, b: Mid Span Plate Detail Drawings (Plates B, D, E, F, G, H, I and J)
17
6.1.3
Upper End Plates
These are the plates located at the upper corners at each end of the bridge. They are
basically the same as the mid-pan plates that connect to 5 trusses, but with a corner cut
off to only connect to 4 trusses.
Again, this was done to eliminate unnecessary
computation of non-load bearing structure.
5
10
40
22
4x 20
45°
4x 10
45°
10
10
10
45
Figure 18: Upper End Plate Detail Drawings (Plates C and K)
6.2 Truss Geometry
The horizontal, vertical and diagonal trusses were sized to span the gaps between plates
while achieving perfect overlap with the partitioned surfaces (which are assigned tie
constraints described in Section 5) and to maintain the overall dimensions of the bridge
described in Section 3. This means that the horizontal trusses are 190”, the vertical
trusses are 200” and the diagonal trusses are 295.411” long. All of the trusses were
modeled to have a solid 12” cross section and the reason for assigning them such a
robust geometry is to reduce error due associated with excessive bending or buckling
and also limit their deflections in general. Therefore, overall deflections of the model
are due to the cumulative deflections of the plates.
18
7. Results
7.1 Carbon Steel Plates
Performing a finite element analysis on a bridge model with carbon steel gusset plates
yielded the following results. The greatest Von-Mises stress in any of the plates was
found to be 12,668 psi. This correlates to a factor of safety of 4.58 based on minimum
UTS of 58 ksi. The steel model serves as a baseline for which the composite models are
compared to. Figure 19 below shows the stress results for the bridge model with A36
carbon steel plates.
Figure 19: Abaqus FEA Von Mises Stress results for A36 Carbon Steel Model
19
Figure 20 shows the overall deflections of the bridge (magnitude) as well as deflections
in specific directions shown on the left with respect to the coordinate system on the
right. Maximum deflection of the structure was 0.447 inches downward and 0.180 inches
sideways, resulting in a maximum magnitude of deflection of 0.454 inches.
U2 (y)
U1 (x)
Figures 20: Abaqus FEA Deflection results for A36 Carbon Steel Model
20
7.2 HexPly 8552 IM7 Plates
Performing a finite element analysis for the models with composite gusset plates yielded
the following results. However unlike the steel plates, factors of safety for the composite
plates cannot be calculated using a Von-Mises stress, their factors of safety are based on
Tsai-Wu failure criterion.
This Tsai-Wu value is calculated automatically by the
CFAILURE field output request based on material properties from table 2. The use of
CFAILURE is discussed in Appendix C and is dependent on the fail stress criteria
defined in the material properties editor shown in Section 5. The results shown are only
for half of the total layers, this is because the layers are symmetric about the mid-plane
and results are the same in layers of the same orientation at the same distance from the
mid-plane.
The lowest peak TSAIW value for the three composite models was 0.286 for the [0 45
90]S layup which corresponds to a factor of safety of 3.50. A TSAIW value equal to or
greater than 1 indicates failure [9].
21
7.2.1
HexPly 8552 IM7 [0 90]S
Figure 21 shows Tsai-Wu results for this model. There are 4 total layers for each plate in
this layup and the results for layers 1 and 2 are shown below. Although these plates are 4
layers thick only half the layers need to be shown, because of symmetry about the midplane. This model generated a maximum TSAIW value of 0.296 found in layer 2
correlating to factor of safety of 3.38.
Figure 21: Abaqus FEA Tsai-Wu results for HexPly [0 90]S. The trusses are shown in
ghost in any layer other than layer 1 because they only have 1 layer.
22
Figure 22 shows the overall deflections of the bridge (magnitude) as well as deflections
in specific directions shown on the left with respect to the coordinate system on the
right. Maximum deflection of the structure was 0.879 inches downward and 0.329 inches
sideways, resulting in a maximum magnitude of deflection of 0.890 inches.
U2 (y)
U1 (x)
Figure 22: Abaqus FEA Deflection results for HexPly [0 90]S
23
7.2.2
HexPly 8552 IM7 [0 45 90]S
Figure 23 shows Tsai-Wu results for this model. There are 6 total layers for each plate in
this layup and the results for layers 1, 2 and 3 are shown below. Although these plates
are 6 layers thick only half the layers need to be shown, because of symmetry about the
mid-plane. This model generated a maximum TSAIW value of 0.286 found in layer 2
correlating to factor of safety of 3.50.
Figure 23: Abaqus FEA Tsai-Wu results for HexPly [0 45 90]S. The trusses are
shown in ghost in any layer other than layer 1 because they only have 1 layer.
24
Figure 24 shows the overall deflections of the bridge (magnitude) as well as deflections
in specific directions shown on the left with respect to the coordinate system on the
right. Maximum deflection of the structure was 0.816 inches downward and 0.331 inches
sideways, resulting in a maximum magnitude of deflection of 0.833 inches.
U2 (y)
U1 (x)
Figure 24: Abaqus FEA Deflection results for HexPly [0 45 90]S
25
7.2.3
HexPly 8552 IM7 [0 15 30 45 60 75 90]S
Figure 25 shows Tsai-Wu results for this model. There are 14 total layers for each plate
in this layup and the results for layers 1 thru 7 are shown below. Although these plates
are 14 layers thick only half the layers need to be shown, because of symmetry about the
mid-plane. This model generated a maximum TSAIW value of 0.400 found in layer 4
correlating to factor of safety of 2.50.
26
Figure 25: Abaqus FEA Tsai-Wu results for HexPly [0 15 30 45 60 75 90]S. The
trusses are shown in ghost in any layer other than layer 1 because they only have 1
layer.
27
Figure 26 shows the overall deflections of the bridge (magnitude) as well as deflections
in specific directions shown on the left with respect to the coordinate system on the
right. Maximum deflection of the structure was 0.921 inches downward and 0.377 inches
sideways, resulting in a maximum magnitude of deflection of 0.944 inches.
U2 (y)
U1 (x)
Figure 26: Abaqus FEA Deflection results for HexPly [0 15 30 45 60 75 90]S
28
7.3 Factors of Safety
The steel gusset plates outperform the best composite ones by approximately 30% based
on factors of safety (4.58 vs. 3.50). The stresses and TSAIW values listed in Table 4 are
the peak values from each plate/layer. The plate with the highest stress or layer with the
highest TSAIW value for that particular model is highlighted. Table 5 lists the factors of
safety for every layer of each plate and the lowest values are highlighted. Factors of
safety are based on the failure criterion of each material and the factor of safety for each
model is taken to be the lowest factor of safety of all the plates or lowest factor of safety
in all the layers for the composite models. This is because each layer of a composite
must be evaluated individually for failure [9]. Therefore, the steel model’s factor of
safety is based on the plate with the highest Von-Mises stress and the composites
models’ factor of safety is based on the plate with the highest TSAIW value. The results
shown are only for half of the total layers, this is because the composites are symmetric
about the mid-plane and results are the same in layers of the same orientation at the same
distance from the mid-plane.
Since the composite models are based on Tsai-Wu criterion (failure), the factors of
safety for the steel model are based on the Ultimate Tensile Strength (UTS). Typically,
factors of safety are based on Yield Strength (YS), but this approach is not appropriate
for a failure analysis. The factors of safety for the composite models are calculated as the
inverse of the TSAIW value.
The Tsai-Wu criterion predicts failure if the left side of equation is equal to or greater
than 1 [9].
F11 12  F22 22  F66 62  F1 1  F2 2  2 F12 1 2  1
Figure 27: Tsai-Wu Equation [9]
29
Table 4 shows the maximum stresses, TSAIW values, and factors of safety for every
model. The stresses and TSAIW values shown are the peak values shown on the figures
from Section 7.1 and 7.2. The factors of safety for the steel plates are based on 58 ksi
UTS divided by the maximum Von Mises stress and factors of safety for the composite
plates are based on the inverse of the TSAIW value.
Table 4: Factors of Safety
Steel Model
Von-Mises Stress
Max allowable
FS
12668
58000
4.58
TSAIW
Max allowable
FS
HexPly [0 90]S
0.296
1
3.38
HexPly [0 45 90]S
0.286
1
3.50
HexPly [0 15 30 45 60 75 90]S
0.400
1
2.50
A36 Carbon Steel
Composite Models
7.4 Deflections
Table 5 compares the maximum deflections of each composite model versus steel and
the lowest deflections of the composite models are highlighted in red.
Based on
deflections the best performing composite models deflected 183% more than the steel
model in every direction.
Table 5: Deflections
Steel Model
U magnitude
U1
U2
0.454
0.180
-0.447
HexPly [0 90]S
0.890
0.329
-0.879
HexPly [0 45 90]S
0.833
0.331
-0.816
HexPly [0 15 30 45 60 75 90]S
0.944
0.377
-0.921
183%
183%
183%
A36 Carbon Steel
Composite Models
Lowest % over steel
30
8. Conclusions
Based on the results of this comparative structural analysis, gusset plates made of
HexPly 8552 IM7 composite material provide no performance advantage versus
conventional A36 Carbon steel plates of equal size. This is due to the orthotropic nature
of composite materials which proved to be disadvantageous in an application where
loading a plate can be in as many as six different directions. Although the composite is
very strong in the longitudinal direction (much stronger than steel) it is significantly
weaker in the transverse direction.
Comparing results between the three composites shows that increasing the number of
different ply orientations within the same thickness in an attempt to increase isotropy
actually decreased overall strength in the case of the [0 15 30 45 60 75 90]S layup. The
reason for this is likely that the number of plies being loaded longitudinally (strong axis)
were reduced.
The difference between the factors of safety for steel and best performing composite was
considerable (approximately 30%) and the deflections of the composite models were
greater still, nearly twice as much as steel. This can be a very undesirable condition as
the larger amount of flex could lead to increased instability under changing load
conditions, larger heave motions and amplify the effects of cyclic loading. This
application is better suited for isotropic materials such as steel.
31
9. References
1. Kulicki, J.M. “Bridge Engineering Handbook.” Boca Raton: CRC Press, 2000.
2. Abaqus Technology Brief TB-09-BRIDGE-1. “Failure Analysis of Minneapolis I35W Bridge Gusset Plates,” Revised: December, 2009.
3. Meyers, M. M. “Safety and Reliability of Bridge Structures.” CRC Press, 2009.
4. Najjar, Walid S., DeOrtentiis, Frank. “Gusset Plates in Railroad Truss Bridges –
Finite Element Analysis and Comparison with Whitmore Testing.” Briarcliff Manor,
New York, 2010. .
5. State of Connecticut Department of Transportation. “Bridge Design Manual.”
Newington, CT 2003.
6. Kinlan, Jeff. “Structural Comparison of a Composite and Steel Truss Bridge.”
Rensselaer
Polytechnic
Institute,
Hartford,
CT,
April,
2012.
http://www.ewp.rpi.edu/hartford/~ernesto/SPR/Kinlan-FinalReport.pdf
7. Budynas, Richard G. and Nisbett, J. Keith. “Shigley’s Mechanical Engineering
Design 9th Edition.” McGraw-Hill, New York, NY, 2011.
8. American Standard for Testing and Materials - Standard Specification for Carbon
Structural Steel, ASTM A36/A36 M. ASTM International, West Conshohocken, PA
2008.
9. Gibson, Ronald F. “Principles of Composite Material Mechanics Second Edition.”
Boca Raton, FL: Taylor and Francis Group, 2007.
32
10. Abaqus/CAE 6.9EF-1. “Abaqus User Manual.” Dassault Systèmes, Providence, RI,
2009.
11. Portland Cement Association. Unit Weights, 2012.
http://www.cement.org/tech/faq_unit_weights.asp
12. Beer, Johnston. “Vector Mechanics for Engineers Statics and Dynamics 7th Edition.”
New York, NY. McGraw-Hill, 2004.
33
10. Appendices
Appendix A. Calculation of Loads
35
Appendix B. Mesh Study
44
Appendix C. CFAILURE
76
34
Appendix A – Calculation of Loads
1. Loading
Atruss  64in
2
Figure A1 - Truss cross section
Figure A2 - Bridge height, length and truss arrangement
1.1 Dead Load
Vertical Warren Truss Section
stl  0.282
lbf
in
A truss  64 in
Density of carbon steel [7]
3
2
Area of the trusses [6]
Wtrusses  ( 15 20ft  6 28.28ft)  Atruss  stl


Wtrusses  101721 lbf
Weight of 1 side of the bridge
Sidewalk
Lbridge  120 ft
Length of the bridge
wsw  5 ft
Width of the sidewalks [5]
hsw  6 in
Height of the sidewalks [5]
lbf
concrete  145
3
ft
Density of concrete [11]
Wsw  Lbridge wsw hsw concrete
Wsw  43500lbf
Weight of 1 sidewalk
35
Roadway
wroad  28 ft
Width of the roadway [5]
wbridge  2 wsw  wroad
Total width of the bridge
wbridge  38ft
Height of the deck, [6]
troad  1ft
lbf
asphalt  45
3
ft
Density of asphalt [6]
Wroadway  wbridge  troad Lbridge asphalt
Wroadway  205200lbf
Weight of the entire roadway
Floor and Roof Joists


Wjoists   12 wbridge  Atruss  stl


Wjoists  98759lbf
Weight of all the floor joists
Total Dead Load
W DL  W trusses  W sw 
W roadway  W joists
2
WDL  297201lbf
Total Dead Load
36
1.2 Live Load
Vehicles
W V 
80000lbf
51ft
 Lbridge
Maximum allowable vehicle
weight for 1 lane [5]
WV  188235lbf
Snow
W snow  40
lbf
ft
2
Snow load [6]
 wbridge  Lbridge
Wsnow  182400lbf
W LL  W V 
W snow
Total Live Load
2
WLL  279435lbf
1.3 Total Load
W  WDL  WLL
W  576636lbf
Total load, this is one-half of the entire
load the bridge will support
W
 115327 lbf
Load applied to each bottom mid-span
plate
W
 5 
 
Surftract 
10in  10in
Load applied to each bottom mid-span
plate as a surface traction
Surftract  1153psi
Surface traction load for Abaqus
5
37
2. Truss Loads - Method of Joints [12]
Feg
Fce
C
Fac
Fcd
Fbc
I
G
E
Fdg
Fde
Fgh
Ffg
  45deg
A
W
R1 
2
D
B
Fab
Fik
Fgi
Fkl
Fhk
Fhi
Fjk
H
F
Ffh
Fdf
Fbd
K
L
J
Fjl
Fhj
W
W
W
W
W
5
5
5
5
5
W
R2 
2
Figure A3 - Bridge FBD
Guess values (Fgxx) for solve blocks, hence the "g".
Fgab  1 lbf
Fgce  1 lbf
Fgde  1 lbf
Fggh  1 lbf
Fghk  1 lbf
Fgac  1 lbf
Fgcd  1 lbf
Fgdf  1 lbf
Fggi  1 lbf
Fghj  1 lbf
Fgbc  1 lbf
Fgeg  1 lbf
Fgfg  1 lbf
Fghi  1 lbf
Fgjk  1 lbf
Fgbd  1 lbf
Fgdg  1 lbf
Fgfh  1 lbf
Fgik  1 lbf
Fgkl  1 lbf
38
F
39
40
41
42
43
Appendix B – Mesh Study
Table of Contents
Results
45
Steel Model
46
Composite Model
60
Plate Locations
C
E
G
I
K
A
B
D
F
H
J
L
R1
W
5
W
5
W
5
W
5
W
5
R2
44
Appendix B – Mesh Study
Mesh Study Results
Plates and trusses were constructed with shell elements and meshed with hex elements
following the guidance provided in [10] for creating composite sections using shell
elements. Shell elements are appropriate for a 2D analysis and the use of hex elements
provides more accurate results than triangular elements.
Steel Model – The following pages document results from the mesh study carried out to
ensure accuracy of the steel model. Mesh density was adjusted by decreasing seed size
(element size) in several increments from a coarse to very fine mesh until a convergence
of stress was observed. It was determined that a seed size of 2 inches provides optimum
results and best modeling efficiency.
Composite Model – Following the same methods as those described in the process to
observe stress convergence in the steel model, convergence of the composite model was
observed by plotting the change in Tsai-Wu failure criterion (TSAIW) as mesh density
was refined. The composite used in this study is a 4 layer laminate symmetric about the
mid plane [0 90]S. Only 2 layers need to be reviewed because results are symmetric
about the mid-plane. Both layers were reviewed in order to observe if there was any
significant difference between the two layers’ ability to converge and identify any
problems, however as the following data shows, both layers followed the same
convergence trend for all plates.
It was determined that a seed size of 2 inches provides optimum results and increases
modeling efficiency for plates A, B, C, E, F, G and a seed size of 1 inch provides best
results for plate D . These are the seed sizes that will be used in all composite models
45
Appendix B – Mesh Study
Steel Model
Plate A
46
Appendix B – Mesh Study
Plate A
Seed Size
10
5
2
Elements
33
155
548
Stress
10090
10779
11037
Plate A
12000
V-M Stress (psi)
10000
8000
6000
4000
2000
0
0
100
200
300
400
500
600
Elem ents
Stress is converging with an increase in element size. A seed size (element size) of 2
inches will provide accurate results.
47
Appendix B – Mesh Study
Plate B
48
Appendix B – Mesh Study
Plate B
Seed Size
10
7
5
4
2
1
Elements
95
152
179
306
1115
4474
Stress
6496
6951
6961
7085
7421
7569
Plate B
8000
V-M Stress (psi)
7000
6000
5000
4000
3000
2000
1000
0
0
1000
2000
3000
4000
5000
Elem ents
Stress is converging with an increase in element size. A seed size (element size) of 2
inches will provide accurate results.
49
Appendix B – Mesh Study
Plate C
50
Appendix B – Mesh Study
Plate C
Seed Size
10
7
5
4
2
Elements
55
89
147
225
962
Stress
9588
9985
10252
10242
10293
Plate C
12000
V-M Stress (psi)
10000
8000
6000
4000
2000
0
0
200
400
600
800
1000
1200
Elem ents
Stress is converging with an increase in element size. A seed size (element size) of 2
inches will provide accurate results.
51
Appendix B – Mesh Study
Plate D
52
Appendix B – Mesh Study
Plate D
Seed Size
10
7
5
4
2
1
Elements
128
163
171
276
1108
4345
Stress
9675
10221
11809
12021
12629
12696
Plate D
14000
V-M Stress (psi)
12000
10000
8000
6000
4000
2000
0
0
1000
2000
3000
4000
5000
Elem ents
Stress is converging with an increase in element size. A seed size (element size) of 2
inches will provide accurate results.
53
Appendix B – Mesh Study
Plate E
54
Appendix B – Mesh Study
Plate E
Seed Size
10
7
5
4
2
Elements
95
152
179
306
1115
Stress
8900
9438
10722
11056
11194
Plate E
12000
V-M Stress (psi)
10000
8000
6000
4000
2000
0
0
200
400
600
800
1000
1200
Elem ents
Stress is converging with an increase in element size. A seed size (element size) of 2
inches will provide accurate results.
55
Appendix B – Mesh Study
Plate F
56
Appendix B – Mesh Study
Plate F
Seed Size
10
7
5
4
2
Elements
95
152
179
306
1115
Stress
10550
11303
12164
12567
12667
Plate F
14000
V-M Stress (psi)
12000
10000
8000
6000
4000
2000
0
0
200
400
600
800
1000
1200
Elem ents
Stress is converging with an increase in element size. A seed size (element size) of 2
inches will provide accurate results.
57
Appendix B – Mesh Study
Plate G
58
Appendix B – Mesh Study
Plate G
Seed Size
10
7
5
4
2
Elements
128
163
171
276
1108
Stress
9999
10353
11224
11335
11950
Plate G
14000
V-M Stress (psi)
12000
10000
8000
6000
4000
2000
0
0
200
400
600
800
1000
1200
Elem ents
Stress is converging with an increase in element size. A seed size (element size) of 2
inches will provide accurate results.
59
Appendix B – Mesh Study
Composite Model – [0 90]S
Plate A
60
Appendix B – Mesh Study
61
Appendix B – Mesh Study
Plate A
Seed Size
10
7
5
2
Elements
33
98
155
548
Layer 1
0.193
0.197
0.227
0.227
Layer 2
0.218
0.241
0.267
0.274
Plate A
0.300
Layer 2
0.250
Layer 1
TSAIW
0.200
0.150
0.100
0.050
0.000
0
100
200
300
400
500
600
Elem ents
TSAI-WU criterion is converging with an increase in element size. A seed size (element size)
of 2 inches will provide accurate results.
62
Appendix B – Mesh Study
Plate B
63
Appendix B – Mesh Study
Plate B
Seed Size
10
5
4
2
Elements
95
179
306
1115
Layer 1
0.084
0.088
0.096
0.107
Layer 2
0.095
0.101
0.104
0.127
Plate B
0.140
Layer 2
0.120
Layer 1
TSAIW
0.100
0.080
0.060
0.040
0.020
0.000
0
200
400
600
800
1000
1200
Elem ents
TSAI-WU criterion is converging with an increase in element size. A seed size (element size)
of 2 inches will provide accurate results.
64
Appendix B – Mesh Study
Plate C
65
Appendix B – Mesh Study
Plate C
Seed Size
10
7
5
4
2
Elements
55
89
147
225
962
Layer 1
0.264
0.218
0.246
0.249
0.266
Layer 2
0.231
0.200
0.229
0.232
0.245
66
Appendix B – Mesh Study
Plate C
0.300
Layer 1
0.250
Layer 2
TSAIW
0.200
0.150
0.100
0.050
0.000
0
200
400
600
800
1000
1200
Elem ents
TSAI-WU criterion is converging with an increase in element size. A seed size (element size)
of 2 inches will provide accurate results.
67
Appendix B – Mesh Study
Plate D
68
Appendix B – Mesh Study
Plate D
Seed Size
10
7
5
4
2
1
Elements
128
163
171
276
1108
4345
Layer 1
0.145
0.151
0.126
0.158
0.208
0.239
Layer 2
0.176
0.184
0.174
0.189
0.249
0.267
Plate D
0.300
Layer 2
0.250
Layer 1
TSAIW
0.200
0.150
0.100
0.050
0.000
0
1000
2000
3000
4000
5000
Elem ents
TSAI-WU criterion is converging with an increase in element size. A seed size (element size)
of 1 inch will provide accurate results.
69
Appendix B – Mesh Study
Plate E
70
Appendix B – Mesh Study
Plate E
Seed Size
7
5
4
2
Elements
152
179
306
1115
Layer 1
0.124
0.129
0.135
0.163
Layer 2
0.110
0.112
0.117
0.145
Plate E
0.180
0.160
Layer 1
TSAIW
0.140
0.120
Layer 2
0.100
0.080
0.060
0.040
0.020
0.000
0
200
400
600
800
1000
1200
Elem ents
TSAI-WU criterion is converging with an increase in element size. A seed size (element size)
of 2 inches will provide accurate results.
71
Appendix B – Mesh Study
Plate F
72
Appendix B – Mesh Study
Plate F
Seed Size
10
4
2
Elements
95
306
1115
Layer 1
0.135
0.140
0.171
Layer 2
0.160
0.163
0.193
Plate F
TSAIW
0.250
0.200
Layer 2
0.150
Layer 1
0.100
0.050
0.000
0
200
400
600
800
1000
1200
Elem ents
TSAI-WU criterion is converging with an increase in element size. A seed size (element size)
of 2 inches will provide accurate results.
73
Appendix B – Mesh Study
Plate G
74
Appendix B – Mesh Study
Plate G
Seed Size
10
4
2
Elements
128
276
1108
Layer 1
0.125
0.141
0.169
Layer 2
0.106
0.123
0.152
Plate G
0.180
Layer 1
0.160
0.140
Layer 2
TSAIW
0.120
0.100
0.080
0.060
0.040
0.020
0.000
0
200
400
600
800
1000
1200
Elem ents
The results for seed sizes 7 and 5 appear to skew results, they are taken to be inaccurate and
therefore, ignored.
TSAI-WU criterion is converging with an increase in element size. A seed size (element size)
of 2 inches will provide accurate results.
75
Appendix C - CFAILURE
1.
Define Fail Stress and/or Fail Strain values in the suboptions menu of the
materials editor. The user defines stress and/or strain depending on which results
they would like to view (MSTRN, MSTRS, TSAIH, TSAIW, etc).
For when defining fail stress for a composite material, the cross-prod term coeff
(f*) is necessary for the F12 term in the Tsai-Wu equation [9] [10]. However, the
stress limit is not required, Abaqus will use f* instead [10].
76
Appendix C –CFAILURE
2. Under Output Field Requests, right click – edit, expand the menu under
“Fracture/Failure” and check the box for CFAILURE.
Manually enter the number of section points in the format: 1,2,3,4,5…n.
Where n is equal to the total number of plies times intergation points. The
default number of integration points is 3 and this can be altered by editing section
properties.
77
Appendix C –CFAILURE
3. The user can now run the analysis and view results for each layer by clicking
“Section Points” under the “Results” menu at the top of the screen.
Select “Plies” and results can be viewed by layer.
78