Finite Element Analysis of Steel and Composite Gusset Plates
in a Warren Truss Bridge using Abaqus
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 2.1 Bridge Design .................................................................................................... 4 2.2 Loading .............................................................................................................. 5 2.2.1 Dead Load .............................................................................................. 5 2.2.2 Live Load ............................................................................................... 5 2.2.3 Total Load (W) ....................................................................................... 5 2.3 Materials............................................................................................................. 6 2.3.1 A36 Carbon Steel ................................................................................... 6 2.3.2 HexPly 8552 IM7 Carbon Fiber............................................................. 7 2.4 FEA Models ..................................................................................................... 10 2.4.1 Plate Geometry ..................................................................................... 16
2.4.1.1 Bottom End Plates ................................................................. 16
17
2.4.1.2 Mid-Span Plates ..................................................................... 16
18
2.4.1.3 Upper End Plates ................................................................... 16
2.4.2 Truss Geometry .................................................................................... 18 3. Results........................................................................................................................ 19 3.1 A36 Carbon Steel Plates .................................................................................. 19 3.2 HexPly 8552 IM7 Carbon Fiber Plates ............................................................ 22 3.2.1 [0 90]S Layup....................................................................................... 23 3.2.2 [0 45 90]S Layup.................................................................................. 25 iii
3.2.3 [0 15 30 45 60 75 90]S Layup.............................................................. 28 3.3 Factors of Safety .............................................................................................. 32 3.4 Deflections ....................................................................................................... 33 4. Conclusions................................................................................................................ 34 5. References.................................................................................................................. 35 6. Appendices ................................................................................................................ 37 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 ................................................................................................ 33
Table 5: Deflections ......................................................................................................... 33
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 HexPly 8552 IM7 ................................................ 8
Figure 4: Abaqus Fail Stress for HexPly 8552 IM7 ........................................................ 8
Figure 5: Abaqus Section Editor for HexPly 8552 IM7 [0 90]S ..................................... 9
Figure 6: Abaqus Section Editor for HexPly 8552 IM7 [0 45 90]S ................................ 9
Figure 7: Abaqus Section Editor for HexPly 8552 IM7 [0 15 30 45 60 75 90]S ............ 10
Figure 8: FEA Bridge showing loads and boundary conditions ...................................... 11
Figure 9: Iso view of a joint with shell thicknesses rendered .......................................... 12
Figure 10: Abaqus model showing all loads and constraints .......................................... 12
Figure 11: Abaqus model with z-constraint (U3) suppressed for clarity......................... 12
Figures 12a, b: Loading applied to the lower mid-span plates ........................................ 13
Figures 13a, b, c, d: Bottom end plate boundary conditions ........................................... 14
Figures 14a, b, c: Side view of bridge and close up showing tie constraints .................. 15
Figure 15: Bottom End Plate Detail Drawing (Plates A and L) ...................................... 16
Figures 16a, b: Mid-Span Plate Detail Drawings (Plates B, D, E, F, G, H, I and J) ....... 17
Figure 17: Upper End Plate Detail Drawings (Plates C and K) ...................................... 18
Figure 18: Abaqus FEA Von Mises Stress results for A36 Carbon Steel Model ............ 20
Figure 19: Abaqus FEA Deflection results for A36 Carbon Steel Model ....................... 21
Figure 20: Abaqus FEA Tsai-Wu results for HexPly [0 90]S Model.............................. 23
Figure 21: Abaqus FEA Deflection results for HexPly [0 90]S Model ........................... 24
Figure 22: Abaqus FEA Tsai-Wu results for HexPly [0 45 90]S Model......................... 26
Figure 23: Abaqus FEA Deflection results for HexPly [0 45 90]S Model ...................... 27
Figure 24: Abaqus FEA Tsai-Wu results for HexPly [0 15 30 45 60 75 90]S Model..... 30
Figure 25: Abaqus FEA Deflection results for HexPly [0 15 30 45 60 75 90]S Model .. 31
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
The purpose of this project is to determine if carbon fiber gusset plates offer superior
strength characteristics compared to those made from structural steel by performing
finite element analyses of a single span Warren Truss bridge using Abaqus/CAE.
Performance of the two materials was evaluated based on failure margin and deflections.
The two materials chosen are A36 Carbon Structural Steel and HexPly brand 8552 IM7
prepreg composite carbon fiber.
Four bridge models were created and analyzed; one consisting of steel plates and three
models using composite plates of different layups: [0 90]S, [0 45 90]S and [0 15 30 45
60 75 90]S. Increasing the number of ply orientations across the same thickness is being
done to optimize the plate’s strength characteristics by increasing isotropy. In order to
make a fair comparison, the dimensions of the gusset plates are identical and the truss
members are identical by material and dimension for all models.
Analysis shows that carbon fiber plates do not offer a performance advantage versus
steel based upon failure margin and deflection. The bridge model using A36 Carbon
Steel plates is calculated to be approximately 30% stronger and deflect only half as
much. This large difference in performance is due to the significantly weaker transverse
properties of HexPly 8552 IM7 despite its superior strength in the longitudinal direction
versus steel. This orthotropic behavior proved to be a pitfall in an application where the
plates are loaded in up to six different directions. That is not to say composites are
inferior, but this is an application that is better suited for isotropic materials.
.
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 Abaqus/CAE Finite Element Analysis (FEA) software.
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 detailed 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 increasing isotropic behavior. For consistency, the plates in all models
are identical in every dimension and trusses are identical by material and dimension
across all models.
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
2.1 Bridge Design
The bridge chosen for this project is a single span Warren Truss with verticals. This
specific 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°. For the purposes
of calculating dead load, all of the vertical, horizontal and diagonal 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
2.2 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.
2.2.1
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 2.1 and calculations
detailed in Appendix A dead load has been determined to be 297,201 lbs.
2.2.2
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.
2.2.3
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 (115327 lbs) is applied to each plate as
shown and converted to a surface traction of 1153 psi for FEA.
A
R1
C
E
G
I
K
B
D
F
H
J
L
W
5
W
5
W
5
W
5
R2
W
= 115327 lbf
5
Figure 2: Bridge Free Body Diagram
5
2.3 Materials
2.3.1
A36 Carbon Steel
A36 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
weldability 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)
Value
30.0
11.5
0.292
0.282
36 min
58-80
6
2.3.2
HexPly 8552 IM7 Carbon Fiber
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. increasing 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 [6]
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
[083,9083]S
[056,4556,9056]S
[024,1524,3024,4524,6024,7524,9024]S
Thickness
0.500 per orientation
0.333 per orientation
0.143 per orientation
7
Total Thickness
2.0
2.0
2.0
Figure 3: Abaqus Material Editor for HexPly 8552 IM7. The required elastic
values from Table 2 are entered here.
Figure 4: 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,
Figures 6, 7 and 8 show how the layups were defined in Abaqus. Abaqus requires a
Abaqus will use f* instead [10].
thickness for each orientation rather than number of plies times a ply thickness. The
8
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 5: Abaqus Section Editor for HexPly 8552 IM7 [0 90]S
Figure 6: Abaqus Section Editor for HexPly 8552 IM7 [0 45 90]S
9
Figure 7: Abaqus Section Editor for HexPly 8552 IM7 [0 15 30 45 60 75 90]S
2.4 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.
10
C
G
E
I
K
y (U2)
i
x (U1)
A
F
D
B
L
J
H
Pinned End
Roller End
W = 115327 lbs
5 (1153 psi)
W
5
W
5
W
5
Figure 8: FEA Bridge showing loads and boundary conditions.
The loads are converted to surface tractions (psi) based on the size
of the area they are assigned to, in this case, 10”x10”.
11
W
5
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 9: Iso view of a joint with shell thicknesses rendered
The Free Body Diagram (FBD) for the bridge is shown below in Figures 11 and 12. The
entire bridge was constrained in the out-of-plane (U3, z-direction) to keep the analysis
2D. 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 2.2 and shown in
Figure 13.
y (U2)
x (U1)
Figure 10: Abaqus model showing all loads and constraints.
y (U2)
x (U1)
Figure 11: Abaqus model with z-constraint (U3) suppressed for clarity.
12
Surface traction
Figures 12a, 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 13a, 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 14a, b, c: Side view of
bridge and close up showing tie
constraints.
15
2.4.1
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 and to mimic real world plates which are sized
differently based on how many trusses they connect. 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.
2.4.1.1 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 2.4) along their
bottom edges. Dimensions for these plates are detailed in Figure 16 and are basically the
same as the mid-span plates cut in half down the 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 15: Bottom End Plate Detail Drawing (Plates A and L)
16
2.4.1.2 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. Dimensions for these plates are
detailed in Figure 17. 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 16a, b: Mid-Span Plate Detail Drawings (Plates B, D, E, F, G, H, I and J)
17
2.4.1.3 Upper End Plates
These are the plates located at the upper corners at each end of the bridge. Dimensions
for these plates are detailed in Figure 18. 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 17: Upper End Plate Detail Drawings (Plates C and K)
2.4.2
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 2.4) and to maintain the overall dimensions of the bridge
described in Section 2.1. 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 10”x12” 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. Note that this cross section does not match the
cross section stated in Section 2.1 (also used to calculate the dead load of the bridge in
Appendix A) and that this difference is inconsequental. Therefore, overall deflections of
the model are due to the cumulative deflections of the plates.
18
3. Results
3.1 A36 Carbon Steel Plates
The steel model serves as a baseline for which the composite models are compared to
and performing a finite element analysis on the bridge model with A36 Carbon Steel
gusset plates yielded the following results:
Von Mises Stress 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. Figure 19 on the following page shows the stress distribution results for
the bridge model with A36 carbon steel plates. As expected, the bottom middle plate
(Plate F) displayed the highest stresses. The stress distributions for plates A thru G are
shown in greater detail in Appendix B as part of the mesh study for the carbon steel
model. Results for a 2 inch seed size coincide with the results of the bridge model
shown in Figure 19.
The trusses with their very robust geometry (10”x12” cross section) show significantly
lower stresses versus the plates (2” thick). Comparing stress results among the trusses
corresponds well with the loads calculated in the Method of Joints calculations in
Appendix A which predicted the two diagonal end trusses to be the most severely
loaded.
19
Figure 18: Abaqus FEA Von Mises Stress results for A36 Carbon Steel Model
20
Deflections: As expected, the deflections in the vertical direction are symmetric about
the vertical centerline of the bridge with the plates midway across the bridge having
deflected the furthest downward. Deflections overall are in accordance with how
boundary conditions were assigned to the bottom end plates in which both are
constrained in the vertical direction and the location of greatest horizontal movement is
at the edge of the right-hand bottom end plate because this plate is allowed to move
laterally. It is also clear that the upper horizontal members are in compression and the
lower horizontal members are in tension which is confirmed by the method of joints
calculations in Appendix A where negative truss loads are compressive and positive
loads are tensile.
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 19: Abaqus FEA Deflection results for A36 Carbon Steel Model
21
3.2 HexPly 8552 IM7 Carbon Fiber 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. Tsai-Wu Criterion has been proven to give better agreement
with experiemental data versus Maximum Stress and Maximum Strain theories for
composites. The Tsai-Wu equation predicts failure if the left hand side is equal to or
greater than 1 [9].
F11 12  F22 22  F66 62  F1 1  F2 2  2 F12 1 2  1
Tsai-Wu Equation [9]
Abaqus refers to the value on the left side of the equation as TSAIW and 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 2.3. 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. Layup [0 45 90]S also
deflected the least among the composites, however both factors of safety and deflection
fared worse than that of the A36 Carbon Steel model.
22
3.2.1
[0 90]S Layup
TSAIW Results: 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 mid-plane. This model generated a maximum TSAIW value of
0.296 found in layer 2 correlating to factor of safety of 3.38, whish is significantly less
than the FS calculated for the A36 Carbon Steel model. Distributions of TSAIW values
for plates A thru G are shown in greater detail in Appendix B as part of the mesh study
for the HexPly 8552 IM7 [0 90]S model. Results for a 2 inch seed size coincide with the
results of the bridge model shown in Figure 21.
The trusses are shown in dark blue (TSAIW = 0, FS = ∞) because they have no fail
stress properties defined for them in layer 1 and they are shown in ghost for any layer
other than layer 1 because they only have 1 layer.
Figure 20: 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.
23
Deflections: As previously stated in the discussion of results for the A36 Carbon Steel
model, the deflections are as expected for the vertical and horizontal directions based on
the nature of the problem and how boundary conditions were assigned. The vertical
deflections are greatest at mid-span across the bridge, horiztonal deflection is greatest at
the edge of bottom end plate on the right-hand side of the model and the way the upper
and lower horizontal trusses have deformed coincide with the compressive and tensile
loads calculated in Appendix A.
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. This model
shows significantly greater deflection in both directions (almost twice) than that of the
A36 Carbon Steel model.
U2 (y)
U1 (x)
Figure 21: Abaqus FEA Deflection results for HexPly [0 90]S
24
3.2.2
[0 45 90]S Layup
TSAIW Results: Figure 23 on the following page shows Tsai-Wu results for layers 1
thru 3 of this model. Although these plates are 6 layers thick only half the layers need to
be shown, because of symmetry about the mid-plane. This model differs from the [0
90]S model by introducing a layer oriented at 45° to each side of the composite’s midplane while limiting total plate thickness to 2 inches. This was done to investigate if
increasing the number of layers of varying orientation within the same total thickness
has a potential strength increase by increasing isotropy. This change proved to be only
slightly beneficial to the strength characteristics of the composite as this model
generated a maximum TSAIW value of 0.286 found in layer 2 (45° ply orientation)
correlating to factor of safety of 3.50 which is slightly greater than the FS calculated for
the [0 90]S model, but still less than the FS calculated for the A36 Carbon Steel. Similar
to the A36 Carbon Steel model, the areas of most severe loading are along the bottom
plates as shown in Layer 2 and 3. Layer 1 appears to contradict this observation, but the
TSAIW values shown there are almost a third lower.
The trusses are shown in dark blue (TSAIW = 0, FS = ∞) because they have no fail
stress properties defined for them in layer 1 and they are shown in ghost for any layer
other than layer 1 because they only have 1 layer.
25
Figure 22: Abaqus FEA Tsai-Wu results for HexPly [0 45 90]S.
26
Deflections: As previously stated in the discussion of results for the A36 Carbon Steel
model, the deflections are as expected for the vertical and horizontal directions based on
the nature of the problem and how boundary conditions were assigned. The vertical
deflections are greatest at mid-span across the bridge, horiztonal deflection is greatest at
the edge of bottom end plate on the right-hand side of the model and the way the upper
and lower horizontal trusses have deformed coincide with the compressive and tensile
loads calculated in Appendix A.
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. Although
this model is the best performing composite in terms of deflection, it still shows
significantly greater deflection in both directions (almost twice) than that of the A36
Carbon Steel model.
U2 (y)
U1 (x)
Figure 23: Abaqus FEA Deflection results for HexPly [0 45 90]S
27
3.2.3
[0 15 30 45 60 75 90]S Layup
TSAIW Results: Figure 25 on the next two pages shows Tsai-Wu results for layers 1
thru 7 of this model. Although these plates are 14 layers thick only half the layers need
to be shown, because of symmetry about the mid-plane. This model differs from the [0
45 90]S model by introducing layers oriented at 15°, 30°, 60° and 75° to each side of the
composite’s mid-plane while limiting total plate thickness to 2 inches. Again, this was
done to investigate if increasing the number of layers of varying orientation within the
same total thickness has a potential strength increase by increasing isotropy. However,
this change actually proved to be detrimental to the strength characteristics of the
composite. This is because the number of plies whose strong (longitudinal) axis is
aligned with loading has now been reduced and the number of plies that are misaligned
with loading have now been increased, creating a less efficient plate.
This model generated a maximum TSAIW value of 0.400 found in layer 4 (45° ply
orientation) correlating to factor of safety of 2.50 which is significantly less than the FS
calculated for the A36 Carbon Steel, [0 90]S and [0 45 90]S models. Similar to the A36
Carbon Steel model, the areas of most severe loading are along the bottom plates as
shown in Layers 3, 4, 5, 6 and 7. Layers 1 and 2 appear to contradict this observation,
but the TSAIW values shown there are significantly lower.
The trusses are shown in dark blue (TSAIW = 0, FS = ∞) because they have no fail
stress properties defined for them in layer 1 and they are shown in ghost for any layer
other than layer 1 because they only have 1 layer.
28
29
Figure 24: Abaqus FEA Tsai-Wu results for HexPly [0 15 30 45 60 75 90]S.
30
Deflections: As previously stated in the discussion of results for the A36 Carbon Steel
model, the deflections are as expected for the vertical and horizontal directions based on
the nature of the problem and how boundary conditions were assigned. The vertical
deflections are greatest at mid-span across the bridge, horiztonal deflection is greatest at
the edge of bottom end plate on the right-hand side of the model and the way the upper
and lower horizontal trusses have deformed coincide with the compressive and tensile
loads calculated in Appendix A.
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. This model
displayed the worst (largest) deflections of any model in any direction.
U2 (y)
U1 (x)
Figure 25: Abaqus FEA Deflection results for HexPly [0 15 30 45 60 75 90]S
31
3.3 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). Stresses, TSAIW values and factors of safety are
listed in Table 4. The stresses and TSAIW values shown are the peak values shown on
the figures from Section 3.1 and 3.2. Factors of safety are based on the failure criterion
of each material and the factor of safety for the composite model is taken to be the is
based on the highest TSAIW value in all the layers. This is because each layer of a
composite must be evaluated individually for failure [9] and the factor of safety for the
model is taken to be the lowest factor of safety for any layer of any plate. 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.
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
Tsai-Wu Equation [9]
Table 4 shows the maximum stresses, TSAIW values, and factors of safety for every
model. . 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.
32
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
3.4 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
33
4. 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 strength by
increasing 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 and the number of plies misaligned with the
loading were increased, subsequently loading those plies in the transverse (weak axis)
direction.
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.
34
5. 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. Web. July, 2012.
<http://imechanica.org/files/Architecture-SIMULIA-Tech-Brief-09-FailureAnalysis-Minneapolis-Full.pdf>
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.
Web. July, 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.
35
10. Abaqus/CAE 6.9EF-1. “Abaqus User Manual.” Dassault Systèmes, Providence, RI,
2009.
11. Portland Cement Association. Unit Weights, 2012. Web. July 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.
36
6. Appendices
Appendix A. Calculation of Loads
38
Appendix B. Mesh Study
47
Appendix C. CFAILURE
79
37
Appendix A – Calculation of Loads
1. Loading
2
A truss  64in
Figure A1 - Truss cross section
Figure A2 - Bridge height, length and truss arrangement
1.1 Dead Load
Vertical Warren Truss Section
lbf
stl  0.282
3
in
Density of carbon steel [7]
2
A truss  64 in
Area of the trusses [6]
W trusses  ( 15 20ft  6 28.28ft)  A truss  stl


Weight of 1 side of the bridge
W trusses  101721lbf
Sidewalk
Lbridge  120 ft
Length of the bridge
wsw  5 ft
Width of the sidewalks [5]
h sw  6 in
Height of the sidewalks [5]
lbf
concrete  145
3
ft
Density of concrete [11]
W sw  Lbridge  wsw  h sw  concrete
Weight of 1 sidewalk
W sw  43500lbf
38
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]
W roadway  wbridge  troad  Lbridge  asphalt
Weight of the entire roadway
W roadway  205200lbf
Floor and Roof Joists


W joists   12 wbridge  A truss  stl


Weight of all the floor joists
W joists  98759lbf
Total Dead Load
W DL  W trusses  W sw 
W roadway  W joists
2
W DL  297201lbf
Total Dead Load
39
1.2 Live Load
Vehicles
W V 
80000lbf
51ft
Maximum allowable vehicle
weight for 1 lane [5]
 Lbridge
W V  188235lbf
Snow
W snow  40
lbf
ft
2
Snow load [6]
 wbridge  Lbridge
W snow  182400lbf
W LL  W V 
W snow
Total Live Load
2
W LL  279435lbf
1.3 Total Load
W  W DL  W LL
W  576636lbf
Total load, this is one-half of the entire
load the bridge will support
W
 115327lbf
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
40
2. Truss Loads - Method of Joints [12]
F eg
F ce
C
Fac
Fcd
Fbc
I
G
E
Fdg
Fde
Fgh
Ffg
  45deg
A
W
R1 
2
Fab
D
B
F ik
F gi
Fbd
Fdf
K
Fjk
H
F
F kl
Fhk
Fhi
Ffh
L
J
F jl
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
41
F
42
43
44
45
Fce = -461309 lbf
Fac = -407743 lbf
Fde = 0 lbf
B
Fab = 288318 lbf
R1 = 288318 lbf
Ffg = 115327 lbf
F
D
Fbd = 288318 lbf
W
= 115327 lbf
5
Ffh = 518972 lbf
Fdf = 518972 lbf
W
= 115327 lbf
5
K
I
Fgh = -81549 lbf
Fdg = -81549 lbf
Fcd = 244646 lbf
Fbc = 115327 lbf
A
G
E
C
Fik = -461309 lbf
Fgi = -461309 lbf
Feg = -461309 lbf
W
= 115327 lbf
5
Fhk = 244646 lbf
Fhi = 0 lbf
Fjk = 115327 lbf
H
J
Fhj = 288318 lbf
W
= 115327 lbf
5
Figure A4 – Bridge FBD labeled with truss loads
46
Fkl = -407743 lbf
L
Fjl = 288318 lbf
W
= 115327 lbf
5
R2 = 288318 lbf
Appendix B – Mesh Study
Table of Contents
Results
48
Steel Model
49
Composite Model
63
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
47
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
48
Appendix B – Mesh Study
Steel Model
Plate A
49
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.
50
Appendix B – Mesh Study
Plate B
51
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.
52
Appendix B – Mesh Study
Plate C
53
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.
54
Appendix B – Mesh Study
Plate D
55
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.
56
Appendix B – Mesh Study
Plate E
57
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.
58
Appendix B – Mesh Study
Plate F
59
Appendix B – Mesh Study
Plate F
Seed Size
10
7
5
4
2
Elements
95
152
179
306
1115
Stress
10550
11303
12164
12567
12668
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.
60
Appendix B – Mesh Study
Plate G
61
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.
62
Appendix B – Mesh Study
Composite Model – [0 90]S
Plate A
63
Appendix B – Mesh Study
64
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.
65
Appendix B – Mesh Study
Plate B
66
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.
67
Appendix B – Mesh Study
Plate C
68
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
69
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.
70
Appendix B – Mesh Study
Plate D
71
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.
72
Appendix B – Mesh Study
Plate E
73
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
TSAIW
Plate E
0.180
0.160
Layer 1
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.
74
Appendix B – Mesh Study
Plate F
75
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.
76
Appendix B – Mesh Study
Plate G
77
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.
78
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].
79
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.
80
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.
81