Structural Comparison of a Composite and Steel Truss Bridge
by
Jeffrey Kinlan
An Engineering Research Project 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:
_________________________________________
Professor Ernesto Gutierrez-Miravete, Engineering Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
April, 2012
© Copyright 2012
by
Jeffrey Kinlan
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................ vi
LIST OF FIGURES ........................................................................................................ viii
LIST OF EQUATIONS .................................................................................................... ix
LIST OF SYMBOLS ........................................................................................................ xi
GLOSSARY ................................................................................................................... xiv
ABSTRACT .................................................................................................................... xv
1. Introduction/Background ............................................................................................. 1
1.1
Truss Geometry .................................................................................................. 1
1.2
Truss Member Geometry ................................................................................... 2
1.3
Truss Gusset Plate Geometry ............................................................................. 3
2. Materials ...................................................................................................................... 5
2.1
Steel Material Selection ..................................................................................... 5
2.2
Composite Material Selection ............................................................................ 6
3. Composite Laminate Theory ....................................................................................... 9
3.1
CLT Assumptions .............................................................................................. 9
3.2
The ABD Matrix .............................................................................................. 10
3.3
Laminate Stress and Strain ............................................................................... 11
3.4
Laminate Failure Criterion ............................................................................... 12
3.5
Laminate Material Properties ........................................................................... 13
3.6
Member Laminate Layup ................................................................................. 14
4. Truss Loads ................................................................................................................ 16
4.1
Dead Load ........................................................................................................ 16
4.2
Live Load ......................................................................................................... 18
4.3
Dynamic Load .................................................................................................. 19
4.4
Truss Free Body Diagram ................................................................................ 19
5. Analysis Methodology ............................................................................................... 22
iii
5.1
5.2
5.3
2D ANSYS Finite Element Model................................................................... 22
5.1.1
Model Geometry, Element Type, and Mesh ........................................ 22
5.1.2
Material Properties, Loads, and Boundary Conditions ........................ 24
Method of Joints............................................................................................... 25
5.2.1
Naming Convention ............................................................................. 25
5.2.2
Nodal Free Body Diagrams.................................................................. 26
5.2.3
Matrix Equation ................................................................................... 26
3D ANSYS Finite Element Model................................................................... 27
5.3.1
Model Geometry and Mesh.................................................................. 28
5.3.2
Material Properties, Loads, and Boundary Conditions ........................ 29
6. Results........................................................................................................................ 30
6.1
2D ANSYS Finite Element Model Results ...................................................... 30
6.2
Method of Joints Results .................................................................................. 33
6.3
3D ANSYS Finite Element Model Results ...................................................... 34
7. Margin of Safety Calculation..................................................................................... 41
7.1
7.2
Steel Truss Margin of Safety............................................................................ 41
7.1.1
Axial Margin of Safety......................................................................... 41
7.1.2
Transverse Margin of Safety ................................................................ 42
7.1.3
Buckling Margin of Safety ................................................................... 43
Composite Truss Margin of Safety .................................................................. 45
7.2.1
Axial Margin of Safety......................................................................... 49
7.2.2
Transverse Margin of Safety ................................................................ 54
7.2.3
Buckling Margin of Safety ................................................................... 58
8. Conclusion ................................................................................................................. 62
9. References.................................................................................................................. 65
10. Appendix.................................................................................................................... 66
10.1 Appendix A – ANSYS Input File Code ........................................................... 66
iv
10.2 Appendix B – MATLAB Composite Laminate Stiffness Calculator .............. 70
10.3 Appendix C – MATLAB Composite Laminate Global Failure Calculator ..... 76
v
LIST OF TABLES
Table 1.1 - Truss Dimensions ............................................................................................ 2
Table 1.2 - Member Cross Section Dimensions ................................................................ 3
Table 2.1 - Steel Alloys [6]................................................................................................ 6
Table 2.2 - 5Cr-Mo-V Steel Properties [6] ........................................................................ 6
Table 2.3 - Composite Materials with Epoxy Resin System [3] ....................................... 7
Table 2.4 - 8552 IM7 Material Properties [3].................................................................... 8
Table 3.1 - Laminate Material Properties ........................................................................ 14
Table 3.2 - Candidate Composite Laminates ................................................................... 15
Table 4.1 - Member Weight Variables ............................................................................ 16
Table 4.2 - Gusset Plate Weight Variables ...................................................................... 17
Table 4.3 - Road Deck Weight Variables ........................................................................ 17
Table 4.4 - Vehicle Weight Variables ............................................................................. 18
Table 4.5 - Snow Load Variables .................................................................................... 19
Table 4.6 - Truss Loads ................................................................................................... 21
Table 6.1 - 2D FEM Member Axial Stresses .................................................................. 31
Table 6.2 - 2D FEM Member Axial Forces ..................................................................... 32
Table 6.3 - Method of Joints Forces ................................................................................ 33
Table 6.4 - Method of Joints Axial Member Stresses ..................................................... 34
Table 6.5 - Truss Group 1 3D FEM Transverse Member Stresses .................................. 35
Table 6.6 - Truss Group 2 3D FEM Transverse Member Stresses .................................. 36
Table 6.7 - Truss Group 3 3D FEM Transverse Member Stresses .................................. 36
Table 6.8 - Truss Group 4 3D FEM Transverse Member Stresses .................................. 37
Table 6.9 - Truss Group 1 3D FEM Transverse Member Forces .................................... 38
Table 6.10 - Truss Group 2 3D FEM Transverse Member Forces .................................. 38
Table 6.11 - Truss Group 3 3D FEM Transverse Member Forces .................................. 39
Table 6.12 - Truss Group 4 3D FEM Transverse Member Forces .................................. 40
Table 7.1 - Steel Material Allowable ............................................................................... 41
Table 7.2 - Steel Truss Axial M.S. .................................................................................. 42
Table 7.3 - Steel Truss Transverse M.S. .......................................................................... 43
Table 7.4 - Steel Truss Buckling M.S.............................................................................. 44
vi
Table 7.5 - Composite Material Allowable ..................................................................... 45
Table 7.6 - Global Axial Ply Failure Load ...................................................................... 47
Table 7.7 - Global Transverse Ply Failure Load .............................................................. 48
Table 7.8 - Composite Laminate Group 1 Axial M.S...................................................... 50
Table 7.9 - Composite Laminate Group 2 Axial M.S...................................................... 51
Table 7.10 - Composite Laminate Group 3 Axial M.S.................................................... 52
Table 7.11 - Composite Laminate Group 4 Axial M.S.................................................... 53
Table 7.12 - Composite Laminate Minimum Axial M.S. ................................................ 54
Table 7.13 - Composite Laminate Group 1 Transverse M.S. .......................................... 55
Table 7.14 - Composite Laminate Group 2 Transverse M.S. .......................................... 56
Table 7.15 - Composite Laminate Group 3 Transverse M.S. .......................................... 57
Table 7.16 - Composite Laminate Group 4 Transverse M.S. .......................................... 57
Table 7.17 - Composite Laminate Minimum Transverse M.S. ....................................... 58
Table 7.18 - Composite Laminate Critical Buckling Loads ............................................ 59
Table 7.19 - Composite Composite Laminate Buckling M.S. ......................................... 60
Table 7.20 - Composite Laminate Minimum Buckling M.S. .......................................... 61
Table 8.1 - Steel vs. Composite Weight Comparison ...................................................... 64
vii
LIST OF FIGURES
Figure 1.1 - Warren Truss .................................................................................................. 1
Figure 1.2 - Truss Wireframe Schematic ........................................................................... 2
Figure 1.3 - Truss Member Cross Section ......................................................................... 2
Figure 1.4 - Node Gusset Plate Type ................................................................................. 3
Figure 1.5 - Gusset Plate Type a ........................................................................................ 4
Figure 1.6 - Gusset Plate Type b ....................................................................................... 4
Figure 1.7 - Gusset Plate Type c ........................................................................................ 4
Figure 3.1 - Composite Ply ................................................................................................ 9
Figure 3.2 - Through Thickness Coordinate System [4] ................................................. 10
Figure 3.3 - Tsai Wu Failure Criterion Ellipsoid [4] ....................................................... 13
Figure 4.1 - Dead and Live Truss Load FBD .................................................................. 20
Figure 4.2 - Dynamic Truss Load FBD ........................................................................... 20
Figure 5.1 - 2D ANSYS FEM Geometry ........................................................................ 23
Figure 5.2 - BEAM188 Element [5] ................................................................................ 23
Figure 5.3 - 2D ANSYS FEM Mesh ............................................................................... 24
Figure 5.4 - Member and Node Designation ................................................................... 25
Figure 5.5 - Edge, Top Side, and Internal Node Free Body Diagrams ............................ 26
Figure 5.6 - 3D CATIA Truss Model .............................................................................. 28
Figure 5.7 - 3D ANSYS FEM Mesh ............................................................................... 29
Figure 6.1 - Steel 2D ANSYS FEM Axial Stress Result ................................................. 30
Figure 6.2 - Tension Compression Members .................................................................. 34
Figure 8.1 - Axial M.S. Comparison ............................................................................... 62
Figure 8.2 - Transverse M.S. Comparison ....................................................................... 63
Figure 8.3 - Buckling M.S. Comparison.......................................................................... 63
viii
LIST OF EQUATIONS
Equation 3.1 - Q Matrix ................................................................................................... 10
Equation 3.2 - Through Thickness Vector ....................................................................... 10
Equation 3.3 - Transformation Matrix ............................................................................. 11
̅ Matrix .................................................................................................. 11
Equation 3.4 - Q
Equation 3.5 - [A] Matrix ................................................................................................ 11
Equation 3.6 - [B] Matrix ................................................................................................ 11
Equation 3.7 - [D] Matrix ................................................................................................ 11
Equation 3.8 - [ABD] Matrix........................................................................................... 11
Equation 3.9 - Laminate Mid-plane Strains ..................................................................... 12
Equation 3.10 - Global Ply Strains .................................................................................. 12
Equation 3.11 - Global Ply Stresses ................................................................................ 12
Equation 3.12 - Local Ply Strains .................................................................................... 12
Equation 3.13 - Local Ply Stresses .................................................................................. 12
Equation 3.14 - Tsai-Wu Failure Criterion ...................................................................... 13
Equation 3.15 - Tsai-Wu Failure Criterion Constants ..................................................... 13
Equation 3.16 - Laminate 0º Tensile Modulus ................................................................ 14
Equation 3.17 - Laminate 90º Tensile Modulus .............................................................. 14
Equation 3.18 - Laminate Through Thickness Tensile Modulus .................................... 14
Equation 3.19 - Laminate In-plane Poisson’s Ratio ........................................................ 14
Equation 3.20 - Laminate xz Poisson’s Ratio .................................................................. 14
Equation 3.21 - Laminate yz Poisson’s Ratio .................................................................. 14
Equation 3.22 - Laminate In-plane Shear modulus ......................................................... 14
Equation 3.23 - Laminate xz Shear Modulus .................................................................. 14
Equation 3.24 - Laminate yz Shear Modulus .................................................................. 14
Equation 4.1 - Member Weight ....................................................................................... 16
Equation 4.2 - Gusset Plate Weight ................................................................................. 16
Equation 4.3 - Road Deck Weight ................................................................................... 17
Equation 4.4 - Vehicle Weight ........................................................................................ 18
Equation 4.5 - Snow Weight ............................................................................................ 18
Equation 4.6 - Wind Drag Force...................................................................................... 19
ix
Equation 5.1 - Beam Slenderness Ratio .......................................................................... 23
Equation 5.2 - Nodal Load to Force Relation .................................................................. 26
Equation 5.3 - Coefficient Matrix .................................................................................... 27
Equation 5.4 - Force Vector............................................................................................. 27
Equation 5.5 - Load Vector ............................................................................................. 27
Equation 6.1 - Member Force .......................................................................................... 31
Equation 7.1 - Margin of Safety ...................................................................................... 41
Equation 7.2 - Critical Buckling Load ............................................................................. 44
Equation 7.3 - Tsai-Wu Failure Criterion ........................................................................ 46
x
LIST OF SYMBOLS
LB – Truss Span [ft]
Lm – Truss Member Span [in]
LT – Truck Length [ft]
Lm – Truss Member Length [in]
wB – Truss Width [in]
tr – Truss Road Deck Thickness [in]
tg – Truss Gusset Plate Thickness [in]
tk – kth Ply thickness [in]
tp – Ply Thickness [in]
A – Cross Sectional Area [in2]
Am – Truss Member Cross Sectional Area [in2]
Ag – Truss Gusset Plate Cross Sectional Area [in2]
g – Weight of Gusset Plate [lb]
m – Weight of Truss Member [lb]
r – Weight of Road Deck [lb]
v – Weight of Vehicles on Bridge [lb]
WT – Weight of Heavy Truck [lb]
s – Weight of Snow on Bridge [lb]
Ps – Snow Design Load [lb/ft2]
P – Weight of Vehicle Load, Snow Load, and Road Deck [lb]
CD – Coefficient of Drag
Fm – Member Force [lb]
Pcr – Critical Buckling Load [lb]
R – Reaction Force [lb]
Fi – Force in Member i [lb]
FD –Drag Force [lb]
FDm – Truss Member Drag Force [lb]
V – Wind Velocity (MPH)
[C] – Coefficient Matrix
{F} – Force Vector [lb]
xi
{P} – Load Vector [lb]
[S] – Stiffness Matrix [1/psi]
[Q] – Reduced Stiffness Matrix [psi]
[𝑄̅ ] – Transformed Reduced Stiffness Matrix [psi]
[T] – Transformation Matrix
[ABD] – Laminate Stiffness Matrix
[A] – Top Left 3x3 Section of [ABD] Matrix [lb/in]
[B] – Bottom Left and Top Right 3x3 Sections of [ABD] Matrix [lb]
[D] – Bottom Right 3x3 Section of [ABD] Matrix [lb-in]
[abd] – Inverse of Laminate Stiffness Matrix
[a] – Top Left 3x3 Section of [abd] Matrix [in/lb]
[b] – Bottom Left and Top Right 3x3 Sections of [abd] Matrix [1/lb]
[d] – Bottom Right 3x3 Section of [abd] Matrix [1/lb-in]
σ – Stress [psi]
σm – Member Stress [psi]
ε – Strain [in/in]
τ – Shear Stress [psi]
γ – Shear Strain [in/in]
κ – Curvature [1/in]
N – Force per unit length of laminate [lb/in]
M – Moment per unit length of laminate [in-lb/in]
 – Poisson’s Ratio
E – Elastic Modulus [psi]
G – Shear Modulus [psi]
I – Moment of inertia [in4]
ρ – Density [lb/in3]
{z} – Through Thickness Laminate Coordinate Vector [in]
H – Total Laminate Thickness [in]
M.S. – Margin of Safety
θ – Ply Orientation angle [radians]
k – Laminate Ply Index
xii
t – Tensile
c – Compressive
u – Ultimate
y – Yield
f – Failure
1 – Local Axial Direction
2 – Local Transverse Direction
3 – Local Through Thickness Direction
x – Global Axial Direction
y – Global Transverse Direction
z – Global Through Thickness Direction
i – Matrix Row Index
j – Matrix Column Index
0 – Indicates mid-plane when used as superscript
s – Steel
cm – Composite Material
F1 – Tsai-Wu Failure Constant
F2 – Tsai-Wu Failure Constant
F11 – Tsai-Wu Failure Constant
F22 – Tsai-Wu Failure Constant
F66 – Tsai-Wu Failure Constant
xiii
GLOSSARY
CAD – Computer Aided Design
ANSYS Workbench and APDL – Finite element program used for structural analysis
CATIA – Computer Aided Three-Dimensional Interactive Application
FEM – Finite Element Model
CLT – Composite Laminate Theory
Composite Laminate – Two or more plies stacked in a sequence
Symmetric Laminate – A laminate of plies that are symmetric about the mid plane
Balanced Laminate – A laminate for which each positive orientation ply has a
corresponding negative orientation ply of the same thickness and material.
Ply – A single layer of composite material
MATLAB – A programming language for technical computing from The MathWorks
Cross Ply – A ply with an orientation between 0 and 90 degrees
Unidirectional Ply – A ply with fibers running in only one direction
2D – Two Dimensional
3D – Three Dimensional
xiv
ABSTRACT
The main purpose of this project is to design and analyze a truss structure bridge using
composite materials that is stronger and lighter than an identical steel bridge. The
geometry and loading conditions of the bridge are sized to mimic a real world
environment. Every attempt was made to adhere to both state and federal regulations.
The layup of the composite material bridge members was optimized to find the laminate
that outperforms to the steel material the most. The analysis was performed using the
method of joints and an ANSYS finite element model. Mesh studies were performed on
all ANSYS finite element models to ensure solution convergence. An identical analysis
was completed for the steel truss bridge. A comparison of the strength was made by
evaluating the minimum margin of safety in all truss bridge members. To make a fair
evaluation both composite material and steel truss bridges have identical geometries.
The intent is to compare which material is more efficient when constructing a truss
bridge.
xv
1. Introduction/Background
A bridge is the solution to a puzzle. It solves the common problem of how to span an
obstacle through the use of basic engineering principles. This solution comes in many
forms and the best is the most efficient, elegant, and safest. One of the more basic types
of bridge is a truss structure. This is comprised of a collection of straight members
organized in such a way that any load is transferred into the surrounding structure. The
members of a truss bridge are connected at gusset plates. Numerous different geometries
are possible in a truss bridge but the one to be analyzed in this report is the Warren truss,
shown in Figure 1.1. This bridge design was first created by James Warren and
Willoughby Monzoni in 1848 and is characterized by alternating equilateral triangles.
Figure 1.1 - Warren Truss
1.1 Truss Geometry
Dimensions of the Warren truss analyzed in this project were sized per the design
standards of the United States Department of Transportation Federal Highway
Administration [10]. This designates the minimum clearance of the bridge be no less
than 16 feet. To meet this requirement and the requirement that the triangles in the
bridge be equilateral the bridge members need to be 18.5 feet long. The design standard
[10] also requires an absolute minimum lane width of 11 feet and strongly recommends
wider. To be conservative and increase the safety of the bridge the lane width was set at
15 feet. The bridge design is for a two lane roadway which results in a total bridge width
of 30 ft. To ensure that a bridge node coincided with the center of the span the number of
triangles in the bridge was set to seven. This results in a bridge span of 74 feet. These
dimensions are summarized in Table 1.1.
1
Dimension
Span
Lane Width
Bridge Width
Member Length
Clearance Height
Length (ft)
74
15
30
18.5
16
Table 1.1 - Truss Dimensions
A wireframe schematic of these dimensions can be seen in Figure 1.2.
Figure 1.2 - Truss Wireframe Schematic
1.2 Truss Member Geometry
All members of the truss were modeled as long axial tension and compression beams
with a hollow square cross section. This cross section type is shown in Figure 1.3.
Figure 1.3 - Truss Member Cross Section
The wall thickness and side length of the members was chosen to uphold the loads
detailed in later chapters. The area and moment of inertia of these members are detailed
in Table 1.2.
2
Dimension
Wall Thickness (in)
Side (in)
Area (in2)
Izz (in4)
Magnitude
2
10
64
725
Table 1.2 - Member Cross Section Dimensions
1.3 Truss Gusset Plate Geometry
The truss members are connected at the nodes by gusset plates. These gusset plates
transfer the truss loads between the truss members. Each member is sandwiched between
two gusset plates. The assembly of two gusset plates and a member is then fastened with
by a bolted connection. In the truss there are three different gusset plate geometries.
These are labeled gusset plate a, gusset plate b, and gusset plate c. The type of gusset
plate at each node is shown in Figure 1.4.
b
a
c
c
c
c
b
c
Figure 1.4 - Node Gusset Plate Type
There are two instances of gusset plate a in the truss. Used only at the very
bottom edges, gusset plate a is a connection between only three members. This type of
gusset plate can be seen in Figure 1.5.
3
a
Figure 1.5 - Gusset Plate Type a
There are also only two instances of gusset plate b in the truss. This type is used
at the top left and right of the truss. Gusset plate b connects four members of the truss.
This type of gusset plate can be seen in Figure 1.6.
Figure 1.6 - Gusset Plate Type b
There are five instances of gusset plate c in the truss. This type is used on the
interior nodes of the truss. Gusset plate c connects five members of the truss. This type
of gusset plate can be seen in Figure 1.7.
Figure 1.7 - Gusset Plate Type c
4
2. Materials
Most truss bridges are constructed out of structural steel but wooden truss bridges are
not uncommon if the loading is minimal. In bridge designs that utilize these materials the
stresses are computed and then compared with a material allowable. If the stress is too
high a designer has only two choices. One is to increase cross sectional area and the
other is to redesign the geometry of the truss to more evenly distribute load. Each of
these choices has unfortunate tradeoffs. Increasing cross sectional area increases weight
which adds additional loading the truss has to carry. It can also cause other geometric
problems which may violate the design parameters of the bridge. Redesigning the truss
geometry adds to the number of connections needed in the truss and possible points of
failure. When these two options are not available the designer has no choice but to
change material which can lead to the need for larger or smaller cross sectional area. To
avoid this problem the designer can choose the only kind of material that gives full
customization ability of stiffness and strength without having to change geometry. These
materials are composite materials.
Composite materials are known for their ability to be tailored to any situation. A
composite layup can have almost any strength and/or stiffness in any direction. In
addition they often have better strength, stiffness, and corrosion properties as well as
lower weight then standard metallic materials. These properties make composite
materials ideal to use when designing truss bridge members. The layup of each member
can be customized to meet the demands of each individual part without significantly
increasing weight or cross sectional area.
2.1 Steel Material Selection
Multiple types of steel were considered for the truss members. These included low,
intermediate, and high alloy steels that are heat treated and otherwise processed into
many different strength levels. These heat treatments and processing methods affect the
alloy microstructure and thus the material properties. The material properties most
important to the steel selected for the member are the tensile strength, stiffness, and
density. These are important for the following reasons. A high tensile strength is
important as it determines how much load the members can withstand before breaking.
5
Having a lower stiffness is important in reducing the buckling load of members and
improving the ductility of the truss. Low density is important to reduce the overall
weight of the truss which is a major factor affecting how much dead load the bridge
must hold up. Table 2.1 is a list of all the candidate steel alloys. They are ranked on a
ratio of ultimate tensile strength, σtu, to density, ρ. Since a high tensile strength and low
density is desired the alloy with the highest ratio of these two properties was selected.
The alloy with the highest ratio is 5Cr-Mo-V which is classified as an intermediate steel
alloy. This means the amount of alloy elements in the steel are above those classified as
low alloy steels but below those in stainless steels.
Alloy Type
Alloy Name
Intermediate Alloy
Low Alloy
High Alloy
High Alloy
High Alloy
High Alloy
Low Alloy
Intermediate Alloy
High Alloy
Low Alloy
5Cr-Mo-V
0.42C 300M
280 Maraging
Ferrium S53
AerMet100
280 Maraging
0.40C 300M
5Cr-Mo-V
AerMet100
AISI 4340
σtu
(ksi)
280
280
280
280
275
275
270
260
262
260
ρ
(lb/in3)
0.281
0.283
0.286
0.288
0.285
0.286
0.283
0.281
0.285
0.283
Et
(Msi)
30
29
26.5
29.6
28
26.5
29
30
28
29
Ec
(Msi)
30
29
28.6
30.7
28.1
28.6
29
30
28.1
29
𝝈𝒕𝒖
𝝆
996
989
979
972
965
962
954
925
919
919
Table 2.1 - Steel Alloys [6]
The full material properties of 5Cr-Mo-V are shown in Table 2.2.
Property
E (Msi)

G (Msi)
σtu (ksi)
σty (ksi)
σcy (ksi)
ρ (lb/in2)
Value
30
0.36
11
280
240
-260
0.281
Table 2.2 - 5Cr-Mo-V Steel Properties [6]
2.2 Composite Material Selection
Many materials fit the description of a composite material. Examples include concrete,
wood, reinforced plastics, and many other materials. The definition of a composite
6
material is the combination of two different substances into one. The generic names for
these substances are the fiber and the matrix. The fibers are a very long and thin material
that is very strong and stiff in one direction while weak in the others. The matrix is an
isotropic glue material that holds the fibers together. In order to achieve the highest and
stiffest material properties all the fibers in a composite material are aligned in one
direction. This direction is often times the direction of the highest load on the part. As a
result composite materials lend themselves well to axially loaded materials. In this
regard they make a very good material for truss members.
As with other materials strength is the most important property of a composite
material for structural applications. Other key properties of the material are density and
stiffness. Low density helps alleviate the weight the truss must sustain and low stiffness
helps members withstand possible buckling. Hexcel Corporation offers a wide selection
of composite material prepreg and specifies their properties on its website [3]. Each uses
a different resin system and fiber type. The resins available from Hexcel are Epoxy, BMI
(Bismaleimide), Cyanate, and Phenolic. The one most appropriate for use in the
composite truss is the Epoxy resin. This is because this type has the highest tensile
strength and lowest density. Further selection of a type of Epoxy prepreg is achieved by
looking at the individual properties of the materials. Table 2.3 is a list of possible
candidate epoxy prepregs offered by Hexcel. This table is ordered by the ratio of tensile
strength to density.
HexPly
Brand
8552
M73
EH04
F515
F593
M76
M74
F155
F263
F161
F185
Fiber
Type
IM7
IM7
M35J
IM6
T2G145
M46J
M55J
T2C145
T3T145
Glass Fabric
Kevlar
σ1t
(ksi)
395
364
377
243
220
315
319
266
198
66
74
Et (Msi)
23.8
23.5
26.5
21.7
18.3
39.6
52.8
18.3
19.2
3.1
3.5
ρ
(lb/in3)
0.0470
0.0466
0.0484
0.0433
0.0441
0.0470
0.0470
0.0482
0.0458
0.0449
0.0465
Table 2.3 - Composite Materials with Epoxy Resin System [3]
7
𝝈𝟏𝒕
𝝆
8404
7812
7789
5606
4992
6708
6794
5516
4316
1470
1584
The HexPly brand chosen from Table 2.3 for the truss was 8552 IM7 prepreg. This has a
very high tensile strength and low density as well as a high stiffness. The full material
properties of 8557 IM7 are shown in Table 2.4.
Property
Value
E1 (Msi)
23.8
E2 (Msi)
1.7
E3 (Msi)
1.7

0.32

0.32

0.0229
G12 (Msi)
0.75
G13 (Msi)
0.75
G23 (Msi)
σ1t (ksi)
σ1c (ksi)
σ2t (ksi)
σ2c (ksi)
τ12f (ksi)
ρ (lb/in2)
tp (in)
0.831
395
-245
16.1
-32.3
17.4
0.047
0.006
Table 2.4 - 8552 IM7 Material Properties [3]
8
3. Composite Laminate Theory
A composite laminate is made up of two or more plies of composite material. Plies are a
combination of fibers and matrix in a very thin sheet of material. In a unidirectional ply
of material the fibers all align in the same direction. The angle these fibers make with the
longitudinal direction of the entire laminate is referred to as the orientation of a ply. The
matrix is a substance that holds the fibers together. The fibers and matrix of a single ply
are depicted in Figure 3.1
Matrix
Figure 3.1 - Composite Ply
Combining plies of varying material, thickness, and orientation can create a laminate
with any type of material properties the designer requires. The method to calculate the
stress and strain in a laminate and in each individual ply when subject to a load is called
Composite Laminate Theory or CLT [4].
3.1 CLT Assumptions
CLT assumes that each ply in a laminate is flawlessly bonded to any adjacent ones. This
prevents any layers from slipping relative to each other and allows the laminate to act as
a single layer of material when stressed and strained. Many of the assumptions of plate
theory are also relevant in CLT. This includes the supposition that plane sections of the
laminate remain plane under deformation; meaning lines perpendicular to the mid-plane
remain so under deformation. This leads the second assumption that the shear strain
perpendicular to the mid-surface, γxz and γyz, is zero. In addition, the stress through the
thickness is assumed to be zero. These assumptions are in line with a plane stress
condition. The third assumption is that the out of plane strain, εz, is also zero. One of the
results of this assumption is that CLT is invalid at edges of laminates. In these regions
the inter-laminar stresses are high and CLT becomes less accurate.
9
3.2 The ABD Matrix
Before the stress and strain in a laminate can be calculated the material properties of
each individual ply must be compiled into the Laminate Stiffness Matrix or [ABD]
matrix. The first step in calculating this matrix is to calculate the [Q] matrix for each ply.
This is done using equation [3.1].
𝐸1
Equation 3.1 - Q Matrix
𝜐12 𝐸2
𝐸
1 − 𝜐12 2 𝐸2
1
𝜐12 𝐸2
[𝑄] =
𝐸
1 − 𝜐12 2 𝐸2
1
[
0
𝐸
1 − 𝜐12 2 𝐸2
0
1
𝐸2
𝐸
1 − 𝜐12 2 𝐸2
1
0
0
[3.1]
𝐺12 ]
The [Q] matrix is also called the reduced stiffness matrix because it is the result of
applying a plane stress condition, σ3 = 0, to the full stiffness matrix, [S].
The second step is to create a coordinate system through the thickness of the
laminate. This coordinate system is input into the {zk} vector. In this vector z is the
distance through the thickness of the kth ply. This can be seen in Figure 3.2.
Figure 3.2 - Through Thickness Coordinate System [4]
The equation used for each value of {zk} is shown in equation [3.2].
Equation 3.2 - Through
Thickness Vector
𝑧𝑘 = −
𝐻
+ 𝑡𝑘
2
[3.2]
The third step is the transform the [Q] matrix into the [𝑄̅ ] matrix. This is done
with the transformation matrix [T]. The transformation matrix is computed for each ply
10
and is determined by a plies orientation angle θ. Equation [3.3] shows how to compute
the transformation matrix.
Equation 3.3 Transformation Matrix
cos2 𝜃
[𝑇] = [ sin2 𝜃
− cos 𝜃 sin 𝜃
sin2 𝜃
cos 2 𝜃
cos 𝜃 sin 𝜃
2 cos 𝜃 sin 𝜃
−2 cos 𝜃 sin 𝜃 ]
cos 2 𝜃 − sin2 𝜃
[3.3]
The [𝑄̅ ] matrix can then be computed using equation [3.4].
̅ ] Matrix
Equation 3.4 - [𝐐
[𝑄̅ ] = [𝑇]−1 [𝑄][𝑇]
[3.4]
Finally the fourth step is the computation of the [A], [B], and [D] matrix and
compilation of them into the [ABD] matrix. The equations for these matrices can be seen
in equation [3.5], [3.6], and [3.7]. These matrices are a compilation of each plies [Q]
matrix and position in the laminate.
𝑁
Equation 3.5 - [A] Matrix
𝐴𝑖𝑗 = ∑(𝑄̅𝑖𝑗 )𝑘 (𝑧𝑘 − 𝑧𝑘−1 )
[3.5]
𝑘=1
𝑁
Equation 3.6 - [B] Matrix
1
𝐵𝑖𝑗 = ∑(𝑄̅𝑖𝑗 )𝑘 (𝑧𝑘 2 − 𝑧𝑘−1 2 )
2
[3.6]
𝑘=1
𝑁
Equation 3.7 - [D] Matrix
1
𝐷𝑖𝑗 = ∑(𝑄̅𝑖𝑗 ) (𝑧𝑘 3 − 𝑧𝑘−1 3 )
𝑘
3
[3.7]
𝑘=1
The [A], [B], and [D] matrix are combined into the 6x6 [ABD] in the following
manner. The upper left 3x3 section of the [ABD] matrix is made up of the [A] matrix,
the lower 3x3 section is made up of the [D] matrix, and the upper right and lower left
3x3 sections are made up by the [B] matrix. This can be seen in equation [3.8].
Equation 3.8 - [ABD]
Matrix
[𝐴𝐵𝐷] = [𝐴
𝐵
𝐵
]
𝐷
[3.8]
3.3 Laminate Stress and Strain
Using the laminate stiffness matrix the stresses and strains in the laminate can be
calculated. The other pieces of information needed to compute these values are the
forces and moments acting on the laminate. The laminate stiffness matrix and force
moment vector can be combined to find the mid-plane strains of the laminate as shown
in equation [3.9].
11
Equation 3.9 - Laminate
Mid-plane Strains
𝑁𝑥
𝜀𝑥𝑜
𝑜
𝑁
𝜀𝑦
𝑦
𝑜
𝑁
𝑥𝑦
𝛾𝑥𝑦
= [𝐴𝐵𝐷]−1
𝑀
𝜅𝑥
𝑥
𝑀𝑦
𝜅𝑦
{𝜅𝑥𝑦 }
{𝑀𝑥𝑦 }
[3.9]
From the mid-plane strains the global strain at the top of each ply in the laminate
coordinate system can be calculated using equation [3.10].
Equation 3.10 - Global Ply
Strains
𝜀𝑥𝑜
𝜀𝑥
𝜅𝑥
𝑜
𝜀
𝜅
𝜀
{ 𝑦 } = { 𝑦 } + 𝑧𝑘 { 𝑦 }
𝑜
𝛾𝑥𝑦
𝜅𝑥𝑦
𝛾𝑥𝑦
𝑘
[3.10]
Then, using Hooke’s Law the global stress at the top of each ply in the laminate
coordinate system can be calculated using equation [3.11].
𝜎𝑥
𝜀𝑥
Equation 3.11 - Global Ply
𝜎
𝜀
̅
{ 𝑦 } = [𝑄 ] { 𝑦 }
Stresses
𝜏𝑥𝑦 𝑘
𝛾𝑥𝑦 𝑘
[3.11]
To calculate the global stress and strain at the bottom of each ply zk in equation [3.10]
should be replaced with zk+1.
To calculate the local stress and strain in the ply coordinate system the global
stresses and strains are transformed using their transformation matrix. This is shown in
equation [3.12] and [3.13].
Equation 3.12 - Local Ply
Strains
{1
2
Equation 3.13 - Local Ply
Stresses
𝜀1
𝜀2
𝛾12
} = [𝑇] {1
𝜀𝑥
𝜀𝑦
𝛾𝑥𝑦
2
𝜎𝑥
𝜎1
𝜎
𝜎
{ 2 } = [𝑇] { 𝑦 }
𝜏𝑥𝑦 𝑘
𝜏12 𝑘
𝑘
}
[3.12]
𝑘
[3.13]
3.4 Laminate Failure Criterion
Failure in composite materials is unique in that each ply of the laminate fails under a
different load. This is due to the varying material properties and orientation of each
layer. As a result each individual ply must be checked. This is done by using the TsaiWu failure criterion [4]. The equation for this criterion is shown in equation [3.14].
12
Equation
3.14 - TsaiWu Failure
2
𝐹1 𝜎1 + 𝐹2 𝜎1 + 𝐹11 𝜎12 +𝐹22 𝜎22 +𝐹66 𝜏12
− √𝐹11 𝐹22 𝜎1 𝜎2 = 1
[3.14]
Criterion
The constants in equation [3.14] and are calculated using equation [3.15]. In this
𝐹
equation 𝜎1𝑇 , 𝜎1𝐶 , 𝜎2𝑇 , 𝜎2𝐶 , and 𝜏12
are properties of the material.
1
1
1
1
𝐹1 = ( 𝑇 + 𝐶 ) 𝐹2 = ( 𝑇 + 𝐶 )
𝜎1 𝜎1
𝜎2 𝜎2
Equation 3.15 - Tsai-Wu
2
Failure Criterion Constants
𝐹11
[3.15]
1
1
1
= − ( 𝑇 𝐶 ) 𝐹22 = − ( 𝑇 𝐶 ) 𝐹66 = ( 𝐹 )
𝜏12
𝜎1 𝜎1
𝜎2 𝜎2
The Tsai-Wu failure criterion for orthotropic materials is similar to the von Mises failure
criterion for isotropic materials. They both can be visualized in 3D space. Since
unidirectional composite materials are very stiff in the longitudinal direction and weak in
the transverse and through thickness direction the Tsai-Wu criterion takes the shape of
an ellipsoid. This ellipsoid is shown in Figure 3.3. The stiff longitudinal direction is the
1 direction whereas the weaker transverse and through thickness directions are 2 and 3
respectively.
Figure 3.3 - Tsai Wu Failure Criterion Ellipsoid [4]
Inside the ellipsoid the ply will not fail but outside the boundary the ply will breakdown.
3.5 Laminate Material Properties
The overall material properties of the laminate can be calculated using values from the
laminate stiffness matrix, [ABD] matrix. These properties are useful to get a sense of
how the laminate as a whole will respond to various forces and moments. The equations
for these properties can be seen in Table 3.1 [4].
13
Laminate Property
Variable
Equation 3.16 - Laminate 0º Tensile Modulus
Ex
Equation 3.17 - Laminate 90º Tensile Modulus
Ey
Equation 3.18 - Laminate Through Thickness Tensile
Modulus
Ez
Equation 3.19 - Laminate In-plane Poisson’s Ratio
xy
Equation 3.20 - Laminate xz Poisson’s Ratio
xz
Equation 3.21 - Laminate yz Poisson’s Ratio
yz
Equation 3.22 - Laminate In-plane Shear modulus
Gxy
Equation 3.23 - Laminate xz Shear Modulus
Gxz
Equation 3.24 - Laminate yz Shear Modulus
Gyz
Equation
1
𝑎11 𝐻
1
𝑎22 𝐻
1
𝑎22 𝐻
𝑎12
−( )
𝑎11
𝑎12
−( )
𝑎22
𝑎12
−( )
𝑎22
1
𝑎33 𝐻
1
𝑎33 𝐻
𝐸𝑦
2(1 + 𝜐𝑦𝑧 )
Table 3.1 - Laminate Material Properties
3.6 Member Laminate Layup
The composite material layup of the members must be optimized to fit the loading
conditions on the truss. This is done by varying the number of plies and their orientation
to the longitudinal direction of the member. As will be described in the truss loads
chapter of this report, each member must be strong enough to withstand all axial and
transverse forces. Table 3.2 is a list of the laminates considered for the truss members in
this project. This list contains laminates that have varying amounts of cross plies or nonzero orientation. They are broken down into four families. The first is laminates
containing 45 and 0 degree plies, the second containing 30 and 0 degree plies, the third
containing 90 and 0 degree plies, and the fourth containing a combination of
orientations. Indicated next to each laminate is the percent of plies that are considered
cross plies. The name of each laminate corresponds to its makeup. For example the [0340]
14
laminate contains 340 zero degree plies and the [3017/-3017/0136]S laminate contains
seventeen 30 degree plies, followed by seventeen -30 degree plies, followed by 136 zero
degree plies and then the mirrored about the mid-plane. The S denotes it is symmetric
about the mid-plane.
Table 3.2 - Candidate Composite Laminates
It should be noted that the list of laminates includes only balanced and symmetric
laminates. This is to avoid any unnecessary bending, twisting, stretching, or shearing
that can be caused by having an anti-symmetric or unbalanced laminate layup. Included
in the table of candidate layups is the material properties of the entire laminate
calculated using the equations in Table 3.1. An analysis of each individual layup will be
performed in later sections to determine which one is best.
15
4. Truss Loads
There are three types of loads all bridges must withstand. These are a dead load, a live
load, and a dynamic load. These three types are treated individually in the following
sections.
4.1 Dead Load
The dead load on a bridge is weight due to its structure. This is made up of the weight of
the truss members, gusset plates, and road deck. These loads never change during the life
of the bridge.
The weight of one truss member was calculated using equation [4.1].
Equation
𝑚 = 𝜌𝐴𝑚 𝐿𝑚
4.1 Member
[4.1]
Weight
There are 15 members per side of the bridge and 9 cross members in the truss for a total
of 39 members. The weight from all members in the truss was calculated using the
values shown in Table 4.1.
Member Variable
ρs (lb/in3)
ρcm (lb/in3)
Am (in2)
Lm (in)
# Members
# Cross Members
Magnitude
0.281
0.047
64
222
15
9
Table 4.1 - Member Weight Variables
Using equation [4.1] and the values of Table 4.1 the weight of each steel member equals
3,992 lbs while the weight of each composite member equals 668 lbs. All 39 members in
the truss add up to a total member weight of 155,705 lbs for the steel truss and 26,043
lbs for the composite truss.
The weight of each gusset plate was calculated using equation [4.2].
Equation
4.2 - Gusset
Plate
𝑔 = 𝜌𝐴𝑔 𝑡𝑔
[4.2]
Weight
Each node of the truss contains 2 gusset plates. Since there are 9 nodes per side of the
truss there are 18 gusset plates per side. This means the entire truss has a total of 36
16
gusset plates. Of the 18 gusset plates per side 4 of them are type a, 4 are type b, and 10
are type c. The weight due to all of these gusset plates was calculated with the values
shown in Table 4.2.
Gusset Plate Variable
ρs (lb/in3)
ρcm (lb/in3)
tg (in)
Ag Type a (in2)
Ag Type b (in2)
Ag Type c (in2)
# of Type a Plates
# of Type b Plates
# of Type c Plates
Magnitude
0.281
0.047
1
500
824
1097
8
8
20
Table 4.2 - Gusset Plate Weight Variables
Using equation [4.2] and the values in Table 4.2 the weight of steel gusset plates a, b,
and c equal 140 lbs, 231 lbs, and 308 lbs. The weight of the corresponding composite
gusset plates equal 23 lbs, 39 lbs, and 52 lbs. All 18 gusset plates in the steel truss weigh
a total of 9,140 lbs and all gusset plates in the composite truss weigh 1,529 lbs.
The weight of the road deck was calculated using the density of asphalt and the
volume of the road as shown in equation [4.3]. The volume of the road deck was
calculated using the length and width of the bridge in combination with the thickness of
the road deck.
Equation
4.3 - Road
Deck
𝑟 = 𝜌𝐴𝑠𝑝ℎ𝑎𝑙𝑡 𝐿𝐵 𝑤𝐵 𝑡𝑟
[4.3]
Weight
The values used to calculate the weight of the road deck are shown in Table 4.3.
Road Deck Variable
ρasphalt (lb/ft3)
LB (ft)
wB (ft)
tr (ft)
Magnitude
45 [7]
74
30
1
Table 4.3 - Road Deck Weight Variables
Using equation [4.3] and the values in Table 4.3 the weight of the road deck equals
99,900 lbs.
17
4.2 Live Load
The live load on a bridge is weight due to items traveling over the bridge or weights that
may temporarily put load on the bridge. This is a combination of the weight of the
vehicles using the bridge and the snow that can accumulate on the road deck. These
loads change and get redistributed over the life of the bridge.
The vehicle weight on the bridge was calculated by assuming each lane of the
bridge is packed end to end with the heaviest allowed vehicles. According to the
department of transportation the heaviest truck allowed on a highway weighs 80,000 lbs
and measures 51 feet in length [11]. The weight from these vehicles is calculated using
equation [4.4].
𝐿𝐵
𝑣 = 2 ( 𝑊𝑇 )
𝐿𝑇
Equation
4.4 Vehicle
Weight
[4.4]
The values used to calculate the weight of the vehicles are shown in Table 4.4.
Vehicle Weight Variable
LB (ft)
LT (ft)
WT (lb)
Magnitude
74
51
80,000
Table 4.4 - Vehicle Weight Variables
Using equation [4.4] and the values in Table 4.4 the weight of the vehicles equals
232,157 lbs.
A snow load must be accounted for in case the bridge is covered with snow. This
is not an insignificant amount and should not be overlooked when calculating bridge
loads. The State of Connecticut building code specifies a minimum snow load all
structures must meet. In Connecticut the most stringent snow load is 40 lb/ft2 as found in
the Connecticut Building Code [8]. Using this design parameter the snow load on the
bridge is calculated using equation [4.5].
Equation
4.5 - Snow
𝑠 = 𝑃𝑠 𝐿𝐵 𝑤𝐵
Weight
The values used to calculate the weight of the snow are shown in Table 4.5.
18
[4.5]
Snow Load Variable
PS (lb/ft2)
LB (ft)
wB (ft)
Magnitude
40
74
30
Table 4.5 - Snow Load Variables
Using equation [4.5] and the values in Table 4.5 the weight of the snow equals 88,800
lbs.
4.3 Dynamic Load
The dynamic load on a bridge is due to temporary loads on a bridge that might perturb
the structure momentarily. The most common type of dynamic load is wind load which
acts in possibly any direction but most often as against the side faces of the truss. This is
generated in the form of a drag force on the truss. The equation to calculate the drag
force is equation [4.6].
Equation
4.6 - Wind
Drag Force
𝐹𝐷 =
1 2
𝜌𝑉 𝐴𝐶𝐷
2
[4.6]
In this equation V is the wind speed. The United States Department of Transportation
designates that all structures be able to withstand a 110 MPH wind in the county of
Hartford, Connecticut [9]. The A is the area of the item withstanding the wind load. The
area of the gusset plates can be found in Table 4.2 and the area of the members is 185
in2. CD is the coefficient of drag [1]. For the gusset plates this is assumed to be a flat
plate which has a CD of 2.0 and the members are assumed to be a cylinder with a CD of
1.2. The result of inputting these values into equation [4.6] was that members are subject
to a drag force of 48 lb and gusset plates are subject to a drag force of 215 lb, 354 lb, and
472 lb respectively for gusset plate type a, b, and c.
4.4 Truss Free Body Diagram
The loads calculated in the previous sections are applied to the truss in the following
manner. The dead and live load are distributed evenly over each side of the truss while
the dynamic load acts only on only one face of the truss. All loads are assumed to be
reacted by the nodes of the truss. Half the weight of each member is distributed between
its connecting nodes. The weight of one member is denoted by the variable m. The
19
weight of each gusset plate is reacted at the node to which it is attached and denoted by
the variable g. The subscript letter for each g denotes the type of gusset plate weight.
The road deck, vehicle, and snow load are denoted by the variable P. P is assumed to be
reacted by the lower three middle nodes. The wind drag force on the gusset plates are
reacted at each node and denoted by the variable FDa, FDb, and FDc. The wind drag force
on each member is distributed evenly between the two nodes it is connected to and
denoted by the variable FDm. Finally, the truss itself is assumed to be simply supported.
The free body diagram of these dead and live bridge loads on the each half of the truss is
shown in Figure 4.1.
Ray
2m
2.5m
2.5m
2m
2gb
2gc
2gc
2gb
Ri
Rax
1.5m
2.5m
2.5m
2.5m
1.5m
2ga
2gc
2gc
2gc
2gc
P/3
P/3
P/3
Figure 4.1 - Dead and Live Truss Load FBD
The free body diagram of the dynamic loads on the front of the bridge can be seen in
Figure 4.2.
1.5FDm
FDb
2FDm
FDa
FDm
FDc
2FDm
FDc
2FDm
FDc
1.5FDm
FDc
2FDm
FDb
FDc
2FDm
FDa
FDm
Figure 4.2 - Dynamic Truss Load FBD
The values of all the loads on the steel and composite truss in Figure 4.1 and Figure 4.2
are summarized in Table 4.6.
20
Load
P
m
ga
gb
gc
FDa
FDb
FDc
FDm
Steel Truss
Weight (lb)
3,992
140
231
308
Composite Truss
Weight (lb)
210,428
668
23
39
52
215
354
472
48
Table 4.6 - Truss Loads
21
5. Analysis Methodology
Application of the dead, live, and dynamic loads on the truss structure will generate
stress and strain in the members. These stresses are distributed throughout the geometry
based on the axial and transverse forces that develop. The calculation of member stresses
and forces was performed using three methods. The first method was a 2D ANSYS finite
element model, the second was by using the method of joints [2] and third was a 3D
ANSYS finite element method. Computation of the stresses and forces using these three
methods are described in this section. The 2D ANSYS FEM and method of joints were
used to calculate the axial stresses in the members under the dead and live load while the
3D ANSYS FEM calculates the transverse member stresses.
5.1 2D ANSYS Finite Element Model
As previously described a 2D ANSYS FEM was used to calculate the axial stresses in
each truss member. This method breaks down a CAD model of the truss geometry into
discrete elements and nodes. The stresses inside these elements are then calculated based
on the specific loads and boundary conditions on each individual node. Plotting the
stress in all the elements of a FEM gives an overall picture of how the load is distributed
throughout the geometry. In this analysis the finite element software that was utilized
was ANSYS APDL. The results were axial stresses in each member. To convert the
axial stresses into axial member forces the axial stresses are multiplied by the cross
sectional area of each member. In the case of the composite truss members the resulting
axial forces were input into CLT as global axial forces on the laminate to compute local
ply stresses.
5.1.1
Model Geometry, Element Type, and Mesh
The CAD model of the truss was generated using lines in ANSYS. The members were
modeled as lines and the nodes were modeled as key points. Figure 5.1 shows this CAD
model.
22
Figure 5.1 - 2D ANSYS FEM Geometry
Once the CAD model was created an element type was selected. BEAM188 was chosen
as the most appropriate element for this analysis. Figure 5.2 shows a BEAM188 element.
Figure 5.2 - BEAM188 Element [5]
BEAM188 elements are used for slender to moderately thick beam structures. This type
of element is based on Timoshenko beam theory and is an ideal type of element for
bridge truss geometry. The members fit the definition of slender beam structures very
well. ANSYS recommends the slenderness ratio, defined in equation [5.1], must be
greater than 30 for beams to be adequately modeled using a BEAM188 element type.
Equation 5.1 - Beam
Slenderness Ratio
Slenderness Ratio 
GAL2
EI
[5.1]
The slenderness ratio of each steel truss member is 1594 while each composite truss
member is 137. These slenderness ratios are significantly above the recommended
minimum. In Timoshenko beam theory the higher the slenderness ratio of a beam, the
more accurate the results of the structural analysis.
BEAM188 element type also allows the definition of beam cross sectional shape.
Possible shapes include quadrilaterals, circles, rings, c-channels, I beams, z shapes, L
brackets, T brackets, box shapes, and many others. Since the beams have a hollow
rectangular cross section the box shape was chosen.
23
The 2D FEM was meshed using quadrilateral mapped meshing. This resulted in a
very nice mesh. The mesh density was refined until it could be shown that adding more
elements to the model did not change the resulting solution. This convergence point was
reached on all 2D finite element models before results were trusted. Figure 5.3 is a view
of the converged mesh. Note the model looks to be 3D because the BEAM188 elements
have the graphical ability to show a cross sectional view of the members but in reality
acts only in two dimensional space.
Figure 5.3 - 2D ANSYS FEM Mesh
5.1.2
Material Properties, Loads, and Boundary Conditions
The steel truss material was defined as linear isotropic and the composite truss material
was defined as linear orthotropic. The inputs for the isotropic material are elastic
modulus and Poisson ratio while the orthotropic material requires elastic modulus,
Poisson ratio, and shear modulus in each of the three material directions.
Loads were applied to the truss per the free body diagram in the previous chapter.
The application point of these loads was at the nodes connecting the beams of the truss.
Dead and live loads were applied in the vertical y direction. The result is stress in the
axial x direction.
The boundary conditions were set such that the truss is simply supported. This
means that node A is fixed in space and node I is set as a pinned support. This translates
to restraining node A from movement in the x, y, and z direction as well as a rotationally
constraining it around the x and y axis. It also sets node I as restrained from movement
in the y and z directions and the rotationally restrained are around the x and y axis.
24
5.2 Method of Joints
The second method of calculating the axial forces due to the dead and live loads on the
truss is the method of joints. Calculating forces and stresses in this manner is a good
check that the FEM is performing as it should. In this method a free body diagram is
drawn at each joint or node. Each side of the Warren truss in this analysis has a total of
nine nodes and fifteen members. Since the truss is simply supported this results in 15
unknown member forces and 3 unknown reaction forces. A sum of forces in the x and y
direction at each of the nine nodes results in a total of eighteen equations. The system of
eighteen unknowns and eighteen equations can be solved using matrix algebra. The
results were axial forces in each member. To convert the axial forces into axial member
stresses the axial forces are divided by the cross sectional area of each member. In the
case of the composite truss members the resulting axial forces were input into CLT as
global axial forces on the laminate to compute local ply stresses.
5.2.1
Naming Convention
In order to draw all nine nodal free body diagrams the naming convention, shown in
Figure 5.4, was created. All members are assigned a number and all nodes a letter.
B
D
F
4
3
1
8
5
2
A
7
12
9
6
C
H
11
13
10
E
15
14
G
I
Figure 5.4 - Member and Node Designation
The convention employed in depicting the free body diagrams at each node was that
each member force is drawn positive in tension. Designating this convention is important
when interpreting the results. Free body diagrams at nodes also include reactions forces
which at node A is a fixed support and at node I is a roller support.
25
5.2.2
Nodal Free Body Diagrams
There are three types of connection nodes in a Warren truss. They include edge nodes,
upper side nodes, and internal nodes shown in Figure 5.5. Nodes A and I are edge nodes,
nodes B and H are top side nodes, and nodes C, D, E, F, and G are internal nodes. The
difference between the three types is the amount members connected at each node. Edge
nodes connect two axial members, upper side nodes connect three, and internal nodes
connect five. A sample of the free body diagram for each type of node is shown in
Figure 5.5. Note that the internal node can be flipped about the y axis to represent nodes
on the lower side of the truss.
y
Fii
y
y
Fi
Fi
Fi
x
x
Fiii
P
P
Fiv
x
Fiii
Fii
P
Fii
Figure 5.5 - Edge, Top Side, and Internal Node Free Body Diagrams
5.2.3
Matrix Equation
After all free body diagrams have been drawn, equilibrium equations for the sum of the
forces in the x and y directions can be written. This results in a total of 18 equations
which can then be combined into matrix form as seen in equation [5.2].
Equation 5.2 - Nodal Load
to Force Relation
CF  P
[5.2]
In this equation [C] is the coefficient matrix, {F} is the vector of member forces, and
{P} is the vector of nodal loads. The coefficient matrix, force vector, and load vector are
shown in equations [5.3], [5.4], and [5.5]. Matrix equation [5.2] was solved using
Microsoft Excel.
26
Equation
5.3 Coefficient
[C] =
Matrix
Equation 5.4 - Force
Vector
Equation 5.5 - Load
Vector
cos(60
sin(60)
-cos(60)
-sin(60)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
-1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
cos(60
-sin(60)
-cos(60)
sin(60)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
-1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
cos(60
sin(60)
-cos(60)
-sin(60)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
-1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
cos(60
-sin(60)
-cos(60)
sin(60)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
-1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
cos(60
sin(60)
-cos(60)
-sin(60)
0
0
0
0
0
0
{F} =
{P} =
0
0
0
0
0
0
0
0
1
0
0
0
-1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
cos(60
-sin(60)
-cos(60)
sin(60)
0
0
0
0
F1
F2
F3
F4
F5
F6
F7
F8
F9
F10
F11
F12
F13
F14
F15
Ra1
Ra2
Ri
0
1.5m + 2g a
0
2m + 2g b
0
2.5m + 2g + + P/3
0
2.5m + 2g c
0
2.5m + 2g + + P/3
0
2.5m + 2g c
0
2.5m + 2g + + P/3
0
2m + 2g b
0
1.5m + 2g a
0
0
0
0
0
0
0
0
0
0
1
0
0
0
-1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
cos(60
sin(60)
-cos(60)
-sin(60)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
-1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
cos(60
-sin(60)
-cos(60)
sin(60)
0
-1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-1
[5.3]
[5.4]
[5.5]
5.3 3D ANSYS Finite Element Model
The method of joints and BEAM188 element used in the 2D FEM cannot calculate
transverse forces and stresses. In order to calculate these values a 3D FEM was needed.
This method expands upon the previous 2D FEM geometry. In the 2D FEM the CAD
model was made up of lines and beam elements. In the 3D FEM this was expanded to
rectangular members and solid elements. In this analysis the finite element software used
was ANSYS workbench. The results were transverse stresses in each member. To
27
convert the transverse stresses into transverse member forces the transverse stresses are
multiplied by the cross sectional area of each member. In the case of the composite truss
members the resulting transverse forces were input into CLT as global transverse forces
on the laminate to compute local ply stresses.
5.3.1
Model Geometry and Mesh
The geometry for this model is more complex than it is for the 2D model so it was
created using CATIA V5. In this model the full cross section of each of the members is
reproduced. Figure 5.6 shows the CAD geometry of this model. This model is comprised
of members and connection solids at the interfaces of the members.
Figure 5.6 - 3D CATIA Truss Model
The 3D FEM was meshed using bricks elements that resulted in a very fine mesh.
Just like in the 2D model the mesh density was refined until it could be shown that
adding more elements to the model did not change the resulting solution. This
convergence point was reached before results were trusted. Figure 5.7 is a view of the
converged mesh.
28
Figure 5.7 - 3D ANSYS FEM Mesh
5.3.2
Material Properties, Loads, and Boundary Conditions
The material property inputs for the 3D ANSYS FEM are identical to those of the 2D
ANSYS FEM. Steel was defined as a linear isotropic material and the composite
material was defined as a linear orthotropic material.
The vertical loads on the truss have an effect on the transverse stresses. To
account for this effect both vertical and transverse forces were applied to the 3D FEM.
Vertical loads were applied to the each side of the truss per the vertical free body
diagram in the previous chapter. Transverse loads were applied to only the front of the
truss as shown in the transverse free body diagram in the previous chapter. The
application point of these loads was at the connection plates of the members of the truss.
To get results from this FEM stresses were plotted in the transverse z direction.
The boundary conditions were set so the truss is simply supported. In the 3D
ANSYS model this meant applying a fixed constraint to the left side of the truss and a
zero displacement constraint in the y and z direction to the right side.
29
6. Results
The results of applying the analysis methodologies detailed in the previous chapter are
presented in the following sections. Results of the steel material truss are presented
along with those from each candidate member laminate layup. The results of this study
will help choose the ideal laminate layup to most efficiently endure the loading on the
truss.
6.1 2D ANSYS Finite Element Model Results
The results of the 2D FEM are stresses in each member in the axial direction. The
distribution of stresses throughout this model did not vary significantly between the steel
and composite models. Members in compression in the steel model were also in
compression in the composite models. This was also true for members in tension. The
only difference was the magnitude of the stresses in each truss. A color plot of the 2D
finite element model axial stress results for the steel truss is shown in Figure 6.1.
Figure 6.1 - Steel 2D ANSYS FEM Axial Stress Result
The difference in stress magnitude between the models can be seen in the summary of
results shown in Table 6.1. ANSYS outputs a minimum and maximum stress for each
member. To be conservative the maximum tensile stress of the members in tension and
the minimum compressive stress of the members in compression are reported in this
table.
30
Table 6.1 - 2D FEM Member Axial Stresses
To convert axial member stresses into axial member forces equation [6.1] was used.
Equation 6.1 - Member
Force
𝐹𝑚 = 𝜎𝑚 𝐴𝑚
This results in the axial member forces shown in Table 6.2.
31
[6.1]
Table 6.2 - 2D FEM Member Axial Forces
32
6.2 Method of Joints Results
The outputs from the method of joints are an axial force in each truss members. The
inputs are the boundary conditions and loading conditions detailed in previous chapters.
The difference between the steel and composite truss member forces are the result of the
decreased weight of the composite truss. The loading on the composite truss is lower so
the axial force in each member is as well. The results for each material are shown in
Table 6.3. Included in this table are the reaction forces at node A and I.
Steel Truss
Composite Truss
Axial Force (lb) Axial Force (lb)
-161,838
-128,239
80,919
64,120
152,084
126,608
-156,961
-127,424
-58,853
-43,567
186,387
149,207
46,616
41,520
-209,695
-169,967
46,616
41,520
186,387
149,207
-58,853
-43,567
-156,961
-127,424
152,084
126,608
80,919
64,120
-161,838
-128,239
-146,426
-112,107
0
0
-146,426
-112,107
Force
F1
F2
F3
F4
F5
F6
F7
F8
F9
F10
F11
F12
F13
F14
F15
Ry
Rx
Ri
Table 6.3 - Method of Joints Forces
As can be seen in Table 6.3 all the composite laminates have identical axial forces. This
is because the only properties that affect the outcome are material density and geometric
shape. These properties are equal in all of the laminates.
The results show that eight of the members are in tension and seven are in
compression. This is visualized in Figure 6.2 where red members are in tension and blue
members are in compression.
33
4
1
8
3
5
2
7
12
9
6
11
10
13
15
14
Figure 6.2 - Tension Compression Members
To convert the axial member forces into axial member stresses equation [6.1] was used.
This results in the axial member stresses in Table 6.4.
Member
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Steel Truss
Composite Truss
Axial Stress (psi) Axial Stress (psi)
-2,529
-2,004
1,264
1,002
2,376
1,978
-2,453
-1,991
-920
-681
2,912
2,331
728
649
-3,276
-2,656
728
649
2,912
2,331
-920
-681
-2,453
-1,991
2,376
1,978
1,264
1,002
-2,529
-2,004
Table 6.4 - Method of Joints Axial Member Stresses
6.3 3D ANSYS Finite Element Model Results
The results of the 3D ANSYS FEM are stresses in each member in the transverse
direction. Each member was investigated to find the maximum tensile and minimum
compressive transverse stress. The results of all 19 truss models are split up into multiple
tables based on model groupings. The first grouping contains the steel, single, and
combined ply orientations laminate models is shown in Table 6.5. The second grouping,
containing the +/-45 degree cross ply laminates, is shown in Table 6.6. The third
grouping, containing the +/-30 degree cross ply laminates, is shown in Table 6.7.
34
Finally, the fourth grouping, containing the 90 degree cross ply laminates, is shown in
Table 6.8.
Member
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Steel
Min Max
-2,081 360
-859 1,535
-318 1,875
-1,397 269
-1,230 316
-405 1,248
-491
988
-1,333 320
-488
982
-405 1,248
-1,169 310
-1,373 239
-318 1,878
-879 1,739
-2,028 354
Member Transverse Stress (psi)
[3042/-3042/9042/042]S [4542/-4542/9042/042]S
[0340]
Min Max
-405 403
-418 366
-241 235
-123 140
-107 244
-254 159
-185 134
-120 125
-184 133
-255 159
-103 246
-123 142
-243 242
-927 895
-433 673
Min
-893
-598
-163
-579
-457
-298
-204
-522
-204
-298
-434
-566
-163
-636
-872
Max
296
705
810
181
291
550
146
230
414
550
290
180
812
863
294
Min
-408
-432
-198
-191
-144
-274
-150
-206
-150
-274
-145
-184
-199
-685
-428
Table 6.5 - Truss Group 1 3D FEM Transverse Member Stresses
35
Max
339
415
311
119
192
244
138
146
140
244
194
119
310
799
337
Member
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
[4517/-4517/0136]S
Min
-551
-466
-191
-208
-174
-243
-151
181
-152
-243
-167
-203
-193
-783
-540
Max
341
656
374
113
217
273
168
116
170
273
219
114
382
992
482
Member Transverse Stress (psi)
[4534/-4534/0102]S [4543/-4543/084]S
Min
-740
-536
-136
-482
-430
-274
-165
-436
-165
-274
-415
-474
-134
-640
-755
Max
257
962
732
126
186
452
308
169
308
453
187
126
733
1,137
256
Min
-876
-553
-181
-685
-628
-303
-226
-636
-225
-303
-605
-675
-178
-647
-899
[4551/-4551/068]S
Max
202
1,172
997
159
201
609
415
200
415
610
212
150
998
1,257
210
Min
-1,033
-622
-238
-917
-869
-328
-293
-873
-292
-329
-837
-906
-234
-738
-1,070
Max
241
1,404
1,311
209
232
793
546
228
544
793
240
181
1,311
1,395
254
Table 6.6 - Truss Group 2 3D FEM Transverse Member Stresses
Member [3017/-3017/0136]S
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Min
-496
-540
-194
-175
-127
-241
-143
-142
-144
-241
-123
-169
-196
-923
-550
Max
350
659
307
111
199
244
162
94
162
244
202
112
311
1,091
658
Member Transverse Stress (psi)
[3034/-3034/0102]S [3043/-3043/084]S [3051/-3051/068]S [3068/-3068/034]S
Min
-602
-626
-155
-346
-284
-237
-110
-291
-111
-237
-274
-342
-156
-891
-667
Max
303
954
524
113
186
363
201
123
200
364
189
113
524
1,288
576
Min
-669
-693
-158
-455
-389
-252
-140
-390
-140
-252
-376
-450
-156
-860
-743
Max
257
1,134
673
130
216
443
250
139
250
445
221
130
673
1,414
513
Min
-738
-746
-199
-558
-503
-267
-171
-494
-171
-267
-486
-552
-196
-823
-824
Max
195
1,316
830
149
250
527
305
153
302
529
258
148
830
1,550
441
Table 6.7 - Truss Group 3 3D FEM Transverse Member Stresses
36
Min
-1,057
-834
-341
-808
-846
-302
-250
-806
-250
-303
-814
-808
-333
-797
-1,064
Max
369
180
1,293
201
369
764
485
192
478
769
383
199
1,292
1,959
356
Member
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
[9017/0153]S
Member Transverse Stress (psi)
[9034/0136]S [9043/0127]S [9051/0119]S [9068/0102]S [9085/085]S
Min
-520
-498
-258
-218
-115
-293
-244
-182
-244
-293
-109
-218
-257
-621
-374
Min
-618
-488
-263
-279
-127
-314
-277
-211
-278
-314
-127
-280
-261
-479
-356
Max
452
380
241
173
302
186
182
215
182
185
304
172
245
665
450
Max
476
420
297
232
335
211
235
283
235
210
338
232
303
590
475
Min
-660
-474
-266
-303
-133
-321
-291
-232
-290
-321
-133
-304
-263
-478
-352
Max
486
437
324
260
349
231
259
315
259
230
352
259
332
563
485
Min
-694
-460
-270
-321
-138
-325
-301
-264
-300
-325
-138
-322
-266
-493
-353
Max
494
452
347
284
360
262
279
343
279
261
363
283
356
543
493
Min
-754
-429
-280
-249
-145
-340
-319
-337
-317
-339
-146
-349
-284
-546
-405
Max
575
502
293
333
380
331
316
403
317
330
383
333
404
570
537
Min Max
-797 698
-401 543
-321 437
-367 384
-161 398
-392 411
-333 349
-421 466
-330 350
-393 409
-163 400
-367 384
-325 451
-627 670
-474 579
Table 6.8 - Truss Group 4 3D FEM Transverse Member Stresses
To convert transverse member stresses into transverse member forces equation [6.1] was
used. This results in the transverse member forces shown in Table 6.9, Table 6.10, Table
6.11, and Table 6.12.
37
Member
Member Transverse Force (lb)
[3042/-3042/9042/042]S [4542/-4542/9042/042]S
[0340]
Steel
Min
-133,171
-55,000
-20,323
-89,402
-78,720
-25,912
-31,453
-85,338
-31,215
-25,890
-74,784
-87,866
-20,377
-56,244
-129,792
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Max
23,035
98,214
120,013
17,205
20,226
79,885
63,210
20,454
62,860
79,866
19,839
15,265
120,186
111,277
22,685
Min
-25,951
-26,778
-15,407
-7,867
-6,826
-16,265
-11,823
-7,700
-11,773
-16,292
-6,618
-7,844
-15,583
-59,349
-27,724
Max
25,814
23,425
15,021
8,943
15,604
10,195
8,545
7,983
8,533
10,147
15,767
9,085
15,508
57,281
43,053
Min
-57,156
-38,301
-10,425
-37,052
-29,235
-19,057
-13,067
-33,385
-13,025
-19,047
-27,798
-36,253
-10,445
-40,730
-55,832
Max
18,938
45,130
51,812
11,590
18,653
35,226
9,322
14,739
26,500
35,216
18,552
11,540
51,962
55,263
18,796
Min
-26,112
-27,648
-12,672
-12,224
-9,216
-17,536
-9,600
-13,184
-9,600
-17,536
-9,280
-11,776
-12,736
-43,840
-27,392
Max
21,696
26,560
19,904
7,616
12,288
15,616
8,832
9,344
8,960
15,616
12,416
7,616
19,840
51,136
21,568
Table 6.9 - Truss Group 1 3D FEM Transverse Member Forces
Member
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
[4517/-4517/0136]S
Member Transverse Force (lb)
[4534/-4534/0102]S [4543/-4543/084]S [4551/-4551/068]S
Min
-35,291
-29,801
-12,208
-13,328
-11,148
-15,556
-9,672
11,597
-9,734
-15,544
-10,687
-12,974
-12,337
-50,115
-34,591
Min
-47,363
-34,319
-8,735
-30,872
-27,542
-17,557
-10,547
-27,919
-10,534
-17,556
-26,533
-30,349
-8,596
-40,984
-48,351
Max
21,819
41,965
23,927
7,261
13,888
17,453
10,777
7,397
10,859
17,484
14,022
7,316
24,431
63,460
30,829
Max
16,472
61,558
46,846
8,070
11,935
28,899
19,690
10,823
19,734
28,981
11,985
8,049
46,901
72,762
16,355
Min
-56,090
-35,363
-11,599
-43,815
-40,195
-19,370
-14,442
-40,703
-14,391
-19,377
-38,750
-43,206
-11,414
-41,421
-57,553
Max
12,907
74,995
63,800
10,177
12,845
38,999
26,567
12,822
26,586
39,031
13,578
9,594
63,853
80,429
13,430
Min
Max
-66,131 15,393
-39,820 89,824
-15,244 83,878
-58,671 13,395
-55,590 14,852
-21,014 50,723
-18,745 34,913
-55,864 14,597
-18,659 34,806
-21,025 50,734
-53,554 15,355
-57,983 11,575
-14,964 83,904
-47,204 89,254
-68,474 16,261
Table 6.10 - Truss Group 2 3D FEM Transverse Member Forces
38
Table 6.11 - Truss Group 3 3D FEM Transverse Member Forces
39
Table 6.12 - Truss Group 4 3D FEM Transverse Member Forces
40
7. Margin of Safety Calculation
The ultimate goal of using composite materials in place of steel for the construction of a
truss bridge is achieving a lower weight structure that has higher strength. Strength can
be quantified by calculating a margin of safety in each truss member. Margin of safety is
defined as in equation [7.1].
Equation 7.1 - Margin of
Safety
𝑀. 𝑆. =
𝐴𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ
−1
𝐴𝑐𝑡𝑢𝑎𝑙 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ
[7.1]
When the M.S. reaches a value less than or equal to zero the structure will fail. An
alternate definition is that it is equal to the factor of safety minus one. This factor is used
to compare the steel and composite material trusses. This is done by computing the M.S
in every member of every truss model for both the axial and transverse directions. Then
a comparison between the lowest axial and transverse M.S. in each truss is made. The
truss with the highest M.S. is deemed the strongest and the one with the lowest is ranked
as the weakest. This results in a simple quantitative parameter to compare between each
truss model.
7.1 Steel Truss Margin of Safety
The intermediate alloy 5Cr-Mo-V steel is an isotropic material. This means it has the
same strength in every direction. The material allowable for this strength only varies
with the type of stress applied. Table 7.1 details the material allowable for 5Cr-Mo-V
steel in compression and tension.
5Cr-Mo-V
σty
σcy
Allowable (ksi)
240
-260
Table 7.1 - Steel Material Allowable
It should be noted that this material is stronger in compression then it is in tension.
7.1.1
Axial Margin of Safety
The axial M.S. of the steel truss was calculated using the results of the 2D ANSYS finite
element model and method of joints. The maximum axial stress in each member as
computed by these methods was compared to the appropriate tension or compression
41
material allowable. The member stresses used in this calculation can found in the
previous chapter. The resulting axial M.S. for each method is shown in Table 7.2.
Member
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
MOJ
2D ANSYS
Axial M.S.
102
93
189
138
100
96
105
88
282
215
81
74
329
275
78
73
329
275
81
74
282
215
105
88
100
96
189
138
102
93
Table 7.2 - Steel Truss Axial M.S.
An example of the M.S. as calculated for member 1 from the method of joints result is
shown below. As can be seen in this example calculation since the stress is compressive
the compressive material allowable is used.
𝑀. 𝑆. =
𝐴𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ
−260,000[𝑝𝑠𝑖]
−1=
− 1 = 102
𝐴𝑐𝑡𝑢𝑎𝑙 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ
−2,529[𝑝𝑠𝑖]
Table 7.2 shows that the lowest M.S. occurs in member 8. In the method of joints this
M.S. is 78 and is 73 in the 2D ANSYS finite element model. Member 8 is the top center
member of the truss.
7.1.2
Transverse Margin of Safety
The transverse M.S. of the steel truss is calculated using the results of the 3D ANSYS
finite element model. These results are in the form of a maximum and minimum
transverse stress in each member. Since the maximum stress is always a tensile stress
and the minimum stress is always a compressive stress the maximum stress is compared
to the tensile material allowable and the minimum stress is compared to the compressive
material allowable. This results in a M.S. for both tension and compression. The lower
42
of these two results is treated as the M.S. for the member. The member stresses used in
this calculation can found in the previous chapter. The minimum transverse M.S. in each
member is presented in Table 7.3.
Member
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Transverse
M.S.
124
155
127
185
210
191
242
194
243
191
222
188
127
137
127
Table 7.3 - Steel Truss Transverse M.S.
An example of the M.S. as calculated for member 1 from the 3D FEM results is shown
below.
𝑇𝑒𝑛𝑠𝑖𝑙𝑒 𝑀. 𝑆. =
𝐴𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 𝑇𝑒𝑛𝑠𝑖𝑙𝑒 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ
240,000[𝑝𝑠𝑖]
−1=
− 1 = 666
𝐴𝑐𝑡𝑢𝑎𝑙 𝑇𝑒𝑛𝑠𝑖𝑙𝑒 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ
360[𝑝𝑠𝑖]
𝐶𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒 𝑀. 𝑆. =
𝐴𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 𝐶𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ
−1
𝐴𝑐𝑡𝑢𝑎𝑙 𝐶𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ
𝐶𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒 𝑀. 𝑆. =
−260,000[𝑝𝑠𝑖]
− 1 = 124
−2,081[𝑝𝑠𝑖]
Since the compressive M.S. is much lower than the tensile this is the minimum M.S. of
the member. Table 7.3 shows that the lowest margin is 124 in the first member of the
truss which is the left most diagonal member.
7.1.3
Buckling Margin of Safety
A potential failure mode of the truss members in addition to yielding is axial buckling of
the members in compression. To check members for buckling the axial force in each is
43
compared with the critical buckling load. The equation for the critical buckling load is
shown in equation [7.2].
Equation 7.2 - Critical
Buckling Load
𝑃𝑐𝑟 =
𝜋 2 𝐸𝐼
𝐿2
[7.2]
This is the buckling load for a pinned-pinned connection of an axially loaded
compression member. If the force in any compression member exceeds this value the
member will buckle. The buckling load of each steel member is 4.36 million pounds
because each member has the same length, stiffness, and moment of inertia. The forces
calculated in the method of joints are used as the actual strength in the M.S. calculation
for buckling. The buckling M.S. for each member is shown in Table 7.4. Members in
tension are N/A since they are not subject to buckling.
Member
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Buckling M.S.
26
N/A
N/A
27
73
N/A
N/A
20
N/A
N/A
73
27
N/A
N/A
26
Table 7.4 - Steel Truss Buckling M.S.
An example of the buckling M.S. as calculated for member 1 from the method of joints
results is shown below.
𝑇𝑒𝑛𝑠𝑖𝑙𝑒 𝑀. 𝑆. =
𝐴𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 𝑇𝑒𝑛𝑠𝑖𝑙𝑒 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ
4,357,653[𝑙𝑏]
−1=
− 1 = 26
𝐴𝑐𝑡𝑢𝑎𝑙 𝑇𝑒𝑛𝑠𝑖𝑙𝑒 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ
161838[𝑙𝑏]
Table 7.4 shows that the lowest buckling M.S. is 20 in member 8 which is the top center
member of the truss.
44
7.2 Composite Truss Margin of Safety
The carbon fiber epoxy IM7/8552 composite material is an orthotropic material. This
means it has a different strength in each material direction. This is because all the fibers
are aligned in one direction which results in a high strength in the fiber direction and a
very weak strength in the transverse matrix direction. This can be seen in the material
allowable for the IM7/8552 carbon fiber epoxy shown in Table 7.5.
8552/IM7 Carbon
Fiber Epoxy
σ1t
σ1c
σ2t
σ2c
Allowable (ksi)
395
-245
16.1
-32.3
Table 7.5 - Composite Material Allowable
The difference in strength between the axial and transverse direction of this material is
the reason plies are stacked up in a laminate at varying orientations. Varying the
orientation of some of the plies in a laminate is a way to strengthen the laminate in the
desired orientation angle direction. This allows for the ability to customize the strength
of a laminate in any direction.
Determining failure in a composite material laminate requires checking every ply
for failure. This is done by employing classical laminate theory and Tsai-Wu failure
criterion. The process is to first find the global axial and global transverse force that will
cause failure in each ply of the laminate. This is done by separately applying an axial
and transverse unit force per length of laminate to the laminate. The length of laminate
used is measured as the length of the laminate in the direction perpendicular to the force.
In the case of the axial direction this is the circumference of the box member which is
equal to 40 inches. In the case of the transverse direction this is equal to 222 inches
which is the length of each member. The resulting local stresses, from the unit forces, in
each ply are calculated using CLT. These local stresses are inserted into the Tsai-Wu
failure criterion equation along with the Tsai-Wu constants. The Tsai-Wu failure
criterion equation and the Tsai-Wu criterion constants can be found in the CLT chapter.
The equation is repeated here in equation [7.3].
45
Equation
7.3 - TsaiWu Failure
Criterion
2
𝐹1 𝜎1 + 𝐹2 𝜎1 + 𝐹11 𝜎12 +𝐹22 𝜎22 +𝐹66 𝜏12
− √𝐹11 𝐹22 𝜎1 𝜎2 = 1
[7.3]
To find the force to cause failure in each ply the unit force on the laminate is increased
until the term on the left hand side of the Tsai-Wu equation is equal to 1. When this
occurs the failure load has been found. Due to the quadratic nature of the Tsai-Wu
failure criterion there is both a tensile and compressive unit force which will cause
failure. The process is repeated until the tensile and compressive failure load has been
found in each ply of the laminate for both the axial and transverse directions. This
procedure was followed for each of the candidate laminates considered in this project.
The resulting axial failure loads are shown in Table 7.6 and the resulting transverse
failure loads are shown in Table 7.7 for each ply of each laminate.
46
Laminate
[0340]
[4517/-4517/0136]S
[4534/-4534/0102]S
[4543/-4543/084]S
[4551/-4551/068]S
[3017/-3017/0136]S
[3034/-3034/0102]S
[3043/-3043/084]S
[3051/-3051/068]S
[3068/-3068/034]S
[9017/0153]S
[9034/0136]S
[9043/0127]S
[9051/0119]S
[9068/0102]S
[9085/085]S
[3042/-3042/9042/042]S
[4542/-4542/9042/042]S
Ply
Orientation
0
45
-45
0
45
-45
0
45
-45
0
45
-45
0
30
-30
0
30
-30
0
30
-30
0
30
-30
0
30
-30
0
90
0
90
0
90
0
90
0
90
0
90
0
30
-30
90
0
45
-45
90
0
Axial
Tension Failure
Load (lb)
32,232,000
21,776,300
21,776,300
24,602,600
16,594,100
16,594,100
18,504,400
14,026,000
14,026,000
15,579,100
11,807,300
11,807,300
13,080,400
25,843,600
25,843,600
25,205,800
19,909,100
19,909,100
19,135,900
17,154,900
17,154,900
16,427,100
14,912,100
14,912,100
14,248,600
10,657,100
10,657,100
10,156,800
16,208,400
30,165,900
14,782,300
27,126,400
13,949,700
25,485,600
13,189,200
24,024,300
11,536,800
20,919,200
9,856,600
17,818,000
16,894,600
16,894,600
8,887,400
17,103,000
10,421,300
10,421,300
6,332,600
12,186,500
Table 7.6 - Global Axial Ply Failure Load
47
Compression Failure
Load (lb)
-19,992,000
-24,045,000
-24,045,000
-13,953,500
-17,819,700
-17,819,700
-10,216,600
-14,931,600
-14,931,600
-8,536,100
-12,496,100
-12,496,100
-7,131,400
-18,785,300
-18,785,300
-13,923,900
-13,647,700
-13,647,700
-10,094,400
-11,544,100
-11,544,100
-8,549,100
-9,911,700
-9,911,700
-7,351,000
-6,955,000
-6,955,000
-5,175,500
-32,862,100
-20,559,200
-29,862,000
-19,068,100
-28,144,000
-18,079,300
-26,585,800
-17,146,100
-23,222,200
-15,065,200
-19,820,700
-12,908,600
-13,938,400
-13,938,400
-18,071,700
-10,607,900
-12,353,000
-12,353,000
-12,876,800
-7,558,500
Laminate
[0340]
[4517/-4517/0136]S
[4534/-4534/0102]S
[4543/-4543/084]S
[4551/-4551/068]S
[3017/-3017/0136]S
[3034/-3034/0102]S
[3043/-3043/084]S
[3051/-3051/068]S
[3068/-3068/034]S
[9017/0153]S
[9034/0136]S
[9043/0127]S
[9051/0119]S
[9068/0102]S
[9085/085]S
[3042/-3042/9042/042]S
[4542/-4542/9042/042]S
Transverse
Ply
Tension Failure
Orientation
Load (lb)
0
7,291,400
45
17,564,200
-45
17,564,200
0
11,538,700
45
23,172,800
-45
23,172,800
0
14,798,300
45
25,390,700
-45
25,390,700
0
15,925,200
45
26,676,100
-45
26,676,100
0
16,434,200
30
10,163,600
-30
10,163,600
0
8,203,400
30
10,928,000
-30
10,928,000
0
8,740,600
30
11,110,000
-30
11,110,000
0
8,839,100
30
11,115,500
-30
11,115,500
0
8,799,200
30
10,529,700
-30
10,529,700
0
8,238,900
90
30,225,800
0
16,824,400
90
47,372,000
0
26,341,000
90
56,453,700
0
31,370,400
90
64,529,200
0
35,834,600
90
81,700,400
0
45,295,000
90
98,889,800
0
54,704,300
30
38,684,200
-30
38,684,200
90
57,414,600
0
30,486,700
45
57,838,000
-45
57,838,000
90
67,635,100
0
35,145,900
Table 7.7 - Global Transverse Ply Failure Load
48
Compression Failure
Load (lb)
-14,628,000
-22,289,600
-22,289,600
-23,288,600
-28,796,300
-28,796,300
-30,030,600
-31,073,600
-31,073,600
-32,382,100
-32,091,800
-32,091,800
-33,438,200
-16,949,000
-16,949,000
-16,529,500
-18,206,700
-18,206,700
-17,683,800
-18,491,000
-18,491,000
-17,917,500
-18,475,500
-18,475,500
-17,863,100
-17,420,600
-17,420,600
-16,760,100
-22,158,000
-33,763,000
-34,659,600
-52,879,200
-41,253,500
-62,989,400
-47,096,800
-71,968,700
-59,440,900
-91,018,500
-71,642,700
-110,005,100
-64,201,500
-64,201,500
-38,022,600
-61,953,800
-68,558,900
-68,558,900
-41,949,800
-71,466,100
The failure loads were calculated using the MATLAB code in appendix C. It should be
noted that plies with equivalent orientation, material, and thickness in different laminates
will have different failure loads. This is because the failure load of a ply is more
determined by its place in the laminate and by the material properties of the surrounding
plies. This is because the stiffness of each laminate is different resulting in the load on
the laminate flowing through it in a different fashion.
7.2.1
Axial Margin of Safety
Using the failure loads in Table 7.6 as the material allowable for each ply and the forces
calculated using the 2D ANSY FEM, in the previous chapter, an axial M.S. can be
calculated. The following tables present the minimum M.S. each ply of each member.
The results of all 18 laminates are split up into multiple tables based on model
groupings. The first grouping contains the single and +/-45 degree cross ply laminates.
The results of which are shown in Table 7.8. The second grouping contains the +/-30
degree cross ply laminates and the results are shown in Table 7.9. The third grouping
contains the 90 degree cross ply laminates and the results are shown in Table 7.10.
Finally, the forth grouping contains the combined orientation laminates and the results
are shown in Table 7.11.
49
-45
172
252
168
161
418
133
456
134
456
133
418
161
168
252
172
0
100
284
190
93
242
151
516
77
516
151
242
93
190
284
100
45
127
191
127
119
308
101
347
99
347
101
308
119
127
191
127
-45
127
191
127
119
308
101
347
99
347
101
308
119
127
191
127
0
72
213
142
68
176
113
387
56
387
113
176
68
142
213
72
45
106
161
108
99
257
85
293
83
293
85
257
99
108
161
106
-45
106
161
108
99
257
85
293
83
293
85
257
99
108
161
106
[4551/-4551/068]S
45
172
252
168
161
418
133
456
134
456
133
418
161
168
252
172
[4543/-4543/084]S
0
233
379
249
135
354
198
679
111
679
198
354
135
249
379
233
[4534/-4534/0102]S
[4517/-4517/0136]S
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
[0340]
Member
0
60
179
120
56
147
95
326
47
326
95
147
56
120
179
60
Table 7.8 - Composite Laminate Group 1 Axial M.S.
50
45
89
135
90
83
215
72
247
69
247
72
215
83
90
135
89
-45
89
135
90
83
215
72
247
69
247
72
215
83
90
135
89
0
50
150
100
47
122
80
273
39
273
80
122
47
100
150
50
30
97
229
153
91
236
122
417
75
417
122
236
91
153
229
97
-30
97
229
153
91
236
122
417
75
417
122
236
91
153
229
97
0
72
220
147
67
174
117
401
56
401
117
174
67
147
220
72
30
82
197
132
77
199
105
359
64
359
105
199
77
132
197
82
-30
82
197
132
77
199
105
359
64
359
105
199
77
132
197
82
0
60
189
126
56
147
100
344
47
344
100
147
56
126
189
60
30
70
171
114
66
171
91
312
54
312
91
171
66
114
171
70
-30
70
171
114
66
171
91
312
54
312
91
171
66
114
171
70
Table 7.9 - Composite Laminate Group 2 Axial M.S.
51
[3068/-3068/034]S
0
99
292
194
93
242
154
529
77
529
154
242
93
194
292
99
[3051/-3051/068]S
-30
135
299
199
126
327
158
542
104
542
158
327
126
199
299
135
[3043/-3043/084]S
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
30
135
299
199
126
327
158
542
104
542
158
327
126
199
299
135
[3034/-3034/0102]S
[3017/-3017/0136]S
Member
0
52
163
109
48
126
87
298
40
298
87
126
48
109
163
52
30
49
102
81
46
119
65
222
38
222
65
119
46
81
102
49
-30 0
49 36
102 97
81 78
46 34
119 88
65 62
222 212
38 28
222 212
65 62
119 88
46 34
81 78
102 97
49 36
90
203
163
107
190
496
85
293
157
293
85
496
190
107
163
203
0
130
298
197
122
318
156
536
100
536
156
318
122
197
298
130
90
192
153
101
179
468
80
277
148
277
80
468
179
101
153
192
0
123
280
185
115
301
147
505
95
505
147
301
115
185
280
123
90
167
134
89
156
407
70
242
129
242
70
407
156
89
134
167
Table 7.10 - Composite Laminate Group 3 Axial M.S.
52
[9085/085]S
0
137
317
210
129
336
166
571
106
571
166
336
129
210
317
137
[9068/0102]S
90
216
173
114
202
527
90
311
166
311
90
527
202
114
173
216
[9051/0119]S
0
148
354
233
139
364
185
635
114
635
185
364
139
233
354
148
[9043/0127]S
90
238
190
125
222
582
99
341
183
341
99
582
222
125
190
238
[9034/0136]S
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
[9017/0153]S
Member
0
108
243
161
101
264
128
439
83
439
128
264
101
161
243
108
90
142
114
75
133
347
60
206
110
206
60
347
133
75
114
142
0
91
203
136
85
221
108
369
71
369
108
221
85
136
203
91
Member
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
[3042/-3042/9042/042]S
[4542/-4542/9042/042]S
30
98
192
129
92
238
102
350
76
350
102
238
92
129
192
98
45
87
118
79
81
211
63
216
68
216
63
211
81
79
118
87
-30
98
192
129
92
238
102
350
76
350
102
238
92
129
192
98
90
128
101
67
119
309
53
184
99
184
53
309
119
67
101
128
0
75
195
130
70
181
103
355
58
355
103
181
70
130
195
75
-45
87
118
79
81
211
63
216
68
216
63
211
81
79
118
87
90
91
71
48
85
220
38
131
70
131
38
220
85
48
71
91
0
53
138
92
49
128
73
252
41
252
73
128
49
92
138
53
Table 7.11 - Composite Laminate Group 4 Axial M.S.
An example calculation of the M.S. in the 0 degree ply of the first member in the [0 340]S
laminate model is shown below. In this example the load is tensile so the tension
allowable is used as the allowable strength.
𝑀. 𝑆. =
𝐴𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ
32,232,000[𝑙𝑏]
−1=
− 1 = 233
𝐴𝑐𝑡𝑢𝑎𝑙 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ
137,501[𝑙𝑏]
The minimum M.S. in each laminate model is shown in Table 7.12. Included in this table
is the member and ply orientation type that results in the minimum M.S.
53
Laminate
[0340]
[4517/-4517/0136]S
[4534/-4534/0102]S
[4543/-4543/084]S
[4551/-4551/068]S
[3017/-3017/0136]S
[3034/-3034/0102]S
[3043/-3043/084]S
[3051/-3051/068]S
[3068/-3068/034]S
[9017/0153]S
[9034/0136]S
[9043/0127]S
[9051/0119]S
[9068/0102]S
[9085/085]S
[3042/-3042/9042/042]S
[4542/-4542/9042/042]S
Minimum
M.S.
111
77
56
47
39
77
56
47
40
28
99
90
85
80
70
60
53
38
Ply Orientation Type
Member # Containing
Containing Minimum M.S.
Minimum M.S.
0
8
0
8
0
8
0
8
0
8
0
8
0
8
0
8
0
8
0
8
90
6, 10
90
6, 10
90
6, 10
90
6, 10
90
6, 10
90
6, 10
90
6, 10
90
6, 10
Table 7.12 - Composite Laminate Minimum Axial M.S.
The results shown in Table 7.12 that for axial strength the [0340] laminate is the strongest
followed by the [9017/0153]S layup. This is because these laminates contain many plies
oriented in the direction of the axial load.
7.2.2
Transverse Margin of Safety
Using the failure loads in Table 7.7 as the material allowable for each ply and the forces
on each member, calculated using the 3D ANSYS FEM in the previous chapter, a
transverse M.S. can be calculated. This analysis method outputs a maximum tensile
force and a minimum compressive force. It was found that the minimum M.S. is
calculated when using the tensile force so the M.S. results use these forces. The results
of all 18 laminates are split up into multiple tables based on model groupings. The first
grouping contains the single and +/-45 degree cross ply laminates. The results of which
are shown in Table 7.13. The second grouping contains the +/-30 degree cross ply
laminates and the results are shown in Table 7.14. The third grouping contains the 90
54
degree cross ply laminates and the results are shown in Table 7.15. Finally, the forth
grouping contains the combined orientation laminates and the results are shown in Table
7.16. An example calculation of the M.S. in the 0 degree ply of the first member in the
[0340]S laminate model is shown below.
𝑀. 𝑆. =
𝐴𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ
7,291,400[𝑙𝑏]
−1 =
− 1 = 281
𝐴𝑐𝑡𝑢𝑎𝑙 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ
25,814[𝑙𝑏]
Table 7.13 - Composite Laminate Group 1 Transverse M.S.
55
Table 7.14 - Composite Laminate Group 2 Transverse M.S.
56
Member
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
[9017/0153]S
90
1,044
1,242
1,958
2,737
1,565
2,532
2,594
2,191
2,593
2,553
1,552
2,742
1,928
709
1,048
0
580
691
1,089
1,523
871
1,409
1,443
1,219
1,443
1,421
863
1,526
1,073
394
583
[9034/0136]S
90
1,553
1,761
2,492
3,191
2,207
3,501
3,151
2,618
3,150
3,520
2,187
3,195
2,439
1,254
1,558
0
863
979
1,385
1,774
1,226
1,946
1,752
1,455
1,751
1,957
1,216
1,776
1,356
697
866
[9043/0127]S
90
1,813
2,017
2,722
3,394
2,525
3,813
3,406
2,800
3,403
3,836
2,503
3,398
2,659
1,565
1,818
0
1,007
1,120
1,512
1,886
1,403
2,119
1,892
1,555
1,890
2,131
1,391
1,888
1,477
869
1,010
[9051/0119]S
90
2,039
2,228
2,905
3,554
2,798
3,845
3,617
2,939
3,612
3,863
2,775
3,558
2,834
1,856
2,043
0
1,132
1,237
1,613
1,973
1,554
2,135
2,008
1,632
2,006
2,145
1,541
1,975
1,573
1,030
1,134
[9068/0102]S
90
2,220
2,541
4,352
3,831
3,354
3,855
4,035
3,170
4,026
3,866
3,331
3,833
3,155
2,237
2,375
0
1,231
1,408
2,412
2,123
1,859
2,137
2,236
1,757
2,232
2,143
1,846
2,125
1,749
1,240
1,316
[9085/085]S
90
2,211
2,846
3,531
4,019
3,881
3,761
4,425
3,316
4,410
3,774
3,861
4,020
3,423
2,305
2,668
Table 7.15 - Composite Laminate Group 3 Transverse M.S.
Member
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
[3042/-3042/9042/042]S
30
2,042
856
746
3,337
2,073
1,097
4,149
2,624
1,459
1,097
2,084
3,351
743
699
2,057
-30
2,042
856
746
3,337
2,073
1,097
4,149
2,624
1,459
1,097
2,084
3,351
743
699
2,057
90
3,031
1,271
1,107
4,953
3,077
1,629
6,158
3,895
2,166
1,629
3,094
4,974
1,104
1,038
3,054
0
1,609
675
587
2,629
1,633
864
3,270
2,067
1,149
865
1,642
2,641
586
551
1,621
[4542/-4542/9042/042]S
45
2,665
2,177
2,905
7,593
4,706
3,703
6,548
6,189
6,454
3,703
4,657
7,593
2,914
1,130
2,681
-45
2,665
2,177
2,905
7,593
4,706
3,703
6,548
6,189
6,454
3,703
4,657
7,593
2,914
1,130
2,681
90
3,116
2,546
3,397
8,880
5,503
4,330
7,657
7,237
7,548
4,330
5,446
8,880
3,408
1,322
3,135
0
1,619
1,322
1,765
4,614
2,859
2,250
3,978
3,760
3,922
2,250
2,830
4,614
1,770
686
1,629
Table 7.16 - Composite Laminate Group 4 Transverse M.S.
The minimum M.S. in each laminate model is shown in Table 7.17. Included in this table
is the member and ply orientation type that results in the minimum M.S.
57
0
1,223
1,574
1,953
2,223
2,146
2,080
2,448
1,834
2,439
2,087
2,135
2,223
1,893
1,275
1,475
Laminate
[0340]
[4517/-4517/0136]S
[4534/-4534/0102]S
[4543/-4543/084]S
[4551/-4551/068]S
[3017/-3017/0136]S
[3034/-3034/0102]S
[3043/-3043/084]S
[3051/-3051/068]S
[3068/-3068/034]S
[9017/0153]S
[9034/0136]S
[9043/0127]S
[9051/0119]S
[9068/0102]S
[9085/085]S
[3042/-3042/9042/042]S
[4542/-4542/9042/042]S
Minimum
M.S.
126
181
202
197
182
117
105
97
88
65
394
697
869
1030
1231
1223
551
686
Ply Orientation Type
Member # Containing
Containing Minimum M.S.
Minimum M.S.
0
14
0
14
0
14
0
14
0
2
0
14
0
14
0
14
0
14
0
14
0
14
0
14
0
14
0
14
0
1
0
1
0
14
0
14
Table 7.17 - Composite Laminate Minimum Transverse M.S.
The results shown in Table 7.17 show that for transverse strength the [9085/9085]S
laminate is the strongest followed by the [9068/90102]S layup. This is because these
laminates contain the most cross plies oriented in the direction of the transverse load.
7.2.3
Buckling Margin of Safety
Similar to the steel truss the members of the composite truss subject to compression
forces must be checked against their critical buckling load. Using equation [7.2] and the
global material properties of each laminate the critical buckling load was calculated and
is shown for each candidate laminate in Table 7.18.
58
Layup
[0340]
[4517/-4517/0136]S
[4534/-4534/0102]S
[4543/-4543/084]S
[4551/-4551/068]S
[3017/-3017/0136]S
[3034/-3034/0102]S
[3043/-3043/084]S
[3051/-3051/068]S
[3068/-3068/034]S
[9017/0153]S
[9034/0136]S
[9043/0127]S
[9051/0119]S
[9068/0102]S
[9085/085]S
[3042/-3042/9042/042]S
[4542/-4542/9042/042]S
Pcr (lb)
3,457,072
2,867,772
2,256,407
1,929,424
1,637,708
3,070,417
2,629,742
2,381,341
2,153,799
1,653,541
3,148,085
2,828,713
2,658,648
2,507,205
2,184,913
1,862,258
1,856,259
1,323,274
Table 7.18 - Composite Laminate Critical Buckling Loads
This results in the buckling M.S. for each member of each truss shown in Table 7.19.
N/A indicates the member is in tension and is not subject to buckling. An example of the
M.S. as calculated for the first member of the [0340]S laminate is shown below.
𝑀. 𝑆. =
𝐴𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ
3,457,072 [𝑙𝑏]
−1=
− 1 = 26
𝐴𝑐𝑡𝑢𝑎𝑙 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ
128,239 [𝑙𝑏]
59
Table 7.19 - Composite Composite Laminate Buckling M.S.
60
The minimum M.S. for member buckling in each laminate is shown in Table 7.20. As
can be seen in Table 7.19 this minimum always occurs in member number eight.
Laminate
[0340]
[4517/-4517/0136]S
[4534/-4534/0102]S
[4543/-4543/084]S
[4551/-4551/068]S
[3017/-3017/0136]S
[3034/-3034/0102]S
[3043/-3043/084]S
[3051/-3051/068]S
[3068/-3068/034]S
[9017/0153]S
[9034/0136]S
[9043/0127]S
[9051/0119]S
[9068/0102]S
[9085/085]S
[3042/-3042/9042/042]S
[4542/-4542/9042/042]S
Minimum
Buckling M.S.
19
16
12
10
9
17
14
13
12
9
18
16
15
14
12
10
10
7
Table 7.20 - Composite Laminate Minimum Buckling M.S.
The laminate that resists buckling the most is the [0340]S laminate. This is a result of this
laminate having the highest stiffness in the axial buckling direction.
61
8. Conclusion
The degree of success in using composite materials in the truss members of a bridge is
measured by the ability of the application to decrease weight and increase strength. The
strength change was found to be very dependent on the layup of the composite members.
Some laminates are stronger while others are weaker. Laminates that have shown
themselves to be stronger than the steel in one direction but are much weaker in another
were also among the candidates. What follows is a summary of all the truss models.
The minimum M.S. in the axial direction for each truss model is shown in Figure
8.1.
Minimum Axial M.S. of Truss Material/Layup
120
100
Axial M.S.
80
60
40
20
0
Truss Material/Layup
Figure 8.1 - Axial M.S. Comparison
As expected the [0340]S layup is the strongest because it has all its fibers aligned in the
axial direction.
A similar figure of the minimum M.S. in the transverse direction is shown in
Figure 8.2.
62
Minimum Transverse M.S. of Truss Material/Layup
1,400
1,200
Transverse M.S.
1,000
800
600
400
200
0
Truss Material/Layup
Figure 8.2 - Transverse M.S. Comparison
This shows that the layups with the 90 degree orientation plies are easily the strongest.
In particular are the [068/90102]S and the [085/9085]S truss laminates. This is because 90
cross plies have all there fibers aligned with the transverse direction.
Finally, a figure comparing the minimum buckling M.S. is shown in Figure 8.3.
Minimum Buckling M.S. of Truss Material/Layup
25
Buckling M.S.
20
15
10
5
0
Truss Material/Layup
Figure 8.3 - Buckling M.S. Comparison
63
This shows the only layup to compare to the steel is the [0340]S layup. The lower margins
across the board are because the axial modulus of elasticity of the composite material is
lower than steel. Since this variable is an important factor in determining buckling these
results are to be expected.
The final important comparison between a steel and composite truss is
calculating the reduction in weight for the composite material truss. Table 8.1 shows this
difference. The structure weight is the sum of the members, gusset plates, and road deck
weight.
Truss
Material/Layup
Steel
Composite
Structure
Weight
264,746
127,472
% Difference
-52%
Table 8.1 - Steel vs. Composite Weight Comparison
As can be seen there is over fifty percent in weight savings when switching to a carbon
fiber epoxy composite material in the overall truss. This is due to an 83% reduction in
density of composite material over the steel.
The results show that switching to IM7/8552 carbon fiber epoxy results in a
higher strength and lower weight truss bridge. The ideal layup is the [9043/0127]S
composite laminate. This layup has much higher axial strength than the steel and a much
improved transverse strength. It only takes a 26% decrease in buckling strength to
achieve these higher axial and transverse margins. It can therefore be concluded that the
switch to composite materials over traditional steel will overall be beneficial to the
construction of a Warren truss bridge structure.
64
9. References
[1]
Array Solutions. Wind Loads. 23 March 2002. 31 March 2012
<http://www.arraysolutions.com/Products/windloads.htm>.
[2]
Camp, Charles. Trusses - Method of Joints. Lecture. Memphis: The University of
Memphis, 2012.
[3]
Hexcel Corporation. Prepreg Data Sheets. 31 March 2012
<http://www.hexcel.com/Resources/prepreg-data-sheets>.
[4]
Hyer, Michael W. Stress Analysis of Fiber-Reinforced Composite Materials.
Lancaster: DEStech Publications, Inc., 2009.
[5]
Kxcad.net. BEAM188 Element.
<http://www.kxcad.net/ansys/ANSYS/ansyshelp/Hlp_E_BEAM188.html>.
[6]
Metallic Materials Properties Development and Standardization (MMPDS).
Washington: Federal Aviation Administration, 2010.
[7]
Reade Advanced Materials. Weight Per Cubic Foot And Specific Gravity. 11
January 2006. 31 March 2012
<http://www.reade.com/Particle_Briefings/spec_gra2.html>.
[8]
State of Connecticut. Connecticut Building Code. 2005. 31 March 2012
<http://www.archive.org/stream/gov.ct.building/ct_building_djvu.txt>.
[9]
U.S. Department of Housing and Development. Wind Zone Comparisons.
Washington, 2006.
[10]
U.S. Department of Transportation Federal Highway Administration. Application
of Design Standards, Uniform Federal Accessibility Standards, and Bridges. 15 4
2009. 31 March 2012 <http://www.fhwa.dot.gov/design/0625sup.cfm>.
[11]
U.S. Department of Transportation Federal Highway Administration Freight
Management and Operations. Bridge Formula Weights Calculator. 27 February
2012. 31 March 2012
<http://www.ops.fhwa.dot.gov/freight/sw/brdgcalc/calc_page.htm>.
65
10.Appendix
10.1 Appendix A – ANSYS Input File Code
! Initialize the model
/FILENAME,file
/TITLE, 2D Steel Truss
/UNITS,BIN
! USCS system of units
! Define parameters for future use
Line_Div = 128
! Line divisions for meshing
Length_T = 74*12
! Sets length of truss
Side = 10
! Sets cross section side length
t = 2.016
! Sets cross section wall thickness
Member_Length = 18.5*12
! Sets the length of each member
Truss_Height = ((Member_Length)**2-(Member_Length/2)**2)**(1/2)
!Truss Height
m = 667.8
! Weight of one member
ga = 23.5
! Weight of gusset plate a
gb = 38.7
! Weight of gusset plate b
gc = 51.6
! Weight of gusset plate c
P = 210428.4
! Weight of vehicles, road deck, and snow
! Model Preprocessor
/PREP7
! Enters the general input data preprocessor
! Element Type
ET,1,BEAM188
! 2D Beam Element
SECTYPE,1,BEAM,HREC
! Selects hollow rectangle as cross section
SECOFFSET,CENT
! Sets centroid
SECDATA,Side,Side,t,t,t,t
! Sets dimension of cross section
! Material properties
66
MP,EX,1,9.11E6
! Elastic modulus x
MP,EY,1,9.11E6
! Elastic modulus y
MP,EZ,1,9.11E6
! Elastic modulus z
MP,PRXY,1,0.3201
! Poisson's ratio xy
MP,PRXZ,1,0.3201
! Poisson's ratio xz
MP,PRYZ,1,0.3201
! Poisson's ratio yz
MP,GXY,1,3.45E6
! Shear modulus xy
MP,GXZ,1,3.45E6
! Shear modulus xz
MP,GYZ,1,3.45E6
! Shear modulus yz
! Geometry
! Creates keypoints at nodes for truss
K,1,0,0,0
K,2,Member_Length*1/2,Truss_Height,0
K,3,Member_Length,0,0
K,4,Member_Length*3/2,Truss_Height,0
K,5,Member_Length*2,0,0
K,6,Member_Length*5/2,Truss_Height,0
K,7,Member_Length*3,0,0
K,8,Member_Length*7/2,Truss_Height,0
K,9,Member_Length*4,0,0
! Creates lines for truss members
L,1,2
L,1,3
L,2,3
L,2,4
L,3,4
L,3,5
L,4,5
L,4,6
L,5,6
67
L,5,7
L,6,7
L,6,8
L,7,8
L,7,9
L,8,9
! Meshing
LESIZE, all,,,Line_Div
! Sets divisons of lines to variable Line_Size
MAT,1
! Selects Material 1 to mesh with
TYPE,1
! Sets the element type to mesh to type 1 (BEAM188)
MSHAPE,0,2D
! Shape of elements are Quad elements
MSHKEY,1
! Sets to a mapped Mesh
LMESH, all
! Meshs all lines
Finish
! Ends preprocessor
! Solution processor
/SOLU
! Enters the solution processor
ANTYPE,STATIC
! Sets the analysis type to static analysis
! Boundary Conditions
KSEL,S,,,1
! Selects node on bottom left
NSLK,S
! Selects node on bottom left
D,All,UX,0
! Applies a pinned constraint to left hand side
D,All,UY,0
! Applies a pinned constraint to left hand side
D,All,UZ,0
! Applies a pinned constraint to left hand side
D,All,ROTX,0
! Applies a pinned constraint to left hand side
D,All,ROTY,0
! Applies a pinned constraint to left hand side
KSEL,S,,,9
! Selects node on bottom right
NSLK,S
! Selects node on bottom right
D,All,UY,0
! Applies a roller constraint to right hand side
68
D,All,UZ,0
! Applies a roller constraint to right hand side
D,All,ROTX,0
! Applies a roller constraint to right hand side
D,All,ROTY,0
! Applies a roller constraint to right hand side
ALLSEL,ALL
! Reselects everything
! Dead and Live Loads
KSEL,S,,,1
! Selects KP for node a
NSLK,S
! Selects node a
F,all,FY,-(1.5*m+2*ga)
! Applies force to node a
KSEL,S,,,2
! Selects KP for node b
NSLK,S
! Selects node b
F,all,Fy,-(2*m+2*gb)
! Applies force to node b
KSEL,S,,,3
! Selects KP for node c
NSLK,S
! Selects node c
F,all,Fy,-(2.5*m+2*gc+P/3)
! Applies force to node c
KSEL,S,,,4
! Selects KP for node d
NSLK,S
! Selects node d
F,all,Fy,-(2.5*m+2*gc)
! Applies force to node d
KSEL,S,,,5
! Selects KP for node e
NSLK,S
! Selects node e
F,all,Fy,-(2.5*m+2*gc+P/3)
! Applies force to node e
KSEL,S,,,6
! Selects KP for node f
NSLK,S
! Selects node f
F,all,Fy,-(2.5*m+2*gc)
! Applies force to node f
KSEL,S,,,7
! Selects KP for node g
NSLK,S
! Selects node g
F,all,Fy,-(2.5*m+2*gc+P/3)
! Applies force to node g
KSEL,S,,,8
! Selects KP for node h
NSLK,S
! Selects node h
F,all,Fy,-(2*m+2*gb)
! Applies force to node h
KSEL,S,,,9
! Selects KP for node i
69
NSLK,S
! Selects node i
F,all,Fy,-(1.5*m+2*ga)
! Applies force to node i
ALLSEL,ALL
! Reselects everything
! Solve
SOLVE
! Solves the current load step
FINISH
! Ends solution processor
! Post processor
/POST1
! Enters the post processor
/ESHAPE,1.0
! Plots beam elements
PLNSOL,S,X,0,1.0
! Plots stress in the x direction
10.2 Appendix B – MATLAB Composite Laminate Stiffness Calculator
% Clears the screen and variables
clear all; clc;
% Material properties for Graphite
E1 = 23.8E6;
E2 = 1.7E6;
v12 = 0.32;
G12 = .750E6;
thickness = 0.006;
% Computes the Q Matrix
Q(1,1) = E1/(1-v12*v12*E2/E1);
Q(1,2) = v12*E2/(1-v12*v12*E2/E1);
Q(1,3) = 0;
Q(2,1) = Q(1,2);
70
Q(2,2) = E2/(1-v12*v12*E2/E1);
Q(2,3) = 0;
Q(3,1) = Q(1,3);
Q(3,2) = Q(2,3);
Q(3,3) = G12;
% # of plies in layup
Num_plies = 336;
theta_1 = 30;
theta_2 = -30;
theta_3 = 90;
theta_4 = 0;
% Creates a matrix of plies of size equal to previous input
for i = 1:Num_plies/8
Ply(i,1) = theta_1;
end
for i = Num_plies/8+1:2*Num_plies/8
Ply(i,1) = theta_2;
end
for i = 2*Num_plies/8+1:3*Num_plies/8
Ply(i,1) = theta_3;
end
for i = 3*Num_plies/8+1:5*Num_plies/8
Ply(i,1) = theta_4;
end
71
for i = 5*Num_plies/8+1:6*Num_plies/8
Ply(i,1) = theta_3;
end
for i = 6*Num_plies/8+1:7*Num_plies/8
Ply(i,1) = theta_2;
end
for i = 7*Num_plies/8+1:Num_plies
Ply(i,1) = theta_1;
end
% Laminate Thickness
H = Num_plies * thickness;
% Creates the z matrix
for i = 1:Num_plies + 1
z(i,1) = -H/2 + (i - 1) * thickness;
end
% Zeroes out the A,B,D,a,b,d matrices
for i = 1:3
for j = 1:3
A(i,j) = 0;
B(i,j) = 0;
D(i,j) = 0;
a(i,j) = 0;
b(i,j) = 0;
d(i,j) = 0;
end
end
72
% Calculates the stresses for each ply
for i = 1:Num_plies
% Calculates m and n for each ply
m = cos(Ply(i,1)/180*pi);
n = sin(Ply(i,1)/180*pi);
% Calculates the Qbar matrix based on Q, m, and n
QBar(1,1,i) = Q(1,1) * m^4 + 2 * (Q(1,2) + 2* Q(3,3)) * n^2 * m^2 + Q(2,2) * n^4;
QBar(1,2,i) = (Q(1,1) + Q(2,2) - 4 * Q(3,3)) * n^2 * m^2 + Q(1,2) * (n^4+m^4);
QBar(1,3,i) = (Q(1,1) - Q(1,2) - 2 * Q(3,3)) * n * m^3 + (Q(1,2) - Q(2,2) + 2 *
Q(3,3)) * n^3 * m;
QBar(2,1,i) = QBar(1,2,i);
QBar(2,2,i) = Q(1,1) * n^4 + 2 * (Q(1,2) + 2* Q(3,3)) * n^2 * m^2 + Q(2,2) * m^4;
QBar(2,3,i) = (Q(1,1) - Q(1,2) - 2 * Q(3,3)) * n^3 * m + (Q(1,2) - Q(2,2) + 2 *
Q(3,3)) * n * m^3;
QBar(3,1,i) = QBar(1,3,i);
QBar(3,2,i) = QBar(2,3,i);
QBar(3,3,i) = (Q(1,1) + Q(2,2) - 2 * Q(1,2) - 2 * Q(3,3)) * n^2 * m^2 + Q(3,3) * (n^4
+ m^4);
% Calculates the ABD matrices
for j = 1:3
for k = 1:3
A(j,k) = A(j,k) + QBar(j,k,i) * (z(i+1) - z(i));
B(j,k) = B(j,k) + QBar(j,k,i) * (z(i+1)^2 - z(i)^2);
D(j,k) = D(j,k) + QBar(j,k,i) * (z(i+1)^3 - z(i)^3);
end
end
% Calculates the transformation matrix
73
T(1,1,i) = m^2;
T(1,2,i) = n^2;
T(1,3,i) = 2*m*n;
T(2,1,i) = n^2;
T(2,2,i) = m^2;
T(2,3,i) = -2*m*n;
T(3,1,i) = -m*n;
T(3,2,i) = m*n;
T(3,3,i) = m^2 - n^2;
end
% Applies the correction coefficient to fully calculate the B & D matrix
B = B/2;
D = D/3;
% Calcutes the 6x6 ABD matrix
for i = 1:3
for j = 1:3
ABD(i,j) = A(i,j);
ABD(i+3,j) = B(i,j);
ABD(i,j+3) = B(i,j);
ABD(i+3,j+3) = D(i,j);
end
end
% Calculates the 6x6 abd matrix
abd = inv(ABD);
% Calculates the a, b, and d matrices
for i = 1:3
for j = 1:3
74
a(i,j) = abd(i,j);
b(i,j) = abd(i,j+3);
d(i,j) = abd(i+3,j+3);
end
end
% Calculates the material properties of the laminate
E_x = 1 / (a(1,1) * H);
E_y = 1 / (a(2,2) * H);
E_z = E_y;
nu_xy = -a(1,2)/a(1,1);
nu_yz = -a(1,2)/a(2,2);
nu_xz = nu_xy;
G_xy = 1 / (a(3,3) * H);
G_xz = G_xy;
G_yz = E_y / (2 * (1 + nu_yz));
% Prints out the properties of the laminate
fprintf('Ex = %.2E Msi\n',E_x/1E6)
fprintf('Ey = %.2E Msi\n',E_y/1E6)
fprintf('Ez = %.2E Msi\n\n',E_z/1E6)
fprintf('nu_xy = %.4f \n',nu_xy)
fprintf('nu_xz = %.4f \n',nu_xz)
fprintf('nu_yz = %.4f \n\n',nu_yz)
fprintf('Gxy = %.2E Msi\n',G_xy/1E6)
fprintf('Gxz = %.2E Msi\n',G_xz/1E6)
fprintf('Gyz = %.2E Msi\n\n',G_yz/1E6)
fprintf('H = % in\n\n',H)
75
10.3 Appendix C – MATLAB Composite Laminate Global Failure
Calculator
% Clears the screen and variables
clear all; clc;
% Length of Laminate in axial direction
L = 10*4;
% Material properties for Graphite
E1 = 23.8E6;
E2 = 1.7E6;
v12 = 0.32;
G12 = .75E6;
thickness = .006;
sigma_1_T = 395E3;
sigma_1_C = -245E3;
sigma_2_T = 16.1E3;
sigma_2_C = -32.3E3;
Tau_12_F = 17.4E3;
% Computes the Q Matrix
Q(1,1) = E1/(1-v12*v12*E2/E1);
Q(1,2) = v12*E2/(1-v12*v12*E2/E1);
Q(1,3) = 0;
Q(2,1) = Q(1,2);
Q(2,2) = E2/(1-v12*v12*E2/E1);
Q(2,3) = 0;
Q(3,1) = Q(1,3);
Q(3,2) = Q(2,3);
Q(3,3) = G12;
76
% Creates a matrix of plies of size equal to previous input
% # of plies in layup
Num_plies = 336;
theta_1 = 45;
theta_2 = -45;
theta_3 = 90;
theta_4 = 0;
% Creates a matrix of plies of size equal to previous input
for i = 1:Num_plies/8
Ply(i,1) = theta_1;
end
for i = Num_plies/8+1:2*Num_plies/8
Ply(i,1) = theta_2;
end
for i = 2*Num_plies/8+1:3*Num_plies/8
Ply(i,1) = theta_3;
end
for i = 3*Num_plies/8+1:5*Num_plies/8
Ply(i,1) = theta_4;
end
for i = 5*Num_plies/8+1:6*Num_plies/8
Ply(i,1) = theta_3;
end
for i = 6*Num_plies/8+1:7*Num_plies/8
Ply(i,1) = theta_2;
end
for i = 7*Num_plies/8+1:Num_plies
Ply(i,1) = theta_1;
77
end
% Information for the Forces
N_M_matrix(1,1) = 1/L;
N_M_matrix(2,1) = 0;
N_M_matrix(3,1) = 0;
N_M_matrix(4,1) = 0;
N_M_matrix(5,1) = 0;
N_M_matrix(6,1) = 0;
% Laminate Thickness
H = Num_plies * thickness;
% Creates the z matrix
for i = 1:Num_plies + 1
z(i,1) = -H/2 + (i - 1) * thickness;
end
% Zeroes out the A,B,D,a,b,d matrices
for i = 1:3
for j = 1:3
A(i,j) = 0;
B(i,j) = 0;
D(i,j) = 0;
a(i,j) = 0;
b(i,j) = 0;
d(i,j) = 0;
end
end
% Calculates the stresses for each ply
78
for i = 1:Num_plies
% Calculates m and n for each ply
m = cos(Ply(i,1)/180*pi);
n = sin(Ply(i,1)/180*pi);
% Calculates the Qbar matrix based on Q, m, and n
QBar(1,1,i) = Q(1,1) * m^4 + 2 * (Q(1,2) + 2* Q(3,3)) * n^2 * m^2 + Q(2,2) * n^4;
QBar(1,2,i) = (Q(1,1) + Q(2,2) - 4 * Q(3,3)) * n^2 * m^2 + Q(1,2) * (n^4+m^4);
QBar(1,3,i) = (Q(1,1) - Q(1,2) - 2 * Q(3,3)) * n * m^3 + (Q(1,2) - Q(2,2) + 2 *
Q(3,3)) * n^3 * m;
QBar(2,1,i) = QBar(1,2,i);
QBar(2,2,i) = Q(1,1) * n^4 + 2 * (Q(1,2) + 2* Q(3,3)) * n^2 * m^2 + Q(2,2) * m^4;
QBar(2,3,i) = (Q(1,1) - Q(1,2) - 2 * Q(3,3)) * n^3 * m + (Q(1,2) - Q(2,2) + 2 *
Q(3,3)) * n * m^3;
QBar(3,1,i) = QBar(1,3,i);
QBar(3,2,i) = QBar(2,3,i);
QBar(3,3,i) = (Q(1,1) + Q(2,2) - 2 * Q(1,2) - 2 * Q(3,3)) * n^2 * m^2 + Q(3,3) * (n^4
+ m^4);
% Calculates the ABD matrices
for j = 1:3
for k = 1:3
A(j,k) = A(j,k) + QBar(j,k,i) * (z(i+1) - z(i));
B(j,k) = B(j,k) + QBar(j,k,i) * (z(i+1)^2 - z(i)^2);
D(j,k) = D(j,k) + QBar(j,k,i) * (z(i+1)^3 - z(i)^3);
end
end
% Calculates the transformation matrix
T(1,1,i) = m^2;
79
T(1,2,i) = n^2;
T(1,3,i) = 2*m*n;
T(2,1,i) = n^2;
T(2,2,i) = m^2;
T(2,3,i) = -2*m*n;
T(3,1,i) = -m*n;
T(3,2,i) = m*n;
T(3,3,i) = m^2 - n^2;
end
% Applies the correction coefficient to fully calculate the B & D matrix
B = B/2;
D = D/3;
% Calcutes the 6x6 ABD matrix
for i = 1:3
for j = 1:3
ABD(i,j) = A(i,j);
ABD(i+3,j) = B(i,j);
ABD(i,j+3) = B(i,j);
ABD(i+3,j+3) = D(i,j);
end
end
% Calculates the 6x6 abd matrix
abd = inv(ABD);
% Calculates the a, b, and d matrices
for i = 1:3
for j = 1:3
80
a(i,j) = abd(i,j);
b(i,j) = abd(i,j+3);
d(i,j) = abd(i+3,j+3);
end
end
% Calculates the epsilon_kappa matrix
epsilon_Kappa_matrix = abd * N_M_matrix;
% Calculates the global strains at the top and bottom of each layer
for i = 1:Num_plies
for j = 1:3
epsilon_global_top(j,:,i)
=
epsilon_Kappa_matrix(j)
+
z(i)
*
z(i+1)
*
epsilon_Kappa_matrix(j+3);
epsilon_global_bot(j,:,i)
=
epsilon_Kappa_matrix(j)
epsilon_Kappa_matrix(j+3);
end
end
% Calculates the global stresses at the top and bottom of each layer
for i = 1:Num_plies
sigma_global_top(:,:,i) = QBar(:,:,i) * epsilon_global_top(:,:,i);
sigma_local_top(:,:,i) = T(:,:,i) * sigma_global_top(:,:,i);
sigma_global_bot(:,:,i) = QBar(:,:,i) * epsilon_global_bot(:,:,i);
sigma_local_bot(:,:,i) = T(:,:,i) * sigma_global_bot(:,:,i);
end
% Calcualtes the Tsai-Wu Failure Criterion Coefficients
F_1 = 1 / sigma_1_T + 1 / sigma_1_C;
F_2 = 1 / sigma_2_T + 1 / sigma_2_C;
81
+
F_11 = -1 / (sigma_1_T * sigma_1_C);
F_22 = -1 / (sigma_2_T * sigma_2_C);
F_66 = (1 / Tau_12_F)^2;
count = 1;
for i = 1:Num_plies
p = eval(solve('F_1 * x * sigma_local_top(1,1,i) + F_2 * x * sigma_local_top(2,1,i) +
F_11 * (x * sigma_local_top(1,1,i))^2 + F_22 * (x * sigma_local_top(2,1,i))^2 + F_66 *
(x * sigma_local_top(3,1,i))^2 - sqrt(F_11 * F_22) * x * sigma_local_top(1,1,i) * x *
sigma_local_top(2,1,i) = 1'));
Answer(count,1) = p(1);
Answer(count + 1,1) = p(2);
count = count +2;
end
for i = 1:Num_plies
Ply_Num(i,1) = i;
end
% Creates table of bottom sigma x, sigma y, and tauxy for each ply
Table_Top_Stress_global(:,1) = Ply_Num;
Table_Top_Stress_global(:,2) = Ply;
count = 1;
for i = 1:Num_plies
Table_Top_Stress_global(i,3) = Answer(count)/1E6;
Table_Top_Stress_global(i,4) = Answer(count+1)/1E6;
count = count + 2;
end
disp('Failure Load of Each Ply (Mlb)')
82
disp(' Ply #
Angle
Tension Compression ')
disp(Table_Top_Stress_global)
% Creates vector of ply numbers
for i = 1:Num_plies
Ply_Num(i,1) = i;
end
% Creates table of bottom sigma x, sigma y, and tauxy for each ply
Table_Top_Stress_global(:,1) = Ply_Num;
Table_Top_Stress_global(:,2) = Ply;
for i = 1:Num_plies
for j = 1:3
Table_Top_Stress_global(i,j+2) = sigma_global_top(j,:,i) / 1E6;
end
end
% Creates table of top sigma 1, sigma 2, and tau12 for each ply
Table_Bot_Stress_global(:,1) = Ply_Num;
Table_Bot_Stress_global(:,2) = Ply;
for i = 1:Num_plies
for j = 1:3
Table_Bot_Stress_global(i,j+2) = sigma_global_bot(j,:,i) / 1E6;
end
end
% Creates table of bottom sigma 1, sigma 2, and tau12 for each ply
Table_Top_Stress_local(:,1) = Ply_Num;
Table_Top_Stress_local(:,2) = Ply;
for i = 1:Num_plies
for j = 1:3
83
Table_Top_Stress_local(i,j+2) = sigma_local_top(j,:,i) / 1E6;
end
end
% Creates table of top sigma 1, sigma 2, and tau12 for each ply
Table_Bot_Stress_local(:,1) = Ply_Num;
Table_Bot_Stress_local(:,2) = Ply;
for i = 1:Num_plies
for j = 1:3
Table_Bot_Stress_local(i,j+2) = sigma_local_bot(j,:,i) / 1E6;
end
end
% Displays the tables
%disp(' Global Stress Top of Plys (Msi)')
%disp(' Ply #
Angle
Sigmax
Sigmay Tauxy')
%disp(Table_Top_Stress_global)
%disp(' Global Stress Bottom of Plys (Msi)')
%disp(' Ply #
Angle
Sigmax
Sigmay Tauxy')
%disp(Table_Bot_Stress_global)
%disp(' Local Stress Top of Plys (Msi)')
%disp(' Ply #
Angle
Sigma1
Sigma2
Tau12')
%disp(Table_Top_Stress_local)
%disp(' Local Stress Bottom of Plys (Msi)')
%disp(' Ply #
Angle
Sigma1
Sigma2
%disp(Table_Bot_Stress_local)
84
Tau12')