Applying Classical Beam Theory to
Twisted Cantilever Beams
and Comparing to the Results of FEA
by
Mitchell S. King
An Engineering 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 Advisor
Rensselaer Polytechnic Institute
Hartford, Connecticut
December, 2012
© Copyright 2012
by
Mitchell S. King
All Rights Reserved
ii
CONTENTS
LIST OF FIGURES ........................................................................................................... v
LIST OF TABLES ........................................................................................................... vii
NOMENCLATURE ....................................................................................................... viii
GLOSSARY ..................................................................................................................... ix
ACKNOWLEDGMENT ................................................................................................... x
ABSTRACT ..................................................................................................................... xi
1. Theory .......................................................................................................................... 1
1.1
Coordinate System ............................................................................................. 1
1.2
Twist Parameters ................................................................................................ 4
1.3
Stress, Strain, and Displacement for a Cantilever Beam ................................... 5
1.4
1.5
1.3.1
Stress Equilibrium .................................................................................. 6
1.3.2
Strain-Displacement Relationships ........................................................ 8
1.3.3
Stress-Strain Relationships .................................................................. 10
Generic Cantilever Beam Stress, Strain, and Displacement Solutions ............ 11
1.4.1
Functionally Varying Moment of Inertia ............................................. 13
1.4.2
Discontinuous Shear Stress Solution for a Rectangular Beam ............ 18
1.4.3
General Stress Distribution for a Twisted Cantilever Beam ................ 21
1.4.4
General Strain Distribution for a Twisted Cantilever Beam ................ 22
1.4.5
General Vertical Displacement of a Twisted Cantilever Beam ........... 23
Specific Theoretical Solution ........................................................................... 32
2. Finite Element Analysis (FEA) Model ...................................................................... 38
2.1
Methodology .................................................................................................... 38
2.2
Analytical Results ............................................................................................ 41
2.2.1
Tensile Stress and Strain Values (x-Direction) .................................... 43
2.2.2
Shear Stress and Strain Values (xy-Direction) ..................................... 47
2.2.3
Vertical Displacement (y-Direction) .................................................... 51
iii
3. Comparison of Beam Theory Formulation Results to FEA Results.......................... 54
3.1
Displacement Values ........................................................................................ 54
3.2
Maximum Tensile Stress Values ...................................................................... 57
3.3
Maximum Shear Stress Values ........................................................................ 58
4. Summary and Conclusion .......................................................................................... 60
5. Appendix A – Theoretical Results at Selected x Values ........................................... 62
6. Appendix B – FEA Results at Selected x Values ...................................................... 67
7. Appendix C – ABAQUS Input File (.inp) ................................................................. 72
8. References................................................................................................................ 100
iv
LIST OF FIGURES
Figure 1 - Example of a Twisted Cantilever Beam ........................................................... 1
Figure 2 – 90° Twisted Cantilever Beam in the xy-Plane.................................................. 2
Figure 3 – 90° Twisted Cantilever Beam in the yz-Plane .................................................. 3
Figure 4 - xyz Axes vs. ABC Axes .................................................................................... 4
Figure 5 – Arbitrary Beam Cross-Section Subjected to Bending Moment Mz(x) ............. 6
Figure 6 - Rotation of Axes and Moment of Inertia ........................................................ 13
Figure 7 - Moment of Inertia Variation within the Twisted Region of a Rectangular
Beam ................................................................................................................................ 17
Figure 8 - Point of Discontinuity in Shear Stress ............................................................ 19
Figure 9 - Width and Thickness Parameters of the Solution ........................................... 32
Figure 10 - Load and Length Parameters of the Solution ................................................ 32
Figure 11 - Midplane Shell of the ABAQUS FEA Model .............................................. 38
Figure 12 - Meshed View of the ABAQUS FEA Model................................................. 39
Figure 13 - Load and Boundary Conditions of the ABAQUS FEA Model..................... 40
Figure 14 - FEA Shell Model, Variation of Principal Axes ............................................ 42
Figure 15 - FEA Tensile Stress at the Shell’s Midplane ................................................. 44
Figure 16 - FEA Tensile Stress at the Shell's Positive Face ............................................ 45
Figure 17 - FEA Indicates an Additional Compressive Stress Peak in Twisted Region,
Shell’s Positive Face ........................................................................................................ 45
Figure 18 - FEA Tensile Stress at the Shell's Negative Face .......................................... 46
Figure 19 - FEA Indicates an Additional Tensile Stress Peak in Twisted Region, Shell’s
Negative Face .................................................................................................................. 46
Figure 20 - FEA Shear Stress Distribution at the Shell's Midplane ................................ 47
Figure 21 - FEA Shear Stress Concentration at Shell's Midplane ................................... 48
Figure 22 - FEA Shear Stress Distribution on Shell’s Positive Face .............................. 49
Figure 23 - FEA Shear Stress Distribution on Shell’s Negative Face ............................. 50
Figure 24 - FEA Displacement Results, Isometric View ................................................ 51
Figure 25 - FEA Displacement Results, xy-Plane ........................................................... 51
Figure 26 - FEA Displacement Results, yz-Plane ............................................................ 52
Figure 27 - FEA Indicates Residual Stiffness and Non-planar Cross-Section at x = x4 .. 53
v
Figure 28 - Theory vs. FEA, Vertical Displacement ....................................................... 54
Figure 29 - Theoretical Displacement Results vs. Beams of Constant Cross-Sectional
Orientation ....................................................................................................................... 56
Figure 30 - Theory vs. FEA, Maximum Tensile Stress ................................................... 57
Figure 31 - Theory vs. FEA, Maximum Shear Stress...................................................... 58
vi
LIST OF TABLES
Table 1 - Summary of FEA Input Parameters ................................................................. 40
vii
NOMENCLATURE
For the following symbols, i ≠ j. Units are in parenthesis.
E - Modulus of Elasticity (psi)
ε - Strain Tensor (in/in)
εii - Extensional Strain (in/in)
εij - Tensor Shear Strain (in/in)
F - Force Applied to the Free End of the Cantilever Beam (lbs)
G - Shear Modulus (psi)
γ - Engineering Shear Strain (in/in)
Iii - Moment of Inertia (in4)
Iij - Product of Inertia (in4)
κ - Curvature (in-1)
L - Length (in)
M - Moment (in-lb)
ν - Poisson’s Ratio (dimensionless)
σ - Stress Tensor (psi)
σii - Tensile Stress (psi)
t - Thickness of the Rectangular Cross-Section, Less than its Width, w (in)
τij - Shear Stress (psi)
ϴ - Angular Measurement Between the B and Y-Axes or the C and Z-Axes (radians or degrees)
u - Longitudinal Displacement (in)
v - Vertical Displacement (in)
w - Transverse Displacement, Only Applicable to Section 1.3.2 (in)
w - Width of the Rectangular Cross-Section, Greater than its Thickness, t (in)
viii
GLOSSARY
ABAQUS
A software application used for both the modeling and analysis of
- mechanical components and assemblies (pre-processing) and
visualizing the finite element analysis results (post-processing).
A projecting beam that is supported at one end and carries a load at
the other end or along its length.
A coordinate system comprised of three mutually orthogonal axes to
represent three-dimensional space.
Cantilever Beam
-
Cartesian Coordinate System
-
Deflection
- A movement of a structural member resultant of an applied force.
Displacement
- See ‘deflection’.
Equilibrium
- A state in which opposing forces or influences are balanced.
FEA
- An acronym for Finite Element Analysis.
Final Orientation
- The cross-sectional orientation of the beam in the region beyond L2.
Global Axes
- of the beam and whose origin is at the fixed end of the beam.
Hooke's Law
Initial Orientation
- related to the applied stress within the elastic limit of the solid.
- The cross-sectional orientation of the beam from its origin to L1.
Isotropic
- Having uniform physical properties in each direction.
Local Axes
- of the beam’s cross-section and whose origin varies to satisfy this
The coordinate system that remains constant throughout the length
A law stating that the strain in a solid is proportional and linearly-
The coordinate system that remains perpendicular to the perimeter
perpendicularity.
Computes an approximation to a definite integral, made by finding
the area of a collection of rectangles whose heights are determined
by the values of the function.
The proportionality constant relating a solid’s extensional strain
value(s) to its longitudinal stress value(s) within the elastic region.
A measurement that quantifies a beam’s ability to resist bending
about a particular axis.
The point in three-dimensional space where all three Cartesian axes
are coincidient, e.g. x = y = z = 0.
Midpoint Rule
-
Modulus of Elasticity
-
Moment of Inertia
-
Origin
-
Poisson's Ratio
- The negative ratio of transverse strain to axial strain.
Product of Inertia
- deformation about a particular axis.
Shear Modulus
- to its shear stress value(s) within the elastic region.
Strain
- material body in which the forces are being applied.
- A measure of the internal forces acting within a deformable body.
Stress
Twisted Cantilever Beam
Varying Orientation
A measurement that quantifies a beam’s ability to resist shear
The proportionality constant relating a solid’s shear strain value(s)
The ratio of total deformation to the initial dimension of the
A cantilever beam whose cross-section remains constant, but is
- rotating about its centroid along the beam’s length.
- The cross-sectional orientation of the beam in the region of L1 to L2.
ix
ACKNOWLEDGMENT
I would like to thank my fiancé, Jessica Rowe, for her unwavering support,
encouragement, and affection during my graduate studies.
x
ABSTRACT
Twisted cantilever beams are defined here as beams whose rectangular cross-sectional
orientation changes along the beam’s length with respect to global axes. These beams
have a twisted or spiral-type geometric feature somewhere along their length. This
project will formulate solutions for twisted cantilever beams in static bending using
classical beam theory, compare the results to numerical solutions, and discuss
inconsistencies.
xi
1. Theory
1.1 Coordinate System
The analytical case of interest is a rectangular cantilever beam in static bending with an
abrupt, 90° twist, like that shown in Figure 1. The beam is rigidly affixed at one end,
preventing any displacements or rotations at that end. The geometric center of the beam
at the affixed end will be defined as the ‘origin’ or the point at which all coordinates
(x,y,z) are equal to zero. Cartesian coordinates will be used in these solutions because
traditional cantilever beams in bending have documented analytical solutions in
Cartesian coordinates as well. Note that the placement of the axes shown in Figure 1
does not place the vertex at the analytical origin.
Figure 1 - Example of a Twisted Cantilever Beam
Moving away from the rigid attachment at (0,0,0), parallel to the length of the beam will
be defined as: ‘moving in the x-direction’, defining the x-axis. The direction of the
y-axis is orthogonal to the x-axis and is chosen to be parallel to one side of the beam’s
cross-section at the origin. The z-axis is orthogonal to both the x and y-axes. These
three axes whose origin occurs at the beam’s geometric center, at the fixed end, are
defined as the ‘global axes’.
1
Moving down the beam’s length, away from the rigid attachment, in the x-direction, the
beam initially has a homogeneous moment of inertia and its ‘initial orientation’. At
some distance, denoted by L1, the beam’s cross-section begins to rotate about its x-axis,
but its moment of inertia remains continuous, where it can be described as having a
‘varying orientation’. The beam’s cross-section continues to rotate until it reaches
another distance, denoted by L2, where the rotation stops but again, the moment of
inertia remains continuous. Beyond L2, the beam’s ‘final orientation’ continues until it
reaches its free end, whose distance from the origin is denoted by the length L3. For the
case of a 90° twist, the final orientation is rotated 90° from the initial orientation. See
Figure 2 and Figure 3.
Y
O
F
X
L1
L12
L3
Figure 2 – 90° Twisted Cantilever Beam in the xy-Plane
2
Y
Z
Figure 3 – 90° Twisted Cantilever Beam in the yz-Plane
Because the cross-sectional orientation of the beam is varying along its length between
L1 and L2, another set of axes are defined in order more easily express twist parameters
and cross-sectional orientations. These ‘local axes’ are also Cartesian, but are denoted
(A,B,C) instead of (x,y,z). The origin of the local axes is not necessarily at the same
location as that of the global axes, but can be anywhere along the beam’s length,
provided that the A-axis remains collinear with the x-axis. At the beam’s fixed end, at
its geometric center, the A-axis is collinear with the x-axis, the B-axis is collinear with
the y-axis, and the C-axis is collinear with the z-axis. Once the beam’s cross-section
begins to rotate (at L1) the B and C-axes develop an angular measurement greater than
zero between their global counterparts (y and z, respectively). The A-axis and x-axis
will always remain collinear, and either can be used interchangeably. See Figure 4.
3
Y
Y
[ x and A-axes both coming out of the page. ]
B
B
Z
ϴ
Z
C
C
Figure 4 - xyz Axes vs. ABC Axes
It is important to describe and define the parameters within the beam’s varying
orientation (L1 ≤ x ≤ L2), because these parameters will contribute to the analytical
solutions that are eventually formulated. The difference between the cross-sections at
x = L1 and L1 ≤ x ≤ L2 can be quantified by the angular difference, ϴ, between the y and
B-axes or the z and C-axes. The rate of twist can then be defined as the change in angle,
ϴ, over the change in length between L1 and L2, or
.
1.2 Twist Parameters
For the problem to be solved, the cross-sectional orientation can be described as:
[1]
[2]
[3]
Let ϴ(x) be a linear function of x, so that the rate of twist from L1 to L2 is a constant
value. Then, by the equalities given by equations [1] through [3]:
4
Substituting the boundary conditions in the expression for ϴ yields:
Solving for the constants M and N, and subtracting these equalities gives:
The function ϴ(x) can then be written as:
[4]
So, the rate of twist is:
[5]
Rewriting equation [1] through [3] with equation [4]:
[6]
[7]
[8]
Together, equations [6] through [8] fully describe the beam’s twist parameters and can
be derived using the values L1 and L2.
1.3 Stress, Strain, and Displacement for a Cantilever Beam
Because the cantilever beam of interest has a functionally varying cross-sectional
orientation along its length, the stress distribution throughout the beam must be solved
for a generic case first so that the twist parameters developed in section 1.2 can be
5
applied. Let a beam of arbitrary cross-section be subjected to a bending moment, Mz(x),
about the z-axis, which is a function of longitudinal (x) position only. The origin of the
cross section is at the beam’s centroid, and the y and z-axes are the principal axes. See
Figure 5.
[ X and A-axes both coming out of the page. ]
Y
Arbitrary
Cross-Section
Z
Mz(x)
Figure 5 – Arbitrary Beam Cross-Section Subjected to Bending Moment Mz(x)
1.3.1
Stress Equilibrium
As a result of the applied load, F, and subsequent moment, Mz(x), the beam experiences
displacements, strains, and a state of stress. Each can be represented by a tensor at each
point throughout its volume. The generic stress tensor, [σ], is given by:
[9]
In order to satisfy equilibrium, the stress tensor must be symmetric such that:
6
Because stresses can be related to displacements (as will be shown in following
discussion), and because the displaced shape of an end-loaded cantilever beam is similar
to that of a beam in pure bending, it is reasonable to assume that the cantilever beam has
a stress tensor similar to that of a beam in pure bending. Beginning with the bending
moment, Mz(x):
[10]
[11]
[12]
That is, the normal stresses, σii, and the x-z and y-z shear stresses, τxz and τyz, are identical
to those for a beam in pure bending, but no specific assumptions are made about the
other shear stress, τxy, only that it is some function of x, y, and z.
The equations of 3-D stress equilibrium with no body forces are now noted to augment
the discussion and simplify the equations above:
[13]
[14]
[15]
Based on equations [10] through [12], equations [13] and [14] reduce to:
[16]
[17]
7
Equations [16] and [17] make up the 3-D stress equilibrium equations for a cantilever
beam experiencing plane stress. Upon inspection, one can see that equation [17] is only
satisfied if τxy is constant in the x-domain. Rewriting equations [10] and [11] for clarity,
equation [12] can be simplified to:
[18]
[19]
[20]
1.3.2
Strain-Displacement Relationships
As mentioned, the beam also experiences a state of strain. The generic stress tensor, [ε],
is given by:
[21]
where εii are normal strains and εij are shear strains.
The strain tensor is symmetric, such that:
8
For small deflections (where sin α ≈ α), the following expressions relate strain to
displacement:
[22]
[23]
[24]
[25]
[26]
[27]
Engineering shear strain is related to tensor shear strain by:
Rewriting equations [22] through [27], the components of the strain tensor, [ε], are:
[28]
[29]
[30]
[31]
[32]
[33]
9
1.3.3
Stress-Strain Relationships
For an isotropic, linear-elastic material (obeying Hooke’s Law), the stress-strain
relationships are as follows:
[34]
[35]
where E is the modulus of elasticity and G is the shear modulus. The two modulii are
related through Poisson’s Ratio, ν, by the following:
[36]
For each component of the strain tensor, the relationships are:
[37]
[38]
[39]
[40]
[41]
[42]
10
1.4 Generic Cantilever Beam Stress, Strain, and Displacement
Solutions
Equation [18] indicates that the normal stress in the x-direction is dependent on the
bending moment, Mz(x), which is a function of x. This bending moment is simply the
force, F, multiplied by the distance from the origin, and can be written as:
[43]
So, the normal stress in the x-direction is:
[44]
Substituting equation [44] in the x-direction stress equilibrium equation ([16]) gives:
[45]
Integrating equation [45]:
P is a constant of integration and can be determined on the basis that in order to satisfy
boundary equilibrium around the perimeter of the beam, the shear force must be equal to
zero at these points.
The total stress distribution is now repeated for clarity:
[46]
[47]
[48]
11
The strains at each point can be found directly from the resultant stresses, by inserting
equations [46] through [48] into equations [37] through [42]. Inserting equation [46]
and [47] into the stress-strain relationships (equation [37]), and that into the straindisplacement relationship (equation [22]) leads to a noteworthy:
[49]
Equation [49] can be integrated in x to solve for the longitudinal displacement of the
beam, once the function of Izz is known.
For small strains and displacements in the elastic range, and assuming that plane sections
remain plane, the curvature of the beam’s neutral surface can be expressed in the
following form:
where κ is the curvature. Inserting equation [43] gives:
[50]
Equation [50] is a second-order linear differential equation, and is the governing
equation for the elastic curve. The product EI is the flexural rigidity of the beam.
Because the moment of inertia, Izz, varies with respect to x, it must be first formulated in
order to integrate equation [50] and solve for the vertical displacement.
Up to this point, each parameter of the stress, strain, and displacement components is
known except the moment of inertia about the neutral axis, Izz, and the constant of
integration, P, for the shear term. This moment of inertia and P are derived in the
following discussion.
12
1.4.1
Functionally Varying Moment of Inertia
Consider the plane area shown in Figure 6 below. The moments and product of inertia
with respect to the local BC-axes are:
[51]
[52]
[53]
The same forms of expressions exist for the global coordinate system, in xyzcoordinates.
[ x and A-axes both coming out of the page. ]
Y
Arbitrary
B
Cross-Section
ϴ
CdA
dA
z
y
BdA
Z
C
Figure 6 - Rotation of Axes and Moment of Inertia
The moments and product of inertia in the BC-plane are constant values, equal to those
of the yz-plane at the origin. However, as the angular measurement θ increases, the
moments and product of inertia in the yz-plane change. To obtain these quantities, the
coordinates of the differential element dA are expressed in terms of the yz-coordinates as
follows:
13
[54]
[55]
Substituting these values in equations [51] through [53] gives:
Using the following trigonometric identities, the form of IYY, IZZ, and IYZ can be
simplified.
14
These are complicated expressions in their most reduced form, unlike the simple and
familiar
(for example). However, if the general expressions for moments
and product of inertia in the local coordinate system (equations [51], [52], and [53]) are
substituted in the above integrals, they take on a more practical form.
[56]
[57]
[58]
IYY, IZZ, and IYZ are the moments and product of inertia in the global coordinate system at
any point along the x-axis, and IBB, ICC, and IBC are the moments and product of inertia in
the local coordinate system.
With the moments and products of inertia defined as such, the reader is referred back to
Figure 2 to be reminded of the cross-sectional orientation at the origin. Specifically, for
this problem, the local moments and product of inertia are the same as those of a
rectangular cross-section in bending, where ICC is the “strong axis” of bending and IBB is
the “weak axis” of bending at the origin. The limits of integration are defined by the
beam’s perimeter. Thus:
15
[59]
[60]
[61]
where w is the width of the beam, and t is the thickness.
For any symmetric cross-section whose centroid is at the origin, IBC will be zero, so any
terms containing the beam’s local product of inertia will drop out of equations [56], [57],
and [58]. Substituting equations [59] through [61] into these equations gives:
By substituting in the relationship between ϴ and x (see equations [6], [7], and [8]),
these equations yield the full form of the functionally varying moments of inertia.
[62]
16
[63]
[64]
Taking (for example) values of t =0.25” and w = 1.00”, the moments and product of
inertia change in the twisted region of the beam as shown in Figure 7 below:
Figure 7 - Moment of Inertia Variation within the Twisted Region of a Rectangular Beam
17
1.4.2
Discontinuous Shear Stress Solution for a Rectangular Beam
The stress distributions will be solved to eventually formulate the strain and
displacement solutions. However, the shear stress, τxy, still has an undefined constant, P,
which must be found to fully formulate the stress in the beam. As stated in section 1.4.1,
P is a constant of integration that can be determined on the basis that in order to satisfy
boundary equilibrium around the perimeter of the beam, the shear force must be equal to
zero at these points. But because the beam’s cross-section is rotating, this constant is
also related to the twist parameters for L1 ≤ x ≤ L2.
For the beam’s initial orientation:
Substituting this value into equation [48] gives:
[65]
For the beam’s final orientation:
Substituting this value into equation [48] gives:
[66]
18
For the beam’s varying orientation, the y-coordinate of the beam’s outermost fiber is
constantly changing. This coordinate is needed to solve the shear stress distribution in
this region. The perimeter of the beam can be represented by four straight lines whose
orientation varies with the x-position, or ϴ. Because the perimeter edges intersect one
another at right angles, however, the function of the outermost y-coordinate will not be a
continuous function. The function changes when the y-axis coincides with the corner of
the beam’s cross-section.
For any rectangular beam, this occurs at an angle of
. See Figure 8.
Y
ϴ = arctan (t/w)
w
Z
t
Figure 8 - Point of Discontinuity in Shear Stress
For the varying orientation, where 0 ≤ ϴ ≤ arctan (t/w), the y-coordinate of the outermost
fiber of the beam’s cross-section is:
19
Substituting this value into equation [48] and using the relationship between ϴ and x
(equation [7]) gives:
[67]
Similarly, for the varying orientation, where arctan (t/w) ≤ ϴ ≤ π/2, the y-coordinate of
the outermost fiber of the beam’s cross-section is:
Substituting this value into equation [48] and using the relationship between ϴ and x
(equation [7]) gives:
[68]
20
1.4.3
General Stress Distribution for a Twisted Cantilever Beam
Consolidating equations [6], [7], [8], [46], and [65] through [68]:
Any
[69]
Orientation:
Any
[70]
Orientation:
Initial
[71]
Orientation:
Varying
[72]
Orientation:
Varying
[73]
Orientation:
Final
[74]
Orientation:
21
1.4.4
General Strain Distribution for a Twisted Cantilever Beam
Inserting equations [69] through [74] into the stress-strain relationships given by
equations [37] through [42]:
Any
[75]
Orientation:
Any
[76]
Orientation:
Any
[77]
Orientation:
Initial
[78]
Orientation:
Varying
[79]
Orientation:
Varying
[80]
Orientation:
Final
[81]
Orientation:
22
1.4.5
General Vertical Displacement of a Twisted Cantilever Beam
Now that the moment of inertia, Izz, has been derived, equation [50] can be integrated to
obtain the y-displacement function.
However, because Izz is dependent upon the
longitudinal position, x, the vertical displacement function must be dissected into three
conditional equations, depending on the magnitude of x. The first solution presented
will be for the case of
.
Because this x-location is within the initial
orientation and the moment of inertia is constant in this region, equation [50] is
integrated twice in x, as would be done for a normal cantilever beam with constant crosssection:
where C1 is a constant of integration. Integrating again in x gives:
The constants of integration, C1 and C2 can be determined by applying the boundary
conditions of the beam. At the fixed end,
beam are equal to 0. Therefore,
, the displacement and the slope of the
. So, for the region of
:
[82]
The displacement function for x values in the varying orientation becomes more
complicated, however. For values of x where
, the moment of inertia cannot
be considered constant and excluded from the integrals, as was done above. Inserting
the equality for Izz into equation [50], for values of x where
once with respect to x gives:
23
, and integrating
The second term of the y-displacement slope solution (above) indicates an inherent flaw
in the application of classical beam theory to this problem. Even a simplified version of
the function
does not stay within the real domain; the varying moment of
inertia function’s presence in the denominator complicates the solution beyond any
reasonable point.
Because the solution is beyond the scope of this document, a piecewise approximation is
instead presented using the midpoint rule. If the second term above is broken into many
separate intervals, the integral can be carried out by approximating the value of x for
each separate interval. That is, the distance from L1 to L2 will be divided by N number
of divisions, and the average longitudinal value between each point will be used in place
of x. The midpoint rule is defined as:
The midpoint of each interval is equal to the variable xn, where N is the number of
predefined intervals chosen:
Because the midpoint rule is only needed for values of x where
, the value
of a is already known and equal to L1. Likewise, the value of b will be set to x.
Therefore, the second term of the exact y-displacement slope solution can be
approximated as:
24
In this form, the first term of the exact solution can be solved directly, and the second
term can be approximated. Of course, as
, the function will converge to the exact
solution, but in this form it is much “easier” to deal with. Thus, the slope of the beam,
, for
can be written as:
The constant of integration, C3, can be determined by applying the boundary condition
provided by equation [82]: at
,
25
Rewriting the expression for the slope of the beam in the region of
:
To solve for the displacements, the equality above must again be integrated in x.
However, the second term again creates problems and yields solutions of non-real
numbers. Thus, the midpoint rule must be applied a second time. Let the second term
above be defined as an arbitrary function of x, β(x), so that:
Integrating the second term using the midpoint rule gives:
26
There have certainly been more elegant expressions derived in engineering. Writing the
total solution for y-displacement when
gives:
The constant of integration, C4, can be found by using equation [82] at
27
:
The y-displacement solution for
becomes:
[83]
Finally, the displacement solution for values of x where
can be formulated
using the same process as the other intervals. Because this x-location is within the final
orientation and the moment of inertia is constant in this region, equation [50] is
integrated twice in x, as was done to derive equations [82] and [83], only the
contribution to the displacement from the varying orientation is given the limits of L1
and L2.
28
The constant of integration, C5, can be determined by recalling the function derived for
the beam’s slope between
, and inserting
29
:
The slope of the beam for
can then be written as:
Notice that in this region only one term of the beam’s slope is dependent on x. This
simplifies things much more than before when deriving the beam’s deflection in the
varying orientation. That is, all integrals henceforth can be computed directly, and no
more approximations are needed. To solve for the y-displacement, the equality above is
integrated in x to give:
30
The final constant of integration can be determined by the boundary condition:
at
. Therefore,
(from equation [83]).
[84]
Equations [83] and [84] are very complicated and tedious to carry out in real
applications. A less cumbersome method is to use the equation of the beam’s slope,
, and integrate the numerical value(s) over x, rather than carrying through to a
closed-form solution. This method will be used to obtain values in section 1.5 and in
Appendix A.
31
1.5 Specific Theoretical Solution
Advancing from the generic case, the specifics of the problem are now defined and solved.
Figure 9 shows the beam’s cross-section at two distinct points and assigns its width (w)
and thickness (t). Let these values be 1.0” and 0.25”, respectively. Figure 10 is a
repetition of Figure 2, but assigns values to the applied force, F, and lengths L1, L2, and L3.
Y
Z
Y
w
Z
t
w
[Final Orientation]
Y
O
t
w = 1.0”
[Initial Orientation]
t = 0.25”
Figure 9 - Width and Thickness Parameters of the Solution
10 lbs
X
3”
1
4.5”
7.5”
Figure 10 - Load and Length Parameters of the Solution
Let the material properties
of the beam be those of mild steel (ms):
32
For comparison to numerical results, the stress, strain, and displacement values at five
distinct values of x will be found. These five x-locations are as follows:
corresponds to the start of the twist,
its twist feature,
corresponds to the midpoint of the beam and
corresponds to the end of the twist, and
corresponds to the end of
the beam. In addition, at each x value the stress, strain, and vertical displacement values
will be formulated for three values of y, corresponding to the positive/negative outermost
beam fiber y value, and at
, the neutral axis. The following solution is that of
. The analytical results for all other points can be found in Appendix A.
In the beam’s local coordinates, the moments and products of inertia are:
For
and
the moment of inertia, IZZ, is:
(See equation [55]).
Therefore, the stress distribution for
and
33
is:
In tensor form:
The strains at
and
can be found by inserting these values into the
stress-strain relationships, given by equations [37] through [42]:
The vertical displacement at this point is found using the slope equation that, when
integrated in x, gives equation [83]. Because the value of x = 3.75” lies in the region of
. For the following solution, let N = 5.
34
To extract a vertical displacement from the beam’s slope at this point,
is simply
integrated again in x with the appropriate limits:
The constant of integration, C7, is determined by the boundary condition provided by the
beam’s vertical displacement in its initial orientation.
At
(from equation [82]),
35
The very outer-most beam fiber that lies on the y-axis is derived in section 1.4.2, and is
given by the equality:
Since
For
at
, the outer-most beam fiber on the y-axis is:
and
the stress distribution is:
In tensor form:
The strains at
and
can be found by inserting these values into
the stress-strain relationships, given by equations [37] through [42]:
The vertical displacement at this point does not depend on y, so it is the same as
previously calculated.
36
For
and
the stress distribution is:
In tensor form:
The strains at
and
can be found by inserting these values into
the stress-strain relationships, given by equations [37] through [42]:
The vertical displacement at this point does not depend on y, so it is the same as
previously calculated.
37
2. Finite Element Analysis (FEA) Model
2.1 Methodology
To compare and validate the theoretical results, the finite element analysis (FEA)
software ABAQUS was used. ABAQUS is a powerful software application used for
both the modeling and analysis of mechanical components and assemblies (preprocessing) and visualizing the finite element analysis results (post-processing).
ABAQUS offers a variety of different modeling approaches for the formulation of a
cantilever beam problem. The user has the option of using 3D-continuum parts, shell
assemblies, axisymmetric models, planar parts, and many other choices.
For this
particular problem, a shell assembly was created and meshed to generate S4R elements
(4-sided shell elements using reduced integration methods). This option was chosen
because it most closely matches the theoretical results of a traditional cantilever beam,
without a twist feature.
First, the midplane shell geometry was created by extruding line-connectors to specific
datum planes. These datum planes represent the x-values chosen for examination of the
theoretical solutions. See Table 1 for the values of x that represent these datum planes.
The figure below shows the shell geometry created in ABAQUS to create an FEA model
of the twisted cantilever beam.
Figure 11 - Midplane Shell of the ABAQUS FEA Model
38
After the geometry was defined, the element was assigned a shell thickness (t = 0.25”)
and material properties (E = 30x106 psi, ν = 0.3) identical to those of the theoretical
solution in Section 1.5. The shell geometry was then meshed to form S4R elements,
with an approximate global seed size of 0.1”. The seed size of the model determines the
coarseness of the mesh. ABAQUS generally selects an appropriate seed size based on
the number of elements it is able to process; with the student version of ABAQUS (used
herein), the number of elements is limited to 1,000.
It is possible to extrude the twisted region of the cantilever beam as one element,
however, Figure 11 above shows that it was created with two separate elements. This
was done to force the meshing to generate element points at the x = 3.75” point. This is
useful in that point results at the middle of the twist feature can later be extracted,
instead of using points nearby and approximating. Figure 12, below, shows the meshed
model of the twisted cantilever beam.
Figure 12 - Meshed View of the ABAQUS FEA Model
Finally, the root of the beam was fixed by creating an initial job step and setting all
displacement and rotations at x = 0” equal to zero (shown as orange cones in Figure 13).
A second job step was created to define the end-load on the beam. To avoid point
effects, a transverse shear, line-load on the far edge was used (shown as red arrows in
39
the negative y direction in Figure 13). This model was submitted for analysis and
completed successfully.
Figure 13 - Load and Boundary Conditions of the ABAQUS FEA Model
Table 1 - Summary of FEA Input Parameters
Variable
Value
L1
3”
L2
4.5”
L3
7.5”
w
1.00”
t
0.25”
E
30 x 106 psi
ν
0.3
F
-10 lbs
40
2.2 Analytical Results
Explicit numerical results for x values corresponding to those analyzed in Section 1.5
and Appendix A can be found in Appendix B. The model used for analysis is a shell
model whose thickness or normal direction defines the z-axis, therefore shear stresses
cannot be directly extracted from ABAQUS in the traditional manner. As shown in
Figure 14 below, the model’s 2 and 3 axes (corresponding to the B and C-axes) evolve
over the length of the beam.
Thus, the numerical results provided by ABAQUS
correspond to the stress states in the ABC-coordinate system (see Figure 4). A direct
comparison can be made to any x-direction values, but to compare shearing components
or any direction other than x, the values must be properly transformed into the global
coordinate system (xyz). To apply this transformation of results, the user must specify a
field output transformation option corresponding to the global axes (under
ResultsOptionsTransformation).
41
Figure 14 - FEA Shell Model, Variation of Principal Axes
42
NOTE:
The following graphical representations extracted from ABAQUS give stress, strain,
and displacement values in index notation. Therefore,
2.2.1
Tensile Stress and Strain Values (x-Direction)
The midplane of the FEA model shows tensile stress values (σxx) peaking at the fixed
end of the beam, as is the case for a traditional cantilever beam. It does not indicate any
specific stress-increasing effect induced by the twist feature. See Figure 15 for a view of
the tensile stress results at the midplane.
At the shell model’s positive face, however, ABAQUS shows a severe increase in
compressive stress as the beam approaches its final orientation. At the model’s negative
face, the same region indicates a severe increase in tensile stress. See Figure 16 and
Figure 17 for the tensile stresses at the positive face of the shell model and Figure 18 and
Figure 19 for the tensile stresses at the negative face of the shell model.
Because the relationship between stress and strain is linear (see equations [34] and [35])
the graphical results of only stress are given. The strain distributions predicted by
ABAQUS are similar to their corresponding stress distributions, shown below.
43
Figure 15 - FEA Tensile Stress at the Shell’s Midplane
44
Figure 16 - FEA Tensile Stress at the Shell's Positive Face
Area of compressive stress peak
Figure 17 - FEA Indicates an Additional Compressive Stress Peak in Twisted Region, Shell’s Positive Face
45
Figure 18 - FEA Tensile Stress at the Shell's Negative Face
Area of tensile stress peak
Figure 19 - FEA Indicates an Additional Tensile Stress Peak in Twisted Region, Shell’s Negative Face
46
2.2.2
Shear Stress and Strain Values (xy-Direction)
Figure 20 shows the shear stress distribution of the shell’s midplane. Notice that the
initial orientation shows a parabolic distribution, as is the case of classical beam theory.
However, Figure 20 also shows that the shear stress has a negative shear stress
concentration at the origin, near the top and bottom fibers (the blue corners). This may
be attributed to point effects and discontinuity extrapolations generated by the numerical
integration routine of ABAQUS. Notice that just adjacent to these points, the shear
stress is positive, and more close to a zero value.
Figure 20 - FEA Shear Stress Distribution at the Shell's Midplane
Figure 20 and Figure 21 also show that at the beam’s midpoint, in the middle of its twist
feature, there is a negative shear stress concentration. Again, note that there is a linear
relationship between stress and strain. Thus, only the distributions for stress are shown.
47
Negative Shear Stress Concentration
Figure 21 - FEA Shear Stress Concentration at Shell's Midplane
The results of the positive and negative faces of the shell model indicate severe shear
stress concentrations after the twist midpoint and before the beam’s final orientation
begins. Most notably, there is a large increase in positive shear in the xy-plane. This
rebounds the color spectrum of the post-processor, and results in the rest of the beam
appearing as a constant shear value. This is not the case, however. The shear stress still
follows a parabolic distribution over the height of the beam, up to the twisted region.
See Figure 22 and Figure 23 for the shear stress distributions on the positive and
negative faces of the shell model.
48
Shear Stress Concentrations
Figure 22 - FEA Shear Stress Distribution on Shell’s Positive Face
49
Shear Stress Concentrations
Figure 23 - FEA Shear Stress Distribution on Shell’s Negative Face
50
2.2.3
Vertical Displacement (y-Direction)
Figure 24 through Figure 26 below show the twisted cantilever beam’s displacement
distribution. Note that in Figure 26, there is a positive z displacement at the end of the
beam. This feature indicates shear coupling within the layers of the shell, which is not
addressed by classical beam theory.
Figure 24 - FEA Displacement Results, Isometric View
Figure 25 - FEA Displacement Results, xy-Plane
51
Δz
Figure 26 - FEA Displacement Results, yz-Plane
When viewing the rotational displacements about the x-axis, an interesting point arises:
the twist feature seems to have imposed a structural anomaly in the beam at the end of its
twist, at x = x4. The displacements across the beam’s width at the end of the twist
(z-direction) follow a parabolic distribution. Classical beam theory suggests that vertical
displacements are independent of this direction. Looking at regions far from this point
(x4), the rotations about the x-axis disappear, corresponding to traditional theory. At
52
x = x4, however, the outer edges of the beam have a higher ‘residual stiffness’ than the
centerline, and experience smaller displacements. See Figure 27.
x4 = 4.5”
Figure 27 - FEA Indicates Residual Stiffness and Non-planar Cross-Section at x = x4
53
3. Comparison of Beam Theory Formulation Results to FEA
Results
3.1 Displacement Values
The vertical displacement functions derived in section 1.4.5 show the closest correlation
to FEA results.
Equation [82] matches the displacements calculated by ABAQUS
almost exactly. Equation [83] does a reasonable job up to the end of the twist but
beyond this (Equation [84]), predicts a slightly stiffer beam.
Figure 28 shows the
displacements of the beam as calculated in this document and by FEA. Notice that each
method plots a smooth, cubic displacement function along the beam’s length.
Figure 28 - Theory vs. FEA, Vertical Displacement
54
Overall, it is shown that for vertical displacements, classical beam theory can be applied
to cantilever beams with functionally varying moments of inertia to obtain reasonable
results. The percent difference of the beam’s vertical deflection at the end of the beam
(x = 7.5”) using beam theory is only 4.5% different than the value predicted by FEA.
Classical beam theory (as applied in the derivation of theoretical values within),
however, does not account for out-of-plane displacements that may occur, as these are
resultant of shear-coupling and non-planar cross-sectional deformations.
shows that FEA results predict out-of-plane displacements.
Figure 26
Abiding by the key
assumptions of traditional beam mechanics (see section 1.3.1) forces one to discount
out-of-plane effects. In the case of a twisted cantilever beam, this may disregard key
aspects of the beam’s internal mechanics. Beam theory does not leave room for out-ofplane displacements but does predict fairly accurate in-plane displacements when
compared to the results of FEA.
When compared to the analytical solutions for similar rectangular beams with constant
cross-sections (see Figure 29), the vertical displacement curve of this particular twisted
cantilever beam seems to relate more closely to the beam experiencing strong-axis
bending, though this correlation will be a function of the length and twist parameters. If
the twist was closer to the fixed end of the beam (near x = 0”), the displacement curve of
the twisted cantilever beam would be expected to more closely follow that of the
constant cross-section beam in weak-axis bending. Conversely, the closer the twist is to
the applied load (near x = L3), the more closely the displacement curve would follow that
of the constant cross-section beam in strong-axis bending.
55
Figure 29 - Theoretical Displacement Results vs. Beams of Constant Cross-Sectional Orientation
56
3.2 Maximum Tensile Stress Values
The theoretical values of tensile stress (σxx) match the numerical results very well outside
of the beam’s twisted region. Within the twisted region, however, the ABAQUS model
generally predicts much higher values, and does not indicate maximum or minimums at
exactly the same value of x. The FEA model also does not follow the smooth curve
based on trigonometric modifications of the moment of inertia in this region. At each
tensile stress’ maximum point, the theoretical value is 9% less than that of the numerical
solution.
See Figure 30 for the maximum tensile stresses predicted by theoretical
methods and by numerical methods. The theoretical method, as presented herein, should
not be used to calculate maximum tensile stress within the varying-orientation region of
a twisted cantilever beam.
Figure 30 - Theory vs. FEA, Maximum Tensile Stress
57
3.3 Maximum Shear Stress Values
Again, beyond the twisted region, the theoretical stress values closely match the
numerical values. However, for the shear stress (τxy), the FEA results vary greatly from
the theoretical results in the twisted region. Both methods allude to a maximum or
minimum at the twist’s midpoint (x = 3.75”), but the theoretical results appear to have an
inverted shape compared to those of the FEA shell model. See Figure 31.
Figure 31 - Theory vs. FEA, Maximum Shear Stress
The FEA model shows the shear stress peaking at the midpoint of the twist, but this
effect is not predicted by the application of classical beam theory.
A possible
explanation is that, contrary to classical beam theory, plane sections are not remaining
plane under deformation (as shown by Figure 27). This effect would cause the shear
deformations to have a much greater impact on the distribution of flexural stresses
58
within the beam than calculated with traditional methods. Figure 31 shows the severity
of error in using traditional beam theory to calculate shear stresses within the beam’s
twisted region.
For the application of failure criterion (such as von Mises, Tresca, maximum stress, etc),
the shear stress equations developed by this document would not accurately demonstrate
maximum shear stress values, and could lead to errors. However, both theoretical and
FEA models indicate a peak of sorts at the twist’s midpoint. To mitigate gross errors
and propagate conservatism, a scalar factor could be applied to the constant shear stress
values of the initial or final orientations formulated by traditional methods to estimate
the maximum shear stress at the midpoint of the twist. In this particular problem, the
scalar factor is approximately 8/3. Additional case studies of beam geometries should be
performed to confirm the validity of this value and method. Refining the tensile and
shear stress equations could be possible by using the methodology of Timoshenko,
where plane sections do not necessarily remain plane, as demonstrated in Chapter 12 of
Reference [d].
59
4. Summary and Conclusion
Based on comparison of theoretical results to FEA results, application of classical beam
theory to twisted cantilever beams of rectangular cross-section is only useful in
calculating vertical displacement values, derived from the curvature of the beam at its
neutral surface. This approach is the beginning of the direct method. The vertical
displacement function developed correlates very well with FEA results. The theoretical
curve follows a smooth, negative-cubic shape along its length as would be expected with
a cantilever beam. However, classical beam theory does not leave room for deformation
coupling, non-planar cross-sections, and gives no method of solving for the out-of-plane
displacements that FEA predicts.
When the indirect method is used to calculate stresses and strains in the twisted region of
the cantilever beam, the results of this document demonstrate that classical beam theory
is not applicable. If the exact stress distribution was needed within the twisted region,
using the indirect method and classical beam theory would not predict reliable values.
The margin of error within the twisted region is too great to be safely applied to real
applications of twisted cantilever beams.
A possible explanation for the stress calculation error is that cross-sections of the beam
that begin plane are not remaining plane under load and subsequent displacement, which
is a critical assumption of classical beam theory. A non-planar section becomes evident
by studying the FEA rotational and displacement results at the end of the twist, presented
in Figure 27. Because the beam’s cross-section is being distorted as a result of the
applied load, a key stipulation of beam theory is obviously not being satisfied.
The theoretical stress and strain functions derived by this document are not
recommended for detailed analyses. However, based on the applied load and beam
geometry, the results presented herein suggest that the method of calculating vertical
displacement gives reasonable results. These displacement results could be applied to
the strain-displacement equations (section 1.3.2), and then to the stress-strain equations
(section 1.3.3) to approximate stress and strain values within the beam’s twisted region.
60
Note that although the vertical displacement functions derived are very close to FEA
results, full application of the direct method to find stresses and strains will still be an
approximation. Under an applied load, FEA shows that the beam experiences out-ofplane displacement.
This z-displacement will contribute to the stress and strain
distribution throughout the beam, though its magnitude is much less than the vertical
displacements’. Therefore, applying the direct method to calculate stresses will not yield
exact results, but could be used to estimate stress and strain values within the beam’s
twisted region.
61
5. Appendix A – Theoretical Results at Selected x Values
x1
x = 0”, y = 0”
Tensile Stress, σxx (Equation [69])
0 psi
Shear Stress, τxy (Equation [71])
-60 psi
Longitudinal Strain, εxx (Equation [75])
0 in/in
Shear Strain, εxy (Equation [78])
-2.600 x 10-6 in/in
Vertical Displacement, v (Equation [82])
0.000 in
x = 0”, y = 0.5”
Tensile Stress, σxx (Equation [69])
1,800 psi
Shear Stress, τxy (Equation [71])
0 psi
Longitudinal Strain, εxx (Equation [75])
6.000 x 10-5 in/in
Shear Strain, εxy (Equation [78])
0 in/in
Vertical Displacement, v (Equation [82])
0.000 in
x = 0”, y = -0.5”
Tensile Stress, σxx (Equation [69])
-1,800 psi
Shear Stress, τxy (Equation [71])
0 psi
Longitudinal Strain, εxx (Equation [75])
-6.000 x 10-5 in/in
Shear Strain, εxy (Equation [78])
0 in/in
Vertical Displacement, v (Equation [82])
0.000 in
62
x2
x = 3.00”, y = 0”
Tensile Stress, σxx (Equation [69])
0 psi
Shear Stress, τxy (Equation [71])
-60 psi
Longitudinal Strain, εxx (Equation [75])
0 in/in
Shear Strain, εxy (Equation [78])
-2.600 x 10-6 in/in
Vertical Displacement, v (Equation [82])
-4.680 x 10-4 in
x = 3.00”, y = 0.5”
Tensile Stress, σxx (Equation [69])
1,080 psi
Shear Stress, τxy (Equation [71])
0 psi
Longitudinal Strain, εxx (Equation [75])
3.600 x 10-5 in/in
Shear Strain, εxy (Equation [78])
0 in/in
Vertical Displacement, v (Equation [82])
-4.680 x 10-4 in
x = 3.00”, y = -0.5”
Tensile Stress, σxx (Equation [69])
-1,080 psi
Shear Stress, τxy (Equation [71])
0 psi
Longitudinal Strain, εxx (Equation [75])
-3.600 x 10-5 in/in
Shear Strain, εxy (Equation [78])
0 in/in
Vertical Displacement, v (Equation [82])
-4.680 x 10-4 in
63
x3
x = 3.75”, y = 0”
Tensile Stress, σxx (Equation [69])
0 psi
Shear Stress, τxy (Equation [73])
-14.12 psi
Longitudinal Strain, εxx (Equation [75])
0 in/in
Shear Strain, εxy (Equation [80])
-6.118 x 10-7 in/in
Vertical Displacement, v (Equation [83])
-7.297x 10-4 in
x = 3.75”, y =
= 0.1768”
Tensile Stress, σxx (Equation [69])
599.0 psi
Shear Stress, τxy (Equation [73])
0 psi
Longitudinal Strain, εxx (Equation [75])
1.997 x 10-5 in/in
Shear Strain, εxy (Equation [80])
0 in/in
Vertical Displacement, v (Equation [83])
-7.297x 10-4 in
x = 3.75”, y = -
= -0.1768”
Tensile Stress, σxx (Equation [69])
-599.0 psi
Shear Stress, τxy (Equation [73])
0 psi
Longitudinal Strain, εxx (Equation [75])
-1.997 x 10-5 in/in
Shear Strain, εxy (Equation [80])
0 in/in
Vertical Displacement, v (Equation [83])
-7.297x 10-4 in
64
x4
x = 4.50”, y = 0”
Tensile Stress, σxx (Equation [69])
0 psi
Shear Stress, τxy (Equation [74])
-60 psi
Longitudinal Strain, εxx (Equation [75])
0 in/in
Shear Strain, εxy (Equation [81])
-2.600 x 10-6 in/in
Vertical Displacement, v (Equation [84])
-1.382 x 10-3 in
x = 4.50”, y = 0.125”
Tensile Stress, σxx (Equation [69])
2,880 psi
Shear Stress, τxy (Equation [74])
0 psi
Longitudinal Strain, εxx (Equation [75])
9.600 x 10-5 in/in
Shear Strain, εxy (Equation [81])
0 in/in
Vertical Displacement, v (Equation [84])
-1.382 x 10-3 in
x = 4.50”, y = -0.125”
Tensile Stress, σxx (Equation [69])
-2,880 psi
Shear Stress, τxy (Equation [74])
0 psi
Longitudinal Strain, εxx (Equation [75])
-9.600 x 10-5 in/in
Shear Strain, εxy (Equation [81])
0 in/in
Vertical Displacement, v (Equation [84])
-1.382 x 10-3 in
65
x5
x = 7.50”, y = 0”
Tensile Stress, σxx (Equation [69])
0 psi
Shear Stress, τxy (Equation [74])
-60 psi
Longitudinal Strain, εxx (Equation [75])
0 in/in
Shear Strain, εxy (Equation [81])
-2.600 x 10-6 in/in
Vertical Displacement, v (Equation [84])
-6.665 x 10-3 in
x = 7.50”, y = 0.125”
Tensile Stress, σxx (Equation [69])
0 psi
Shear Stress, τxy (Equation [74])
0 psi
Longitudinal Strain, εxx (Equation [75])
0 in/in
Shear Strain, εxy (Equation [81])
0 in/in
Vertical Displacement, v (Equation [84])
-6.665 x 10-3 in
x = 7.50”, y = -0.125”
Tensile Stress, σxx (Equation [69])
0 psi
Shear Stress, τxy (Equation [74])
0 psi
Longitudinal Strain, εxx (Equation [75])
0 in/in
Shear Strain, εxy (Equation [81])
0 in/in
Vertical Displacement, v (Equation [84])
-6.665 x 10-3 in
66
6. Appendix B – FEA Results at Selected x Values
x1
x = 0”, y = 0”
Tensile Stress, σxx
-3.691 psi
Shear Stress, τxy
-62.26 psi
Longitudinal Strain, εxx
-1.096 x 10-11 in/in
Shear Strain, εxy
-3.676 x 10-6 in/in
Vertical Displacement, v
0.000 in
x = 0”, y = 0.5”
Tensile Stress, σxx
1,717 psi
Shear Stress, τxy
-150.7 psi
Longitudinal Strain, εxx
5.527 x 10-5 in/in
Shear Strain, εxy
-1.306 x 10-5 in/in
Vertical Displacement, v
0.000 in
x = 0”, y = -0.5”
Tensile Stress, σxx
-1,719 psi
Shear Stress, τxy
-150.0 psi
Longitudinal Strain, εxx
-5.533 x 10-5 in/in
Shear Strain, εxy
-1.300 x 10-5 in/in
Vertical Displacement, v
0.000 in
67
x2
x = 3.00”, y = 0”
Tensile Stress, σxx
-204.8 psi
Shear Stress, τxy
-51.12 psi
Longitudinal Strain, εxx
-7.470 x 10-8in/in
Shear Strain, εxy
-4.460 x 10-6 in/in
Vertical Displacement, v
-4.812 x 10-6 in
x = 3.00”, y = 0.5”
Tensile Stress, σxx
818.1 psi
Shear Stress, τxy
35.16 psi
Longitudinal Strain, εxx
3.225 x 10-5 in/in
Shear Strain, εxy
-1.502 x 10-6 in/in
Vertical Displacement, v
-4.840 x 10-6 in
x = 3.00”, y = -0.5”
Tensile Stress, σxx
-850.3 psi
Shear Stress, τxy
-68.46 psi
Longitudinal Strain, εxx
-3.230 x 10-5 in/in
Shear Strain, εxy
-1.453 x 10-6 in/in
Vertical Displacement, v
-4.840 x 10-6 in
68
x3
x = 3.75”, y = 0”
Tensile Stress, σxx
-9.876 psi
Shear Stress, τxy
-162.9 psi
Longitudinal Strain, εxx
-5.360 x 10-7 in/in
Shear Strain, εxy
-1.411 x 10-5 in/in
Vertical Displacement, v
-7.589 x 10-4 in
x = 3.75”, y =
= 0.1768”
Tensile Stress, σxx
2,719 psi
Shear Stress, τxy
-852.9 psi
Longitudinal Strain, εxx
8.139 x 10-5 in/in
Shear Strain, εxy
-7.471 x 10-5 in/in
Vertical Displacement, v
-7.561 x 10-4 in
x = 3.75”, y = -
= -0.1768”
Tensile Stress, σxx
-2,576 psi
Shear Stress, τxy
-844.8 psi
Longitudinal Strain, εxx
-8.267 x 10-5 in/in
Shear Strain, εxy
-8.848 x 10-5 in/in
Vertical Displacement, v
-7.560 x 10-4 in
69
x4
x = 4.50”, y = 0”
Tensile Stress, σxx
-0.7396 psi
Shear Stress, τxy
-66.70 psi
Longitudinal Strain, εxx
-2.047 x 10-8 in/in
Shear Strain, εxy
-5.780 x 10-6 in/in
Vertical Displacement, v
-1.352 x 10-3 in
x = 4.50”, y = 0.125”
Tensile Stress, σxx
2,929 psi
Shear Stress, τxy
-70.85 psi
Longitudinal Strain, εxx
9.758 x 10-5 in/in
Shear Strain, εxy
-6.052 x 10-6 in/in
Vertical Displacement, v
-1.352 x 10-3 in
x = 4.50”, y = -0.125”
Tensile Stress, σxx
-2,930 psi
Shear Stress, τxy
-62.54 psi
Longitudinal Strain, εxx
-9.762 x 10-5 in/in
Shear Strain, εxy
-5.509 x 10-6 in/in
Vertical Displacement, v
-1.352 x 10-3 in
70
x5
x = 7.50”, y = 0”
Tensile Stress, σxx
8.282 x 10-6 psi
Shear Stress, τxy
-58.87 psi
Longitudinal Strain, εxx
-5.136 x 10-6 in/in
Shear Strain, εxy
x 10-6 in/in
Vertical Displacement, v
-6.986 x 10-3 in
x = 7.50”, y = 0.125”
Tensile Stress, σxx
47.86 psi
Shear Stress, τxy
0.3965 psi
Longitudinal Strain, εxx
2.559 x 10-6 in/in
Shear Strain, εxy
3.436 x 10-8 in/in
Vertical Displacement, v
-6.986 x 10-3 in
x = 7.50”, y = -0.125”
Tensile Stress, σxx
-47.86 psi
Shear Stress, τxy
-0.3965 psi
Longitudinal Strain, εxx
-2.559 x 10-6 in/in
Shear Strain, εxy
-3.436 x 10-8 in/in
Vertical Displacement, v
-6.986 x 10-3 in
71
7. Appendix C – ABAQUS Input File (.inp)
*Heading
ProjectLoad
** Job name: TFBshell Model name: TFBshell
*Preprint, echo=YES, model=YES, history=NO, contact=NO
**
** PARTS
**
*Part, name=TFBshell
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82
759,
760,
761,
762,
763,
764,
765,
766,
767,
768,
769,
770,
771,
772,
773,
774,
775,
776,
777,
778,
779,
780,
781,
782,
783,
784,
785,
786,
787,
788,
789,
790,
791,
792,
793,
794,
795,
796,
797,
798,
799,
800,
801,
802,
803,
804,
805,
806,
807,
808,
809,
810,
811,
812,
813,
814,
815,
816,
817,
818,
819,
820,
821,
822,
823,
824,
825,
826,
827,
828,
0., -0.111995712, 2.69161034
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0., -0.225062832, 2.0287528
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0., -0.138680741, 0.844175637
0., 0.128171429, 0.785207331
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0., 0.128032967, 1.78364468
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0., -0.139103472, 1.34257293
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0., 0.124930024, 2.38964415
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0., 0.207913443, 0.402775168
83
829,
0., -0.106790975, 0.428245097
830,
0., -0.139619067, 1.0437212
831,
0., -0.139435574, 0.943990827
832,
0., 0.126098394, 1.18370545
833,
0., 0.126410872, 1.08279943
834,
0., 0.126635373, 1.28353918
835,
0., 0.126996785, 1.38337946
836,
0., -0.137339339, 1.74347484
837,
0., -0.138040796, 1.64307106
838,
0., -0.135593861, 2.04296684
839,
0., 0.131080419, 2.0822401
840,
0., -0.141757384, 2.33473706
841,
0., -0.137638956, 2.23934984
842,
0., 0.130813032, 2.18239903
843,
0., 0.0374400765, 0.971184552
844,
0., 0.0424266867, 0.676610529
845,
0., 0.1210225, 2.49502325
846,
0., -0.0483214706, 0.759751499
847,
0., 0.0390555114, 1.77085972
848,
0., 0.0385941193, 1.67027366
849,
0., 0.00142542843, 0.444741756
850,
0., -0.0510413013, 0.957865596
851,
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852,
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853,
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854,
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855,
0., 0.0412924252, 2.27165127
856,
0., -0.0522827879, 2.56730556
857,
0., -0.0511544086, 1.1571939
858,
0., 0.0355980098, 2.37631655
859,
0., -0.0471575223, 1.95803285
860,
0., 0.0614641905, 0.510598898
861,
0., -0.0507636443, 1.35715365
862,
0., 0.0412930027, 1.97084665
863,
0., -0.0474235639, 2.25546408
864,
0., 0.0190080386, 2.49383402
865,
0., -0.0497976653, 1.65674829
866,
0., -0.0329956189, 0.558593094
867,
0., -0.0432668515, 0.660010099
868,
0., -0.0481717065, 1.85769224
869,
0., 0.0402927399, 1.87028468
870,
0., 0.0484860949, 0.584182382
871,
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872,
0., 0.0378011167, 1.47087109
873,
0., -0.0905951932, 2.50771475
874,
0., -0.051157292, 1.0576961
875,
0., -0.050102476, 0.858643651
876,
0., 0.0386689529, 0.871499777
877,
0., 0.0375370905, 1.07052016
878,
0., -0.0506241843, 1.45674109
879,
0., 0.0379467569, 1.37040997
880,
0., 0.0382454097, 1.57009625
881,
0., -0.0502610542, 1.55650496
882,
0., -0.0462949984, 2.05721116
883,
0., 0.0425187051, 2.16989231
884,
0., 0.0421838202, 2.06983495
885,
0., -0.0526436791, 2.35360408
886,
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887,
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888,
0., -0.0667613223, 2.44523573
*Element, type=S3
1, 383, 399, 473
2, 468, 426, 455
3, 441, 436, 440
4, 460, 442, 458
325, 542, 500, 499
326, 540, 509, 501
403, 601, 566, 602
404, 603, 555, 562
*Element, type=S4R
84
5, 75, 76, 224, 200
6, 249, 224, 76, 77
7, 19, 207, 262, 18
8, 66, 67, 218, 202
9, 79, 80, 204, 251
10, 263, 253, 205, 322
11, 56, 211, 264, 55
12, 208, 20, 21, 283
13, 20, 208, 207, 19
14, 22, 283, 21, 2
15, 50, 3, 51, 234
16, 234, 51, 52, 233
17, 233, 232, 265, 302
18, 49, 303, 266, 48
19, 267, 47, 48, 266
20, 304, 268, 235, 477
21, 235, 57, 58, 477
22, 62, 63, 212, 285
23, 433, 400, 404, 452
24, 267, 327, 269, 209
25, 214, 212, 63, 64
26, 46, 209, 213, 45
27, 202, 216, 65, 66
28, 67, 68, 239, 218
29, 307, 240, 220, 310
30, 222, 244, 72, 73
31, 312, 244, 222, 314
32, 247, 222, 73, 74
33, 36, 245, 246, 35
34, 203, 249, 77, 78
35, 32, 201, 225, 31
36, 30, 250, 226, 29
37, 253, 263, 272, 204
38, 254, 26, 27, 227
39, 229, 228, 273, 274
40, 24, 229, 230, 23
41, 4, 210, 231, 54
42, 53, 232, 233, 52
43, 5, 59, 477, 58
44, 57, 235, 211, 56
45, 304, 236, 347, 268
46, 217, 43, 44, 215
47, 237, 42, 43, 217
48, 289, 305, 217, 215
49, 40, 219, 241, 39
50, 39, 241, 242, 38
51, 244, 243, 71, 72
52, 315, 247, 200, 294
53, 37, 221, 245, 36
54, 200, 247, 74, 75
55, 223, 34, 35, 246
56, 316, 249, 203, 278
57, 319, 251, 204, 272
58, 29, 226, 252, 28
59, 81, 82, 205, 253
60, 83, 84, 257, 255
61, 321, 299, 254, 227
62, 17, 206, 256, 16
63, 255, 279, 322, 205
64, 206, 262, 328, 280
65, 15, 258, 259, 14
66, 259, 258, 281, 282
67, 14, 259, 261, 13
68, 87, 1, 13, 261
69, 201, 32, 33, 248
70, 248, 33, 34, 223
71, 215, 44, 45, 213
72, 78, 79, 251, 203
73, 80, 81, 253, 204
74, 250, 30, 31, 225
85
75, 82, 83, 255, 205
76, 28, 252, 227, 27
77, 206, 17, 18, 262
78, 26, 254, 228, 25
79, 25, 228, 229, 24
80, 209, 46, 47, 267
81, 336, 363, 385, 327
82, 210, 4, 55, 264
83, 387, 386, 353, 346
84, 284, 330, 329, 301
85, 269, 287, 213, 209
86, 64, 65, 216, 214
87, 355, 364, 289, 287
88, 215, 213, 287, 289
89, 331, 332, 290, 291
90, 218, 306, 291, 202
91, 220, 240, 69, 70
92, 219, 40, 41, 238
93, 238, 41, 42, 237
94, 70, 71, 243, 220
95, 221, 37, 38, 242
96, 276, 293, 246, 245
97, 315, 294, 375, 376
98, 223, 246, 293, 295
99, 333, 334, 293, 276
100, 201, 248, 296, 297
101, 317, 318, 250, 225
102, 298, 321, 227, 252
103, 300, 335, 207, 208
104, 274, 300, 230, 229
105, 230, 300, 208, 283
106, 232, 231, 301, 265
107, 54, 231, 232, 53
108, 336, 327, 267, 266
109, 50, 234, 303, 49
110, 59, 60, 304, 477
111, 60, 61, 236, 304
112, 61, 62, 285, 236
113, 268, 347, 387, 346
114, 327, 385, 356, 269
115, 239, 270, 306, 218
116, 240, 239, 68, 69
117, 243, 271, 310, 220
118, 275, 308, 219, 238
119, 308, 275, 338, 368
120, 241, 219, 308, 309
121, 314, 222, 247, 315
122, 309, 311, 242, 241
123, 294, 200, 224, 277
124, 297, 296, 361, 377
125, 295, 293, 334, 340
126, 379, 378, 320, 318
127, 279, 255, 257, 324
128, 280, 323, 256, 206
129, 16, 256, 258, 15
130, 281, 323, 342, 384
131, 343, 384, 342, 392
132, 260, 257, 84, 85
133, 260, 85, 86, 326
134, 326, 261, 259, 282
135, 87, 261, 326, 86
136, 389, 344, 352, 390
137, 268, 346, 211, 235
138, 231, 210, 284, 301
139, 266, 303, 302, 336
140, 236, 285, 354, 347
141, 290, 216, 202, 291
142, 310, 271, 371, 367
143, 270, 239, 240, 307
144, 271, 243, 244, 312
86
145, 225, 201, 297, 317
146, 262, 207, 335, 328
147, 228, 254, 299, 273
148, 238, 237, 292, 275
149, 245, 221, 313, 276
150, 248, 223, 295, 296
151, 277, 224, 249, 316
152, 278, 203, 251, 319
153, 380, 272, 263, 341
154, 325, 384, 343, 324
155, 299, 321, 345, 352
156, 325, 324, 257, 260
157, 323, 281, 258, 256
158, 260, 326, 282, 325
159, 283, 22, 23, 230
160, 264, 353, 284, 210
161, 286, 212, 214, 288
162, 286, 354, 285, 212
163, 288, 214, 216, 290
164, 364, 365, 305, 289
165, 355, 287, 269, 356
166, 433, 452, 472, 453
167, 237, 217, 305, 292
168, 292, 305, 365, 357
169, 348, 366, 270, 307
170, 374, 369, 312, 314
171, 407, 423, 359, 360
172, 296, 295, 340, 361
173, 377, 362, 317, 297
174, 252, 226, 320, 298
175, 392, 342, 383, 473
176, 233, 302, 303, 234
177, 363, 336, 302, 265
178, 366, 358, 306, 270
179, 367, 348, 307, 310
180, 369, 371, 271, 312
181, 309, 308, 368, 370
182, 371, 369, 419, 394
183, 338, 401, 416, 368
184, 221, 242, 311, 313
185, 370, 372, 311, 309
186, 314, 315, 376, 374
187, 360, 359, 277, 316
188, 359, 375, 294, 277
189, 340, 391, 396, 361
190, 362, 379, 318, 317
191, 380, 341, 478, 397
192, 226, 250, 318, 320
193, 319, 272, 380, 350
194, 298, 320, 378, 381
195, 279, 351, 382, 322
196, 351, 279, 324, 343
197, 280, 383, 342, 323
198, 282, 281, 384, 325
199, 265, 301, 329, 363
200, 274, 273, 344, 349
201, 284, 353, 386, 330
202, 288, 290, 332, 337
203, 291, 306, 358, 331
204, 456, 393, 412, 440
205, 276, 313, 373, 333
206, 339, 360, 316, 278
207, 300, 274, 349, 335
208, 344, 389, 399, 349
209, 400, 387, 347, 354
210, 286, 288, 337, 404
211, 275, 292, 357, 338
212, 414, 406, 366, 348
213, 278, 319, 350, 339
214, 334, 402, 391, 340
87
215, 263, 322, 382, 341
216, 473, 399, 389, 449
217, 349, 399, 328, 335
218, 299, 352, 344, 273
219, 321, 298, 381, 345
220, 345, 403, 390, 352
221, 264, 211, 346, 353
222, 400, 433, 386, 387
223, 404, 400, 354, 286
224, 406, 412, 358, 366
225, 464, 465, 421, 423
226, 396, 445, 466, 425
227, 397, 409, 350, 380
228, 385, 363, 329, 405
229, 355, 388, 411, 364
230, 440, 436, 457, 456
231, 481, 413, 357, 365
232, 414, 442, 441, 406
233, 394, 415, 367, 371
234, 441, 442, 460, 436
235, 374, 376, 424, 417
236, 370, 395, 418, 372
237, 417, 419, 369, 374
238, 443, 422, 420, 444
239, 313, 311, 372, 373
240, 423, 421, 375, 359
241, 391, 437, 445, 396
242, 361, 396, 425, 377
243, 447, 438, 407, 408
244, 378, 410, 427, 381
245, 377, 425, 426, 362
246, 430, 448, 429, 478
247, 431, 470, 471, 432
248, 383, 280, 328, 399
249, 405, 329, 330, 453
250, 398, 332, 331, 393
251, 355, 356, 474, 388
252, 403, 345, 381, 427
253, 402, 334, 333, 422
254, 331, 358, 412, 393
255, 462, 419, 417, 463
256, 395, 370, 368, 416
257, 408, 407, 360, 339
258, 434, 388, 475, 435
259, 337, 332, 398, 479
260, 452, 404, 337, 479
261, 401, 338, 357, 413
262, 422, 333, 373, 420
263, 463, 451, 443, 462
264, 382, 431, 432, 341
265, 386, 433, 453, 330
266, 365, 364, 411, 481
267, 339, 350, 409, 408
268, 454, 467, 429, 448
269, 455, 438, 447, 482
270, 410, 378, 379, 428
271, 428, 379, 362, 426
272, 440, 412, 406, 441
273, 457, 413, 481, 434
274, 415, 414, 348, 367
275, 416, 460, 458, 395
276, 394, 450, 459, 415
277, 401, 436, 460, 416
278, 372, 418, 420, 373
279, 395, 458, 461, 418
280, 462, 443, 444, 450
281, 376, 375, 421, 424
282, 465, 445, 437, 446
283, 407, 438, 464, 423
284, 438, 455, 466, 464
88
285, 467, 468, 480, 429
286, 427, 439, 469, 403
287, 471, 470, 469, 439
288, 431, 382, 351, 476
289, 341, 432, 430, 478
290, 385, 405, 474, 356
291, 434, 481, 411, 388
292, 435, 398, 393, 456
293, 449, 389, 390, 470
294, 476, 392, 473, 449
295, 474, 405, 453, 472
296, 436, 401, 413, 457
297, 437, 391, 402, 451
298, 451, 402, 422, 443
299, 408, 409, 482, 447
300, 439, 427, 410, 454
301, 454, 410, 428, 467
302, 442, 414, 415, 459
303, 444, 420, 418, 461
304, 424, 446, 463, 417
305, 419, 462, 450, 394
306, 464, 466, 445, 465
307, 446, 424, 421, 465
308, 446, 437, 451, 463
309, 409, 397, 480, 482
310, 468, 455, 482, 480
311, 430, 432, 471, 448
312, 439, 454, 448, 471
313, 343, 392, 476, 351
314, 403, 469, 470, 390
315, 459, 461, 458, 442
316, 475, 472, 452, 479
317, 428, 426, 468, 467
318, 455, 426, 425, 466
319, 475, 388, 474, 472
320, 434, 435, 456, 457
321, 461, 459, 450, 444
322, 449, 470, 431, 476
323, 398, 435, 475, 479
324, 397, 478, 429, 480
327, 522, 98, 99, 534
328, 521, 522, 514, 513
329, 514, 507, 541, 513
330, 505, 506, 498, 497
331, 497, 90, 91, 505
332, 505, 91, 92, 512
333, 538, 537, 513, 541
334, 97, 521, 520, 96
335, 521, 97, 98, 522
336, 89, 90, 497, 490
337, 497, 498, 491, 490
338, 498, 499, 492, 491
339, 499, 500, 493, 492
340, 500, 539, 494, 493
341, 495, 501, 502, 535
342, 503, 496, 535, 502
343, 526, 504, 108, 109
344, 526, 529, 489, 496
345, 110, 5, 58, 529
346, 4, 54, 486, 533
347, 493, 494, 533, 486
348, 483, 88, 89, 490
349, 490, 491, 484, 483
350, 491, 492, 485, 484
351, 492, 493, 486, 485
352, 54, 53, 485, 486
353, 484, 485, 53, 52
354, 52, 51, 483, 484
355, 92, 93, 519, 512
356, 96, 520, 532, 95
89
357, 519, 532, 520, 537
358, 519, 93, 94, 532
359, 95, 532, 94, 6
360, 534, 99, 100, 523
361, 522, 534, 515, 514
362, 540, 508, 515, 516
363, 540, 501, 539, 508
364, 500, 507, 508, 539
365, 514, 515, 508, 507
366, 516, 515, 534, 523
367, 530, 527, 105, 106
368, 527, 518, 525, 531
369, 528, 511, 536, 530
370, 528, 504, 503, 511
371, 107, 108, 504, 528
372, 102, 525, 524, 101
373, 7, 104, 531, 103
374, 487, 488, 56, 55
375, 489, 488, 535, 496
376, 503, 504, 526, 496
377, 529, 526, 109, 110
378, 489, 529, 58, 57
379, 488, 489, 57, 56
380, 494, 495, 487, 533
381, 501, 495, 494, 539
382, 533, 487, 55, 4
383, 488, 487, 495, 535
384, 506, 542, 499, 498
385, 510, 502, 501, 509
386, 510, 518, 536, 511
387, 502, 510, 511, 503
388, 538, 506, 505, 512
389, 542, 541, 507, 500
390, 510, 509, 517, 518
391, 518, 517, 524, 525
392, 521, 513, 537, 520
393, 517, 509, 540, 516
394, 517, 516, 523, 524
395, 527, 530, 536, 518
396, 101, 524, 523, 100
397, 88, 483, 51, 3
398, 102, 103, 531, 525
399, 104, 105, 527, 531
400, 107, 528, 530, 106
401, 519, 537, 538, 512
402, 538, 541, 542, 506
405, 549, 556, 591, 585
406, 586, 547, 548, 594
407, 100, 99, 592, 586
408, 543, 111, 112, 550
409, 112, 113, 557, 550
410, 558, 557, 564, 565
411, 574, 573, 581, 596
412, 596, 582, 575, 574
413, 122, 582, 596, 9
414, 582, 122, 123, 583
415, 124, 584, 583, 123
416, 577, 576, 583, 584
417, 570, 569, 576, 577
418, 605, 563, 562, 561
419, 603, 562, 563, 556
420, 556, 549, 548, 603
421, 594, 593, 102, 101
422, 598, 592, 546, 553
423, 99, 98, 546, 592
424, 98, 97, 545, 546
425, 95, 6, 111, 543
426, 557, 113, 114, 564
427, 564, 571, 600, 565
428, 573, 572, 580, 581
90
429, 121, 581, 580, 120
430, 581, 121, 9, 596
431, 564, 114, 115, 571
432, 599, 572, 601, 600
433, 579, 119, 120, 580
434, 115, 116, 578, 571
435, 119, 579, 595, 118
436, 578, 595, 579, 599
437, 578, 116, 117, 595
438, 118, 595, 117, 8
439, 584, 124, 125, 587
440, 127, 590, 587, 126
441, 126, 587, 125, 10
442, 129, 589, 588, 128
443, 130, 591, 589, 129
444, 584, 587, 590, 577
445, 570, 588, 589, 563
446, 589, 591, 556, 563
447, 585, 597, 593, 549
448, 7, 103, 597, 132
449, 96, 95, 543, 544
450, 549, 593, 594, 548
451, 97, 96, 544, 545
452, 544, 543, 550, 551
453, 552, 551, 558, 559
454, 567, 604, 602, 566
455, 568, 567, 574, 575
456, 576, 575, 582, 583
457, 545, 544, 551, 552
458, 560, 553, 552, 559
459, 560, 568, 561, 554
460, 569, 568, 575, 576
461, 546, 545, 552, 553
462, 553, 560, 554, 598
463, 570, 563, 605, 569
464, 592, 598, 547, 586
465, 555, 554, 561, 562
466, 547, 555, 603, 548
467, 551, 550, 557, 558
468, 559, 558, 565, 602
469, 567, 566, 573, 574
470, 554, 555, 547, 598
471, 560, 604, 567, 568
472, 604, 560, 559, 602
473, 561, 568, 569, 605
474, 588, 570, 577, 590
475, 572, 573, 566, 601
476, 580, 572, 599, 579
477, 586, 594, 101, 100
478, 585, 131, 132, 597
479, 593, 597, 103, 102
480, 590, 127, 128, 588
481, 591, 130, 131, 585
482, 571, 578, 599, 600
483, 601, 602, 565, 600
484, 634, 187, 188, 658
485, 658, 188, 189, 608
486, 614, 191, 192, 662
487, 664, 194, 195, 666
488, 628, 178, 179, 648
489, 612, 121, 120, 615
490, 121, 612, 611, 9
491, 118, 8, 133, 672
492, 640, 639, 673, 713
493, 11, 162, 887, 161
494, 162, 163, 674, 887
495, 164, 165, 618, 617
496, 617, 714, 739, 674
497, 644, 717, 675, 643
498, 173, 620, 621, 172
91
499, 717, 752, 777, 675
500, 158, 641, 642, 157
501, 175, 624, 622, 174
502, 177, 610, 626, 176
503, 156, 623, 625, 155
504, 648, 179, 180, 649
505, 184, 632, 653, 183
506, 653, 632, 724, 678
507, 185, 656, 632, 184
508, 654, 147, 148, 631
509, 143, 607, 635, 142
510, 660, 614, 696, 679
511, 140, 636, 661, 139
512, 664, 666, 689, 680
513, 137, 663, 638, 136
514, 136, 638, 639, 135
515, 118, 672, 613, 119
516, 616, 160, 161, 887
517, 159, 697, 641, 158
518, 676, 619, 643, 675
519, 619, 618, 165, 166
520, 167, 168, 644, 643
521, 677, 682, 622, 624
522, 172, 621, 645, 171
523, 621, 683, 699, 645
524, 170, 12, 171, 645
525, 627, 154, 155, 625
526, 647, 646, 684, 685
527, 178, 628, 610, 177
528, 151, 629, 650, 150
529, 150, 650, 651, 149
530, 629, 647, 685, 721
531, 633, 655, 706, 707
532, 186, 606, 656, 185
533, 633, 145, 146, 655
534, 696, 732, 750, 741
535, 139, 661, 637, 138
536, 193, 609, 662, 192
537, 138, 637, 663, 137
538, 666, 195, 196, 668
539, 668, 736, 689, 666
540, 9, 611, 665, 122
541, 695, 711, 742, 760
542, 755, 736, 737, 794
543, 123, 667, 669, 124
544, 669, 667, 691, 692
545, 124, 669, 671, 125
546, 199, 10, 125, 671
547, 725, 723, 758, 743
548, 606, 186, 187, 634
549, 607, 143, 144, 657
550, 657, 144, 145, 633
551, 608, 189, 190, 660
552, 660, 190, 191, 614
553, 609, 193, 194, 664
554, 690, 611, 612, 694
555, 615, 120, 119, 613
556, 639, 638, 695, 673
557, 135, 639, 640, 134
558, 141, 659, 636, 140
559, 614, 662, 732, 696
560, 163, 164, 617, 674
561, 160, 616, 697, 159
562, 166, 167, 643, 619
563, 677, 744, 776, 682
564, 174, 622, 620, 173
565, 624, 175, 176, 626
566, 715, 700, 623, 642
567, 623, 156, 157, 642
568, 740, 745, 681, 676
92
569, 626, 610, 703, 701
570, 814, 747, 764, 800
571, 704, 705, 753, 764
572, 154, 627, 646, 153
573, 181, 630, 649, 180
574, 705, 704, 628, 648
575, 629, 151, 152, 647
576, 647, 152, 153, 646
577, 630, 181, 182, 652
578, 652, 182, 183, 653
579, 631, 148, 149, 651
580, 743, 748, 693, 725
581, 693, 706, 655, 654
582, 655, 146, 147, 654
583, 767, 749, 707, 706
584, 142, 635, 659, 141
585, 657, 633, 707, 708
586, 732, 733, 791, 750
587, 711, 734, 772, 742
588, 713, 712, 615, 613
589, 672, 640, 713, 613
590, 618, 681, 714, 617
591, 641, 697, 751, 698
592, 644, 168, 169, 699
593, 682, 716, 620, 622
594, 170, 645, 699, 169
595, 683, 716, 752, 717
596, 700, 702, 625, 623
597, 722, 719, 630, 652
598, 719, 720, 649, 630
599, 796, 779, 778, 805
600, 678, 722, 652, 653
601, 650, 629, 721, 686
602, 632, 656, 726, 724
603, 631, 651, 723, 725
604, 687, 727, 606, 634
605, 679, 728, 608, 660
606, 728, 679, 788, 754
607, 607, 657, 708, 709
608, 688, 731, 636, 659
609, 662, 609, 733, 732
610, 680, 733, 609, 664
611, 731, 710, 661, 636
612, 668, 196, 197, 670
613, 665, 611, 690, 735
614, 122, 665, 667, 123
615, 689, 736, 755, 773
616, 670, 197, 198, 738
617, 738, 671, 669, 692
618, 199, 671, 738, 198
619, 672, 133, 134, 640
620, 638, 663, 711, 695
621, 712, 694, 612, 615
622, 616, 887, 674, 739
623, 619, 676, 681, 618
624, 624, 626, 701, 677
625, 804, 745, 740, 849
626, 686, 723, 651, 650
627, 734, 711, 663, 637
628, 739, 714, 798, 751
629, 716, 683, 621, 620
630, 644, 699, 683, 717
631, 646, 627, 718, 684
632, 704, 703, 610, 628
633, 721, 685, 781, 757
634, 757, 783, 686, 721
635, 634, 658, 729, 687
636, 659, 635, 730, 688
637, 755, 794, 759, 809
638, 760, 799, 774, 775
93
639, 737, 736, 668, 670
640, 735, 691, 667, 665
641, 670, 738, 692, 737
642, 654, 631, 725, 693
643, 710, 734, 637, 661
644, 688, 730, 769, 789
645, 715, 642, 641, 698
646, 746, 700, 715, 756
647, 746, 763, 702, 700
648, 627, 625, 702, 718
649, 819, 876, 843, 816
650, 648, 649, 720, 705
651, 718, 702, 763, 765
652, 818, 795, 870, 844
653, 706, 693, 748, 767
654, 836, 787, 768, 821
655, 709, 708, 770, 771
656, 771, 770, 803, 815
657, 749, 770, 708, 707
658, 635, 607, 709, 730
659, 761, 792, 710, 731
660, 760, 742, 845, 799
661, 713, 673, 775, 712
662, 829, 849, 740, 777
663, 777, 752, 776, 829
664, 678, 724, 784, 782
665, 720, 719, 779, 780
666, 778, 779, 719, 722
667, 765, 766, 684, 718
668, 783, 758, 723, 686
669, 782, 806, 805, 778
670, 727, 726, 656, 606
671, 687, 729, 768, 787
672, 727, 687, 787, 785
673, 784, 807, 806, 782
674, 658, 608, 728, 729
675, 741, 788, 679, 696
676, 789, 769, 808, 839
677, 689, 773, 790, 680
678, 792, 772, 734, 710
679, 692, 691, 794, 737
680, 616, 739, 751, 697
681, 740, 676, 675, 777
682, 776, 752, 716, 682
683, 792, 761, 842, 810
684, 774, 694, 712, 775
685, 793, 774, 799, 811
686, 701, 762, 744, 677
687, 701, 703, 747, 762
688, 798, 828, 698, 751
689, 703, 704, 764, 747
690, 754, 768, 729, 728
691, 802, 812, 767, 748
692, 888, 864, 858, 885
693, 705, 720, 780, 753
694, 876, 875, 850, 843
695, 691, 735, 759, 794
696, 715, 698, 828, 756
697, 781, 766, 819, 816
698, 782, 778, 722, 678
699, 735, 690, 793, 759
700, 774, 793, 690, 694
701, 695, 760, 775, 673
702, 731, 688, 789, 761
703, 746, 795, 818, 763
704, 765, 801, 819, 766
705, 766, 781, 685, 684
706, 786, 820, 807, 784
707, 848, 823, 812, 880
708, 730, 709, 771, 769
94
709, 840, 791, 790, 813
710, 823, 848, 847, 803
711, 741, 822, 838, 788
712, 873, 826, 809, 856
713, 733, 680, 790, 791
714, 773, 826, 813, 790
715, 681, 745, 798, 714
716, 860, 827, 804, 849
717, 744, 797, 829, 776
718, 779, 796, 830, 780
719, 874, 830, 796, 857
720, 724, 726, 786, 784
721, 833, 877, 851, 832
722, 853, 852, 861, 879
723, 726, 727, 785, 786
724, 832, 851, 853, 834
725, 832, 834, 758, 783
726, 854, 847, 848, 865
727, 815, 808, 769, 771
728, 885, 858, 855, 863
729, 761, 789, 839, 842
730, 759, 793, 856, 809
731, 795, 746, 756, 827
732, 851, 857, 852, 853
733, 797, 744, 762, 817
734, 817, 762, 747, 814
735, 804, 828, 798, 745
736, 825, 772, 792, 810
737, 855, 810, 842, 883
738, 800, 764, 753, 831
739, 831, 753, 780, 830
740, 765, 763, 818, 801
741, 748, 743, 835, 802
742, 770, 749, 823, 803
743, 869, 868, 859, 862
744, 816, 833, 757, 781
745, 820, 786, 785, 837
746, 743, 758, 834, 835
747, 869, 815, 803, 847
748, 773, 755, 809, 826
749, 864, 811, 799, 845
750, 885, 840, 813, 888
751, 825, 858, 864, 845
752, 749, 767, 812, 823
753, 836, 854, 865, 837
754, 814, 846, 867, 817
755, 859, 824, 838, 882
756, 866, 870, 860, 849
757, 844, 871, 801, 818
758, 837, 785, 787, 836
759, 865, 848, 880, 881
760, 821, 768, 754, 824
761, 824, 754, 788, 838
762, 822, 741, 750, 841
763, 841, 750, 791, 840
764, 742, 772, 825, 845
765, 873, 856, 864, 888
766, 805, 852, 857, 796
767, 871, 846, 875, 876
768, 833, 832, 783, 757
769, 816, 843, 877, 833
770, 879, 861, 878, 872
771, 834, 853, 879, 835
772, 881, 880, 872, 878
773, 837, 865, 881, 820
774, 862, 859, 882, 884
775, 863, 855, 883, 886
776, 808, 862, 884, 839
777, 886, 883, 884, 882
778, 877, 874, 857, 851
95
779, 870, 866, 867, 844
780, 846, 814, 800, 875
781, 875, 800, 831, 850
782, 824, 859, 868, 821
783, 880, 812, 802, 872
784, 810, 855, 858, 825
785, 888, 813, 826, 873
786, 849, 829, 797, 866
787, 866, 797, 817, 867
788, 850, 831, 830, 874
789, 852, 805, 806, 861
790, 861, 806, 807, 878
791, 854, 836, 821, 868
792, 883, 842, 839, 884
793, 840, 885, 863, 841
794, 856, 793, 811, 864
795, 843, 850, 874, 877
796, 882, 838, 822, 886
797, 815, 869, 862, 808
798, 795, 827, 860, 870
799, 878, 807, 820, 881
800, 841, 863, 886, 822
801, 872, 802, 835, 879
802, 844, 867, 846, 871
803, 847, 854, 868, 869
804, 819, 801, 871, 876
805, 804, 827, 756, 828
*Nset, nset=_PickedSet42, internal, generate
1, 888, 1
*Elset, elset=_PickedSet42, internal, generate
1, 805, 1
*Nset, nset=_PickedSet51, internal, generate
1, 888, 1
*Elset, elset=_PickedSet51, internal, generate
1, 805, 1
*Nset, nset=_PickedSet52, internal
6, 7, 8, 9, 10, 95, 96, 97, 98, 99, 100, 101, 102, 103, 111, 112
113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128
129, 130, 131, 132, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554
555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570
571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586
587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602
603, 604, 605
*Elset, elset=_PickedSet52, internal, generate
403, 483, 1
*Orientation, name=Ori-1
0.,
0.,
1.,
0.,
1.,
0.
3, 0.
** Region: (TFBsection:Picked), (Material Orientation:Picked)
*Elset, elset=_I1, internal
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48
49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64
65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96
97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112
113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128
129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144
145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160
161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176
177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192
193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208
209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224
225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240
241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256
257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272
273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288
289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304
305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320
96
321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336
337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352
353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368
369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384
385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400
401, 402, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497
498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513
514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529
530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545
546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561
562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577
578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593
594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609
610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625
626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641
642, 643, 644, 645, 646, 647, 648, 649, 650, 651, 652, 653, 654, 655, 656, 657
658, 659, 660, 661, 662, 663, 664, 665, 666, 667, 668, 669, 670, 671, 672, 673
674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688, 689
690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704, 705
706, 707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 720, 721
722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737
738, 739, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750, 751, 752, 753
754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 765, 766, 767, 768, 769
770, 771, 772, 773, 774, 775, 776, 777, 778, 779, 780, 781, 782, 783, 784, 785
786, 787, 788, 789, 790, 791, 792, 793, 794, 795, 796, 797, 798, 799, 800, 801
802, 803, 804, 805
** Section: TFBsection
*Shell Section, elset=_I1, orientation=Ori-1, material=Steel
0.25, 5
*Orientation, name=Ori-2
0.,
0.,
1.,
0.,
1.,
0.
3, 0.
** Region: (TFBsection:Picked), (Material Orientation:Picked)
*Elset, elset=_I2, internal, generate
403, 483, 1
** Section: TFBsection
*Shell Section, elset=_I2, orientation=Ori-2, material=Steel
0.25, 5
*End Part
**
**
** ASSEMBLY
**
*Assembly, name=Assembly
**
*Instance, name=TFBshell-1, part=TFBshell
*End Instance
**
*Nset, nset=Set-1, instance=TFBshell-1
11, 12, 162, 163, 164, 165, 166, 167, 168, 169, 170
*Elset, elset=Set-1, instance=TFBshell-1
493, 494, 495, 519, 520, 524, 560, 562, 592, 594
*Nset, nset=Set-2, instance=TFBshell-1
8, 9, 10, 118, 119, 120, 121, 122, 123, 124, 125
*Elset, elset=Set-2, instance=TFBshell-1
413, 414, 415, 429, 430, 433, 435, 438, 439, 441, 489, 490, 491, 515, 540, 543
545, 546, 555, 614
*Nset, nset=Set-3, instance=TFBshell-1
6, 7, 95, 96, 97, 98, 99, 100, 101, 102, 103
*Elset, elset=Set-3, instance=TFBshell-1
327, 334, 335, 356, 359, 360, 372, 373, 396, 398, 407, 421, 423, 424, 425, 448
449, 451, 477, 479
*Nset, nset=Set-4, instance=TFBshell-1
3, 4, 5, 51, 52, 53, 54, 55, 56, 57, 58
*Elset, elset=Set-4, instance=TFBshell-1
11, 15, 16, 21, 41, 42, 43, 44, 82, 107, 345, 346, 352, 353, 354, 374
378, 379, 382, 397
*Nset, nset=Set-5, instance=TFBshell-1
1, 2, 13, 14, 15, 16, 17, 18, 19, 20, 21
*Elset, elset=Set-5, instance=TFBshell-1
97
7, 12, 13, 14, 62, 65, 67, 68, 77, 129
*Nset, nset=_PickedSet33, internal, instance=TFBshell-1
11, 12, 162, 163, 164, 165, 166, 167, 168, 169, 170
*Elset, elset=_PickedSet33, internal, instance=TFBshell-1
493, 494, 495, 519, 520, 524, 560, 562, 592, 594
*Elset, elset=__PickedSurf32_E4, internal, instance=TFBshell-1
7, 13, 62, 65, 67, 129
*Elset, elset=__PickedSurf32_E2, internal, instance=TFBshell-1
12, 68, 77
*Elset, elset=__PickedSurf32_E3, internal, instance=TFBshell-1
14,
*Surface, type=ELEMENT, name=_PickedSurf32, internal
__PickedSurf32_E4, E4
__PickedSurf32_E2, E2
__PickedSurf32_E3, E3
*Nset, nset=_T-TFBshell-1-ProjectCoords, internal
_PickedSet33,
*Transform, nset=_T-TFBshell-1-ProjectCoords
0.,
0.,
1.,
0.,
1.,
0.
*End Assembly
**
** MATERIALS
**
*Material, name=Steel
** Mild Steel
*Elastic
3e+07, 0.3
**
** BOUNDARY CONDITIONS
**
** Name: FixedEnd Type: Displacement/Rotation
*Boundary
_PickedSet33, 1, 1
_PickedSet33, 2, 2
_PickedSet33, 3, 3
_PickedSet33, 4, 4
_PickedSet33, 5, 5
_PickedSet33, 6, 6
** ---------------------------------------------------------------**
** STEP: EndLoad
**
*Step, name=EndLoad
10lbs on Free End
*Static
1., 1., 1e-05, 1.
**
** LOADS
**
** Name: Endload10lbs Type: Shell edge load
*Dsload, constant resultant=YES
_PickedSurf32, EDTRA, 10.
**
** OUTPUT REQUESTS
**
*Restart, write, frequency=0
**
** FIELD OUTPUT: whole model
**
*Output, field
*Node Output
RF, TF, U
*Element Output, directions=YES
1, 3, 5
E, S
**
** FIELD OUTPUT: x-1 values
**
*Node Output, nset=Set-1
U,
98
*Element Output, elset=Set-1, directions=YES
1, 3, 5
E, S
**
** FIELD OUTPUT: x-2 values
**
*Node Output, nset=Set-2
U,
*Element Output, elset=Set-2, directions=YES
1, 3, 5
E, S
**
** FIELD OUTPUT: x-3 values
**
*Node Output, nset=Set-3
U,
*Element Output, elset=Set-3, directions=YES
1, 3, 5
E, S
**
** FIELD OUTPUT: x-4 values
**
*Node Output, nset=Set-4
U,
*Element Output, elset=Set-4, directions=YES
1, 3, 5
E, S
**
** FIELD OUTPUT: x-5 values
**
*Node Output, nset=Set-5
U,
*Element Output, elset=Set-5, directions=YES
1, 3, 5
E, S
**
** HISTORY OUTPUT: H-Output-1
**
*Output, history, variable=PRESELECT
*End Step
99
8. References
[a] Shames, Irving H. and Francis A. Cozzarelli. 1997. Elastic and Inelastic Stress
Analysis, Revised Printing. Boca Raton, Florida: Taylor and Francis.
[b] Gere, James A. 2003. Mechanics of Materials, 6th Edition. Tampa, Florida:
Thomson-Engineering.
[c] Gibson, Ronald F. 2012. Principles of Composite Material Mechanics, 3rd
Edition. Boca Raton, Florida: Taylor and Francis.
[d] Timoshenko, Stephen P.
1970.
Theory of Elasticity.
McGraw-Hill.
100
Albany, New York: