Finite Element Analysis
Of Composite Layered Structures
Connor Kaufmann – B. Sc. ‘14
Neola Putnam – M. Eng. ‘14
Ethan Seo – M. Eng. ‘14
Ju Hwan (Jay) Shin – B. Sc. ‘14
Cornell University
Sibley School of Mechanical & Aerospace Engineering
Spring 2014 – Professor N. Zabaras
Objective
C. Kaufmann, N. Putnam, E. Seo, J. Shin
 Develop a linear 3D finite element analysis from scratch using MATLAB.
 Consider a uniaxial loading of a symmetric laminate.
 Verify the results with expected results, namely the state of out-of-plane stresses near the free edges.
 Observe the effect of h-refinement (convergence of results through mesh refinements).
 Perform a simple sanity check by doing a force reaction balance with the applied traction.
 Compare the numerical results with a commercial FE software, or Ansys Composite PrepPost (ACP).
Stress_yz for [0 90]s laminate
14000
12000
𝜏𝑦𝑧
yz [Pa]
10000
𝑃0
8000
6000
4000
2000
0
0
20
40
60
Along y-direction [m]
80
100
2
Some stress contours!
C. Kaufmann, N. Putnam, E. Seo, J. Shin
𝝈𝒙𝒙
𝝈𝒚𝒚
𝝈𝒛𝒛
𝝉𝒚𝒛
𝝉𝒙𝒛
𝝉𝒙𝒚
These results refer to a 0°/90°/90°/0° cross-ply laminate.
3
Overview of composite materials
C. Kaufmann, N. Putnam, E. Seo, J. Shin
 Composite materials are commonly used in aerospace structures to minimize mass.
 We considered symmetric, unidirectional, fiber-reinforced composites.
 Composite lamina (sheets) can be stacked to form high strength laminates.
 Laminate stack-ups are characterized by the orientation angles of the fibers, and the materials used.
 Anisotropy of the laminates allows one to tailor designs for stiffness and strength in specific directions.
i.e. 0°/+45°/−45°/90°
s
4
Complexities of layered structures
C. Kaufmann, N. Putnam, E. Seo, J. Shin
 Composite materials can give a much more complicated mechanical response than monolithic materials.
 Stress equilibrium must be satisfied in the laminate by way of interlaminar stresses.
 Special care must be taken to consider the free-edge and free-corner effect in composite samples!
 Non-intuitive effects, such as normal-shear coupling, can occur in anisotropic materials.
 As a result, finite element analysis is often found useful for predicting the behavior of complicated composites.
Normal-shear
coupling
Delamination
5
FEM Formulation
C. Kaufmann, N. Putnam, E. Seo, J. Shin
 Pre-processing:
 Define the required size dimensions, material properties, and the laminate configuration.
 Discretize the model into finite elements.
 Consider a tri-linear hexahedron element.
 Apply any bias factor when discretizing (optional).
 Calculate the 3D elasticity matrix, 𝑫𝑒 .  i.e. 𝜎 𝑥 = 𝑫 𝜀 𝑥
 Compute the elemental matrices necessary in developing the stiffness equation.
 Specify the boundary conditions (includes the external load).
 Processing:
 “Globalize” and assemble the local stiffness matrices and the local load vectors.
 Partition and rearrange the global stiffness equation.
 Solve for the nodal displacement field!
 Post-processing:
 Compute the strain field by applying the kinematic equation (displacement ↔ strain).
 Compute the stress field by applying the constitutive equation (Hooke’s Law).
6
Pre-processing (1/7)
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C. Kaufmann, N. Putnam, E. Seo, J. Shin
Define the required size dimensions, material properties, and the laminate configuration.
Discretize the model into finite elements.
Consider a 8-noded hexahedron, or tri-linear element (3 translational DOF per node).
Apply any bias factor when discretizing (optional).
Calculate the 3D elasticity matrix, 𝑫𝑒 .  i.e. 𝜎 𝑥 = 𝑫 𝜀 𝑥
Compute the elemental matrices necessary in developing the stiffness equation.
Specify the boundary conditions (includes the external load).
 Define the coordinate axes, based on fiber orientations.
 x-axis: longitudinal direction
 y-axis: transverse direction
 z-axis: normal direction (or thickness direction)
𝑵𝒙
 Specify the size dimensions.  𝐿, 𝐵, and 𝐻
 Specify the material properties.  𝐸1 , 𝐸2 , 𝐸3 , 𝜈12 , 𝜈13 , 𝜈23 , 𝐺12 , 𝐺13 , and 𝐺23
 Specify the fiber orientations of the off-axis plies.
7
Pre-processing (2/7)
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C. Kaufmann, N. Putnam, E. Seo, J. Shin
Define the required size dimensions, material properties, and the laminate configuration.
Discretize the model into finite elements.
Consider a 8-noded hexahedron, or tri-linear element (3 translational DOF per node).
Apply any bias factor when discretizing (optional).
Calculate the 3D elasticity matrix, 𝑫𝑒 .  i.e. 𝜎 𝑥 = 𝑫 𝜀 𝑥
Compute the elemental matrices necessary in developing the stiffness equation.
Specify the boundary conditions (includes the external load).
 Store the global nodes as local nodes for each element.
 Adhere to the given node-numbering scheme to ensure that the determinant of the Jacobian matrix is positive.
 Guarantee invertible mapping to natural coordinate system.
8
Pre-processing (3/7)
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C. Kaufmann, N. Putnam, E. Seo, J. Shin
Define the required size dimensions, material properties, and the laminate configuration.
Discretize the model into finite elements.
Consider a tri-linear hexahedron element.
Apply any bias factor when discretizing (optional).
Calculate the 3D elasticity matrix, 𝑫𝑒 .  i.e. 𝜎 𝑥 = 𝑫 𝜀 𝑥
Compute the elemental matrices necessary in developing the stiffness equation.
Specify the boundary conditions (includes the external load).
 Use an eight-noded hexahedron element.
 Each node has three translational degrees-of-freedom.
 𝑢𝑥 , 𝑢𝑦 , and 𝑢𝑧
9
Pre-processing (4/7)
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C. Kaufmann, N. Putnam, E. Seo, J. Shin
Define the required size dimensions, material properties, and the laminate configuration.
Discretize the model into finite elements.
Consider a tri-linear hexahedron element.
Apply any bias factor when discretizing (optional).
Calculate the 3D elasticity matrix, 𝑫𝑒 .  i.e. 𝜎 𝑥 = 𝑫 𝜀 𝑥
Compute the elemental matrices necessary in developing the stiffness equation.
Specify the boundary conditions (includes the external load).
 Bias Factor: Allows us to have more concentrated mesh density near a particular region of interest.
 The spacing between the node becomes a geometrical series.
Node Profile
100
80
Finer mesh
y-axis
60
40
20
0
0
100
200
300
x-axis
400
500
10
Pre-processing (5/7)
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C. Kaufmann, N. Putnam, E. Seo, J. Shin
Define the required size dimensions, material properties, and the laminate configuration.
Discretize the model into finite elements.
Consider a tri-linear hexahedron element.
Apply any bias factor when discretizing (optional).
Calculate the 3D elasticity matrix, 𝑫𝑒 .  i.e. 𝜎 𝑥 = 𝑫 𝜀 𝑥
Compute the elemental matrices necessary in developing the stiffness equation.
Specify the boundary conditions (includes the external load).
 The three-dimensional elasticity (stiffness) matrix is defined by applying the generalized Hooke’s Law.
 Take into account the anisotropy, assuming a transversely isotropic layer.
 Material nonlinearity (plasticity) is neglected!
𝜎 = 𝑻1 −𝜃
𝜎𝑥𝑥
𝜎𝑦𝑦
𝜎𝑧𝑧
𝜎 = 𝜏
𝑦𝑧
𝜏𝑥𝑧
𝜏𝑥𝑦
𝑪 𝑻2 𝜃
𝜀𝑥𝑥
𝜀𝑦𝑦
𝜀𝑧𝑧
𝜀 = 𝛾
𝑦𝑧
𝛾𝑥𝑧
𝛾𝑥𝑦
𝜀
1 − 𝜈23 𝜈32
𝐸2 𝐸3 Δ
𝜈21 + 𝜈23 𝜈31
𝐸2 𝐸3 Δ
𝑪 = 𝜈31 + 𝜈21 𝜈32
𝐸2 𝐸3 Δ
0
0
0
Δ≡
𝜈21 + 𝜈23 𝜈31
𝐸2 𝐸3 Δ
1 − 𝜈13 𝜈31
𝐸1 𝐸3 Δ
𝜈32 + 𝜈12 𝜈31
𝐸1 𝐸3 Δ
0
0
0
𝜈31 + 𝜈21 𝜈32
𝐸2 𝐸3 Δ
𝜈32 + 𝜈12 𝜈31
𝐸1 𝐸3 Δ
1 − 𝜈12 𝜈21
𝐸1 𝐸2 Δ
0
0
0
0
0
0
0
0
0
0
0
0
𝐺23
0
0
0
𝐺13
0
0
0
𝐺12
1 − 𝜈12 𝜈21 − 𝜈23 𝜈32 − 𝜈13 𝜈31 − 2𝜈21 𝜈32 𝜈13
𝐸1 𝐸2 𝐸3
11
Pre-processing (6/7)
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Define the required size dimensions, material properties, and the laminate configuration.
Discretize the model into finite elements.
Consider a tri-linear hexahedron element.
Apply any bias factor when discretizing (optional).
Calculate the 3D elasticity matrix, 𝑫𝑒 .  i.e. 𝜎 𝑥 = 𝑫 𝜀 𝑥
Compute the elemental matrices necessary in developing the stiffness equation.
Specify the boundary conditions (includes the external load).
𝑵
𝑒
𝑁1𝑒
= 0
0
𝜕𝑁1𝑒
𝜕𝑥
0
𝑩
𝑒
C. Kaufmann, N. Putnam, E. Seo, J. Shin
𝑒
≡ 𝛁s 𝑵
0
𝑁1𝑒
0
0
𝜕𝑁1𝑒
𝜕𝑦
0
0
0
𝜕𝑁1𝑒
𝜕𝑧
=
𝜕𝑁1𝑒
𝜕𝑧
𝜕𝑁1𝑒
𝜕𝑦
0
𝜕𝑁1𝑒
𝜕𝑥
0
0
𝑁1𝑒
0
0
𝜕𝑁1𝑒
𝜕𝑧
𝜕𝑁1𝑒
𝜕𝑦
𝜕𝑁1𝑒
𝜕𝑥
0
𝑁2𝑒
0
0
0
𝑁2𝑒
0
𝜕𝑁2𝑒
𝜕𝑥
0
0
0
𝑁2𝑒
0
𝜕𝑁2𝑒
𝜕𝑦
0
0
0
𝜕𝑁2𝑒
𝜕𝑧
𝜕𝑁2𝑒
𝜕𝑧
𝜕𝑁2𝑒
𝜕𝑦
0
𝜕𝑁2𝑒
𝜕𝑥
𝑒
⋯ 𝑁nen
⋯
0
⋯
0
0
0
𝜕𝑁2𝑒
𝜕𝑧
𝜕𝑁2𝑒
𝜕𝑦
𝜕𝑁2𝑒
𝜕𝑥
0
0
𝑒
𝑁nen
0
𝑒
𝜕𝑁nen
⋯
𝜕𝑥
⋯
⋯
⋯
0
0
0
𝑒
𝑁nen
0
𝑒
𝜕𝑁nen
𝜕𝑦
0
0
0
𝑒
𝜕𝑁nen
𝜕𝑧
𝑒
𝜕𝑁nen
⋯
𝜕𝑧
𝑒
𝜕𝑁nen
⋯
𝜕𝑦
0
𝑒
𝜕𝑁nen
𝜕𝑥
𝑁𝑖𝑒 = 𝐿𝑒𝐼 𝜉 𝐿𝑒𝐽 𝜂 𝐿𝑒𝐾 𝜁
𝑝+1
𝐿𝑒𝑚
𝜉 =
𝑗≠𝑚
0
0
𝜉 − 𝜉𝑗𝑒
𝑒
𝜉𝑚
− 𝜉𝑗𝑒
0
𝐿𝑒𝑚
𝜂 =
𝑗≠𝑚
𝑝+1
𝜂 − 𝜂𝑗𝑒
𝑒
𝜂𝑚
− 𝜂𝑗𝑒
𝐿𝑒𝑚
𝜁 =
𝑗≠𝑚
𝑝+1
d𝐿𝑒𝑚 𝜉
=
d𝜉
1
𝑒
𝜉𝑚 − 𝜉ℎ𝑒
𝑝+1
𝜉 − 𝜉𝑗𝑒
𝑒
𝜉𝑚
− 𝜉𝑗𝑒
𝑗≠ℎ ∧ 𝑗≠𝑚
ℎ≠𝑚
𝑒
𝜕𝑁nen
𝜕𝑧
𝑒
𝜕𝑁nen
𝜕𝑦
𝑒
𝜕𝑁nen
𝜕𝑥
𝑝+1
𝑝+1
d𝐿𝑒𝑚 𝜂
=
d𝜂
1
𝑒
𝜂𝑚 − 𝜂ℎ𝑒
𝑝+1
𝜂 − 𝜂𝑗𝑒
𝑗≠ℎ ∧ 𝑗≠𝑚
𝑒
𝜂𝑚
− 𝜂𝑗𝑒
ℎ≠𝑚
𝑝+1
d𝐿𝑒𝑚 𝜁
=
d𝜁
1
𝑒
𝜁𝑚 − 𝜁ℎ𝑒
ℎ≠𝑚
𝑝+1
𝑗≠ℎ ∧ 𝑗≠𝑚
𝜁 − 𝜁𝑗𝑒
𝑒
𝜁𝑚
− 𝜁𝑗𝑒
12
𝜁 − 𝜁𝑗𝑒
𝑒
𝜁𝑚
− 𝜁𝑗𝑒
Pre-processing (7/7)
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C. Kaufmann, N. Putnam, E. Seo, J. Shin
Define the required size dimensions, material properties, and the laminate configuration.
Discretize the model into finite elements.
Consider a tri-linear hexahedron element.
Apply any bias factor when discretizing (optional).
Calculate the 3D elasticity matrix, 𝑫𝑒 .  i.e. 𝜎 𝑥 = 𝑫 𝜀 𝑥
Compute the elemental matrices necessary in developing the stiffness equation.
Specify the boundary conditions (includes the external load).
 Essential Boundary Condition
 𝑢𝑥 𝑥 = 0, 𝑦, 𝑧 = 0
 𝑢𝑦 𝑥, 𝑦 = 0, 𝑧 = 0
 𝑢𝑧 𝑥, 𝑦, 𝑧 = 0 = 0
 Natural Boundary Condition
 Pressure-based load, 𝑃0 , at 𝑥 = 𝐿
13
Processing (1/3)
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C. Kaufmann, N. Putnam, E. Seo, J. Shin
“Globalize” and assemble the local stiffness matrices and the local load vectors.
Partition and rearrange the global stiffness equation.
Solve for the nodal displacement field!
 The weak form of our finite element formulation is given below.
 Use Gauss Quadrature rule to numerically evaluate the local integration.
 Apply the transformation rule to 𝑲𝑒 and 𝑓 𝑒 using the connectivity matrix.
 𝐾 𝑒 glb = 𝑳𝑒 ⊤ 𝑲𝑒 𝑳𝑒
 𝑓 𝑒 glb = 𝑳𝑒 ⊤ 𝑓 𝑒
 Sum individual matrices and vectors for global assembly.
 𝑲 = 𝑲𝑒 glb
 𝑓 = 𝑓 𝑒 glb
nel
nel
+1 +1 +1
𝑳𝑒
⊤
⊤
𝑩𝑒 𝑫𝑒 𝑩𝑒 𝑱𝑒 d𝜉 d𝜂 d𝜁 𝑳𝑒 𝑑 =
𝑳𝑒
−1 −1 −1
𝑒=1
𝑒=1
𝑲
⊤
⊤
𝑵𝑒 𝑡 𝑒 dΓ
Γ𝑒t
𝑓
14
Processing (2/3)
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C. Kaufmann, N. Putnam, E. Seo, J. Shin
“Globalize” and assemble the local stiffness matrices and the local load vectors.
Partition and rearrange the global stiffness equation.
Solve for the nodal displacement field!
 Partition the global stiffness equation into the known and unknown components.
 Apply the transformation rule to rearrange them as shown below.
𝑲 𝑑 = 𝑓
𝑲𝐸
𝑲⊤
𝐸𝐹
𝑲𝐸𝐹 𝑑𝐸
𝑓𝐸
=
𝑓𝐹
𝑲𝐹 𝑑𝐹
15
Processing (3/3)
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C. Kaufmann, N. Putnam, E. Seo, J. Shin
“Globalize” and assemble the local stiffness matrices and the local load vectors.
Partition and rearrange the global stiffness equation.
Solve for the nodal displacement field!
 Solve the system of equations efficiently by using the Gaussian elimination method.
 In MATLAB, a built-in function, d=K\f can be employed.
𝑲⊤
𝐸𝐹 𝑑𝐸 + 𝑲𝐹 𝑑𝐹 = 𝑓𝐹
𝑑𝐹 = 𝑲−1
𝐹 𝑓𝐹 , ∀ 𝑑𝐸 = 0
𝑓𝐸 = 𝑲𝐸𝐹 𝑑𝐹 = 𝑲𝐸𝐹 𝑲−1
𝐹 𝑓𝐹
16
Post-processing (1/2)
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
C. Kaufmann, N. Putnam, E. Seo, J. Shin
Compute the strain field by applying the kinematic equation (displacement ↔ strain).
Compute the stress field by applying the constitutive equation (Hooke’s Law).
 The elemental strain vector can be computed as shown below.
 Weighted average of the strain values, evaluated the Gauss points.
𝜀 = 𝛁s 𝑢
𝜕
𝜕𝑥
0
𝜀𝑥𝑥
𝜀𝑦𝑦
0
𝜀𝑧𝑧
𝛾𝑦𝑧 =
0
𝛾𝑥𝑧
𝛾𝑥𝑦
𝜕
𝜕𝑧
𝜕
𝜕𝑦
0
0
𝜕
𝜕𝑦
0
0
𝜕
𝜕𝑧
0
𝜕
𝜕𝑥
𝜕
𝑢𝑥
𝜕𝑧 𝑢
𝑦
𝜕
𝑢𝑧
𝜕𝑦
𝜕
𝜕𝑥
0
17
Post-processing (2/2)
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
C. Kaufmann, N. Putnam, E. Seo, J. Shin
Compute the strain field by applying the kinematic equation (displacement ↔ strain).
Compute the stress field by applying the constitutive equation (Hooke’s Law).
 The elemental stress vector can be computed as shown below.
 Weighted average of the stress values, evaluated the Gauss points.
𝜎 = 𝑫 𝜀 = 𝑻1 −𝜃
𝜎vm =
1
2
𝜎𝑥𝑥 − 𝜎𝑦𝑦
2
+ 𝜎𝑦𝑦 − 𝜎𝑧𝑧
2
𝑪 𝑻2 𝜃
+ 𝜎𝑥𝑥 − 𝜎𝑧𝑧
2
𝜀
2
2
2
+ 6 𝜏𝑦𝑧
+ 𝜏𝑥𝑧
+ 𝜏𝑥𝑦
det 𝝈 − 𝜆𝑰 = 0
𝜆3 − 𝐼1 𝜆2 − 𝐼2 𝜆 − 𝐼3 = 0
𝐼1 = 𝜎𝑥𝑥 + 𝜎𝑦𝑦 + 𝜎𝑧𝑧
2
2
2
𝐼2 = 𝜎𝑦𝑦 𝜎𝑧𝑧 + 𝜎𝑥𝑥 𝜎𝑧𝑧 + 𝜎𝑥𝑥 𝜎𝑦𝑦 − 𝜏𝑦𝑧
− 𝜏𝑥𝑧
− 𝜏𝑥𝑦
2
2
2
𝐼3 = 𝜎𝑥𝑥 𝜎𝑦𝑦 𝜎𝑧𝑧 − 𝜎𝑥𝑥 𝜏𝑦𝑧
− 𝜎𝑦𝑦 𝜏𝑥𝑧
− 𝜎𝑧𝑧 𝜏𝑥𝑦
+ 2𝜏𝑦𝑧 𝜏𝑥𝑧 𝜏𝑥𝑦
𝜆1 = 𝜎p1
𝜏max = MAX
𝜆2 = 𝜎p2
𝜆3 = 𝜎p3
𝜎p2 − 𝜎p3 𝜎p1 − 𝜎p3 𝜎p1 − 𝜎p2
,
,
2
2
2
18
Pathwise-results!
C. Kaufmann, N. Putnam, E. Seo, J. Shin
These results refer to a −35°/+35°/+35°/−35° angle-ply laminate.
Stress_yz for [-35 35]s laminate
6
x 10
Stress_xz for [-35 35]s laminate
7
x 10
Stress_yz for [-35 35]s laminate
1
-2000
-0.5
-4000
0.5
0
-1.5
-2
-0.5
-1
-6000
yz [Pa]
0
xz [Pa]
yz [Pa]
-1
𝜏𝑦𝑧
5
-3
10
15
20
25
Along z-direction [m]
30
35
0
𝜏𝑥𝑧
5
-10000
-12000
Stress
singularity!
-2.5
Ply interface
-8000
-14000
-16000
-18000
10
15
20
25
Along z-direction [m]
30
35
0
𝜏𝑦𝑧
20
40
60
Along y-direction [m]
80
100
16800 seconds ≈ 4.5 hours!
19
Contour Plots!
C. Kaufmann, N. Putnam, E. Seo, J. Shin
𝝈𝒙𝒙
𝝈𝒚𝒚
𝝈𝒛𝒛
𝝉𝒚𝒛
𝝉𝒙𝒛
𝝉𝒙𝒚
These results refer to a −35°/+35° s angle-ply laminate.
20
More contours…
C. Kaufmann, N. Putnam, E. Seo, J. Shin
𝝈𝐩𝟐
𝝈𝐩𝟏
𝝈𝐩𝟑
𝝉𝐦𝐚𝐱
𝝈𝐯𝐦
These results refer to a −35°/+35°
s
angle-ply laminate.
21
Sanity Check!
C. Kaufmann, N. Putnam, E. Seo, J. Shin
 A force reaction balance check would indicate that our analysis was modeled correctly!
 External load ≟ Reaction load
• Reaction load is extracted from nodes (DOF to be more specific), where an essential BC is specified.
Nodal
Reaction
Forces, 𝑅𝑥
External
Pressure
Load, 𝑃0
𝑲𝐸 𝑑𝐸 + 𝑲𝐸𝐹 𝑑𝐹 = 𝑓𝐸
𝑓𝐸 = 𝑲𝐸𝐹 𝑑𝐹
Input
Laminate width, 𝐵
Laminate thickness, 𝐻
Pressure Load, 𝑃0
𝟏𝟎
𝟑. 𝟔
𝟓 × 𝟏𝟎𝟕
𝐹𝑥 = 𝑓ext + 𝑓r = 0

22
Error Analysis
C. Kaufmann, N. Putnam, E. Seo, J. Shin
• L2 and Energy error norms are considered.
• In the below formula, 𝑝 is equal to 1, since a linear element is considered.
𝑒
𝑒
en
L2
= 𝑢 𝑥 − 𝑢h 𝑥
= 𝑢 𝑥 − 𝑢h 𝑥
𝑒
L2
L2
en
=
𝑥2𝑒
=
𝑥1𝑒
1
2
𝑥2𝑒
𝑥1𝑒
1
𝑢 𝑥 − 𝑢h 𝑥
2
d𝑥
1
𝐸𝑒 𝜀 𝑥 − 𝜀 h 𝑥
≤ 𝐶ℎ𝑝+1
𝑒
ℎ=
2
en
2
2
d𝑥
≤ 𝐶ℎ𝑝
ℎ𝑥2 + ℎ𝑦2 + ℎ𝑧2
23
Comparison to Ansys?
C. Kaufmann, N. Putnam, E. Seo, J. Shin
Averaged over an element
5.3539e7 Pa
(Ansys ACP)
𝜎𝑥𝑥
5.358e7 Pa
(MATLAB)
24