Shell property module (`pyfe3d.shellprop`)#

Highly based on the composites. module.

class pyfe3d.shellprop.GradABDE#

Container to store the gradients of the ABDE matrices with respect to the lamination parameters

Attributes:

gradAij, gradBij, gradDij, gradEijtuple of 2D np.array objects

The shapes of these gradient matrices are:

gradAij: (6, 5) gradBij: (6, 5) gradDij: (6, 5) gradEij: (3, 3)

They contain the gradients of each laminate stiffness with respect to the thickness and respective lamination parameters. The rows and columns correspond to:

Methods

calc_LP_grad(self, double thickness, ...)

Gradients of the shell stiffnesses with respect to the thickness and lamination parameters

calc_LP_grad(self, double thickness, MatLamina mat, LaminationParameters lp) → void#

Gradients of the shell stiffnesses with respect to the thickness and lamination parameters

Parameters:

thicknessfloat: The total thickness of the laminate
matMatLamina object: Material object
lpLaminationParameters object: The container class with all lamination parameters already defined

Returns:

None: The attributes of the object are updated.

gradAij#: gradAij: ‘double[:, ::1]’

gradBij#: gradBij: ‘double[:, ::1]’

gradDij#: gradDij: ‘double[:, ::1]’

gradEij#: gradEij: ‘double[:, ::1]’

class pyfe3d.shellprop.Lamina#

Attributes:

plyidint: plyid: ‘int’
matlaminaMatLamina object: matlamina: pyfe3d.shellprop.MatLamina
hfloat: h: ‘double’
thetadegfloat: thetadeg: ‘double’

Methods

`get_constitutive_matrix`(self)	Return the constitutive matrix
`get_transf_matrix_displ_to_laminate`(self)	Return displacement transformation matrix from lamina to laminate
`get_transf_matrix_stress_to_lamina`(self)	Return stress transformation matrix from laminate to lamina
`get_transf_matrix_stress_to_laminate`(self)	Return stress transformation matrix from lamina to laminate
`rebuild`(self)	Update constitutive matrices

cos2t#: cos2t: ‘double’

cos4t#: cos4t: ‘double’

cost#: cost: ‘double’

get_constitutive_matrix(self) → double[:, ::1]#: Return the constitutive matrix

get_transf_matrix_displ_to_laminate(self) → double[:, ::1]#: Return displacement transformation matrix from lamina to laminate

get_transf_matrix_stress_to_lamina(self) → double[:, ::1]#: Return stress transformation matrix from laminate to lamina

get_transf_matrix_stress_to_laminate(self) → double[:, ::1]#: Return stress transformation matrix from lamina to laminate

h#: h: ‘double’

matlamina#: matlamina: pyfe3d.shellprop.MatLamina

plyid#: plyid: ‘int’

q11L#: q11L: ‘double’

q12L#: q12L: ‘double’

q16L#: q16L: ‘double’

q22L#: q22L: ‘double’

q26L#: q26L: ‘double’

q44L#: q44L: ‘double’

q45L#: q45L: ‘double’

q55L#: q55L: ‘double’

q66L#: q66L: ‘double’

rebuild(self) → void#

Update constitutive matrices

Reference:

Reddy, J. N., Mechanics of Laminated Composite Plates and Shells - Theory and Analysys. Second Edition. CRC PRESS, 2004.

sin2t#: sin2t: ‘double’

sin4t#: sin4t: ‘double’

sint#: sint: ‘double’

thetadeg#: thetadeg: ‘double’

class pyfe3d.shellprop.LaminationParameters#

Lamination parameters

Attributes:

xiA1, xiA2, xiA3, xiA4float: Lamination parameters \(\xi_{Ai}\) (in-plane)
xiB1, xiB2, xiB3, xiB4float: Lamination parameters \(\xi_{Bi}\) (in-plane coupling with bending)
xiD1, xiD2, xiD3, xiD4float: Lamination parameters \(\xi_{Di}\) (bending)
xiE1, xiE2float: Lamination parameters \(\xi_{Ei}\) (transverse shear)

xiA1#: xiA1: ‘double’

xiA2#: xiA2: ‘double’

xiA3#: xiA3: ‘double’

xiA4#: xiA4: ‘double’

xiB1#: xiB1: ‘double’

xiB2#: xiB2: ‘double’

xiB3#: xiB3: ‘double’

xiB4#: xiB4: ‘double’

xiD1#: xiD1: ‘double’

xiD2#: xiD2: ‘double’

xiD3#: xiD3: ‘double’

xiD4#: xiD4: ‘double’

xiE1#: xiE1: ‘double’

xiE2#: xiE2: ‘double’

xiE3#: xiE3: ‘double’

xiE4#: xiE4: ‘double’

class pyfe3d.shellprop.MatLamina#

Orthotropic material lamina

Attributes:

e1float: e1: ‘double’
e2float: e2: ‘double’
g12float: g12: ‘double’
g13float: g13: ‘double’
g23float: g23: ‘double’
nu12: nu12: ‘double’
nu13: nu13: ‘double’
nu23: nu23: ‘double’
nu21: nu21: ‘double’
nu31: nu31: ‘double’
nu32: nu32: ‘double’
rho: rho: ‘double’
a1: a1: ‘double’
a2: a2: ‘double’
a3: a3: ‘double’
tref: tref: ‘double’
st1,st2: allowable tensile stresses for directions 1 and 2
sc1,sc2: allowable compressive stresses for directions 1 and 2
ss12: ss12: ‘double’
q11: q11: ‘double’
q12: q12: ‘double’
q13: q13: ‘double’
q21: q21: ‘double’
q22: q22: ‘double’
q23: q23: ‘double’
q31: q31: ‘double’
q32: q32: ‘double’
q33: q33: ‘double’
q44: q44: ‘double’
q55: q55: ‘double’
q66: q66: ‘double’
ci: lamina stiffness constants
ui: lamina material invariants

Methods

`get_constitutive_matrix`(self)	Return the constitutive matrix
`get_invariant_matrix`(self)	Return the invariant matrix
`rebuild`(self)	Update constitutive and invariant terms
`trace_normalize_plane_stress`(self)	Trace-normalize the lamina properties for plane stress

Notes

For isotropic materials when the user defines \(\nu\) and \(E\), \(G\) will be recaculated based on equation: \(G = E/(2 \times (1+\nu))\); in a lower priority if the user defines \(\nu\) and \(G\), \(E\) will be recaculated based on equation: \(E = 2 \times (1+\nu) \times G\).

a1#: a1: ‘double’

a2#: a2: ‘double’

a3#: a3: ‘double’

c11#: c11: ‘double’

c12#: c12: ‘double’

c13#: c13: ‘double’

c22#: c22: ‘double’

c23#: c23: ‘double’

c33#: c33: ‘double’

c44#: c44: ‘double’

c55#: c55: ‘double’

c66#: c66: ‘double’

e1#: e1: ‘double’

e2#: e2: ‘double’

e3#: e3: ‘double’

g12#: g12: ‘double’

g13#: g13: ‘double’

g23#: g23: ‘double’

get_constitutive_matrix(self) → double[:, ::1]#: Return the constitutive matrix

get_invariant_matrix(self) → double[:, ::1]#: Return the invariant matrix

nu12#: nu12: ‘double’

nu13#: nu13: ‘double’

nu21#: nu21: ‘double’

nu23#: nu23: ‘double’

nu31#: nu31: ‘double’

nu32#: nu32: ‘double’

q11#: q11: ‘double’

q12#: q12: ‘double’

q13#: q13: ‘double’

q21#: q21: ‘double’

q22#: q22: ‘double’

q23#: q23: ‘double’

q31#: q31: ‘double’

q32#: q32: ‘double’

q33#: q33: ‘double’

q44#: q44: ‘double’

q55#: q55: ‘double’

q66#: q66: ‘double’

rebuild(self) → void#

Update constitutive and invariant terms

Reference:

Reddy, J. N., Mechanics of laminated composite plates and shells. Theory and analysis. Second Edition. CRC Press, 2004.

rho#: rho: ‘double’

sc1#: sc1: ‘double’

sc2#: sc2: ‘double’

ss12#: ss12: ‘double’

st1#: st1: ‘double’

st2#: st2: ‘double’

trace_normalize_plane_stress(self) → void#

Trace-normalize the lamina properties for plane stress

Modify the original MatLamina object with a trace-normalization performed after calculating the trace according to Eq. 1 of reference:

Melo, J. D. D., Bi, J., and Tsai, S. W., 2017, “A Novel Invariant-Based Design Approach to Carbon Fiber Reinforced Laminates,” Compos. Struct., 159, pp. 44–52.

The trace calculated as \(tr = Q_{11} + Q_{22} + 2Q_{66}\). The universal in-plane stress stiffness components \(Q_{11},Q_{12},Q_{22},Q_{44},Q_{55},Q_{66}\) are divided by \(tr\), and the invariants \(U_1,U_2,U_3,U_4,U_5,U_6,U_7\) are calculated with the normalized stiffnesses, such they also become trace-normalized invariants. These can be accessed using the u1,u2,u3,u4,u5,u6,u7 attributes.

tref#: tref: ‘double’

u1#: u1: ‘double’

u2#: u2: ‘double’

u3#: u3: ‘double’

u4#: u4: ‘double’

u5#: u5: ‘double’

u6#: u6: ‘double’

u7#: u7: ‘double’

class pyfe3d.shellprop.ShellProp#

Attributes:

plieslist: plies: list
stacklist: stack: list
hfloat: h: ‘double’
offsetfloat: offset: ‘double’
e1, e2float: Equivalent laminate moduli in directions 1 and 2
g12float: g12: ‘double’
nu12, nu21float: Equivalent laminate Poisson ratios in the 12 and 21 directions
scf_k13, scf_k23float: Shear correction factor in the 13 and 23 directions
intrhofloat: intrho: ‘double’
intrhozfloat: intrhoz: ‘double’
intrhoz2float: intrhoz2: ‘double’

Methods

`calc_constitutive_matrix`(self)	Calculate the laminate constitutive terms
`calc_equivalent_properties`(self)	Calculate the equivalent laminate properties
`calc_lamination_parameters`(self)	Calculate the lamination parameters.
`calc_scf`(self)	Update shear correction factors of the `ShellProp` object
`force_balanced`(self)	Force a balanced laminate
`force_orthotropic`(self)	Force an orthotropic laminate
`force_symmetric`(self)	Force a symmetric laminate

A11#: A11: ‘double’

A12#: A12: ‘double’

A16#: A16: ‘double’

A22#: A22: ‘double’

A26#: A26: ‘double’

A66#: A66: ‘double’

B11#: B11: ‘double’

B12#: B12: ‘double’

B16#: B16: ‘double’

B22#: B22: ‘double’

B26#: B26: ‘double’

B66#: B66: ‘double’

D11#: D11: ‘double’

D12#: D12: ‘double’

D16#: D16: ‘double’

D22#: D22: ‘double’

D26#: D26: ‘double’

D66#: D66: ‘double’

E44#: E44: ‘double’

E45#: E45: ‘double’

E55#: E55: ‘double’

calc_constitutive_matrix(self) → void#

Calculate the laminate constitutive terms

This is the commonly called ABD matrix with shape=(6, 6) when the classical laminated plate theory is used, or the ABDE matrix when the first-order shear deformation theory is used, containing the transverse shear terms.

calc_equivalent_properties(self) → void#

Calculate the equivalent laminate properties

The following attributes are updated:

e1, e2, g12, `u12, nu21

calc_lamination_parameters(self) → LaminationParameters#

Calculate the lamination parameters.

The following attributes are calculated:

xiA, xiB, xiD, xiE

calc_scf(self) → void#

Update shear correction factors of the ShellProp object

Reference:

Vlachoutsis, S. “Shear correction factors for plates and shells”, Int. Journal for Numerical Methods in Engineering, Vol. 33, 1537-1552, 1992.

http://onlinelibrary.wiley.com/doi/10.1002/nme.1620330712/full

Using “one shear correction factor” (see reference), assuming:

constant G13, G23, E1, E2, nu12, nu21 within each ply
g1 calculated using z at the middle of each ply
zn1 = ShellProp offset attribute

Returns:

k13, k23tuple: Shear correction factors. Also updates attributes: scf_k13 and scf_k23.

e1#: e1: ‘double’

e2#: e2: ‘double’

force_balanced(self) → void#

Force a balanced laminate

The attributes \(A_{16}\), \(A_{26}\), \(B_{16}\), \(B_{26}\) are set to zero to force a balanced laminate.

force_orthotropic(self) → void#

Force an orthotropic laminate

The attributes \(A_{16}\), \(A_{26}\), \(B_{16}\), \(B_{26}\), \(D_{16}\), \(D_{26}\) are set to zero to force an orthotropic laminate.

force_symmetric(self) → void#

Force a symmetric laminate

The \(B_{ij}\) terms of the constitutive matrix are set to zero.

g12#: g12: ‘double’

h#: h: ‘double’

intrho#: intrho: ‘double’

intrhoz#: intrhoz: ‘double’

intrhoz2#: intrhoz2: ‘double’

nu12#: nu12: ‘double’

nu21#: nu21: ‘double’

offset#: offset: ‘double’

plies#: plies: list

scf_k13#: scf_k13: ‘double’

scf_k23#: scf_k23: ‘double’

stack#: stack: list

pyfe3d.shellprop.force_balanced_LP(LaminationParameters lp) → LaminationParameters#

Force balanced lamination parameters

The lamination parameters \(\xi_{A2}\) and \(\xi_{A4}\) are set to null to force a balanced laminate.

pyfe3d.shellprop.force_orthotropic_LP(LaminationParameters lp) → LaminationParameters#

Force orthotropic lamination parameters

The lamination parameters \(\xi_{A2}\), \(\xi_{A4}\), \(\xi_{B2}\), \(\xi_{B4}\), \(\xi_{D2}\) and \(\xi_{D4}\) are set to null to force an orthotropic laminate. The \(\xi_{D2}\) and \(\xi_{D4}\) are related to the bend-twist coupling and become often very small for balanced laminates with a large amount of plies.

pyfe3d.shellprop.force_symmetric_LP(LaminationParameters lp) → LaminationParameters#

Force symmetric lamination parameters

The lamination parameters \(\xi_{Bi}\) are set to null to force a symmetric laminate.

pyfe3d.shellprop.shellprop_from_LaminationParameters(double thickness, MatLamina mat, LaminationParameters lp) → ShellProp#

Return a ShellProp object based in the thickness, material and lamination parameters

Parameters:

thicknessfloat: The total thickness of the laminate
matMatLamina object: Material object
lpLaminationParameters object: The container class with all lamination parameters already defined

Returns:

lamShellProp: laminate with the ABD and ABDE matrices already calculated

pyfe3d.shellprop.shellprop_from_lamination_parameters(double thickness, MatLamina matlamina, double xiA1, double xiA2, double xiA3, double xiA4, double xiB1, double xiB2, double xiB3, double xiB4, double xiD1, double xiD2, double xiD3, double xiD4, double xiE1=0, double xiE2=0) → ShellProp#

Return a ShellProp object based in the thickness, material and lamination parameters

Note that \(\xi_{E1}\) and \(\xi_{E2}\) are optional and usually equal to zero, becoming important only when the transverse shear modulus is different in the two directions, i.e. when \(G_{13} \ne G{23}\).

Parameters:

thicknessfloat: The total thickness of the plate
matlaminaMatLamina object: Material object
xiAj, xiBj, xiDj, xiEjfloat: The 14 lamination parameters according to the first-order shear deformation theory: \(\xi_{A1} \cdots \xi_{A4}\), \(\xi_{B1} \cdots \xi_{B4}\), \(\xi_{D1} \cdots \xi_{D4}\), \(\xi_{E1}\) and \(\xi_{E2}\)

Returns:

lamShellProp: Shell property with the ABD and ABDE matrices already calculated

Shell property module (pyfe3d.shellprop)#

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Shell property module (`pyfe3d.shellprop`)#