Property definitions#

Beam property module (`pyfe3d.beamprop`)#

class pyfe3d.beamprop.BeamProp#

General class to represent beam properties.

About shear correction factors, usually one should only apply the shear correction factor to the shear modulus G. It is, however, possible to have more complex scenarios where the different cross section parameters Ay, Az, J have different corrections applied due to geometric properties. It is the user’s responsability to apply these correction factors properly to achieve a general representation of the beam element behaviour by means of this property class. No shear correction factors or geometric correction factors are applied internally during the calculation of the structural matrices.

Attributes:

A,float: Area of the cross section
E,float: Young modulus
G,float: Shear modulus
Iyy,float: Second moment of area about the y axis \(\int_y \int_z z^2 dy dz\)
Izz,float: Second moment of area about the z axis \(\int_y \int_z y^2 dy dz\)
Iyz,float: Product moment of area \(\int_y \int_z y z dy dz\)
J,float: Torsion stiffness or torsion constant, see https://en.wikipedia.org/wiki/Torsion_constant.
Ay,float: Integral \(\int_y \int_z y dy dz\)
Az,float: Integral \(\int_y \int_z z dy dz\)
intrho,float: Integral \(\int_{y_e} \int_{z_e} \rho(y, z) dy dz\), where \(\rho\) Is the density
intrhoy,float: Integral \(\int_y \int_z y \rho(y, z) dy dz\)
intrhoz,float: Integral \(\int_y \int_z z \rho(y, z) dy dz\)
intrhoy2,float: Integral \(\int_y \int_z y^2 \rho(y, z) dy dz\)
intrhoz2,float: Integral \(\int_y \int_z z^2 \rho(y, z) dy dz\)
intrhoyz,float: Integral \(\int_y \int_z y z \rho(y, z) dy dz\)

Notes

For beams with homogeneous material along the cross section some of the quantities above can be calculated as follows, assuming that rho is the material density:

intrho = A*rho intrhoy2 = Izz*rho intrhoz2 = Iyy*rho intrhoyz = Iyz*rho

A#: A: ‘double’

Ay#: Ay: ‘double’

Az#: Az: ‘double’

E#: E: ‘double’

G#: G: ‘double’

Iyy#: Iyy: ‘double’

Iyz#: Iyz: ‘double’

Izz#: Izz: ‘double’

J#: J: ‘double’

intrho#: intrho: ‘double’

intrhoy#: intrhoy: ‘double’

intrhoy2#: intrhoy2: ‘double’

intrhoyz#: intrhoyz: ‘double’

intrhoz#: intrhoz: ‘double’

intrhoz2#: intrhoz2: ‘double’

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 utils module (`pyfe3d.shellprop_utils`)#

Highly based on the composites. module.

pyfe3d.shellprop_utils.isotropic_plate(thickness, E, nu, offset=0.0, calc_scf=True, rho=0.0)#

Read data for an isotropic plate

ShellProp object is returned based on the inputs given.

Parameters:

thicknessfloat: Plate thickness.
Efloat: Young modulus.
nufloat, optional: Poisson’s ratio.
rhofloat, optional: Material density
offsetfloat, optional: Offset along the normal axis about the mid-surface, which influences the extension-bending coupling (B matrix).
calc_scfbool, optional: If True, use ShellProp.calc_scf() to compute shear correction factors, otherwise the default value of 5/6 is used.

pyfe3d.shellprop_utils.laminated_plate(stack, plyt=None, laminaprop=None, rho=0.0, plyts=None, laminaprops=None, rhos=None, offset=0.0, calc_scf=True)#

Read a laminate stacking sequence data.

ShellProp object is returned based on the inputs given.

Parameters:

stacklist: Angles of the stacking sequence in degrees.
plytfloat, optional: When all plies have the same thickness, plyt can be supplied.
laminaproptuple, optional: When all plies have the same material properties, laminaprop can be supplied.
rhofloat, optional: Uniform material density to be used for all plies.
plytslist, optional: A list of floats with the thickness of each ply.
laminapropslist, optional: A list of tuples with a laminaprop for each ply.
rhoslist, optional: A list of floats with the material density of each ply.
offsetfloat, optional: Offset along the normal axis about the mid-surface, which influences the laminate properties.
calc_scfbool, optional: If True, use ShellProp.calc_scf() to compute shear correction factors, otherwise the default value of 5/6 is used

Notes

plyt or plyts must be supplied laminaprop or laminaprops must be supplied

For orthotropic plies, the laminaprop should be:

laminaprop = (E11, E22, nu12, G12, G13, G23)

For isotropic plies, the laminaprop should be:

laminaprop = (E, nu)

pyfe3d.shellprop_utils.read_laminaprop(laminaprop, rho=0)#

Returns a MatLamina object based on an input laminaprop tuple

Parameters:

laminaproplist or tuple

For the most general case of tri-axial stress, use a tuple containing the folliwing entries:

laminaprop = (e1, e2, nu12, g12, g13, g23, e3, nu13, nu23)

For isotropic materials aiming calculations with tri-axial stresses, use:

g = e/(2*(1+nu))
laminaprop = (e, e, nu, g, g, g, e, nu, nu)

For othotropic materials with in-plane stresses the user can only supply:

laminaprop = (e1, e2, nu12, g12, g13, g23)

For isotropic materials with in-plane stresses the user can only supply:

laminaprop = (e, nu) # new

symbol	value
e1	Young Module in direction 1
e2	Young Module in direction 2
nu12	12 Poisson’s ratio
g12	12 Shear Modulus
g13	13 Shear Modulus
g23	13 Shear Modulus
e3	Young Module in direction 3
nu13	13 Poisson’s ratio
nu23	23 Poisson’s ratio

rhofloat, optional

Material density

Returns:

matlamMatLamina: A MatLamina object.

Property definitions#

Beam property module (pyfe3d.beamprop)#

Shell property module (pyfe3d.shellprop)#

Shell property utils module (pyfe3d.shellprop_utils)#

Beam property module (`pyfe3d.beamprop`)#

Shell property module (`pyfe3d.shellprop`)#

Shell property utils module (`pyfe3d.shellprop_utils`)#