weac.mixins module¶

Mixins for the elastic analysis of layered snow slabs.

class weac.mixins.AnalysisMixin[source]¶

Bases: object

Mixin for the analysis of model outputs.

Provides methods for the analysis of layered slabs on compliant elastic foundations.

Sxx(Z, phi, dz=2, unit='kPa')[source]¶

Compute axial normal stress in slab layers.

Parameters:

Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x), psi’(x)]^T
phi (float) – Inclination (degrees). Counterclockwise positive.
dz (float, optional) – Element size along z-axis (mm). Default is 2 mm.
unit ({'kPa', 'MPa'}, optional) – Desired output unit. Default is ‘kPa’.

Returns:

Axial slab normal stress in specified unit.

Return type:

ndarray, float

Szz(Z, phi, dz=2, unit='kPa')[source]¶

Compute transverse normal stress in slab layers.

Parameters:

Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x), psi’(x)]^T
phi (float) – Inclination (degrees). Counterclockwise positive.
dz (float, optional) – Element size along z-axis (mm). Default is 2 mm.
unit ({'kPa', 'MPa'}, optional) – Desired output unit. Default is ‘kPa’.

Returns:

Transverse normal stress at grid points in the slab in specified unit.

Return type:

ndarray, float

Txz(Z, phi, dz=2, unit='kPa')[source]¶

Compute shear stress in slab layers.

Parameters:

Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x), psi’(x)]^T
phi (float) – Inclination (degrees). Counterclockwise positive.
dz (float, optional) – Element size along z-axis (mm). Default is 2 mm.
unit ({'kPa', 'MPa'}, optional) – Desired output unit. Default is ‘kPa’.

Returns:

Shear stress at grid points in the slab in specified unit.

Return type:

ndarray

gdif(C, phi, li, ki, unit='kJ/m^2', **kwargs)[source]¶

Compute differential energy release rate of all crack tips.

Parameters:

C (ndarray) – Free constants of the solution.
phi (float) – Inclination (degress).
li (ndarray) – List of segment lengths.
ki (ndarray) – List of booleans indicating whether segment lies on a foundation or not in the cracked configuration.

Returns:

List of total, mode I, and mode II energy release rates.

Return type:

ndarray

get_zmesh(dz=2)[source]¶

Get z-coordinates of grid points and corresponding elastic properties.

Parameters:: dz (float, optional) – Element size along z-axis (mm). Default is 2 mm.
Returns:: mesh – Mesh along z-axis. Columns are a list of z-coordinates (mm) of grid points along z-axis with at least two grid points (top, bottom) per layer, Young’s modulus of each grid point, shear modulus of each grid point, and Poisson’s ratio of each grid point.
Return type:: ndarray

ginc(C0, C1, phi, li, ki, k0, **kwargs)[source]¶

Compute incremental energy relase rate of of all cracks.

Parameters:

C0 (ndarray) – Free constants of uncracked solution.
C1 (ndarray) – Free constants of cracked solution.
phi (float) – Inclination (degress).
li (ndarray) – List of segment lengths.
ki (ndarray) – List of booleans indicating whether segment lies on a foundation or not in the cracked configuration.
k0 (ndarray) – List of booleans indicating whether segment lies on a foundation or not in the uncracked configuration.

Returns:

List of total, mode I, and mode II energy release rates.

Return type:

ndarray

principal_stress_slab(Z, phi, dz=2, unit='kPa', val='max', normalize=False)[source]¶

Compute maxium or minimum principal stress in slab layers.

Parameters:

Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x), psi’(x)]^T
phi (float) – Inclination (degrees). Counterclockwise positive.
dz (float, optional) – Element size along z-axis (mm). Default is 2 mm.
unit ({'kPa', 'MPa'}, optional) – Desired output unit. Default is ‘kPa’.
val (str, optional) – Maximum ‘max’ or minimum ‘min’ principal stress. Default is ‘max’.
normalize (bool) – Toggle layerwise normalization to strength.

Returns:

Maximum or minimum principal stress in specified unit.

Return type:

ndarray

Raises:

ValueError – If specified principal stress component is neither ‘max’ nor ‘min’, or if normalization of compressive principal stress is requested.

principal_stress_weaklayer(Z, sc=2.6, unit='kPa', val='min', normalize=False)[source]¶

Compute maxium or minimum principal stress in the weak layer.

Parameters:

Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x), psi’(x)]^T
sc (float) – Weak-layer compressive strength. Default is 2.6 kPa.
unit ({'kPa', 'MPa'}, optional) – Desired output unit. Default is ‘kPa’.
val (str, optional) – Maximum ‘max’ or minimum ‘min’ principal stress. Default is ‘min’.
normalize (bool) – Toggle layerwise normalization to strength.

Returns:

Maximum or minimum principal stress in specified unit.

Return type:

ndarray

Raises:

ValueError – If specified principal stress component is neither ‘max’ nor ‘min’, or if normalization of tensile principal stress is requested.

rasterize_solution(C: ndarray, phi: float, li: list[float] | bool, ki: list[bool] | bool, num: int = 250, **kwargs)[source]¶

Compute rasterized solution vector.

Parameters:

C (ndarray) – Vector of free constants.
phi (float) – Inclination (degrees).
li (ndarray) – List of segment lengths (mm).
ki (ndarray) – List of booleans indicating whether segment lies on a foundation or not.
num (int) – Number of grid points.

Returns:

xq (ndarray) – Grid point x-coordinates at which solution vector is discretized.
zq (ndarray) – Matrix with solution vectors as colums at grid points xq.
xb (ndarray) – Grid point x-coordinates that lie on a foundation.

class weac.mixins.FieldQuantitiesMixin[source]¶

Bases: object

Mixin for field quantities.

Provides methods for the computation of displacements, stresses, strains, and energy release rates from the solution vector.

Gi(Ztip, unit='kJ/m^2')[source]¶

Get mode I differential energy release rate at crack tip.

Parameters:

Ztip (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T at the crack tip.
unit ({'N/mm', 'kJ/m^2', 'J/m^2'}, optional) – Desired output unit. Default is kJ/m^2.

Returns:

Mode I differential energy release rate (N/mm = kJ/m^2 or J/m^2) at the crack tip.

Return type:

float

Gii(Ztip, unit='kJ/m^2')[source]¶

Get mode II differential energy release rate at crack tip.

Parameters:

Ztip (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T at the crack tip.
unit ({'N/mm', 'kJ/m^2', 'J/m^2'}, optional) – Desired output unit. Default is kJ/m^2 = N/mm.

Returns:

Mode II differential energy release rate (N/mm = kJ/m^2 or J/m^2) at the crack tip.

Return type:

float

M(Z)[source]¶

Get bending moment M = B11 u’ + D11 psi’ in the slab.

Parameters:: Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T.
Returns:: Bending moment M (Nmm) in the slab.
Return type:: float

N(Z)[source]¶

Get the axial normal force N = A11 u’ + B11 psi’ in the slab.

Parameters:: Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T.
Returns:: Axial normal force N (N) in the slab.
Return type:: float

V(Z)[source]¶

Get vertical shear force V = kA55(w’ + psi).

Parameters:: Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T.
Returns:: Vertical shear force V (N) in the slab.
Return type:: float

dpsi_dx(Z)[source]¶

Get first derivative psi’ of the midplane rotation.

Parameters:: Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T.
Returns:: First derivative psi’ of the midplane rotation (radians/mm) of the slab.
Return type:: float

dpsi_dxdx(z, phi)[source]¶

Get second derivative of the cross-section rotation psi’’(x).

Parameters:

z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T.
phi (float) – Inclination (degrees). Counterclockwise positive.

Returns:

Second derivative of the cross-section rotation psi’’(x) (1/mm^2).

Return type:

ndarray, float

dpsi_dxdxdx(z, phi)[source]¶

Get third derivative of the cross-section rotation psi’’’(x).

Parameters:

z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T.
phi (float) – Inclination (degrees). Counterclockwise positive.

Returns:

Third derivative of the cross-section rotation psi’’’(x) (1/mm^3).

Return type:

ndarray, float

du0_dxdx(z, phi)[source]¶

Get second derivative of the horiz. centerline displacement u0’’(x).

Parameters:

z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T.
phi (float) – Inclination (degrees). Counterclockwise positive.

Returns:

Second derivative of the horizontal centerline displacement u0’’(x) (1/mm).

Return type:

ndarray, float

du0_dxdxdx(z, phi)[source]¶

Get third derivative of the horiz. centerline displacement u0’’’(x).

Parameters:

z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T.
phi (float) – Inclination (degrees). Counterclockwise positive.

Returns:

Third derivative of the horizontal centerline displacement u0’’’(x) (1/mm^2).

Return type:

ndarray, float

du_dx(Z, z0)[source]¶

Get first derivative of the horizontal displacement.

Parameters:

Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T.
z0 (float) – Z-coordinate (mm) where u is to be evaluated.

Returns:

First derivative u’ = u0’ + z0 psi’ of the horizontal displacement of the slab.

Return type:

float

dw_dx(Z)[source]¶

Get first derivative w’ of the centerline deflection.

Parameters:: Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T.
Returns:: First derivative w’ of the deflection of the slab.
Return type:: float

dz_dx(z, phi)[source]¶

Get first derivative z’(x) = K*z(x) + q of the solution vector.

z’(x) = [u’(x) u’’(x) w’(x) w’’(x) psi’(x), psi’’(x)]^T

Parameters:

z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x), psi’(x)]^T
phi (float) – Inclination (degrees). Counterclockwise positive.

Returns:

First derivative z’(x) for the solution vector (6x1).

Return type:

ndarray, float

dz_dxdx(z, phi)[source]¶

Get second derivative z’’(x) = K*z’(x) of the solution vector.

z’’(x) = [u’’(x) u’’’(x) w’’(x) w’’’(x) psi’’(x), psi’’’(x)]^T

Parameters:

z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x), psi’(x)]^T
phi (float) – Inclination (degrees). Counterclockwise positive.

Returns:

Second derivative z’’(x) = (K*z(x) + q)’ = K*z’(x) = K*(K*z(x) + q) of the solution vector (6x1).

Return type:

ndarray, float

eps(Z)[source]¶

Get weak-layer normal strain.

Parameters:: Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T.
Returns:: Weak-layer normal strain epsilon.
Return type:: float

gamma(Z)[source]¶

Get weak-layer shear strain.

Parameters:: Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T.
Returns:: Weak-layer shear strain gamma.
Return type:: float

int1(x, z0, z1)[source]¶

Get mode I crack opening integrand at integration points xi.

Parameters:

x (float, ndarray) – X-coordinate where integrand is to be evaluated (mm).
z0 (callable) – Function that returns the solution vector of the uncracked configuration.
z1 (callable) – Function that returns the solution vector of the cracked configuration.

Returns:

Integrant of the mode I crack opening integral.

Return type:

float or ndarray

int2(x, z0, z1)[source]¶

Get mode II crack opening integrand at integration points xi.

Parameters:

x (float, ndarray) – X-coordinate where integrand is to be evaluated (mm).
z0 (callable) – Function that returns the solution vector of the uncracked configuration.
z1 (callable) – Function that returns the solution vector of the cracked configuration.

Returns:

Integrant of the mode II crack opening integral.

Return type:

float or ndarray

psi(Z, unit='rad')[source]¶

Get midplane rotation psi.

Parameters:

Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T.
unit ({'deg', 'degrees', 'rad', 'radians'}, optional) – Desired output unit. Default is radians.

Returns:

psi – Cross-section rotation psi (radians) of the slab.

Return type:

float

sig(Z, unit='MPa')[source]¶

Get weak-layer normal stress.

Parameters:

Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T.
unit ({'MPa', 'kPa'}, optional) – Desired output unit. Default is MPa.

Returns:

Weak-layer normal stress sigma (in specified unit).

Return type:

float

tau(Z, unit='MPa')[source]¶

Get weak-layer shear stress.

Parameters:

Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T.
unit ({'MPa', 'kPa'}, optional) – Desired output unit. Default is MPa.

Returns:

Weak-layer shear stress tau (in specified unit).

Return type:

float

u(Z, z0, unit='mm')[source]¶

Get horizontal displacement u = u0 + z0 psi.

Parameters:

Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T.
z0 (float) – Z-coordinate (mm) where u is to be evaluated.
unit ({'m', 'cm', 'mm', 'um'}, optional) – Desired output unit. Default is mm.

Returns:

Horizontal displacement u (unit) of the slab.

Return type:

float

w(Z, unit='mm')[source]¶

Get centerline deflection w.

Parameters:

Z (ndarray) – Solution vector [u(x) u’(x) w(x) w’(x) psi(x) psi’(x)]^T.
unit ({'m', 'cm', 'mm', 'um'}, optional) – Desired output unit. Default is mm.

Returns:

Deflection w (in specified unit) of the slab.

Return type:

float

class weac.mixins.OutputMixin[source]¶

Bases: object

Mixin for outputs.

Provides convenience methods for the assembly of output lists such as rasterized displacements or rasterized stresses.

external_potential(C, phi, L, **segments)[source]¶

Compute total external potential (pst only).

Parameters:

C (ndarray) – Matrix(6xN) of solution constants for a system of N segements. Columns contain the 6 constants of each segement.
phi (float) – Inclination of the slab (°).
L (float, optional) – Total length of model (mm).
segments (dict) – Dictionary with lists of touchdown booleans (tdi), segement lengths (li), skier weights (mi), and foundation booleans in the cracked (ki) and uncracked (k0) configurations.

Returns:

Pi_ext – Total external potential (Nmm).

Return type:

float

get_slab_deflection(x, z, unit='mm')[source]¶

Compute vertical slab displacement.

Parameters:

x (ndarray) – Discretized x-coordinates (mm) where coordinates of unsupported (no foundation) segments are NaNs.
z (ndarray) – Solution vectors at positions x as columns of matrix z. Default is mid.
unit ({'m', 'cm', 'mm', 'um'}, optional) – Displacement output unit. Default is mm.

Returns:

x (ndarray) – Horizontal coordinates (cm).
ndarray – Vertical deflections (unit input).

get_slab_displacement(x, z, loc='mid', unit='mm')[source]¶

Compute horizontal slab displacement.

Parameters:

x (ndarray) – Discretized x-coordinates (mm) where coordinates of unsupported (no foundation) segments are NaNs.
z (ndarray) – Solution vectors at positions x as columns of matrix z.
loc ({'top', 'mid', 'bot'}) – Get displacements of top, midplane or bottom of slab. Default is mid.
unit ({'m', 'cm', 'mm', 'um'}, optional) – Displacement output unit. Default is mm.

Returns:

x (ndarray) – Horizontal coordinates (cm).
ndarray – Horizontal displacements (unit input).

get_slab_rotation(x, z, unit='degrees')[source]¶

Compute slab cross-section rotation angle.

Parameters:

x (ndarray) – Discretized x-coordinates (mm) where coordinates of unsupported (no foundation) segments are NaNs.
z (ndarray) – Solution vectors at positions x as columns of matrix z. Default is mid.
unit ({'deg', degrees', 'rad', 'radians'}, optional) – Rotation angle output unit. Default is degrees.

Returns:

x (ndarray) – Horizontal coordinates (cm).
ndarray – Cross section rotations (unit input).

get_weaklayer_normalstress(x, z, unit='MPa', removeNaNs=False)[source]¶

Compute weak-layer normal stress.

Parameters:

x (ndarray) – Discretized x-coordinates (mm) where coordinates of unsupported (no foundation) segments are NaNs.
z (ndarray) – Solution vectors at positions x as columns of matrix z.
unit ({'MPa', 'kPa'}, optional) – Stress output unit. Default is MPa.
keepNaNs (bool) – If set, do not remove

Returns:

x (ndarray) – Horizontal coordinates (cm).
sig (ndarray) – Normal stress (stress unit input).

get_weaklayer_shearstress(x, z, unit='MPa', removeNaNs=False)[source]¶

Compute weak-layer shear stress.

Parameters:

x (ndarray) – Discretized x-coordinates (mm) where coordinates of unsupported (no foundation) segments are NaNs.
z (ndarray) – Solution vectors at positions x as columns of matrix z.
unit ({'MPa', 'kPa'}, optional) – Stress output unit. Default is MPa.
keepNaNs (bool) – If set, do not remove

Returns:

x (ndarray) – Horizontal coordinates (cm).
sig (ndarray) – Normal stress (stress unit input).

internal_potential(C, phi, L, **segments)[source]¶

Compute total internal potential (pst only).

Parameters:

C (ndarray) – Matrix(6xN) of solution constants for a system of N segements. Columns contain the 6 constants of each segement.
phi (float) – Inclination of the slab (°).
L (float, optional) – Total length of model (mm).
segments (dict) – Dictionary with lists of touchdown booleans (tdi), segement lengths (li), skier weights (mi), and foundation booleans in the cracked (ki) and uncracked (k0) configurations.

Returns:

Pi_int – Total internal potential (Nmm).

Return type:

float

total_potential(C, phi, L, **segments)[source]¶

Returns total differential potential

Parameters:

C (ndarray) – Matrix(6xN) of solution constants for a system of N segements. Columns contain the 6 constants of each segement.
phi (float) – Inclination of the slab (°).
L (float, optional) – Total length of model (mm).
segments (dict) – Dictionary with lists of touchdown booleans (tdi), segement lengths (li), skier weights (mi), and foundation booleans in the cracked (ki) and uncracked (k0) configurations.

Returns:

Pi – Total differential potential (Nmm).

Return type:

float

class weac.mixins.SlabContactMixin[source]¶

Bases: object

Mixin for handling the touchdown situation in a PST.

Provides Methods for the calculation of substitute spring stiffnesses, cracklength-tresholds and element lengths.

calc_a1()[source]¶

Calc transition lengths a1 (aAB).

Returns:: a1 – Length of the crack for transition of stage A to stage B (mm).
Return type:: float

calc_a2()[source]¶

Calc transition lengths a2 (aBC).

Returns:: a2 – Length of the crack for transition of stage B to stage C (mm).
Return type:: float

calc_lA()[source]¶: Calculate the length of the touchdown element in mode A.

calc_lB()[source]¶: Calculate the length of the touchdown element in mode B.

calc_lC()[source]¶: Calculate the length of the touchdown element in mode C.

calc_qn()[source]¶

Calc total surface normal load.

Returns:: Total surface normal load (N/mm).
Return type:: float

calc_qt()[source]¶

Calc total surface normal load.

Returns:: Total surface normal load (N/mm).
Return type:: float

calc_touchdown_length()[source]¶: Calculate touchdown length

calc_touchdown_mode()[source]¶: Calculate touchdown-mode from thresholds

calc_touchdown_system(L, a, cf, phi, ratio=1000)[source]¶: Calculate touchdown

set_columnlength(L)[source]¶

Set cracklength.

Parameters:: L (float) – Column length of a PST (mm).

set_cracklength(a)[source]¶

Set cracklength.

Parameters:: a (float) – Cracklength in a PST (mm).

set_phi(phi)[source]¶

Set inclination of the slab.

Parameters:: phi (float) – Inclination of the slab (°).

set_stiffness_ratio(ratio=1000)[source]¶

Set ratio between collapsed and uncollapsed weak-layer stiffness.

Parameters:: ratio (int, optional) – Stiffness ratio between collapsed and uncollapsed weak layer. Default is 1000.

set_tc(cf)[source]¶

Set height of the crack.

Parameters:: cf (float) – Collapse-factor. Ratio of the crack height to the uncollapsed weak-layer height.

set_touchdown_attributes(L, a, cf, phi, ratio)[source]¶: Set class attributes for touchdown consideration

substitute_stiffness(L, support='rested', dof='rot')[source]¶

Calc substitute stiffness for beam on elastic foundation.

Parameters:

L (float) – Total length of the PST-column (mm).
support (string) – Type of segment foundation. Defaults to ‘rested’.
dof (string) – Type of substitute spring, either ‘rot’ or ‘trans’. Defaults to ‘rot’.

Returns:

Return type:

stiffness of substitute spring.

class weac.mixins.SolutionMixin[source]¶

Bases: object

Mixin for the solution of boundary value problems.

Provides methods for the assembly of the system of equations and for the computation of the free constants.

assemble_and_solve(phi, li, mi, ki)[source]¶

Compute free constants for arbitrary beam assembly.

Assemble LHS from supported and unsupported segments in the form [ ] [ zh1 0 0 … 0 0 0 ][ ] [ ] [ ] left [ ] [ zh1 zh2 0 … 0 0 0 ][ ] [ ] [ ] mid [ ] [ 0 zh2 zh3 … 0 0 0 ][ ] [ ] [ ] mid [z0] = [ … … … … … … … ][ C ] + [ zp ] = [ rhs ] mid [ ] [ 0 0 0 … zhL zhM 0 ][ ] [ ] [ ] mid [ ] [ 0 0 0 … 0 zhM zhN ][ ] [ ] [ ] mid [ ] [ 0 0 0 … 0 0 zhN ][ ] [ ] [ ] right and solve for constants C.

Parameters:

phi (float) – Inclination (degrees).
li (ndarray) – List of lengths of segements (mm).
mi (ndarray) – List of skier weigths (kg) at segement boundaries.
ki (ndarray) – List of one bool per segement indicating whether segement has foundation (True) or not (False).

Returns:

C – Matrix(6xN) of solution constants for a system of N segements. Columns contain the 6 constants of each segement.

Return type:

ndarray

bc(z, k=False, pos='mid')[source]¶

Provide equations for free (pst) or infinite (skiers) ends.

Parameters:

z (ndarray) – Solution vector (6x1) at a certain position x.
l (float, optional) – Length of the segment in consideration. Default is zero.
k (boolean) – Indicates whether segment has foundation(True) or not (False). Default is False.
pos ({'left', 'mid', 'right', 'l', 'm', 'r'}, optional) – Determines whether the segement under consideration is a left boundary segement (left, l), one of the center segement (mid, m), or a right boundary segement (right, r). Default is ‘mid’.

Returns:

bc – Boundary condition vector (lenght 3) at position x.

Return type:

ndarray

Assemble lists defining the segments.

This includes length (li), foundation (ki, k0), and skier weight (mi).

Parameters:

li (squence, optional) – List of lengths of segements(mm). Used for system ‘skiers’.
mi (squence, optional) – List of skier weigths (kg) at segement boundaries. Used for system ‘skiers’.
ki (squence, optional) – List of one bool per segement indicating whether segement has foundation (True) or not (False) in the cracked state. Used for system ‘skiers’.
k0 (squence, optional) – List of one bool per segement indicating whether segement has foundation(True) or not (False) in the uncracked state. Used for system ‘skiers’.
L (float, optional) – Total length of model (mm). Used for systems ‘pst-’, ‘-pst’, ‘vpst-’, ‘-vpst’, and ‘skier’.
a (float, optional) – Crack length (mm). Used for systems ‘pst-’, ‘-pst’, ‘pst-‘, ‘-pst’, and ‘skier’.
phi (float, optional) – Inclination (degree).
m (float, optional) – Weight of skier (kg) in the axial center of the model. Used for system ‘skier’.
cf (float, optional) – Collapse factor. Ratio of the crack height to the uncollapsed weak-layer height. Used for systems ‘pst-’, ‘-pst’. Default is 0.5.
ratio (float, optional) – Stiffness ratio between collapsed and uncollapsed weak layer. Default is 1000.

Returns:

segments – Dictionary with lists of touchdown booleans (tdi), segement lengths (li), skier weights (mi), and foundation booleans in the cracked (ki) and uncracked (k0) configurations.

Return type:

dict

eqs(zl, zr, k=False, pos='mid')[source]¶

Provide boundary or transmission conditions for beam segments.

Parameters:

zl (ndarray) – Solution vector (6x1) at left end of beam segement.
zr (ndarray) – Solution vector (6x1) at right end of beam segement.
k (boolean) – Indicates whether segment has foundation(True) or not (False). Default is False.
pos ({'left', 'mid', 'right', 'l', 'm', 'r'}, optional) – Determines whether the segement under consideration is a left boundary segement (left, l), one of the center segement (mid, m), or a right boundary segement (right, r). Default is ‘mid’.

Returns:

eqs – Vector (of length 9) of boundary conditions (3) and transmission conditions (6) for boundary segements or vector of transmission conditions (of length 6+6) for center segments.

Return type:

ndarray