| Type: | Package |
| Title: | Finite Element Modeling for R |
| Version: | 0.0.2 |
| Author: | Henna D. Bhramdat |
| Maintainer: | Henna D. Bhramdat <bhramdath@ufl.edu> |
| Description: | Finite element modeling of beam structures and 2D geometries using constant strain triangles. Applies material properties and boundary conditions (load and constraint) to generate a finite element model. The model produces stress, strain, and nodal displacements; a heat map is available to demonstrate regions where output variables are high or low. Also provides options for creating a triangular mesh of 2D geometries. Package developed with reference to: Bathe, K. J. (1996). Finite Element Procedures.[ISBN 978-0-9790049-5-7] – Seshu, P. (2012). Textbook of Finite Element Analysis. [ISBN-978-81-203-2315-5] – Mustapha, K. B. (2018). Finite Element Computations in Mechanics with R. [ISBN 9781315144474]. |
| License: | GPL-2 | GPL-3 |
| Encoding: | UTF-8 |
| RoxygenNote: | 7.2.1 |
| Imports: | geometry, geosphere, ptinpoly, sp, MASS, graphics |
| Depends: | R (≥ 3.5.0) |
| LazyData: | true |
| NeedsCompilation: | no |
| Packaged: | 2023-01-10 22:19:46 UTC; Henna |
| Repository: | CRAN |
| Date/Publication: | 2023-01-10 22:33:10 UTC |
ApplyBC.2d
Description
Boundary constraint for element centroids based on coordinate points. For the x & y direction per centroid create matrix with boundary 1(unfixed) or 0(fixed).
Usage
ApplyBC.2d(meshP, BoundConx, BoundCony)
Arguments
meshP |
Matrix (2 x n) containing coordinate points of the mesh |
BoundConx |
Boundary constraint for nodes in the x-direction |
BoundCony |
Boundary constraint for nodes in the y-direction |
Value
A data frame with constraint parameters applied to each node in the x and y directions. Formatted for use in reduced element matrix.
NodeKnownL |
Constraint parameters |
Examples
data(triMesh)
meshP = triMesh$MeshPts$p
BoundConx = BoundCony = numeric(NROW(meshP))
BoundConx[1:NROW(meshP)] = BoundCony[1:NROW(meshP)] = 1
BoundConx[c(10, 11, 12)] = BoundCony[c(10, 11, 12)] = 0
bound = ApplyBC.2d(meshP, BoundConx, BoundCony)
AutoAdjust.2d
Description
Allows for automatic refinement of the triangular mesh generated based on given parameters. Will remove elements that are outside the margin of the geometry.
Usage
AutoAdjust.2d(meshP, meshT, edge, centroid, AspectR, AR)
Arguments
meshP |
Matrix (2 x n) containing coordinate points of the mesh nodes. |
meshT |
Matrix (3 x n) containing the number of the coordinate point that forms a given triangle within the mesh. |
edge |
Coordinate points of the initial geometry. |
centroid |
Matrix (2 x n) of triangle elements. |
AspectR |
Aspect ratio of each triangle element. |
AR |
maximum desired aspect ratio, numeric value. |
Value
Generates new mesh and centroid tables
Meshpts |
Includes both new mesh coordinate points and triangulation of points. |
Centroids |
Centroid positions for each triangle element. |
Examples
data(triMesh)
data(polyshape)
data(dime)
meshP = triMesh$MeshPts$p
meshT = triMesh$MeshPts$T
edge = polyshape$Line
centroid = triMesh$Centroids
AspectR = dime$AspectRatio
AR = 10
auto = AutoAdjust.2d(meshP, meshT, edge, centroid, AspectR, AR)
Sample geometry. Matrix with x and y coordinates for initial shape.
Description
Sample geometry. Matrix with x and y coordinates for initial shape.
Usage
Cart
Format
An object of class matrix (inherits from array) with 11 rows and 2 columns.
Dimensions.2d
Description
Calculates dimensional values for each triangular element, including truss length & angles, distance from nodal point to centroid, aspect ratio of each triangle element, and area of the triangle.
Usage
Dimensions.2d(meshP, meshT, centroid)
Arguments
meshP |
Matrix (2 x n) containing coordinate points of the mesh nodes. |
meshT |
Matrix (3 x n) containing the number of the coordinate point that forms a given triangle within the mesh. |
centroid |
Matrix (2 x n) containing coordinate points of the centroid of each triangular element. |
Value
Evaluation of triangle elements truss, angle, and area.
Truss |
Nodal pairs that form each truss. |
TrussLength |
Distance between each paired nodes forming a truss, its length. |
Dist2Cent |
Shortest distance from truss to triangle centroid. |
Truss angle |
Angles of the triangle created from truss meeting. |
AspectRatio |
Aspect ratio of triangle elements. |
Area |
Area within triangle elements. |
Examples
data(triMesh)
data(polyshape)
meshP = triMesh$MeshPts$p
meshT = triMesh$MeshPts$T
centroid = triMesh$Centroids
dime = Dimensions.2d(meshP, meshT, centroid)
ElementMat.2d
Description
Generates an element stiffness matrix
Usage
ElementMat.2d(meshP, meshT, Nu, Y, Thick)
Arguments
meshP |
Matrix (2 x n) containing coordinate points of the mesh nodes. |
meshT |
Matrix (3 x n) containing the number of the coordinate point that forms a given triangle within the mesh. |
Nu |
Value of Poisson's ratio for each element |
Y |
Value of Young's (Elastic) modulus for each element |
Thick |
Value of the thickness of the mesh, a positive value must be given. |
Value
Generates initial element matrix needed for the finite element model.
EMPStress |
An element matrix of the geometry under stress. |
EMPStrain |
An element matrix of the geometry under strain. |
Examples
data(triMesh)
meshP = triMesh$MeshPts$p
meshT = triMesh$MeshPts$T
Y = matrix(20e9, nrow = NROW(meshT))
Nu = matrix(0.45, nrow = NROW(meshT))
Thick = 0.001
DOF = 6
fea_EM = ElementMat.2d(meshP, meshT, Nu, Y, Thick)
EulerBeamFEA
Description
Calculates stress in beam structures using the Euler-Bernoulli beam theory.
Usage
EulerBeamFEA(Y, beamP, beamT, fx, fy, BCtran, BCrot, Length, MoI, RotAng)
Arguments
Y |
Elastic modulus value for material (Pa). |
beamP |
Matrix (2 x n) of beam coordinates. |
beamT |
Matrix (2 x n) containing the number of the coordinate point as shown in beamP that connect to form a given beam (Discretization table). |
fx |
Load value (newtons) in the x direction. |
fy |
Load value (newtons) in the y direction. |
BCtran |
Boundary constraint for nodes to translate in x or y directions. Input as a non-matrix column. |
BCrot |
Boundary constraint for nodes to rotate. Input as a non-matrix column. |
Length |
Length of beam. |
MoI |
Moment of inertia for each beam segment. |
RotAng |
Angle of rotation |
Value
Calculates local forces and stresses, as well as bending stress for beams following the Euler-Bernoulli beam theory.
Stress |
Local stress at node |
LocalLoad |
Local load at node |
BendingStress |
Bending Stress |
Examples
data(beamGeo)
data(beamDime)
Length = beamDime$Length
MoI = beamDime$MomentofInertia
RotAng = beamDime$Angle
beamFEA = EulerBeamFEA(beamGeo$Y, beamGeo$beamP, beamGeo$beamT, beamGeo$fx, beamGeo$fy,
beamGeo$BCtran, beamGeo$BCrot, Length, MoI, RotAng)
ExpandEM.2d
Description
Generates the expanded element matrix, which represents the contribution of individual finite elements towards the global structural matrix
Usage
ExpandEM.2d(meshP, meshT, centroid, EMatrixlist)
Arguments
meshP |
Matrix (2 x n) containing coordinate points of the mesh nodes. |
meshT |
Matrix (3 x n) containing the number of the coordinate point that forms a given triangle within the mesh. |
centroid |
Matrix (2 x n) containing coordinate points of the centroid of each triangular element. |
EMatrixlist |
EMPStress or EMPStrain generated from ElementMat function. List of element matrices. |
Value
Produces large (n x n) matrix.
ExpandedMat |
The expanded element matrix |
Examples
data(triMesh)
data(fea_EM)
meshP = triMesh$MeshPts$p
meshT = triMesh$MeshPts$T
centroid = triMesh$Centroids
EMatrixlist = fea_EM$EMPStress
fea_ExEM = ExpandEM.2d(meshP, meshT, centroid, EMatrixlist)
ExpandSFT.2d
Description
Generates expanded surface force element matrix from SurfaceTraction function
Usage
ExpandSFT.2d(meshP, meshT, SurfTrac)
Arguments
meshP |
Matrix (2 x n) containing coordinate points of the mesh nodes. |
meshT |
Matrix (3 x n) containing the number of the coordinate point that forms a given triangle within the mesh. |
SurfTrac |
List of surface forces. |
Value
Produces a large (n x n) element matrix of surface forces.
ExpandedSurf |
Expanded surface force element matrix. |
Examples
data(triMesh)
data(SurfTrac)
meshT = triMesh$MeshPts$T
meshP = triMesh$MeshPts$p
expSurf = ExpandSFT.2d(meshP, meshT, SurfTrac)
FEMStrain.2d
Description
Creates a complete finite element model using strain for a given 2D mesh under specified boundary conditions (constrain and load).
Usage
FEMStrain.2d(meshP, meshT, centroid, BoundConx, BoundCony, SFShear,
SFTensile, Length, area, Fx, Fy, Y, Nu, Thick)
Arguments
meshP |
Matrix (2 x n) containing coordinate points of the mesh nodes. |
meshT |
Matrix (3 x n) containing the number of the coordinate point that forms a given triangle within the mesh. |
centroid |
Matrix (2 x n) containing coordinate points of the centroid of each triangular element. |
BoundConx |
Boundary constraint for nodes in the x-direction |
BoundCony |
Boundary constraint for nodes in the y-direction |
SFShear |
Magnitude of positive shear traction; if there is no surface traction then SFShear = 0 |
SFTensile |
Magnitude of tensile surface traction; if there is no surface traction then SFTensile = 0 |
Length |
Truss length |
area |
Triangle element area |
Fx |
Load vector for the x-direction |
Fy |
Load vector for the y-direction |
Y |
Value of Young's (Elastic) modulus |
Nu |
Value of Poisson's ratio |
Thick |
Value of the thickness of the mesh, a value must be given. |
Value
Completes the FEM to generate values of stress and strain and nodal displacement.
NodeDisplacement |
Node displacement on each axis |
LocalStress |
Stress as calucated from stress, strain, and stress from strain. Three (3) [3 x n] matrices where [x, y, tau] |
Examples
data(triMesh)
data(dime)
meshP = triMesh$MeshPts$p
meshT = triMesh$MeshPts$T
centroid = triMesh$Centroids
Y = matrix(20e9, nrow = NROW(meshT))
Nu = matrix(0.45, nrow = NROW(meshT))
Thick = 0.001
DOF = 6
BoundConx = BoundCony = numeric(NROW(meshP))
BoundConx[1:NROW(meshP)] = BoundCony[1:NROW(meshP)] = 1
BoundConx[c(10, 11, 12)] = BoundCony[c(10, 11, 12)] = 0
SFShear = 0
SFTensile = 0
Length = dime$TrussLength
area = dime$Area
Fx = 10
Fy = 10
fea_strain = FEMStrain.2d(meshP, meshT, centroid, BoundConx, BoundCony, SFShear, SFTensile,
Length, area, Fx, Fy, Y, Nu, Thick)
FEMStress.2d
Description
Creates a complete finite element model using stress for a given 2D mesh under specified boundary conditions (constrain and load).
Usage
FEMStress.2d(meshP, meshT, centroid, BoundConx, BoundCony, SFShear,
SFTensile, Length, area, Fx, Fy, Y, Nu, Thick)
Arguments
meshP |
Matrix (2 x n) containing coordinate points of the mesh nodes. |
meshT |
Matrix (3 x n) containing the number of the coordinate point that forms a given triangle within the mesh. |
centroid |
Matrix (2 x n) containing coordinate points of the centroid of each triangular element. |
BoundConx |
Boundary constraint for nodes in the x-direction |
BoundCony |
Boundary constraint for nodes in the y-direction |
SFShear |
Magnitude of positive shear traction; if there is no surface traction then SFShear = 0 |
SFTensile |
Magnitude of tensile surface traction; if there is no surface traction then SFTensile = 0 |
Length |
Truss length |
area |
Triangle element area |
Fx |
Load vector for the x-direction |
Fy |
Load vector for the y-direction |
Y |
Value of Young's (Elastic) modulus |
Nu |
Value of Poisson's ratio |
Thick |
Value of the thickness of the mesh, a value must be given. |
Value
Completes the FEM to generate values of stress and strain and nodal displacement.
NodeDisplacement |
Node displacement on each axis |
LocalStress |
Stress as calucated from stress, strain, and stress from strain. Three (3) [3 x n] matrices where [x, y, tau] |
Examples
data(triMesh)
data(dime)
meshP = triMesh$MeshPts$p
meshT = triMesh$MeshPts$T
centroid = triMesh$Centroids
Y = matrix(20e9, nrow = NROW(meshT))
Nu = matrix(0.45, nrow = NROW(meshT))
Thick = 0.001
DOF = 6
BoundConx = BoundCony = numeric(NROW(meshP))
BoundConx[1:NROW(meshP)] = BoundCony[1:NROW(meshP)] = 1
BoundConx[c(10, 11, 12)] = BoundCony[c(10, 11, 12)] = 0
SFShear = 0
SFTensile = 0
Length = dime$TrussLength
area = dime$Area
Fx = 10
Fy = 10
fea_stress = FEMStress.2d(meshP, meshT, centroid, BoundConx, BoundCony, SFShear, SFTensile,
Length, area, Fx, Fy, Y, Nu, Thick)
ForceVector.2d
Description
Creates a matrix of loads in the x & y direction for each load unconstrained node.
Usage
ForceVector.2d(Fx, Fy, RSF, meshP, NodeKnownL)
Arguments
Fx |
Load vector for the x-direction |
Fy |
Load vector for the y-direction |
RSF |
If surface traction is present assign value as the ReducedSF matrix; if there is no surface traction set RSF = 0 |
meshP |
Matrix (2 x n) containing coordinate points of the mesh nodes. |
NodeKnownL |
data frame with constraint parameters applied to each node in the x and y directions. Formatted for use in reduced element matrix. Generated from ApplyBC function. |
Value
Produces a matrix with loading parameters for each node.
ReducedFV |
Reduced force vector matrix containing the model load parameters. |
Examples
data(triMesh)
data(reduc_SF)
data(bound)
meshP = triMesh$MeshPts$p
RSF = reduc_SF
Fx = 10
Fy = 10
NodeKnownL = bound
load = ForceVector.2d(Fx, Fy, RSF, meshP, NodeKnownL)
GLForces.2d
Description
Uses nodal displacements to determine global and local forces at each node
Usage
GLForces.2d(meshP, meshT, GMat, GlobalND, EMatrixlist)
Arguments
meshP |
Matrix (2 x n) containing coordinate points of the mesh nodes. |
meshT |
Matrix (3 x n) containing the number of the coordinate point that forms a given triangle within the mesh. |
GMat |
Global matrix |
GlobalND |
Global nodal displacement |
EMatrixlist |
Element matrix list |
Value
Matrices of global and local forces
GForce |
Large global force matrix. |
Lforce |
Large local force matrix. |
Examples
data(triMesh)
data(gloMat)
data(displacN)
data(fea_EM)
meshP = triMesh$MeshPts$p
meshT = triMesh$MeshPts$T
GMat = gloMat
GlobalND = displacN$GlobalND
EMatrixlist = fea_EM$EMPStress
glfor = GLForces.2d(meshP, meshT, GMat, GlobalND, EMatrixlist)
GlobalMat.2d
Description
Generates global stiffness matrix - once established, the expanded element matrix must be combined to create the global structural stiffness matrix by adding the expanded matrices.
Usage
GlobalMat.2d(meshP, meshT, ExEM)
Arguments
meshP |
Matrix (2 x n) containing coordinate points of the mesh nodes. |
meshT |
Matrix (3 x n) containing the number of the coordinate point that forms a given triangle within the mesh. |
ExEM |
Expanded element matrix |
Value
Produces large (n x n) global matrix
GlobalMat |
Global matrix |
Examples
data(triMesh)
data(fea_ExEM)
meshP = triMesh$MeshPts$p
meshT = triMesh$MeshPts$T
ExEM = fea_ExEM
gloMat = GlobalMat.2d(meshP, meshT, ExEM)
LocalStress.2d
Description
Calculates local stress and strain for triangular elements of the mesh
Usage
LocalStress.2d(meshP, meshT, Y, Nu, GlobalND)
Arguments
meshP |
Matrix (2 x n) containing coordinate points of the mesh nodes. |
meshT |
Matrix (3 x n) containing the number of the coordinate point that forms a given triangle within the mesh. |
Y |
Value of Young's (Elastic) modulus |
Nu |
Value of Poisson's ratio |
GlobalND |
Global nodal displacement, return from function NodeDis |
Value
Completes FEM by calculating values of stress and strain, produces three (3) [3 x n] matrix.
Strain |
Calculated strain. [x, y, tau] |
Stress |
Calculated stress in pascals. [x, y, tau] |
StressStrain |
Stress as calucated from strain. [x, y, tau] |
Examples
data(triMesh)
data(displacN)
meshP = triMesh$MeshPts$p
meshT = triMesh$MeshPts$T
Y = matrix(20e9, nrow = NROW(meshT))
Nu = matrix(0.45, nrow = NROW(meshT))
GlobalND = displacN$GlobalND
fea_result = LocalStress.2d(meshP, meshT, Y, Nu, GlobalND)
ManualAdjust.2d
Description
Allows for manual refinement of the triangular mesh generated based on given parameters. Will remove triangle elements that are identified in the input (loc).
Usage
ManualAdjust.2d(meshP, meshT, edge, centroid, loc)
Arguments
meshP |
Matrix (2 x n) containing coordinate points of the mesh nodes. |
meshT |
Matrix (3 x n) containing the number of the coordinate point that forms a given triangle within the mesh. |
edge |
Coordinate points of the initial geometry. |
centroid |
Matrix (2 x n) of triangle elements. |
loc |
String containing the number of the meshT matrix row of the triangle chosen to be removed. |
Value
Generates new mesh and centroid tables
Meshpts |
Includes both new mesh coordinate points and triangulation of points. |
Centroids |
Centroid positions for each triangle element. |
Examples
data(triMesh)
data(polyshape)
meshP = triMesh$MeshPts$p
meshT = triMesh$MeshPts$T
edge = polyshape$Line
centroid = triMesh$Centroids
loc = c(7, 35, 17)
ManualAdjust.2d(meshP, meshT, edge, centroid, loc)
NodeDis.2d
Description
Calculates global nodal displacements
Usage
NodeDis.2d(meshP, REM, ForceV, NodeKnownL)
Arguments
meshP |
Matrix (2 x n) containing coordinate points of the mesh nodes. |
REM |
Reduced element matrix, returned from function ReducedEM. |
ForceV |
Reduced force vector matrix containing the model load parameters. Returned from function ForceVector. |
NodeKnownL |
data frame with constraint parameters applied to each node in the x and y directions. Formatted for use in reduced element matrix. Generated from ApplyBC function. |
Value
Produces tables with new node coordinates that are produced by the geometry under an applied load.
NodeDis |
Nodal displacement |
GlobalND |
Nodal displacement in the global environment |
Examples
data(triMesh)
data(load)
data(reduc_EM)
data(bound)
meshP = triMesh$MeshPts$p
REM = reduc_EM
ForceV = load
NodeKnownL = bound
displacN = NodeDis.2d(meshP, REM, ForceV, NodeKnownL)
PlotSystem.2d
Description
Generates heat map for given stress or strain on the geometry. Threshold values for the color must be assigned.
Usage
PlotSystem.2d(meshP, meshT, PlotVal, a, b, c, d, e, f, g, h, i, j,
Oc, ac, bc, cc, dc, ec, fc, gc, hc, ic, jc)
Arguments
meshP |
Matrix (2 x n) containing coordinate points of the mesh nodes. |
meshT |
Matrix (3 x n) containing the number of the coordinate point that forms a given triangle within the mesh. |
PlotVal |
Value to be plotted, either stress or strain, return from function LocalStress function. |
a |
Threshold 1 |
b |
Threshold 2 |
c |
Threshold 3 |
d |
Threshold 4 |
e |
Threshold 5 |
f |
Threshold 6 |
g |
Threshold 7 |
h |
Threshold 8 |
i |
Threshold 9 |
j |
Threshold 10 |
Oc |
Color 0 |
ac |
Color 1 |
bc |
Color 2 |
cc |
Color 3 |
dc |
Color 4 |
ec |
Color 5 |
fc |
Color 6 |
gc |
Color 7 |
hc |
Color 8 |
ic |
Color 9 |
jc |
Color 10 |
Value
Plot of colored polygon with mesh colored based on the plot value
Examples
data(triMesh)
data(fea_result)
meshP = triMesh$MeshPts$p
meshT = triMesh$MeshPts$T
PlotVal = abs(fea_result$Stress[,1])
Oc = "slateblue"; ac = "steelblue2"; bc = "cyan2"; cc = "palegreen2";
dc = "darkolivegreen1"; ec = "lemonchiffon"; fc = "lightgoldenrod1"; gc = "gold";
hc= "lightsalmon"; ic= "tomato"; jc= "firebrick3"
a = 1e5; b = 5e5; c = 1e6; d = 5e6; e = 1e7; f = 5e7; g = 1e8; h = 5e8; i = 1e9; j =5e9
PlotSystem.2d(meshP, meshT, PlotVal, a, b, c, d, e, f, g, h, i, j,
Oc, ac, bc, cc, dc, ec, fc, gc, hc, ic, jc)
ReducedEM.2d
Description
Reduced stiffness matrix - use boundary condition to reduce matrix to smaller form by removing systems that are bound.
Usage
ReducedEM.2d(GMat, NodeKnownL)
Arguments
GMat |
Global stiffness matrix |
NodeKnownL |
data frame with constraint parameters applied to each node in the x and y directions. Formatted for use in reduced element matrix. Generated from ApplyBC function. |
Value
Produces a large matrix.
ReducedEM |
Reduced element matrix. |
Examples
data(gloMat)
data(bound)
GMat = gloMat
NodeKnownL = bound
reduc_EM = ReducedEM.2d(GMat, NodeKnownL)
ReducedSF.2d
Description
Reduced matrix of surface forces
Usage
ReducedSF.2d(meshP, ExSurf)
Arguments
meshP |
Matrix (2 x n) containing coordinate points of the mesh nodes. |
ExSurf |
Expanded surface matrix, output from ExpandSFT |
Value
Produces a large matrix.
RSF |
Produces a large, reduced surface force matrix |
Examples
data(triMesh)
data(expSurf)
meshP = triMesh$MeshPts$p
ExSurf = expSurf
reduc_SF = ReducedSF.2d(meshP, ExSurf)
SinglePoly.2d
Description
Generates a mesh for polygon with a single continuous geometry
Usage
SinglePoly.2d(x, y, ptDS, ptDL)
Arguments
x |
X-coordinates for geometry. |
y |
Y-coordinates for geometry. |
ptDS |
Density of points desired within the geometry. |
ptDL |
Density of points desired at the perimeter of the geometry. |
Value
Coordinate points of nodes distributed within and on the line of a given geometry.
AllCoords |
all coordinate points distributed across the geometry. |
Within |
all coordinate points within the geometry ONLY. |
Line |
all coordinate points that lay on the perimeter of the geometry ONLY. |
Examples
data(Cart)
x = Cart[,1]
y= Cart[,2]
ptDS = 30
ptDL = 20
polyshape = SinglePoly.2d(x, y, ptDS, ptDL)
List of element matrices with surface traction. Obtained from function: SurfaceTraction
Description
List of element matrices with surface traction. Obtained from function: SurfaceTraction
Usage
SurfTrac
Format
An object of class list of length 50.
SurfaceTraction.2d
Description
Element Surface Traction - generates the column matrix for uniformly distributed surface traction. If surface traction is not present, assign SFTensile and SFShear a value of 0.
Usage
SurfaceTraction.2d(meshP, SFTensile, SFShear, Length, Thick, area)
Arguments
meshP |
Matrix (2 x n) containing coordinate points of the mesh nodes. |
SFTensile |
Magnitude of tensile surface traction |
SFShear |
Magnitude of positive shear traction |
Length |
Truss length |
Thick |
Triangle element thickness |
area |
Triangle element area |
Value
List of element matrices containing surface forces.
SurfT |
List of surface forces for each element. |
Examples
data(triMesh)
data(dime)
meshP = triMesh$MeshPts$p
SFShear = 0
SFTensile = 0
Thick = 0.001
Length = dime$TrussLength
area = dime$Area
SurfTrac = SurfaceTraction.2d(meshP, SFTensile, SFShear, Length, Thick, area)
ThreshPts.2d
Description
Clean node distribution within or outside of geometry. Optional function for complex geometries.
Usage
ThreshPts.2d(coords, thresh, edge)
Arguments
coords |
Nodal coordinates |
thresh |
Threshold for point removal. Ranges include: 500000-50000000 |
edge |
Coordinate points of the initial geometry. |
Value
Coordinate points of valid nodes.
CleanedNodes |
Matrix of new nodes that abide by given threshold rules. |
NodeReport |
Report identifying with nodes were kept and which were removed. |
Examples
data(polyshape)
coords = polyshape$Within
thresh = 5000000
edge = polyshape$Line
cleanpoly = ThreshPts.2d(coords, thresh, edge)
beamApplyBC
Description
Boundary constraint for element centroids based on coordinate points. For the x & y direction per centroid create matrix with boundary 1(unfixed) or 0(fixed).
Usage
beamApplyBC(beamP, BCtran, BCrot)
Arguments
beamP |
Matrix (2 x n) of beam coordinates. |
BCtran |
Boundary constraint for nodes to translate in x or y directions. Input as a non-matrix column. |
BCrot |
Boundary constraint for nodes to rotate. Input as a non-matrix column. |
Value
A data frame with constraint parameters applied to each node for directional translation and rotation. Formatted for use in reduced element matrix.
NodeKnownL |
Matrix (1 x n) of constraint parameters |
Examples
data(beamGeo)
beamBC = beamApplyBC(beamGeo$beamP, beamGeo$BCtran, beamGeo$BCrot)
Boundary conditions applied to each node. Obtained from function: beamApplyBC
Description
Boundary conditions applied to each node. Obtained from function: beamApplyBC
Usage
beamBC
Format
An object of class matrix (inherits from array) with 8 rows and 1 columns.
Dimensional data for beam elements. Includes area, length, aspect ratio, angles and lengths of elements. Obtained from function: beamDimensions
Description
Dimensional data for beam elements. Includes area, length, aspect ratio, angles and lengths of elements. Obtained from function: beamDimensions
Usage
beamDime
Format
An object of class list of length 7.
beamDimensions
Description
Calculates input dimensions needed for beam finite element.
Usage
beamDimensions(Y, G, Nu, beamP, beamT, thick, fx, fy)
Arguments
Y |
Elastic modulus value for material (Pa). |
G |
Shear modulus value for material (Pa). If using Euler-Bernoulli model, G = 0. |
Nu |
Poisson's ratio value for material. |
beamP |
Matrix (2 x n) of beam coordinates. |
beamT |
Matrix (2 x n) containing the number of the coordinate point as shown in beamP that connect to form a given beam (Discretization table). |
thick |
Thickness of the beam |
fx |
Load value (newtons) in the x direction. |
fy |
Load value (newtons) in the y direction. |
Value
Calculates values needed for both Timoshenko-Ehrenfest and Euler-Bernoulli beam theories.
k |
Timoshenko-Ehrenfest correction |
Length |
Beam length |
Angle |
Beam angle within the plane |
MomentofInertia |
Moment of Inertia for each beam |
Displacement |
Displacement under Timoshenko-Ehernfest beam theory |
RotationAngle |
Angle of rotation |
StiffnessAngle |
Stiffness angle |
Examples
data(beamGeo)
DOF = 4
n = NROW(beamGeo$beamT)
thick = matrix(c(0.039149, 0.03, 0.0246625), ncol = 1, nrow = n) #height(thickness) of beam
beamDime = beamDimensions(beamGeo$Y, beamGeo$G, beamGeo$Nu, beamGeo$beamP, beamGeo$beamT,
beamGeo$thick, beamGeo$fx, beamGeo$fy)
beamElementMat
Description
Generates element stiffness matrix for beams.
Usage
beamElementMat(beamP, beamT, Y, Length, MoI)
Arguments
beamP |
Matrix (2 x n) of beam coordinates. |
beamT |
Matrix (2 x n) containing the number of the coordinate point as shown in beamP that connect to form a given beam (Discretization table). |
Y |
Elastic modulus value for material. |
Length |
Length of beams. |
MoI |
Moment of inertia for each beam segment. |
Value
Generates initial element matrix needed for the finite element model.
beamEmat |
An element matrix of the beam |
Examples
data(beamGeo)
data(beamDime)
Length = beamDime$Length
MoI = beamDime$MomentofInertia
beamEmat = beamElementMat(beamGeo$beamP, beamGeo$beamT, beamGeo$Y, Length, MoI)
List of element matrices for each element. Obtained from function: beamElementMat
Description
List of element matrices for each element. Obtained from function: beamElementMat
Usage
beamEmat
Format
An object of class list of length 3.
List of element matrices for each element. Obtained from function: beamElementMat
Description
List of element matrices for each element. Obtained from function: beamElementMat
Usage
beamExMat
Format
An object of class list of length 3.
beamExpandEM
Description
Expanded element matrix for beam.
Usage
beamExpandEM(beamP, beamT, ElementMat)
Arguments
beamP |
Matrix (2 x n) of beam coordinates. |
beamT |
Matrix (2 x n) containing the number of the coordinate point as shown in beamP that connect to form a given beam (Discretization table). |
ElementMat |
Element stiffness matrix list. |
Value
produces large (n x n) element matrix from initial element matrix.
beamExMat |
The expanded element matrix |
Examples
data(beamGeo)
data(beamEmat)
ElementMat = beamEmat
beamExMat = beamExpandEM(beamGeo$beamP, beamGeo$beamT, ElementMat)
Load vector produced from function function: beamForceVector
Description
Load vector produced from function function: beamForceVector
Usage
beamFV
Format
An object of class matrix (inherits from array) with 5 rows and 1 columns.
beamForceVector
Description
Creates a matrix of loads for beams in the x & y direction for each load unconstrained node.
Usage
beamForceVector(beamP, fx, fy, NodeKnownL)
Arguments
beamP |
Matrix (2 x n) of beam coordinates. |
fx |
Load vector (newtons) in the x-direction. |
fy |
Load vector (newtons) in the y-direction. |
NodeKnownL |
Data frame with constraint parameters applied to each node in the x and y directions. Formatted for use in reduced element matrix. Generated from ApplyBC function. |
Value
Produces a matrix with loading parameters for each node.
ReducedFV |
Reduced force vector matrix containing the model load parameters. |
Examples
data(beamGeo)
data(beamUDL)
NodeKnownL = beamBC
FV = beamForceVector(beamGeo$beamP, beamGeo$fx, beamGeo$fy, NodeKnownL)
beamGLForces
Description
Uses nodal displacements to determine global and local forces at each node
Usage
beamGLForces(beamP, beamT, Y, MoI, Length, GMat, BUDL, BND)
Arguments
beamP |
Matrix (2 x n) of beam coordinates. |
beamT |
Matrix (2 x n) containing the number of the coordinate point as shown in beamP that connect to form a given beam (Discretization table). |
Y |
Elastic Modulus of material |
MoI |
Moment of Inertia |
Length |
Length of beam |
GMat |
Global stiffness matrix |
BUDL |
Column matrix for beam distributed load |
BND |
beam nodal displacement, output from function "beamNodeDis" |
Value
Matrices of global and local forces. (Results in kN)
GForce |
Large global force matrix. |
Lforce |
Large local force matrix. |
Examples
data(beamGeo)
data(beamDime)
data(beamsUDL)
data(beamND)
data(beamGloMat)
Length = beamDime$Length
MoI = beamDime$MomentofInertia
BUDL = beamsUDL
BND = beamND
GMat = beamGloMat
GLforce = beamGLForces(beamGeo$beamP, beamGeo$beamT, beamGeo$Y, MoI, Length, GMat, BUDL, BND)
Global and Local loading force matrices obtained from function: beamGLForces
Description
Global and Local loading force matrices obtained from function: beamGLForces
Usage
beamGLforce
Format
An object of class list of length 2.
Sample geometry for beam. Includes shape, discretization table, boundary conditions, thickness, and material details.
Description
Sample geometry for beam. Includes shape, discretization table, boundary conditions, thickness, and material details.
Usage
beamGeo
Format
An object of class list of length 12.
Global element matrix, obtained from function: beamGlobalEM
Description
Global element matrix, obtained from function: beamGlobalEM
Usage
beamGloMat
Format
An object of class matrix (inherits from array) with 8 rows and 8 columns.
beamGlobalEM
Description
Generates global stiffness matrix for beams.
Usage
beamGlobalEM(beamExEM)
Arguments
beamExEM |
Expanded Element Matrix |
Value
Produces large (n x n) global matrix
GlobalMat |
Global matrix |
Examples
data(beamExMat)
beamExEM = beamExMat
GMat = beamGlobalEM(beamExEM)
Global nodal displacement, obtained from function: beamNodeDis
Description
Global nodal displacement, obtained from function: beamNodeDis
Usage
beamND
Format
An object of class list of length 3.
beamNodeDis
Description
Calculates global nodal displacements of beam.
Usage
beamNodeDis(beamP, BCtran, BCrot, REM, NodeKnownL, ForceV)
Arguments
beamP |
Matrix (2 x n) of beam coordinates. |
BCtran |
Boundary constraint for nodes to translate in x or y directions. |
BCrot |
Boundary constraint for nodes to rotate. |
REM |
Reduced element matrix, returned from function ReducedEM. |
NodeKnownL |
data frame with constraint parameters applied to each node in the x and y directions. Formatted for use in reduced element matrix. Generated from ApplyBC function. |
ForceV |
Reduced force vector matrix containing the model load parameters. Returned from function ForceVector. |
Value
Produces tables with new node coordinates that are produced by the geometry under an applied load.
NodeDis |
Nodal displacement |
GlobalND |
Nodal displacement in the global environment |
GlobalNDMatrix |
Nodal displacement in the global environment as a reduced matrix |
Examples
data(beamGeo)
data(beamFV)
data(beamREM)
data(beamBC)
ForceV = beamFV
REM = beamREM
NodeKnownL = beamBC
beamND = beamNodeDis(beamGeo$beamP, beamGeo$BCtran, beamGeo$BCrot, REM, NodeKnownL, ForceV)
beamPlotSystem
Description
Generates heat map for given stress or strain on the beam geometry. Threshold values for the color must be assigned.
Usage
beamPlotSystem(beamP, beamT, PlotVal, a, b, c, d, e, f, g, h, i, j,
Oc, ac, bc, cc, dc, ec, fc, gc, hc, ic, jc, LWD)
Arguments
beamP |
Matrix (2 x n) of beam coordinates. |
beamT |
Matrix (2 x n) containing the number of the coordinate point as shown in beamP that connect to form a given beam (Discretization table). |
PlotVal |
Value to be plotted, either stress or strain, return from function beamLocalStress function. |
a |
Threshold 1 |
b |
Threshold 2 |
c |
Threshold 3 |
d |
Threshold 4 |
e |
Threshold 5 |
f |
Threshold 6 |
g |
Threshold 7 |
h |
Threshold 8 |
i |
Threshold 9 |
j |
Threshold 10 |
Oc |
Color for all zero values |
ac |
Color 1 |
bc |
Color 2 |
cc |
Color 3 |
dc |
Color 4 |
ec |
Color 5 |
fc |
Color 6 |
gc |
Color 7 |
hc |
Color 8 |
ic |
Color 9 |
jc |
Color 10 |
LWD |
Line (beam) width |
Value
Plot of colored beam based on the plot value
Examples
data(beamGeo)
data(beamStressResult)
PlotVal = beamStressResult
Oc = "slateblue"; ac = "steelblue2"; bc = "cyan2"; cc = "palegreen2";
dc = "darkolivegreen1"; ec = "lemonchiffon"; fc = "lightgoldenrod1";
gc = "gold"; hc= "lightsalmon"; ic= "tomato"; jc= "firebrick3"
a = 1e5; b = 5e5; c = 1e6; d = 5e6; e = 1e7; f = 5e7; g = 1e8; h = 5e8; i = 1e9; j =5e9
beamPlotSystem(beamGeo$beamP, beamGeo$beamT, PlotVal, a, b, c, d, e, f, g, h, i, j, Oc,
ac, bc, cc, dc, ec, fc, gc, hc, ic, jc, LWD = 4)
Reduced element matrix calculated from the expanded element matrix. Obtained from function: beamReducedEM
Description
Reduced element matrix calculated from the expanded element matrix. Obtained from function: beamReducedEM
Usage
beamREM
Format
An object of class matrix (inherits from array) with 5 rows and 5 columns.
beamReducedEM
Description
Reduced stiffness matrix - use boundary condition to reduce matrix to smaller form by removing systems that are bound.
Usage
beamReducedEM(GMat, NodeKnownL)
Arguments
GMat |
Global stiffness matrix |
NodeKnownL |
data frame with constraint parameters applied to each node in the x and y directions. Formatted for use in reduced element matrix. Generated from ApplyBC function. |
Value
Produces a large matrix.
ReducedEM |
Reduced element matrix. |
Examples
data(beamBC)
data(beamGloMat)
NodeKnownL = beamBC
GMat = beamGloMat
beamREM = beamReducedEM(GMat, NodeKnownL)
beamStress
Description
Calculates local stress and strain for beam elements
Usage
beamStress(beamP, beamT, Y, Length, MoI, RotAng, BND)
Arguments
beamP |
Matrix (2 x n) of beam coordinates. |
beamT |
Matrix (2 x n) containing the number of the coordinate point as shown in beamP that connect to form a given beam (Discretization table). |
Y |
Value of Young's (Elastic) modulus |
Length |
Length of beam |
MoI |
Moment of Inertia |
RotAng |
Angle of rotation |
BND |
Global nodal displacement matrix, return from function beamNodeDis |
Value
Completes FEM by calculating values of stress and strain, produces three (3) [3 x n] matrix.
BendingStress |
Bending Stress |
Examples
data(beamGeo)
data(beamGLforce)
Length = beamDime$Length
MoI = beamDime$MomentofInertia
RotAng = beamDime$Angle
BND = beamND
beamBendStress = beamStress(beamGeo$beamP, beamGeo$beamT, beamGeo$Y, Length, MoI, RotAng, BND)
FEA results for the beam model. Obtained from function: beamStress
Description
FEA results for the beam model. Obtained from function: beamStress
Usage
beamStressResult
Format
An object of class numeric of length 3.
beamUDL
Description
Uniformly distributes load over the length of the beam.
Usage
beamUDL(beamP, beamT, Length, fx, fy)
Arguments
beamP |
Matrix (2 x n) of beam coordinates. |
beamT |
Matrix (2 x n) containing the number of the coordinate point as shown in beamP that connect to form a given beam (Discretization table). |
Length |
Length of beam. |
fx |
Load value (newtons) in the x direction. |
fy |
Load value (newtons) in the y direction. |
Value
Produces matrix representing uniformly distributed load on beam
DLMatrix |
Column matrix for beam distributed load |
ExpandedDLMatrix |
Expanded beam distribution load |
ReductedDLMatrix |
Reduced beam distribution load |
Examples
data(beamGeo)
data(beamDime)
Length = beamDime$Length
beamUDL = beamUDL(beamGeo$beamP, beamGeo$beamT, Length, beamGeo$fx, beamGeo$fy)
Uniformly distributed load on beam surface
Description
Uniformly distributed load on beam surface
Usage
beamsUDL
Format
An object of class list of length 3.
Boundary conditions applied to each node. Obtained from function: ApplyBC
Description
Boundary conditions applied to each node. Obtained from function: ApplyBC
Usage
bound
Format
An object of class matrix (inherits from array) with 100 rows and 1 columns.
Cleaned nodal distribution in and on polygon. Obtained from function: Threshpts
Description
Cleaned nodal distribution in and on polygon. Obtained from function: Threshpts
Usage
cleanpoly
Format
An object of class list of length 2.
Dimensional data for mesh elements. Includes area, length, aspect ratio, angles and lengths of elements. Obtained from function: Dimensions
Description
Dimensional data for mesh elements. Includes area, length, aspect ratio, angles and lengths of elements. Obtained from function: Dimensions
Usage
dime
Format
An object of class list of length 6.
Global nodal displacement, obtained from function: NodeDis
Description
Global nodal displacement, obtained from function: NodeDis
Usage
displacN
Format
An object of class list of length 2.
Expanded element matrix for surface forces. Obtained from function: ExpandSFT
Description
Expanded element matrix for surface forces. Obtained from function: ExpandSFT
Usage
expSurf
Format
An object of class list of length 50.
List of element matrices for each element. Obtained from function: ElementMat
Description
List of element matrices for each element. Obtained from function: ElementMat
Usage
fea_EM
Format
An object of class list of length 2.
List of large expanded element matrices calculated from the element matrix. Obtained from function: ExpandEM
Description
List of large expanded element matrices calculated from the element matrix. Obtained from function: ExpandEM
Usage
fea_ExEM
Format
An object of class list of length 78.
FEA results. Produces list with results from local stresses including Stress, Strain, and Stress from Strain. Obtained from function: LocalStress
Description
FEA results. Produces list with results from local stresses including Stress, Strain, and Stress from Strain. Obtained from function: LocalStress
Usage
fea_result
Format
An object of class list of length 3.
Global and Local loading force matrices obtained from function: GLForces
Description
Global and Local loading force matrices obtained from function: GLForces
Usage
glfor
Format
An object of class list of length 2.
Global element matrix, obtained from function: GlobalMat
Description
Global element matrix, obtained from function: GlobalMat
Usage
gloMat
Format
An object of class matrix (inherits from array) with 100 rows and 100 columns.
Load vector produced from function function: ForceVector
Description
Load vector produced from function function: ForceVector
Usage
load
Format
An object of class matrix (inherits from array) with 94 rows and 1 columns.
Sample geometry converted into a 2D polygon. Polygon data that specifies all coordinate, coordinates that are within the geometry and coordinates that construct the lines of the geometry. Obtained from function: SinglePoly
Description
Sample geometry converted into a 2D polygon. Polygon data that specifies all coordinate, coordinates that are within the geometry and coordinates that construct the lines of the geometry. Obtained from function: SinglePoly
Usage
polyshape
Format
An object of class list of length 3.
Reduced element matrix calculated from the expanded element matrix. Obtained from function: ReducedEM
Description
Reduced element matrix calculated from the expanded element matrix. Obtained from function: ReducedEM
Usage
reduc_EM
Format
An object of class matrix (inherits from array) with 94 rows and 94 columns.
Reduced surface force matrix calculated from expanded element matrix. Obtained from function: ReducedSF
Description
Reduced surface force matrix calculated from expanded element matrix. Obtained from function: ReducedSF
Usage
reduc_SF
Format
An object of class matrix (inherits from array) with 100 rows and 1 columns.
Meshed coordinate points obtained from function: triangulate0
Description
Meshed coordinate points obtained from function: triangulate0
Usage
triMesh
Format
An object of class list of length 2.
triangulate0.2d
Description
Triangulation by Delaunayn algorithm. Automatically generates a triangular mesh for a geometry containing nodal points.
Usage
triangulate0.2d(u0, edge)
Arguments
u0 |
Matrix (2 x n) of node coordinates within the geometry. |
edge |
Matrix (2 x n) of coordinate points on the perimeter of the geometry. |
Value
Produces data for generated mesh.
Meshpts |
Includes both new mesh coordinate points and triangulation of points. |
Centroids |
Centroid positions for each triangle element. |
Examples
data(cleanpoly)
data(polyshape)
u0 = cleanpoly$CleanedNodes
edge = polyshape$Line
triMesh = triangulate0.2d(u0, edge)