public StrainTensor GetBendingInternalStrain(double localX, double localY, LoadCombination cmb)
{
//step 1 : get transformation matrix
//step 2 : convert globals points to locals
//step 3 : convert global displacements to locals
//step 4 : calculate B matrix
//step 5 : e=B*U
//Note : Steps changed...
var trans = this.GetTransformationMatrix();
var lp = GetLocalPoints();
var g2l = new Func <Vector, Vector>(glob => (trans.Transpose() * glob.ToMatrix()).ToVector());
//var l2g = new Func<Vector, Vector>(local => (trans*local.ToMatrix()).ToPoint());
var d1g = this.nodes[0].GetNodalDisplacement(cmb);
var d2g = this.nodes[1].GetNodalDisplacement(cmb);
var d3g = this.nodes[2].GetNodalDisplacement(cmb);
//step 3
var d1l = new Displacement(g2l(d1g.Displacements), g2l(d1g.Rotations));
var d2l = new Displacement(g2l(d2g.Displacements), g2l(d2g.Rotations));
var d3l = new Displacement(g2l(d3g.Displacements), g2l(d3g.Rotations));
var uDkt =
new Matrix(new[]
{ d1l.DZ, d1l.RX, d1l.RY, /**/ d2l.DZ, d2l.RX, d2l.RY, /**/ d3l.DZ, d3l.RX, d3l.RY });
var b = DktElement.GetBMatrix(localX, localY,
lp.Select(i => i.X).ToArray(),
lp.Select(i => i.Y).ToArray());
var mDkt = b * uDkt;
var buf = new StrainTensor();
buf.S11 = mDkt[0, 0];
buf.S22 = mDkt[1, 0];
buf.S12 = mDkt[2, 0];
return(buf);
}
/// <inheritdoc />
public override Matrix GetGlobalStifnessMatrix()
{
//step 1 : get points in local system
//step 2 : get local stiffness matrix
//step 3 : expand local stiffness matrix
//step 4 : get global stiffness matrix
//step 1
var ls = GetLocalPoints();
var xs = new double[] { ls[0].X, ls[1].X, ls[2].X };
var ys = new double[] { ls[0].Y, ls[1].Y, ls[2].Y };
//step 2
var kl = DktElement.GetStiffnessMatrix(xs, ys, this._thickness, this.ElasticModulus, this.PoissonRatio);
//step 3
var currentOrder = new FluentElementPermuteManager.ElementLocalDof[]
{
new FluentElementPermuteManager.ElementLocalDof(0, DoF.Dz),
new FluentElementPermuteManager.ElementLocalDof(0, DoF.Rx),
new FluentElementPermuteManager.ElementLocalDof(0, DoF.Ry),
new FluentElementPermuteManager.ElementLocalDof(1, DoF.Dz),
new FluentElementPermuteManager.ElementLocalDof(1, DoF.Rx),
new FluentElementPermuteManager.ElementLocalDof(1, DoF.Ry),
new FluentElementPermuteManager.ElementLocalDof(2, DoF.Dz),
new FluentElementPermuteManager.ElementLocalDof(2, DoF.Rx),
new FluentElementPermuteManager.ElementLocalDof(2, DoF.Ry),
};
var kle = FluentElementPermuteManager.FullyExpand(kl, currentOrder, 3);
var lambda = GetTransformationMatrix();
var t = Matrix.DiagonallyRepeat(lambda.Transpose(), 6); // eq. 5-16 page 78 (87 of file)
//step 4 : get global stiffness matrix
var buf = t.Transpose() * kle * t; //eq. 5-15 p77
return(buf);
}
public Matrix GetLocalPlateBendingStiffnessMatrix()
{
//dkt
//step 1 : get points in local system
//step 2 : get local stiffness matrix
//step 3 : expand local stiffness matrix
//step 4 : get global stiffness matrix
//step 1
var ls = GetLocalPoints();
var xs = new double[] { ls[0].X, ls[1].X, ls[2].X };
var ys = new double[] { ls[0].Y, ls[1].Y, ls[2].Y };
//step 2
var kl = DktElement.GetStiffnessMatrix(xs, ys, this._thickness, this.ElasticModulus, this.PoissonRatio);
//step 3
var currentOrder = new ElementPermuteHelper.ElementLocalDof[]
{
new ElementPermuteHelper.ElementLocalDof(0, DoF.Dz),
new ElementPermuteHelper.ElementLocalDof(0, DoF.Rx),
new ElementPermuteHelper.ElementLocalDof(0, DoF.Ry),
new ElementPermuteHelper.ElementLocalDof(1, DoF.Dz),
new ElementPermuteHelper.ElementLocalDof(1, DoF.Rx),
new ElementPermuteHelper.ElementLocalDof(1, DoF.Ry),
new ElementPermuteHelper.ElementLocalDof(2, DoF.Dz),
new ElementPermuteHelper.ElementLocalDof(2, DoF.Rx),
new ElementPermuteHelper.ElementLocalDof(2, DoF.Ry),
};
var kle = ElementPermuteHelper.FullyExpand(kl, currentOrder, 3);
return(kle);
}
/// <summary>
/// Gets the 3x3 transformation matrix which converts local and global coordinates to each other.
/// </summary>
/// <returns>Transformation matrix.</returns>
/// <remarks>
/// GC = T * LC
/// where
/// GC: Global Coordinates [X;Y;Z] (3x1) (same as lambda in "Development of Membrane, Plate and Flat Shell Elements in Java" thesis by Kaushalkumar Kansara available on the web).
/// LC: Local Coordinates [x;y;z] (3x1) (same as xyz in "Development of Membrane, Plate and Flat Shell Elements in Java" thesis by Kaushalkumar Kansara available on the web).
/// T: Transformation Matrix (from this method) (3x3) (same as XYZ in "Development of Membrane, Plate and Flat Shell Elements in Java" thesis by Kaushalkumar Kansara available on the web).
/// </remarks>
private Matrix GetTransformationMatrix()
{
return(DktElement.GetTransformationMatrix(nodes[0].Location, nodes[1].Location, nodes[2].Location));
}
private MembraneStressTensor GetMembraneInternalForce(LoadCombination combination)
{
//Note: membrane internal force is constant
//step 1 : get transformation matrix
//step 2 : convert globals points to locals
//step 3 : convert global displacements to locals
//step 4 : calculate B matrix and D matrix
//step 5 : M=D*B*U
//Note : Steps changed...
var trans = this.GetTransformationMatrix();
var lp = GetLocalPoints();
var g2l = new Func <Vector, Vector>(glob => (trans.Transpose() * glob.ToMatrix()).ToVector());
//var l2g = new Func<Vector, Vector>(local => (trans*local.ToMatrix()).ToPoint());
var d1g = this.nodes[0].GetNodalDisplacement(combination);
var d2g = this.nodes[1].GetNodalDisplacement(combination);
var d3g = this.nodes[2].GetNodalDisplacement(combination);
//step 3
var d1l = new Displacement(g2l(d1g.Displacements), g2l(d1g.Rotations));
var d2l = new Displacement(g2l(d2g.Displacements), g2l(d2g.Rotations));
var d3l = new Displacement(g2l(d3g.Displacements), g2l(d3g.Rotations));
var uCst =
new Matrix(new[]
{ d1l.DX, d1l.DY, d2l.DX, d2l.DY, /**/ d3l.DX, d3l.DY });
var dbCst = CstElement.GetDMatrix(_elasticModulus, _poissonRatio, _formulationType);
var bCst = CstElement.GetBMatrix(lp.Select(i => i.X).ToArray(),
lp.Select(i => i.Y).ToArray());
var sCst = dbCst * bCst * uCst;
var buf = new MembraneStressTensor();
buf.Sx = sCst[0, 0];
buf.Sy = sCst[1, 0];
buf.Txy = sCst[2, 0];
return(buf);
}
private PlateBendingStressTensor GetBendingInternalForce(double localX, double localY, LoadCombination cmb)
{
//step 1 : get transformation matrix
//step 2 : convert globals points to locals
//step 3 : convert global displacements to locals
//step 4 : calculate B matrix and D matrix
//step 5 : M=D*B*U
//Note : Steps changed...
var trans = this.GetTransformationMatrix();
var lp = GetLocalPoints();
var g2l = new Func <Vector, Vector>(glob => (trans.Transpose() * glob.ToMatrix()).ToVector());
//var l2g = new Func<Vector, Vector>(local => (trans*local.ToMatrix()).ToPoint());
var d1g = this.nodes[0].GetNodalDisplacement(cmb);
var d2g = this.nodes[1].GetNodalDisplacement(cmb);
var d3g = this.nodes[2].GetNodalDisplacement(cmb);
//step 3
var d1l = new Displacement(g2l(d1g.Displacements), g2l(d1g.Rotations));
var d2l = new Displacement(g2l(d2g.Displacements), g2l(d2g.Rotations));
var d3l = new Displacement(g2l(d3g.Displacements), g2l(d3g.Rotations));
var uDkt =
new Matrix(new[]
{ d1l.DZ, d1l.RX, d1l.RY, /**/ d2l.DZ, d2l.RX, d2l.RY, /**/ d3l.DZ, d3l.RX, d3l.RY });
var dbDkt = DktElement.GetDMatrix(this._thickness, this._elasticModulus, this._poissonRatio);
var b = DktElement.GetBMatrix(localX, localY,
lp.Select(i => i.X).ToArray(),
lp.Select(i => i.Y).ToArray());
var mDkt = dbDkt * b * uDkt; //eq. 32, batoz article
var buf = new PlateBendingStressTensor();
buf.Mx = mDkt[0, 0];
buf.My = mDkt[1, 0];
buf.Mxy = mDkt[2, 0];
return(buf);
}
