/// <summary>
///Calculates the element stiffness matrix used for warping analysis
///and the torsion load vector.
/// </summary>
/// <param name="mat"></param>
/// <param name="coords"></param>
/// <returns>Element stiffness matrix *(k_el)* and element torsion load vector* (f_el) *</returns>
internal void torsion_properties(SectionMaterial mat, ExtendedTri tri, out double[,] k_el, out double[] f_el)
{
//# initialise stiffness matrix and load vector
k_el = new double[6, 6];
f_el = new double[6];
//# Gauss points for 6 point Gaussian integration
var gps = ShapeFunctionHelper.gauss_points(6);
var Nxy = new double[2];
for (int i = 0; i < gps.RowCount(); i++)
{
var gp = gps.Row(i);
var B = tri.ShapeInfo[i].B;
double[] N = tri.ShapeInfo[i].N;
double j = tri.ShapeInfo[i].j;
//# determine x and y position at Gauss point
var Nx = N.Dot(tri.coords.Row(0));
var Ny = N.Dot(tri.coords.Row(1));
var B_Transpose = B.Transpose();
//# calculated modulus weighted stiffness matrix and load vector
k_el.Append(B_Transpose.Dot(B).Dot(gp[0] * j * mat.elastic_modulus));
Nxy[0] = Ny;
Nxy[1] = -Nx;
f_el.Append(B_Transpose.Dot(Nxy).Dot(gp[0] * j * mat.elastic_modulus));
}
}
shear_warping_integrals(SectionMaterial mat, ExtendedTri tri, double ixx, double iyy, double ixy, double[] omega)
{
//# initialise integrals
var sc_xint = 0.0;
var sc_yint = 0.0;
var q_omega = 0.0;
var i_omega = 0.0;
var i_xomega = 0.0;
var i_yomega = 0.0;
var gps = ShapeFunctionHelper.gauss_points(6);
for (int i = 0; i < gps.RowCount(); i++)
{
var gp = gps.Row(i);
double[] N = tri.ShapeInfo[i].N;
double j = tri.ShapeInfo[i].j;
//# determine x and y position at Gauss point
var Nx = N.Dot(tri.coords.Row(0));
var Ny = N.Dot(tri.coords.Row(1));
var Nomega = N.Dot(omega);
sc_xint += gp[0] * (iyy * Nx + ixy * Ny) * (Nx * Nx + Ny * Ny) * j * mat.elastic_modulus;
sc_yint += gp[0] * (ixx * Ny + ixy * Nx) * (Nx * Nx + Ny * Ny) * j * mat.elastic_modulus;
q_omega += gp[0] * Nomega * j * mat.elastic_modulus;
i_omega += gp[0] * Nomega * Nomega * j * mat.elastic_modulus;
i_xomega += gp[0] * Nx * Nomega * j * mat.elastic_modulus;
i_yomega += gp[0] * Ny * Nomega * j * mat.elastic_modulus;
}
return(sc_xint, sc_yint, q_omega, i_omega, i_xomega, i_yomega);
}
/// <summary>
/// Calculates the integrals used to evaluate the monosymmetry constant about both global axes and both prinicipal axes.
/// </summary>
/// <param name="mat"></param>
/// <param name="coords"></param>
/// <param name="phi">Principal bending axis angle</param>
/// <returns></returns>
internal (double int_x, double int_y, double int_11, double int_22) monosymmetry_integrals(SectionMaterial mat, ExtendedTri tri, double phi)
{
//# initialise integrals
var int_x = 0.0;
var int_y = 0.0;
var int_11 = 0.0;
var int_22 = 0.0;
var gps = ShapeFunctionHelper.gauss_points(6);
for (int i = 0; i < gps.RowCount(); i++)
{
var gp = gps.Row(i);
double[] N = tri.ShapeInfo[i].N;
double j = tri.ShapeInfo[i].j;
//# determine x and y position at Gauss point
var Nx = N.Dot(tri.coords.Row(0));
var Ny = N.Dot(tri.coords.Row(1));
//# determine 11 and 22 position at Gauss point
(var Nx_11, var Ny_22) = principal_coordinate(phi, Nx, Ny);
//# weight the monosymmetry integrals by the section elastic modulus
int_x += (gp[0] * (Nx * Nx * Ny + Ny * Ny * Ny) * j * mat.elastic_modulus);
int_y += (gp[0] * (Ny * Ny * Nx + Nx * Nx * Nx) * j * mat.elastic_modulus);
int_11 += (gp[0] * (Nx_11 * Nx_11 * Ny_22 + Ny_22 * Ny_22 * Ny_22) * j * mat.elastic_modulus);
int_22 += (gp[0] * (Ny_22 * Ny_22 * Nx_11 + Nx_11 * Nx_11 * Nx_11) * j * mat.elastic_modulus);
}
return(int_x, int_y, int_11, int_22);
}
/// <summary>
/// Calculates the variables used to determine the shear deformation coefficients.
/// </summary>
/// <param name="mat"></param>
/// <param name="coords"></param>
/// <param name="ixx">Second moment of area about the centroidal x-axis</param>
/// <param name="iyy">Second moment of area about the centroidal y-axis</param>
/// <param name="ixy">Second moment of area about the centroidal xy-axis</param>
/// <param name="psi_shear">Values of the psi shear function at the element nodes</param>
/// <param name="phi_shear">Values of the phi shear function at the element nodes</param>
/// <param name="nu">Effective Poisson's ratio for the cross-section</param>
/// <returns></returns>
internal (double kappa_x, double kappa_y, double kappa_xy) shear_coefficients(SectionMaterial mat,
ExtendedTri tri, double ixx, double iyy, double ixy, double[] psi_shear, double[] phi_shear, double nu)
{
//# initialise integrals
var kappa_x = 0.0;
var kappa_y = 0.0;
var kappa_xy = 0.0;
var gps = ShapeFunctionHelper.gauss_points(6);
var d = new double[2];
var h = new double[2];
//var psi_shear = new Vector(psi_shear2);
//var phi_shear = new Vector(phi_shear2);
for (int i = 0; i < gps.RowCount(); i++)
{
var gp = gps.Row(i);
var B = tri.ShapeInfo[i].B;
double[] N = tri.ShapeInfo[i].N;
double j = tri.ShapeInfo[i].j;
//# determine x and y position at Gauss point
var Nx = N.Dot(tri.coords.Row(0));
var Ny = N.Dot(tri.coords.Row(1));
//# determine shear parameters
var r = Nx * Nx - Ny * Ny;
var q = 2 * Nx * Ny;
var d1 = ixx * r - ixy * q;
var d2 = ixy * r + ixx * q;
var h1 = -ixy * r + iyy * q;
var h2 = -iyy * r - ixy * q;
d[0] = d1;
d[1] = d2;
h[0] = h1;
h[1] = h2;
var B_Transpose = B.Transpose();
var psi_shearXB_Transpose = psi_shear.Dot(B_Transpose).Subtract(d.Dot(nu / 2));
var BXphi_shear = B.Dot(phi_shear).Subtract(h.Dot(nu / 2));
kappa_x += psi_shearXB_Transpose.Dot(B.Dot(psi_shear).Subtract(d.Dot(nu / 2))) * (gp[0] * j * mat.elastic_modulus);
kappa_y += phi_shear.Dot(B_Transpose).Subtract(h.Dot(nu / 2)).Dot(BXphi_shear) * gp[0] * j * mat.elastic_modulus;
kappa_xy += psi_shearXB_Transpose.Dot(BXphi_shear) * (gp[0] * j * mat.elastic_modulus);
}
return(kappa_x, kappa_y, kappa_xy);
}
/// <summary>
/// Calculates the element shear load vectors used to evaluate the shear functions.
/// </summary>
/// <param name="mat"></param>
/// <param name="coords"></param>
/// <param name="ixx">Second moment of area about the centroidal x-axis</param>
/// <param name="iyy">Second moment of area about the centroidal y-axis</param>
/// <param name="ixy">Second moment of area about the centroidal xy-axis</param>
/// <param name="nu">Effective Poisson's ratio for the cross-section</param>
/// <returns>Element shear load vector psi *(f_psi)* and phi *(f_phi)*</returns>
internal (double[] f_psi, double[] f_phi) shear_load_vectors(SectionMaterial mat,
ExtendedTri tri, double ixx, double iyy, double ixy, double nu)
{
//# initialise stiffness matrix and load vector
var f_psi = new double[6];
var f_phi = new double[6];
//# Gauss points for 6 point Gaussian integration
var gps = ShapeFunctionHelper.gauss_points(6);
var d = new double[2];
var h = new double[2];
for (int i = 0; i < gps.RowCount(); i++)
{
var gp = gps.Row(i);
// shape_function(coords, gp, out N, ref B, out j);
var B = tri.ShapeInfo[i].B;
double[] N = tri.ShapeInfo[i].N;
double j = tri.ShapeInfo[i].j;
//# determine x and y position at Gauss point
var Nx = N.Dot(tri.coords.Row(0));
var Ny = N.Dot(tri.coords.Row(1));
//# determine shear parameters
var r = Nx * Nx - Ny * Ny;
var q = 2 * Nx * Ny;
var d1 = ixx * r - ixy * q;
var d2 = ixy * r + ixx * q;
var h1 = -ixy * r + iyy * q;
var h2 = -iyy * r - ixy * q;
//d[0, 0] = d1;
//d[1, 0] = d2;
//h[0, 0] = h1;
//h[1, 0] = h2;
//var B_Transpose = B.Transpose;
//Vector tmp = new Vector(N);
//tmp *= 2 * (1 + nu);
//f_psi += gp[0] * (nu / 2 * (B_Transpose * d).Transpose.Row(0) +
// tmp * (ixx * Nx - ixy * Ny)) * j * mat.elastic_modulus;
//f_phi += gp[0] * (nu / 2 * (B_Transpose * h).Transpose.Row(0) +
// tmp * (iyy * Ny - ixy * Nx)) * j * mat.elastic_modulus;
d[0] = d1;
d[1] = d2;
h[0] = h1;
h[1] = h2;
var B_Transpose = B.Transpose();
var NN = N.Dot(2 * (1 + nu));
f_psi.Append(B_Transpose.Dot(d).Dot(nu / 2).Append(NN.Dot(ixx * Nx - ixy * Ny)).Dot(gp[0] * j * mat.elastic_modulus));
f_phi.Append(B_Transpose.Dot(h).Dot(nu / 2).Append(NN.Dot(iyy * Ny - ixy * Nx)).Dot(gp[0] * j * mat.elastic_modulus));
}
return(f_psi, f_phi);
}