The element and its degrees of freedom#

BFSCPlate2D is a four-node rectangle of sides lex along \(x\) and ley along \(y\), parallel to the axes, of the classical laminated plate theory: the deflection \(w\) and the in-plane displacements \(u\) and \(v\) are interpolated with products of cubic Hermite polynomials, whose nodal values are the degrees of freedom below. The kinematics, the interpolation functions and the integration of each matrix are given in the Notes of bfscplate2d.bfscplate2d.

Node ordering#

The nodes are numbered counterclockwise from the corner with the smallest \(x\) and \(y\), as the CQUAD4 element of Nastran:

Node

Natural coordinates

Position

1

\(\xi = -1, \eta = -1\)

\((x_1, y_1)\)

2

\(\xi = +1, \eta = -1\)

\((x_1 + lex, y_1)\)

3

\(\xi = +1, \eta = +1\)

\((x_1 + lex, y_1 + ley)\)

4

\(\xi = -1, \eta = +1\)

\((x_1, y_1 + ley)\)

with \(x = x_1 + lex (\xi + 1)/2\) and \(y = y_1 + ley (\eta + 1)/2\). The element knows neither \(x_1\) nor \(y_1\): its matrices depend on lex, ley, its properties and its degrees of freedom, not on its position.

Degrees of freedom of a node#

Each node has DOF = 10 degrees of freedom, at the positions c to c + 9 of the global vectors and matrices, with c the attribute c1, c2, c3 or c4 of the element for the nodes 1 to 4:

Position

Degree of freedom

Dimension

c + 0

\(u\)

length

c + 1

\(u_{,x}\)

dimensionless

c + 2

\(u_{,y}\)

dimensionless

c + 3

\(v\)

length

c + 4

\(v_{,x}\)

dimensionless

c + 5

\(v_{,y}\)

dimensionless

c + 6

\(w\)

length

c + 7

\(w_{,x}\)

dimensionless

c + 8

\(w_{,y}\)

dimensionless

c + 9

\(w_{,xy}\)

one over length

\(u\), \(v\) and \(w\) are the displacements of the reference surface along \(x\), \(y\) and \(z\). The cross derivative \(w_{,xy}\) is a degree of freedom of \(w\) only, as in the element of Bogner, Fox and Schmit; \(u\) and \(v\) have no cross derivative. With c = DOF*i for the node of index i, the usual choice of the tests, the slice u[6::DOF] of a global vector gives \(w\) at all the nodes, u[0::DOF] gives \(u\), and so on, which is how the boundary conditions and the loads are applied, see Assembling a model.

The 40 degrees of freedom of the element, e.g. its attribute u, are those of the nodes 1, 2, 3 and 4 in sequence, 10 per node in the order of the table.

Continuity between elements#

Along an edge, \(u\), \(v\), \(w\) and their first derivatives depend only on the degrees of freedom of the two nodes of that edge, which makes them continuous between neighbouring elements: the element is conforming for the classical plate theory. Along an edge \(x\) = constant, for instance, \(w\) is interpolated from \(w\) and \(w_{,y}\) of the two nodes, and the normal slope \(w_{,x}\) from \(w_{,x}\) and \(w_{,xy}\). A displacement vanishes along a whole edge only when both its value and its derivative along the edge are constrained at the nodes of the edge, see Assembling a model.

Properties of the element#

Before the functions of the module are called, the attributes below are set for each element, see BFSCPlate2D for their definitions:

  • c1, c2, c3, c4: positions of the nodes in the global vectors;

  • lex, ley: lengths of the sides;

  • ABD: the (6, 6) constitutive matrix of the laminate, e.g. the ABD attribute of composites.isotropic_plate() or composites.laminated_plate(), constant over the element;

  • h and rho: thickness and density, used only by the mass matrix of update_M();

  • init_k_KC0, init_k_KCNL, init_k_KG, init_k_M, init_k_KCfollower: positions of the element in the sparse arrays of each matrix, see Assembling a model.

n1, n2, n3 and n4 are identification numbers for the user, not read by the functions. The methods BFSCPlate2D.update_Su() to BFSCPlate2D.update_Sw_y(), BFSCPlate2D.update_Bm() and BFSCPlate2D.update_Bb() evaluate the interpolation functions and the strain operators at a point \((\xi, \eta)\), e.g. to integrate a load or to recover the strains; the update_* functions of the module do not use them.

Integration points#

All the functions except update_M() integrate over the element with nint by nint Gauss-Legendre points, the products of the one-dimensional points and weights of bfscplate2d.quadrature.get_points_weights(), given to each function. The number of points that integrates each matrix exactly is given in the Notes of bfscplate2d.bfscplate2d.