Types of Flat Finite Elements

SCAD includes various flat finite elements with a triangular and quadrangular shape. The number of nodes in some types of elements can be greater than the number of vertices. In this case additional nodes lie on one or several sides of the element and their numbers follow after the numbers of vertices in any order.

Elements for the Analysis of Thin Flexural Plates

These FEs are always located in the XOY plane and there are three degrees of freedom in each of their nodes: W — vertical displacement (deflection), and U_X, U_Y — rotation angles about the X and Y axes. The FEs can be used in the models with indices 3, 5, 8 and 9 and have an isotropic, orthotropic, or anisotropic material. Moments M_X, M_Y, M_XY and shear forces Q_X and Q_Y are calculated. When the subsoil parameter С₁ is specified, R_Z is calculated.

Types of elements for the analysis of thin plates (Kirchhoff-Love theory)

Type	Name	Number of nodes	Node numbering order and local axes	Comments
11	Rectangular	4	Fig. 1, a	semi-compatible [18, 22, 23]
12, 14	Triangular	3	Fig. 1, b	incompatible [18, 22, 23]
13	Rectangular	4	Fig. 1, a	incompatible [28]
15	Triangular	3-6	Fig. 1, d	SubAreas¹, option 1, [24, 23]
16	Quadrangular	4-8	Fig. 1, e	SubAreas, option 2 [25, 23]
18	Triangular	3-6	Fig. 1, d	SubAreas, option 2 [24, 23]
19	Quadrangular	4	Fig. 1, c	SubAreas, [25, 23]
20	Quadrangular	4-8	Fig. 1, e	SubAreas, [24, 23], option 1

¹ SubAreas – subdomain method: triangular and quadrangular elements are divided, respectively, by medians and diagonals into triangles. Approximation polynomials of the appropriate degree are used on each of these triangles in such a way that compatibility is ensured. We obtain piecewise polynomial approximations.

Bending of Plates according to the Reissner-Mindlin Theory

These elements are used for the analysis of medium thickness plates and implement the Reissner-Mindlin theory. They are completely similar to the elements used for the analysis of thin plates in terms of the specification of the initial data. They differ from the elements given in Table 1 only in their type number, which is greater by 100. For example, element 120 is a quadrangular element with the number of nodes from 4 to 8, similarly to the element 20.

Each node of the elements has three degrees of freedom: w — vertical displacement (deflection), the positive direction of which coincides with the direction of the Z axis, and the rotation angles UX and UY about the X and Y axes.

Types of elements for the analysis of medium-thickness plates (Reissner-Mindlin theory)

Type	Name	Number of nodes	Node numbering order and local axes	Comments
111	Rectangular	4	Fig. 1, a	JIDR4² [91, 23]
112	Triangular	3	Fig. 1, b	JIDR3 [91, 23]
115	Triangular	3-6	Fig. 1, d	JIDR3-6 [91, 23]
116	Quadrangular	4-8	Fig. 1, g	isoparametric, JIDR [91, 23]
118	Triangular	3-6	Fig. 1, f	isoparametric, JIDR [91, 23]
119	Quadrangular	4	Fig. 2, b	JIDR [91, 23]
120	Quadrangular	4-8	Fig. 1, e	JIDR, SubAreas [91, 23]
512	Triangular	3	Fig. 1, b	modified, DSG3³,[23]
517	Quadrangular	4	Fig. 1, c	isoparametric, MITC4⁴ [2, 75]
518	Triangular	3	Fig. 1, b	DSG3 [76]

The list of FEs for the analysis of plates according to the Reissner-Mindlin theory is given in Table 2.

Elements for Solving the Plane Stress and Plane Strain Problems of the Theory of Elasticity

All elements considered in this section enable to analyze both plane-stress and plane-strain systems (depending on the index, which is specified when describing the stiffness properties of the elements).

Types of elements for the plane problem of the theory of elasticity

Degrees of freedom of nodes	Type	Name	Number of nodes	Node numbering order and local axes	Comments
X, Z	21	Rectangular	4	Fig. 2, a	polylinear shape functions
	22	Triangular	3	Fig. 2, b	linear shape functions
	25	Triangular	3-6	Fig. 2, d	SubAreas [16, 23]
	29	Quadrangular	4-12	Fig. 2, h	SubAreas [26, 23]
	30	Quadrangular	4-8	Fig. 2, e	SubAreas [26, 23]
X, Y, Z	23	Rectangular	4	Fig. 2, a	polylinear shape functions
	24	Triangular	3	Fig. 2, b	linear shape functions
	26	Quadrangular	4-8	Fig. 2, g	isoparametric
	27	Quadrangular	4-8	Fig. 2, e	SubAreas [26, 23]
	28	Triangular	3-6	Fig. 2, f	isoparametric
Elements with drilling (DDF) and quasi-rotational degrees of freedom (QRDF)
X, Y, U_Y	121	Rectangular	4	Fig. 2, a	DDF⁶, incompatible [90, 23]
	122	Triangular	3	Fig. 2, b	DDF, incompatible [90, 23]
	125	Triangular	3-6	Fig. 2, d	DDF, SubAreas [90, 23]
	129	Quadrangular	4-8	Fig. 2, e	DDF, SubAreas, incompatible [90, 23]
	130	Quadrangular	4-8	Fig. 2, e	DDF, SubAreas [90, 23]
	526	Quadrangular	4	Fig. 2, c	QRDF⁷, isoparametric [90, 23]
	527	Quadrangular	4	Fig. 2, c	QRDF, SubAreas [90, 23]
	528	Triangular	3	Fig. 2, b	QRDF [90, 23]

All elements can have isotropic, orthotropic or anisotropic material, as well as transverse isotropic for plane deformation.

The following stresses are calculated — N_X, N_Z, N_XZ, as well as N_Y — at the analysis of structures in the plane strain state. When the subsoil parameter С_uv is specified, R_x and R_z are calculated.

Finite Elements for the Analysis of Thin Shallow Shells

Finite elements used for the analysis of thin shallow shells can take any position in space. Six degrees of freedom are determined in the nodes of the elements — U, V, W, UX, UY and UZ (three linear displacements along and three rotation angles about the coordinate axes). The degrees of freedom U, V correspond to the membrane deformations, and W, UX, UY correspond to the bending ones.

Elements for the analysis of thin shells (Kirchhoff-Love theory)

Type	Name	Number of nodes	Node numbering order and local axes	Degrees of freedom	Comments
41	Rectangular	4	Fig. 3, a	U, V	polylinear shape functions
41	Rectangular	4	Fig. 3, a	W, U_X, U_Y	semi-compatible [18, 22, 23]
42	Triangular	3	Fig. 3, b	U, V	linear shape functions
42	Triangular	3	Fig. 3, b	W, U_X, U_Y	incompatible [18, 22, 23]
43	Rectangular	4	Fig. 3, a	U, V	polylinear shape functions
43	Rectangular	4	Fig. 3, a	W, U_X, U_Y	incompatible [28, 23]
44	Quadrangular	4	Fig. 3, c	U, V	SubAreas [26, 23]
44	Quadrangular	4	Fig. 3, c	W, U_X, U_Y	SubAreas [25, 23]
45	Triangular	3-6	Fig. 3, d	U, V	SubAreas [16, 23]
45	Triangular	3-6	Fig. 3, d	W, U_X, U_Y	SubAreas [24, 23], option 1
46	Quadrangular	4-8	Fig. 3, c	U, V	SubAreas [26, 23]
46	Quadrangular	4-8	Fig. 3, c	W, U_X, U_Y	SubAreas, option 2 [25, 23]
48	Triangular	3-6	Fig. 3, d	U, V	SubAreas [16, 23]
48	Triangular	3-6	Fig. 3, d	W, U_X, U_Y	SubAreas, option 2 [24, 23]
50	Quadrangular	4-8	Fig. 3, e	U, V	SubAreas [26, 23]
50	Quadrangular	4-8	Fig. 3, e	W, U_X, U_Y	SubAreas, option 1 [26, 23]
Elements with drilling (DDF) and quasi-rotational degrees of freedom (QRDF)
91	Rectangular	4	Fig. 3, a	U, V, U_Z	DDF, incompatible [90, 23]
91	Rectangular	4	Fig. 3, a	W, U_X, U_Y	semi-compatible [18, 22, 23]
92	Triangular	3	Fig. 3, b	U, V, U_Z	DDF, incompatible [90, 23]
92	Triangular	3	Fig. 3, b	W, U_X, U_Y	incompatible [18, 22, 23]
93	Rectangular	4	Fig. 3, a	U, V, U_Z	DDF, incompatible [90, 23]
93	Rectangular	4	Fig. 3, a	W, U_X, U_Y	incompatible [28, 23]
94	Quadrangular	4	Fig. 3, e	U, V, U_Z	DDF, SubAreas, incompatible [90, 23]
94	Quadrangular	4	Fig. 3, e	W, U_X, U_Y	SubAreas, [25, 23]
95	Triangular	3-6	Fig. 3, d	U, V, U_Z	DDF, SubAreas [90, 23]
95	Triangular	3-6	Fig. 3, d	W, U_X, U_Y	SubAreas [25, 23]
96	Quadrangular	4-8	Fig. 3, e	U, V, U_Z	DDF, SubAreas [90, 22]
96	Quadrangular	4-8	Fig. 3, e	W, U_X, U_Y	SubAreas [25, 23]
97	Quadrangular	4-8	Fig. 3, e	U, V, U_Z	DDF, SubAreas, incompatible [90, 23]
97	Quadrangular	4-8	Fig. 3, e	W, U_X, U_Y	SubAreas [25, 23]
591	Rectangular	4	Fig. 3, a	U, V, U_Z	QRDF4 [90, 23]
591	Rectangular	4	Fig. 3, a	W, U_X, U_Y	semi-compatible [18, 22, 23]
592	Triangular	3	Fig. 3, b	U, V, U_Z	QRDF3 [90, 23]
592	Triangular	3	Fig. 3, b	W, U_X, U_Y	incompatible [18, 22, 23]
593	Rectangular	4	Fig. 3, c	U, V, U_Z	QRDF4, [90, 23]
593	Rectangular	4	Fig. 3, c	W, U_X, U_Y	incompatible [28, 23]
594	Quadrangular	4	Fig. 3, c	U, V, U_Z	QRDF4, SubAreas [23]
594	Quadrangular	4	Fig. 3, c	W, U_X, U_Y	SubAreas [25, 23]

Stresses N_X, N_Y, N_XY, moments M_X, M_Y, M_XY and shear forces Q_X and Q_Y are calculated. When the subsoil parameter С_Y specified, R_z is calculated, and when Сuv is specified, R_x and R_y are calculated.

The graphic postprocessor calculates also the forces in the plate section SN_X, SNZ_Z, SN_XZ.

Finite Elements for the Analysis of Shells according to the Reissner-Mindlin Theory

These elements are used for the analysis of shallow shells allowing for shear and implement the Reissner-Mindlin theory. They are completely similar to the elements used for the analysis of thin shells in terms of the specification of the initial data.

Elements for the analysis of medium-thickness shells (Reissner-Mindlin theory)

Type	Name	Number of nodes	Node numbering order and local axes	Degrees of freedom	Comments
141	Rectangular	4	Fig. 3, а	U, V	polylinear shapes
141	Rectangular	4	Fig. 3, а	W, U_X, U_Y	JIDR, incompatible [91, 23]
142	Triangular	3	Fig. 3, b	U, V	linear shape functions
142	Triangular	3	Fig. 3, b	W, U_X, U_Y	JIDR3 [91, 23]
143	Quadrangular isoparametric	4	Fig. 3, с	U, V	polylinear shape functions
143	Quadrangular isoparametric	4	Fig. 3, с	W, U_X, U_Y	MITC4, isoparametric [3, 75, 23]
144	Quadrangular	4	Fig. 3, с	U, V	SubAreas [26, 23]
144	Quadrangular	4	Fig. 3, с	W, U_X, U_Y	JIDR4, SubAreas [91, 23]
145	Triangular	3-6	Fig. 3, d	U, V	SubAreas [26, 23]
145	Triangular	3-6	Fig. 3, d	W, U_X, U_Y	JIDR, SubAreas [91, 23]
146	Quadrangular isoparametric	4-8	Fig. 3, e	U, V	isoparametric
146	Quadrangular isoparametric	4-8	Fig. 3, e	W, U_X, U_Y	JIDR, isoparametric [91, 23]
147	Triangular, isoparametric	3-6	Fig. 3, f	U, V	isoparametric
147	Triangular, isoparametric	3-6	Fig. 3, f	W, U_X, U_Y	JIDR, isoparametric [91, 23]
148	Triangular	3	Fig. 3, b	U, V	linear shape functions
148	Triangular	3	Fig. 3, b	W, U_X, U_Y	DSG3M [23]
149	Triangular	3	Fig. 3, b	U, V	linear shape functions
149	Triangular	3	Fig. 3, b	W, U_X, U_Y	DSG3 [76, 23]
150	Quadrangular	4-8	Fig. 3, e	U, V	JIDR, SubAreas [26, 23]
150	Quadrangular	4-8	Fig. 3, e	W, U_X, U_Y	JIDR, SubAreas [91, 23]
Elements with drilling (DDF) and quasi-rotational degrees of freedom (QRDF)
191	Rectangular	4	Fig. 3, a	U, V, U_Z	DDF, incompatible [90, 23]
191	Rectangular	4	Fig. 3, a	W, U_X, U_Y	JIDR [91, 23]
192	Triangular	3	Fig. 3, b	U, V, U_Z	DDF, SubAreas, [90, 23]
192	Triangular	3	Fig. 3, b	W, U_X, U_Y	JIDR, SubAreas [91, 23]
194	Quadrangular	4	Fig. 3, с	U, V, U_Z	DDF, SubAreas [90, 23]
194	Quadrangular	4	Fig. 3, с	W, U_X, U_Y	JIDR, SubAreas [91, 23]
195	Triangular	3-6	Fig. 3, d	U, V, U_Z	DDF, SubAreas [90, 23]
195	Triangular	3-6	Fig. 3, d	W, U_X, U_Y	JIDR, SubAreas [91, 23]
196	Quadrangular	4-8	Fig. 3, e	U, V, U_Z	DDF, SubAreas [90, 23]
196	Quadrangular	4-8	Fig. 3, e	W, U_X, U_Y	JIDR, SubAreas, [91, 23]
197	Quadrangular	4-8	Fig. 3, e	U, V, U_Z	DDF, incompatible, SubAreas [90, 23]
197	Quadrangular	4-8	Fig. 3, e	W, U_X, U_Y	JIDR, SubAreas, [91, 2QRDF3]
542	Triangular	3	Fig. 3, b	U, V, U_Z	[90, 23]
542	Triangular	3	Fig. 3, b	W, U_X, U_Y	JIDR [91, 23]
543	Quadrangular	4	Fig. 3, с	U, V, U_Z	QRDF, isoparametric [90, 23]
543	Quadrangular	4	Fig. 3, с	W, U_X, U_Y	MITC4, isoparametric [3, 75, 23]
544	Quadrangular	4	Fig. 3, а	U, V, U_Z	QRDF, SubAreas [90, 23]
544	Quadrangular	4	Fig. 3, а	W, U_X, U_Y	JIDR, SubAreas [91, 23]
546	Quadrangular	4	Fig. 3, с	U, V, U_Z	QRDF, isoparametric [90, 23]
546	Quadrangular	4	Fig. 3, с	W, U_X, U_Y	JIDR, isoparametric [91, 23]
547	Triangular	4	Fig. 3, b	U, V, U_Z	QRDF [90, 23]
547	Triangular	4	Fig. 3, b	W, U_X, U_Y	DSG3M [23]
548	Triangular	3	Fig. 3, b	U, V, U_Z	QRDF [90, 23]
548	Triangular	3	Fig. 3, b	W, U_X, U_Y	DSG3 [76, 23]