Twenty Noded Brick element Formulation

FORMULATION OF 20-NODED HEXAHEDRAL ELEMENT

INTRODUCTION

20–Noded Hexahedral Element is a solid 3-dimensional isoparametric element having a hexahedron or brick-shaped geometry containing 8 corner nodes and 12 mid-edge nodes as shown in Figure 1. It uses quadratic shape functions, which are higher-order elements that give better stress gradients. The number of degrees of freedom is 3 (X, Y, Z displacements), and it uses 3 × 3 × 3 Gauss quadrature for numerical integration.

Figure 1: 20-Noded Hexahedral Element showing nodes

Coordinates for each node:

Node	X	Y	Z
1	-1	-1	-1
2	1	-1	-1
3	1	1	-1
4	-1	1	-1
5	-1	-1	1
6	1	-1	1
7	1	1	1
8	-1	1	1
9	0	-1	-1
10	1	0	-1
11	0	1	-1
12	-1	0	-1
13	0	-1	1
14	1	0	1
15	0	1	1
16	-1	0	1
17	-1	-1	0
18	1	-1	0
19	1	1	0
20	-1	1	0

SHAPE FUNCTION

The shape functions for 20-noded hexahedral elements are defined for corner nodes N1–N8 and mid-edge nodes N9–N20. (Equations omitted here; you can paste them as images or in LaTeX if Blogger supports MathJax.)

ELEMENT STIFFNESS MATRIX

The element stiffness matrix expression is given by equation (81), where the size of K_e is the number of degrees of freedom × number of degrees of freedom, B is the strain-displacement matrix, and J is the Jacobian matrix.

Strain Displacement Matrix

The strain-displacement matrix is formed from the partial derivatives of the shape functions with respect to global coordinates (x, y, z) for each node.

Constitutive Matrix

For isotropic 3D materials, the constitutive matrix is defined in terms of Young’s modulus (E) and Poisson’s ratio (ν).

Jacobian Matrix

The Jacobian matrix is calculated by summing the derivatives of each shape function with respect to natural coordinates, multiplied by their corresponding node coordinates, over all 20 nodes of the element.

Numerical Integration

The integration is performed using 3 × 3 × 3 Gauss quadrature points, totaling 27 sampling points, to evaluate the element stiffness matrix accurately.

VERIFICATION OF CODE

Static analysis of a straight cantilever beam

The cantilever beam, 200 cm long with a cross-section of 20 cm × 30 cm, carries a 10 kN point load at the free end. E = 2×10³ kN/cm² and ν = 0. The beam was analyzed using 20-noded elements with 3 and 5 elements, with displacements at the free end compared to textbook results.

• No of elements 3: 0.29983 (MATLAB), 0.29629 (Textbook)
• No of elements 5: 0.2970 (MATLAB), 0.29732 (Textbook)

Static analysis of a tapered cantilever beam

The tapered beam is 125 cm long, width 5 cm, height tapering from 8 cm to 24 cm, with a 15 kN load at the free end, E = 2.1×10⁴ kN/cm², ν = 0.0. Results compared for 5 elements:

• 0.1728 (MATLAB)
• 0.17083 (Textbook)

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