MacNeal-Harder Test This is a straight cantilever beam
solved with solid and shell elements. Three models (rectangular, parallelogram, trapezoidal)
are made with each element's type to investigate the effect of distorted elements with a
high aspect ratio.

Model Files

Before you begin, copy the file(s) used in this problem
to your working directory.

Six types of elements are used for this problem. They are tria-shell, quad-shell, and hexa-solid
elements, each with 1st and 2nd order. Four loading cases, extension, in-plane
bending, transverse bending, and twist, are used for each model. For the extension
and bending load cases, unit loads are applied in a consistent fashion over all of
the nodes at the tip of the beam. For the twist load cases, a unit moment is applied
at the tip.

Theoretical solutions for the deflections at the tip, computed by beam theory, are:

Load Type

Component

Value

Extension

UX

0.00003

In-plane bending

UZ

0.1081

Transverse bending

UY

0.4321

Twist

ROTX

0.03208

Linear Static Analysis Results

All results are normalized with the target value.

Table 1. (a) Rectangular

In-plane Extension

In-plane Bending

Transverse Bending

Twist

QUAD4

1.000

0.992

0.981

0.941

QUAD8

1.006

1.000

1.016

0.953

TRI3

1.000

0.032

0.973

1.072

TRI6

1.006

0.994

1.001

0.950

HEX8

0.988

0.978

0.973

0.892

HEX20

1.008

0.992

0.992

0.905

Table 2. (b) Parallelogram

In-plane Extension

In-plane Bending

Transverse Bending

Twist

QUAD4

1.000

0.712

0.981

0.905

QUAD8

1.008

0.999

1.015

0.937

TRI3

1.000

0.012

0.955

0.931

TRI6

1.005

0.962

0.995

0.982

HEX8

1.012

0.624

0.529

0.820

HEX20

1.008

0.976

0.977

0.905

Table 3. (c) Trapezoidal

In-plane Extension

In-plane Bending

Transverse Bending

Twist

QUAD4

1.000

0.173

0.964

0.869

QUAD8

1.005

0.981

1.015

0.950

TRI3

1.000

0.019

0.965

1.175

TRI6

1.006

0.972

0.999

0.947

HEX8

1.010

0.047

0.030

0.563

HEX20

1.008

0.902

0.950

0.905

Reference

MacNeal, R.H., and Harder, R.L., A
Proposed Standard Set of Problems to Test Finite Element Accuracy, Finite Elements in
Analysis and Design, 1 (1985) 3-20.