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.
図 1. FE Model of the Beam with Boundary Conditions and Loadcases
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.
表 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
表 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
表 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.
