Numerical Results for shear-lock free finite elements based on
Mindlin-Reissner plate and Timoshenko beam theories
Plate Elements
NxN
Mesh Size (Full Plate)
Exact Solution
Srinivasa Rao and AK Rao and Theory of Plates and Shells
CF
Convergence Factor
ASR
Aspect Ratio
MR_FE_1
MR_FE_2
MR_FE_3
MR_FE_4
Finite Elements based on Mindlin-Reissner theory using new
shape functions
Material:
E=1.092E06; 𝜐 = 0.3;
Geometry:
Square plate a=10.0;
h=2.0, 1.4, 1.0, 0.5;
Type of plate
Simply supported (SS)
Load:
UDL=10.0
Timoshenko beam elements
N
Number of Elements (Full beam)
Exact Solution
Theory of Elasticity
CF
Convergence Factor
ASR
Aspect Ratio
Timo_FE_1
Timo_FE_2
Timo_FE_3
Timo_FE_4
Finite Elements based on Timoshenko beam theory using new
shape functions
Material:
29,000; 𝜐 = 0.3;
Geometry:
L=5,10, 25,100, 200, 400;
h=1.0; b=1.0;
Load:
Concentrated Load q=100.0
Higher order beam elements (capable of accurately predicting three dimensional stresses) based on the
higher order shear deformation theories developed by me
HFE_1 HFE_3 FE_NSF_1 FE_NSF_3 -
based on Lagrangian polynomials
based on Lagrangian polynomials
based on new shape functions
based on new shape functions
Advantages of the present new finite elements
1.
The primary aim of this research work is to replace the finite elements based on the first order shear
deformation theory available in the general purpose finite element packages by these new finite
elements.
2.
Presently all general purpose finite element packages like MSC-NASTRAN, NX- NASTRAN,
ABAQUS, LS-DYNA, and ANSYS
use the finite elements based on Timoshenko beam theory,
Reissner-Mindlin plate theory and Kirchhoff-Love shell theory ( all are called first order shear
deformation theories). The drawback of these finite elements is that they can not be used for the
analysis of thin structures due to shear lock problem. To eliminate this problem, special integration
scheme must be used.
The new finite elements based on the Timoshenko beam theory, Reissner-Mindlin Plate theory
developed by me using special shape functions and standard finite element procedure are applied to the
analysis of beams and plates. The numerical results show that accurate solution is obtained for less
number of elements. The specialties of these finite elements are that (i) thick and thin structures can be
analyzed, (ii) No special integration scheme is required, (iii) a new concept, Convergence Factor (CF),
is introduced in the formulation of these elements to accelerate convergence, and (iv) Accurate
solution is obtained by keeping the number of elements as constant and increasing the value of the CF.
Hence, this procedure reduces modeling effort, computational time and increasing accuracy.
3.
Arbitrary higher order continuity can be achieved in this development. This is very essential for the
problems related to crack propagation where high stress gradient exists near the crack tip.
4.
A comparison study was carried out among the finite elements based on the new shape functions and
Lagrangian shape functions using two higher order shear deformation theories developed by me for the
analysis of simply supported beam under transverse load. The convergence is achieved faster than that
of the finite elements based on the Lagrangian shape functions.
5.
Considering the above points, study related to the development of triangular finite elements based on
the new shape functions is in progress.
6.
Compare to the development of other thin plate finite elements like Discrete Kirchhoff Theory (DKT)
finite element and Mixed Interpolated Tensorial Component (MITC4) element, the development of
these finite elements is very simple.
7.
Since these new shape functions are very effective in accurately predicting displacements, strains and
stresses, finite elements based on these shape functions can be developed for various applications, for
example, multiscale modelling, crack propagation and gradient-enhanced damage models.
Simply supported plate with UDL for ASR = 5.0
error in deflection at centre
80.00
Aspect Ratio = 5
Mesh size = 2 x 2
30.00
-20.00 1.00
10.00
100.00
1,000.00
10,000.00
% Error
Convergence Factor
-70.00
Element 1
-120.00
Element 2
Element 3
-170.00
Element 4
-220.00
Fig.1
80.00
Aspect Ratio = 5
Mesh size = 4 x 4
60.00
40.00
20.00
0.00
% Error
1.00
-20.00
-40.00
10.00
100.00
Convergence Factor
Element 1
-60.00
Element 2
-80.00
Element 3
-100.00
1,000.00
Element 4
-120.00
Fig.2
10,000.00
Aspect Ratio = 5
Mesh size = 6 x 6
65.00
45.00
25.00
% Error
5.00
-15.00
1.00
10.00
100.00
1,000.00
Convergence Factor
10,000.00
-35.00
Element 1
-55.00
Element 2
Element 3
-75.00
Element 4
-95.00
-115.00
Fig.3
80.00
Aspect Ratio = 5
Mesh size = 10 x 10
60.00
40.00
% Error
20.00
0.00
1.00
10.00
100.00
-20.00
-40.00
Convergence Factor
Element 1
Element 2
-60.00
-80.00
1,000.00
Element 3
Element 4
-100.00
Fig.4
10,000.00
80.00
Aspect Ratio = 5
Mesh size = 14 x 14
60.00
40.00
% Error
20.00
0.00
1.00
10.00
100.00
-20.00
-40.00
-60.00
Convergence Factor
Element 1
Element 3
Element 2
-80.00
1,000.00
Element 4
-100.00
Fig.5
10,000.00
Simply supported plate with UDL for ASR = 7.14
error in deflection at centre
75
Aspect Ratio = 7.14
Mesh size = 2 x 2
55
35
% Error
15
-5
1.00
10.00
100.00
1,000.00
10,000.00
Convergence Factor
-25
Element 1
Element 2
-45
Element 3
-65
Element 4
-85
Fig.1
80
Aspect Ratio = 7.14
Mesh size = 4 x 4
Element 1
Element 2
60
Element 3
Element 4
% Error
40
20
0
1.00
10.00
100.00
1,000.00
Convergence Factor
-20
-40
Fig.2
10,000.00
75
Aspect Ratio = 7.14
Mesh size = 6 x 6
Element 1
Element 2
55
Element 3
Element 4
35
% Error
15
-5 1.00
10.00
100.00
1,000.00
Convergence Factor
10,000.00
-25
-45
Fig.3
70
Aspect Ratio = 7.14
Mesh size = 10 x 10
Element 1
Element 2
Element 3
50
Element 4
% Error
30
10
-10
1.00
10.00
100.00
Convergence Factor
-30
-50
Fig.4
1,000.00
10,000.00
70
Aspect Ratio = 7.14
Mesh size = 14 x 14
Element 1
Element 2
Element 3
50
Element 4
% Error
30
10
-10
1.00
10.00
100.00
Convergence Factor
-30
-50
Fig.5
1,000.00
10,000.00
Simply supported plate with UDL for ASR =10.0
error in deflection at centre
70
Aspect Ratio = 10.0
Mesh size = 2 x 2
Element 1
Element 2
50
Element 3
Element 4
% Error
30
10
-10
1.00
10.00
100.00
1,000.00
Convergence Factor
10,000.00
-30
-50
Fig.1
70
Aspect Ratio = 10.0
Mesh size = 4 x 4
Element 1
Element 2
Element 3
50
Element 4
% Error
30
10
-10
1.00
10.00
100.00
Convergence Factor
-30
-50
Fig.2
1,000.00
10,000.00
70
Aspect Ratio = 10.0
Mesh size = 6 x 6
Element 1
Element 2
Element 3
50
% Error
Element 4
30
10
-10
1.00
10.00
100.00
1,000.00
10,000.00
Convergence Factor
-30
-50
Fig.3
70
Aspect Ratio = 10.0
Mesh size = 10 x 10
Element 1
Element 2
Element 3
50
Element 4
% Error
30
10
-10
1.00
10.00
100.00
Convergence Factor
-30
-50
Fig.4
1,000.00
10,000.00
70
Aspect Ratio = 10.0
Mesh size = 14 x 14
Element 1
Element 2
Element 3
% Error
50
Element 4
30
10
-10
1.00
10.00
100.00
Convergence Factor
-30
-50
Fig.5
1,000.00
10,000.00
Simply supported plate with UDL for ASR =20.0
error in deflection at centre
70
Aspect Ratio = 20.0
Mesh size = 2 x 2
Element 1
Element 2
50
Element 3
Element 4
% Error
30
10
-10
1.00
10.00
100.00
1,000.00
Convergence Factor
10,000.00
-30
-50
Fig.1
70
Aspect Ratio = 20.0
Mesh size = 4 x 4
Element 1
Element 2
Element 3
50
Element 4
% Error
30
10
-10
1.00
10.00
100.00
Convergence Factor
-30
-50
Fig.2
1,000.00
10,000.00
70
Aspect Ratio = 20.0
Mesh size = 6 x 6
Element 1
Element 2
Element 3
50
% Error
Element 4
30
10
-10
1.00
10.00
100.00
1,000.00
10,000.00
Convergence Factor
-30
-50
Fig.3
70
Aspect Ratio = 20.0
Mesh size = 10 x 10
Element 1
Element 2
Element 3
50
Element 4
% Error
30
10
-10
1.00
10.00
100.00
Convergence Factor
-30
-50
Fig.4
1,000.00
10,000.00
70
Aspect Ratio = 20.0
Mesh size = 14 x 14
Element 1
Element 2
Element 3
50
Element 4
% Error
30
10
-10
1.00
10.00
100.00
Convergence Factor
-30
-50
Fig.5
1,000.00
10,000.00
Cantilever beam with tip load for ASR = 5.0
error in deflection at centre
75
Element 1
Element 2
% Error
55
Aspect Ratio = 5
No of Elements = 2
Element 3
Element 4
35
15
-5 1.00
10.00
100.00
1,000.00
Convergence Factor
10,000.00
-25
-45
Fig.1
85.00
Element 1
65.00
Element 2
Aspect Ratio = 5
No of Elements = 4
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
Convergence Factor
-35.00
-55.00
Fig.2
10,000.00
85.00
Element 1
65.00
Element 2
Aspect Ratio = 5
No of Elements = 6
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
10,000.00
Convergence Factor
-35.00
-55.00
Fig.3
85.00
Element 1
65.00
Element 2
Aspect Ratio = 5
No of Elements = 10
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
Convergence Factor
-35.00
-55.00
Fig.4
10,000.00
85.00
Element 1
65.00
Element 2
Aspect Ratio = 5
No of Elements = 20
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
Convergence Factor
-35.00
-55.00
Fig.5
10,000.00
Cantilever beam with tip load for ASR = 10.0
error in deflection at centre
75.00
Element 1
Element 2
55.00
Aspect Ratio = 10
No of Elements = 2
Element 3
Element 4
% Error
35.00
15.00
-5.00 1.00
10.00
100.00
1,000.00
Convergence Factor
10,000.00
-25.00
-45.00
Fig.1
85.00
Element 1
65.00
Element 2
Aspect Ratio = 10
No of Elements = 4
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
Convergence Factor
-35.00
-55.00
Fig.2
10,000.00
85.00
Element 1
65.00
Element 2
Aspect Ratio = 10
No of Elements = 6
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
10,000.00
Convergence Factor
-35.00
-55.00
Fig.3
85
Element 1
% Error
65
Element 2
Aspect Ratio = 10
No of Elements = 10
Element 3
45
Element 4
25
5
1.00
10.00
100.00
-15
1,000.00
Convergence Factor
-35
-55
Fig.4
10,000.00
85.00
Element 1
65.00
Element 2
Aspect Ratio = 10
No of Elements = 20
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
Convergence Factor
-35.00
-55.00
Fig.5
10,000.00
Cantilever beam with tip load for ASR =25.0
error in deflection at centre
75
Element 1
Element 2
% Error
55
Aspect Ratio = 25
No of Elements = 2
Element 3
Element 4
35
15
-5 1.00
10.00
100.00
1,000.00
Convergence Factor
10,000.00
-25
-45
Fig.1
85.00
Element 1
65.00
Element 2
Aspect Ratio = 25
No of Elements = 4
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
Convergence Factor
-35.00
-55.00
Fig.2
10,000.00
85.00
Element 1
65.00
Element 2
Aspect Ratio = 25
No of Elements = 6
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
10,000.00
Convergence Factor
-35.00
-55.00
Fig.3
85.00
Element 1
65.00
Element 2
Aspect Ratio = 25
No of Elements = 10
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
Convergence Factor
-35.00
-55.00
Fig.4
10,000.00
85.00
Element 1
65.00
Element 2
Aspect Ratio = 25
No of Elements = 20
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
Convergence Factor
-35.00
-55.00
Fig.4
10,000.00
Cantilever beam with tip load for ASR =100.0
error in deflection at centre
75.00
Aspect Ratio = 100
No of Elements = 2
Element 1
Element 2
% Error
55.00
Element 3
Element 4
35.00
15.00
-5.00 1.00
10.00
100.00
1,000.00
Convergence Factor
10,000.00
-25.00
-45.00
Fig.1
85.00
Element 1
65.00
Element 2
Aspect Ratio = 100
No of Elements = 4
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
Convergence Factor
-35.00
-55.00
Fig.2
10,000.00
85.00
Aspect Ratio = 100
No of Elements = 6
Element 1
65.00
Element 2
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
10,000.00
Convergence Factor
-35.00
-55.00
Fig.3
85.00
65.00
Element 1
Aspect Ratio = 100
No of Elements = 10
Element 2
45.00
Element 3
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
Convergence Factor
-35.00
-55.00
Fig.4
10,000.00
85.00
Element 1
65.00
Element 2
Aspect Ratio = 100
No of Elements = 20
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
Convergence Factor
-35.00
-55.00
Fig.5
10,000.00
Cantilever beam with tip load for ASR = 200.0
error in deflection at centre
75.00
Element 1
Element 2
55.00
Aspect Ratio = 200
No of Elements = 2
Element 3
Element 4
% Error
35.00
15.00
-5.00 1.00
10.00
100.00
1,000.00
Convergence Factor
10,000.00
-25.00
-45.00
Fig.1
85.00
Element 1
65.00
Element 2
Aspect Ratio = 200
No of Elements = 4
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
Convergence Factor
-35.00
-55.00
Fig.2
10,000.00
85.00
Element 1
65.00
Element 2
Aspect Ratio = 200
No of Elements = 6
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
10,000.00
Convergence Factor
-35.00
-55.00
Fig.3
85.00
65.00
Element 1
Element 2
45.00
Aspect Ratio = 200
No of Elements = 10
Element 3
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
Convergence Factor
-35.00
-55.00
Fig.4
10,000.00
85.00
Element 1
65.00
Element 2
Aspect Ratio = 200
No of Elements = 20
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
Convergence Factor
-35.00
-55.00
Fig.5
10,000.00
Cantilever beam with tip load for ASR =400.0
error in deflection at centre
75.00
Element 1
Element 2
55.00
Aspect Ratio = 400
No of Elements = 2
Element 3
Element 4
% Error
35.00
15.00
-5.00 1.00
10.00
100.00
1,000.00
Convergence Factor
10,000.00
-25.00
-45.00
Fig.1
85.00
Element 1
65.00
Element 2
Aspect Ratio = 400
No of Elements = 4
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
Convergence Factor
-35.00
-55.00
Fig.2
10,000.00
85.00
Element 1
65.00
Element 2
Aspect Ratio = 400
No of Elements = 6
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
10,000.00
Convergence Factor
-35.00
-55.00
Fig.3
85.00
65.00
Element 1
Element 2
45.00
Aspect Ratio = 400
No of Elements = 10
Element 3
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
Convergence Factor
-35.00
-55.00
Fig.4
10,000.00
85.00
Element 1
65.00
Element 2
Aspect Ratio = 400
No of Elements = 20
Element 3
45.00
Element 4
% Error
25.00
5.00
1.00
10.00
100.00
-15.00
1,000.00
Convergence Factor
-35.00
-55.00
Fig.5
10,000.00
2. Rhombic Plate under UDL
y
E=10.5e5 psi
υ=0.3
h=0.125 in
q=0.25066 psi
∙
∙
∙
2
3
∙
1
∙4
∙5
5
6
12”
450
x
12”
Element
Mesh
DOF
4x4
75
MR_FE_11
4x4
75
MR_FE_21
4x4
75
MR_FE_31
4x4
75
MR_FE_41
4x4
75
DKT2
HSM
4x4
75
ACM
8x6
189
HCT
8x6
189
Experimental Value
1
one point integration;
2
1
0.3132
0.3133
0.3132
0.3133
0.304
0.264
0.296
0.281
0.297
2
0.2059
0.2010
0.2059
0.2060
0.198
0.173
0.198
0.188
0.204
3
0.1144
0.1144
0.1144
0.1144
0.113
0.100
0.114
0.111
0.121
4
0.1255
0.1256
0.1255
0.1256
0.121
0.095
0.114
0.111
0.129
5
0.05239
0.05241
0.05239
0.05241
0.055
0.043
0.052
0.049
0.056
6
0.02053
0.02054
0.02053
0.02054
0.028
0.021
0.020
0.018
0.022
Three point integration;
DKT – Discrete Krichhoff Theory element , Jean-Louis Batoz et al, A study of three-node
triangular plate bending elements,
international journal for numerical methods in
engineering, Vol. 15, 1771-1812 (1980).
3. Twisting of a square plate
5.0lb
A
1
1
B
6
∙
∙
∙
∙
2
4
9
8
∙
9
6
7
2
∙
1
∙
2
Mesh (b)
Mesh (b)
Node No
X-co
Y-co
Mesh (c)
Element
Type
Point C*
Node No
X-co
Y-co
∙
∙
4
1
0
0
2
0
4
1 2
0 0
0 2
3
0
6
3
0
8
4
0
8
4
5
0
5
4
0
5
5
4
6
3
3
DKT
HSM
ACM
HCT
0.24960
0.24960
.24972
0.25002
*
0.24960
Three point integration
7
1
∙
Mesh
8
∙9
1
14
∙
∙ 13
4
∙
∙8
7
5 Mesh (c)
7
8
0
8
4
8
∙15
∙
5
∙
6
6.2
8
7
3
5
∙
11
6∙
3
1
∙ 16
∙
3
5
∙
12
∙
3∙
8
3
Mesh (a)
D
8.0”
4
2
4
8.0”
E=10000 psi
h=1.0”
υ=0.3
3
2
C
8
8
6.2
9
6
0
9
8
8
10
6
2
MR_FE_1
11
7
5
12
6
8
MR_FE_2
a
0.25740
0.25740
b
0.18963
0.18963
c
0.19229
0.19229
(Exact thin plate solution)
13
8
0
14
8
1
15
8
5
16
8
8
MR_FE_3
MR_FE_4
0.25742
0.18945
0.19218
0.25739
0.19218
0.19239
4.
Cantilever Beam (Tip Load)
z
q=100
x
2h
L
N
HFE_1
HFE_2
2
4
8
12
20
24
30
36
Reddy *
Exact Solution
1.8136
6.2233
15.8701
22.2602
28.0410
29.3511
30.5177
31.1911
1.8136
6.2233
15.8701
22.2602
28.0410
29.3511
30.5177
31.1911
N
HFE_1
HFE_2
2
4
8
12
20
24
30
36
0.2626
0.4302
0.5118
0.5305
0.5406
0.5424
0.5439
0.5447
0.2626
0.4302
0.5118
0.5305
0.5406
0.5424
0.5439
0.5447
Reddy *
Exact Solution
Timo_FE_1
L/2h=160/12
Timo_FE_3
FE_NSF_1
FE_NSF_3
32.2712
32.2391
32.2712
32.2712
(6 elements
with 1000 CF)
(6 elements
with 1000
CF)
(6 elements
with 1000
CF)
(6 elements
with 1000 CF)
HFE_1
HFE_2
0.7867
2.0073
3.2797
3.7158
3.9874
4.0381
4.0806
4.1040
0.7867
2.0073
3.2797
3.7158
3.9874
4.0381
4.0806
4.1040
Timo_FE_1
L/2h=40/12
Timo_FE_2
FE_NSF_3
4.1100
4.1258
4.1218
(6 elements
with 1000
CF)
(6 elements
with 1000 CF
(6 elements
with 1000 CF)
(6 elements
with 1000
CF)
FE_NSF_1
FE_NSF_3
4.1567
4.1317
FE_NSF_1
FE_NSF_3
0.5366
0.5362
0.5424
0.5419
( 6 elements
with 1000
CF)
( 6 elements
with 1000
CF)
( 6 elements
with 1000
CF)
( 6 elements
with 1000 CF)
0.54588
0.5333
FE_NSF_1
4.1140
32.823
32.7844
Timo_FE_1
L/2h=80/12
Timo_FE_3
HFE_1
HFE_2
0.02194
0.02367
0.02414
0.02423
0.02428
0.02194
0.02367
0.02414
0.02423
0.02428
Timo_FE_1
L/2h=12/12
Timo_NFE_3
0.02264
0.02264
0.02419
0.02418
( 6 elements
with 1000
CF)
( 6 elements
with 1000 CF)
( 6 elements
with 1000 CF)
( 6
elements
with 1000
CF)
0.02393
0.02052
5.
Simply Supported Beam (Uniformly distributed load)
z
Δ
N
2
4
8
12
20
24
30
36
HFE_1
0.8254
3.6431
9.8570
13.9791
17.7108
18.5568
19.3102
19.7452
HFE_2
0.8254
3.6431
9.8570
13.9791
17.7108
18.5568
19.3102
19.7452
q=10
x
L
Timo_FE_1
20.326
( 10
elements
with 1000
CF)
Reddy *
Exact Solution
L/2h=160/12
Timo_FE_3
2h
FE_NSF_1
FE_NSF_3
20.306
20.374
20.354
(10 elements
with
1000
CF)
(10 elements
with 1000
CF)
( 10 elements
with 1000 CF)
HFE_1
0.1849
0.6046
1.0482
1.2009
1.2960
1.3138
1.3287
1.3369
HFE_2
0.1849
0.6046
1.0482
1.2009
1.2960
1.3138
1.3287
1.3369
L/2h=80/12
Timo_FE_1 Timo_FE_3
HFE_1
HFE_2
2
4
8
12
20
24
30
36
0.03475
0.07202
0.09064
0.09491
0.09723
0.09764
-----------------
0.03475
0.07202
0.09064
0.09491
0.09723
0.09764
-----------------
L/2h=40/12
Timo_FE_1 Timo_FE_3
FE_NSF_3
1.3139
1.3271
1.3259
( 10
elements
with 1000
CF)
( 10
elements
with 1000
CF)
( 10
elements
with 1000
CF
( 10
elements
with 1000
CF
FE_NSF_1
FE_NSF_3
1.3486
1.3408
FE_NSF_1
FE_NSF_3
0.09339
0.09333
0.09634
0.09628
( 10
elements
with 1000
CF)
( 10
elements
with 1000
CF)
( 10
elements
with 1000
CF)
( 10
elements
with 1000
CF)
0.09770
0.09576
FE_NSF_1
1.3151
20.717
20.6892
N
Reddy *
Exact Solution
Δ
HFE_1
HFE_2
0.0016486
0.0021069
0.0022298
0.0022529
0.0022649
-------------------------------
0.0016486
0.0021069
0.0022298
0.0022529
0.0022649
-------------------------------
L/2h=12/12
Timo_FE_1 Timo_FE_3
0.001979
0.001980
0.002230
0.002210
( 10
elements
with 1000
CF)
( 10
elements
with 1000
CF)
( 10
elements
with 1000
CF)
( 10
elements
with 1000
CF)
0.002220
0.002082