A tapered bar with a solid circular cross section is held rigidly at either end. As shown in Figure Q1a, axial loads of 30 kN and 20 kN are applied to the bar. The bar material has a Young's modulus of 200 GN/m². The bar is to be modelled using a single one dimensional cubic element. The shape functions and their derivatives for this type of element are shown in figure 1b:
A tapered bar with a solid circular cross section is held rigidly at either end. As shown in Figure Q1a, axial loads of 30 kN and 20 kN are applied to the bar. The bar material has a Young's modulus of 200 GN/m². The bar is to be modelled using a single one dimensional cubic element. The shape functions and their derivatives for this type of element are shown in figure 1b:
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
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![A tapered bar with a solid circular cross section is held rigidly at either end. As shown in Figure
Q1a, axial loads of 30 kN and 20 kN are applied to the bar. The bar material has a Young's
modulus of 200 GN/m². The bar is to be modelled using a single one dimensional cubic element.
The shape functions and their derivatives for this type of element are shown in figure 1b:
0.2 m
X
N₁
N4 =
9
30 kN
(1-5)(-5²)
N₂-271-5)(1-²)
№3 = 27+5)(1-5²)
1 + 5) (1 − 5²)
0.2 m
16(1
20 KN
0.2 m
Figure Q1a.
b) The stiffness matrix is found to be:
ƏN₁
JE 16
=
ƏN ₂
d
=
ƏN 3
JE
ƏN4
JE
=
CROSS
SECTIONAL
AREA
X
A = 0.03- 40
A in m², x in m
(185-2752+1)
(-185+8152-27)
(27-
1
7/6 (-1 + 185+275²)
7-185-8152)
Figure Q1b
a) Determine the strain shape function matrix, [B], for this element.
11.625 -14.6875 3.875 -0.8125
K = 106-14.6875 30.375 -18.5625 2.875
3.875 -18.5625 23.625 -8.9375
-0.8125 2.875 -8.9375 6.875
By applying appropriate boundary conditions, use this stiffness matrix to find the
displacements of the points where the loads are applied.
c) Using the displacements found in part (b) and the strain shape function matrix found in part
(a), determine the strain in the bar at the point where the 20 kN load is applied.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae4556bc-f96c-4422-810a-f9be18ed4561%2Fe32d9051-c60d-4b29-80b0-8629ac1de771%2Fah12czn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A tapered bar with a solid circular cross section is held rigidly at either end. As shown in Figure
Q1a, axial loads of 30 kN and 20 kN are applied to the bar. The bar material has a Young's
modulus of 200 GN/m². The bar is to be modelled using a single one dimensional cubic element.
The shape functions and their derivatives for this type of element are shown in figure 1b:
0.2 m
X
N₁
N4 =
9
30 kN
(1-5)(-5²)
N₂-271-5)(1-²)
№3 = 27+5)(1-5²)
1 + 5) (1 − 5²)
0.2 m
16(1
20 KN
0.2 m
Figure Q1a.
b) The stiffness matrix is found to be:
ƏN₁
JE 16
=
ƏN ₂
d
=
ƏN 3
JE
ƏN4
JE
=
CROSS
SECTIONAL
AREA
X
A = 0.03- 40
A in m², x in m
(185-2752+1)
(-185+8152-27)
(27-
1
7/6 (-1 + 185+275²)
7-185-8152)
Figure Q1b
a) Determine the strain shape function matrix, [B], for this element.
11.625 -14.6875 3.875 -0.8125
K = 106-14.6875 30.375 -18.5625 2.875
3.875 -18.5625 23.625 -8.9375
-0.8125 2.875 -8.9375 6.875
By applying appropriate boundary conditions, use this stiffness matrix to find the
displacements of the points where the loads are applied.
c) Using the displacements found in part (b) and the strain shape function matrix found in part
(a), determine the strain in the bar at the point where the 20 kN load is applied.
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