SME
Finite Element Analysis
Shape functions
Use this for “A bar element runs from x = 100 to 200 mm with nodal displacements 0.02 and 0.05 mm. Find u at x = 130 mm and the strain.”
Chart
N₁(130) = 0.7
Ringed: the nodes (100, 1), (200, 0)
This is an example. Type your own values to start a new problem.
u (mm)
=
N₁
×
u₁ (mm)
+
N₂
×
u₂ (mm)
ε
=
(
u₂ (mm)
−
u₁ (mm)
)
÷
L (mm)
u₁ and u₂ are how far nodes 1 and 2 move, at x₁ and x₂; the graph shows N₁, and N₂ = 1 − N₁ is its mirror image.
x₁
Node 1 position
example
x₂
Node 2 position
example
x
Point position
example
Formulas
L
=
x₂
−
x₁
100 = 200 − 100
N₁
=
x₂
−
x
L
0.7
=
200 − 130
100
N₂
=
x
−
x₁
L
0.3
=
130 − 100
100
u
=
N₁
×
u₁
+
N₂
×
u₂
0.029 = 0.7 × 0.02 + 0.3 × 0.05
ε
=
u₂
−
u₁
L
3 ×
10
-4
=
0.05 − 0.02
100
Assumptions
•
A linear two-node element: N₁ and N₂ are straight lines, each 1 at its own node and 0 at the other.
•
N₁ + N₂ = 1 at every point, so a rigid shift of the element gives no strain.
•
The strain du/dx is the same everywhere in the element.
Step-by-step
WE KNOW
x₁ = 100 mm, x₂ = 200 mm, u₁ = 0.02 mm, u₂ = 0.05 mm, x = 130 mm
FIND
element length (L), shape function 1 (N₁), shape function 2 (N₂), displacement at x (u), strain (ε)
Step 1 · Find element length (L)
Use L = x₂ − x₁
The element runs from node 1 to node 2.
L
=
200
−
100
L
=
100
mm
Step 2 · Find shape function 1 (N₁)
Use N₁ = (x₂ − x) ÷ L
N₁
falls
in
a
straight
line
from
1
at
node
1
to
0
at
node
2.
N₁
=
200 − 130
100
N₁
=
70
100
N₁
=
0.7
Step 3 · Find shape function 2 (N₂)
Use N₂ = (x − x₁) ÷ L
N₂
rises
in
a
straight
line
from
0
at
node
1
to
1
at
node
2.
N₂
=
130 − 100
100
N₂
=
30
100
N₂
=
0.3
Step 4 · Find displacement at x (u)
Use u = N₁ × u₁ + N₂ × u₂
Each nodal displacement is weighted by its shape function, then the two are added.
u
=
0.7
×
0.02
+
0.3
×
0.05
u
=
0.014
+
0.015
u
=
0.029
mm
Step 5 · Find strain (ε)
Use ε = (u₂ − u₁) ÷ L
Strain
is
du/dx;
u
is
a
straight
line,
so
its
slope
is
the
change
in
u
over
L
.
ε
=
0.05 − 0.02
100
ε
=
0.03
100
ε
=
3 ×
10
-4
Check
Put the numbers back in. Both sides should match.
100 = 200 − 100 ✓
0.7
=
200 − 130
100
✓
0.3
=
130 − 100
100
✓
0.029 = 0.7 × 0.02 + 0.3 × 0.05 ✓
3 ×
10
-4
=
0.05 − 0.02
100
✓
Topic check
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