SME
Finite Element Analysis · Problem type
Strains in a constant-strain triangle
Use this for “A right-angled CST has node 1 at the origin pinned, node 2 at (40, 0) and node 3 at (0, 30) mm. Node 2 moves (12, 4) μm and node 3 (6, 9) μm. Find the strains.”
Diagram
Node 2 = ⟨12, 4⟩
|Node
2|
=
4
√
10
≈
12.65
μm.
Node 3 = ⟨6, 9⟩
|Node
3|
=
3
√
13
≈
10.82
μm.
This is an example. Type your own values to start a new problem.
εₓ (με)
=
u₂ (μm)
÷
a (mm)
εᵧ (με)
=
v₃ (μm)
÷
b (mm)
γₓᵧ (μrad)
=
u₃ (μm)
÷
b (mm)
+
v₂ (μm)
÷
a (mm)
a and b are the triangle’s legs: node 2 sits at (a, 0) and node 3 at (0, b). The arrows are how far nodes 2 and 3 move, (u, v).
A
Element area
worked out (not typed)
Formulas
A
=
a
×
b
2
600,000,000
=
40,000
×
30,000
2
εₓ
=
u₂
a
+0.0003
=
12
40,000
εᵧ
=
v₃
b
+0.0003
=
9
30,000
γₓᵧ
=
u₃
b
+
v₂
a
+0.0003
=
6
30,000
+
4
40,000
Formulas are worked in μm, μm², mm/mm, rad. The step-by-step shows the conversions.
Assumptions
•
Displacements vary linearly, so each strain is the same all over the element.
•
Node 1 sits at the right angle and is held: u₁ = v₁ = 0.
•
Node 2 lies along x and node 3 along y; strains are small.
Step-by-step
WE KNOW
a = 40 mm, b = 30 mm, u₂ = +12 μm, v₂ = +4 μm, u₃ = +6 μm, v₃ = +9 μm
FIND
element area (A), strain along x (εₓ), strain along y (εᵧ), shear strain (γₓᵧ)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to μm, μm², mm/mm, rad).
a = 40 mm = 40,000 μm (1 mm = 1000 μm)
b = 30 mm = 30,000 μm (1 mm = 1000 μm)
Step 2 · Find element area (A)
Use A = a × b ÷ 2
A right triangle is half the rectangle on its two legs.
A
=
40,000
×
30,000
2
A
=
1,200,000,000
2
A
=
600,000,000
μm²
Step 3 · Find strain along x (εₓ)
Use εₓ = u₂ ÷ a
Only
node
2
sits
off
the
y-axis,
so
u
climbs
from
0
to
u₂
across
the
leg
a
:
that
slope
is
εₓ.
εₓ
=
12
40,000
εₓ = 0.0003 mm/mm
Step 4 · Find strain along y (εᵧ)
Use εᵧ = v₃ ÷ b
Only
node
3
sits
off
the
x-axis,
so
v
climbs
from
0
to
v₃
up
the
leg
b
:
that
slope
is
εᵧ.
εᵧ
=
9
30,000
εᵧ = 0.0003 mm/mm
Step 5 · Find shear strain (γₓᵧ)
Use γₓᵧ = u₃ ÷ b + v₂ ÷ a
Shear
strain
is
∂u/∂y
+
∂v/∂x:
node
3’s
sideways
shift
over
b
,
plus
node
2’s
upward
shift
over
a
.
γₓᵧ
=
6
30,000
+
4
40,000
γₓᵧ = 0.0002 + 0.0001
γₓᵧ = 0.0003 rad
Step 6 · Convert the answers
Change each answer back to the unit you chose.
A = 600,000,000 μm² = 600 mm² (1 mm² = 10⁶ μm²)
εₓ = 0.0003 mm/mm = +300 με (1 mm/mm = 10⁶ με)
εᵧ = 0.0003 mm/mm = +300 με (1 mm/mm = 10⁶ με)
γₓᵧ = 0.0003 rad = +300 μrad (1 rad = 10⁶ μrad)
Check
Put the numbers back in. Both sides should match.
600,000,000
=
40,000
×
30,000
2
✓
+0.0003
=
12
40,000
✓
+0.0003
=
9
30,000
✓
+0.0003
=
6
30,000
+
4
40,000
✓
← Previous lesson
Isoparametric 2D elements (CST and Q4)
Next lesson →
The Jacobian of a parallelogram Q4
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.