SME
Finite Element Analysis
Truss, beam and 2D elements
Use this for “A 2 m truss member at 30°, 500 mm², E = 200 GPa: node 2 moves 0.5 mm in x and 0.2 mm in y past node 1. Find its force and stress.”
Diagram
δ = Δu cos θ + Δv sin θ
δ = 0.5 cos 30° + 0.2 sin 30°
δ = 0.533 mm
f
=
(
AE
L
)δ
=
26.7
kN;
σ
=
f
A
=
53.3
MPa.
Node 2’s move is drawn 1,130 times the bar’s scale.
This is an example. Type your own values to start a new problem.
δ (mm)
=
Δu (mm)
×
cos(
θ
°)
+
Δv (mm)
×
sin(
θ
°)
f (kN)
=
A (mm²)
×
E (GPa)
×
δ (mm)
÷
L (m)
σ (MPa)
=
f (kN)
÷
A (mm²)
Formulas
δ
=
Δu
×
cos(
θ
)
+
Δv
×
sin(
θ
)
+0.533 = 0.5 × cos(30°) + 0.2 × sin(30°)
f
=
A
×
E
×
δ
1,000 ×
L
+26.6506
=
500
×
200
×
0.533
1,000 × 2
σ
=
1,000
×
f
A
+53.3013
=
1,000
×
26.6506
500
Assumptions
•
A two-node bar element: only the stretch along the bar, δ, makes force.
•
θ runs from the global x axis; Δu and Δv are node 2’s moves minus node 1’s.
•
Number the nodes so θ lies from 0° to 180°; tension is +, compression −.
•
mm², GPa, m and mm give f in kN with a factor of 1000; σ = f ÷ A in MPa.
Step-by-step
WE KNOW
A = 500 mm², E = 200 GPa, L = 2 m, θ = 30°, Δu = +0.5 mm, Δv = +0.2 mm
FIND
elongation (δ), axial force (f), axial stress (σ)
Step 1 · Find elongation (δ)
Use δ = Δu × cos(θ) + Δv × sin(θ)
Only
the
part
of
node
2’s
move
along
the
bar
stretches
it:
Δu
cos
θ
plus
Δv
sin
θ
.
δ
=
0.5
×
cos(30°)
+
0.2
×
sin(30°)
δ
=
0.5
×
0.866
+
0.2
×
0.5
δ
=
0.433
+
0.1
δ
=
+0.533
mm
Step 2 · Find axial force (f)
Use f = A × E × δ ÷ (1,000 × L)
The
element
stiffness
AE
L
times
the
stretch;
mm²
×
GPa
×
mm
m
is
N,
so
divide
by
1000
for
kN.
f
=
500
×
200
×
0.533
1,000 × 2
f
=
500
×
200
×
0.533
2,000
f
=
100,000
×
0.533
2,000
f
=
53,300
2,000
f
=
+26.6506
kN
Step 3 · Find axial stress (σ)
Use σ = 1,000 × f ÷ A
Force
over
area;
kN
mm²
times
1000
is
N/mm²,
which
is
MPa.
σ
=
1,000
×
26.6506
500
σ
=
26,650.6
500
σ
=
+53.3013
MPa
Check
Put the numbers back in. Both sides should match.
+0.533 = 0.5 × cos(30°) + 0.2 × sin(30°) ✓
+26.6506
=
500
×
200
×
0.533
1,000 × 2
✓
+53.3013
=
1,000
×
26.6506
500
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
More problem types
Truss element stiffness matrix
Use this for “A truss element has AE ÷ L = 50,000 N/mm and lies at 30°. Find the terms of its 4 × 4 global stiffness matrix.”
One-element cantilever beam
Use this for “Model a 2 m cantilever, E = 200 GPa, I = 8 × 10⁶ mm⁴, as one beam element with 5 kN at its free end. Solve for v and θ.”