SME
Finite Element Analysis · Problem type
One bar element: stiffness, force and stress
Use this for “A steel bar element 1,000 mm long has an area of 100 mm² and E = 200 GPa. Its nodes move 0.02 mm and 0.12 mm. Find k, the element force and the stress.”
Diagram
Bar 1: A = 100 mm², L = 1,000 mm.
f = 2,000 N.
Bar 1 is in tension.
Each free node balances: its load equals the left element’s force less the right one’s.
σ = 20 MPa: the force over the bar’s area.
Elastic modulus E = 200 GPa
This is an example. Type your own values to start a new problem.
k (N/mm)
=
A (mm²)
×
E (GPa)
L (mm)
f (N)
=
k (N/mm)
×
(
u₂ (mm)
−
u₁ (mm)
)
σ (MPa)
=
f (N)
A (mm²)
L is the element’s length from node 1 to node 2 and A its cross-section area; with E they give the stiffness k labelled on the element.
Formulas
k
=
A
×
E
L
20,000
=
100
×
200,000
1,000
f
=
k
×
(
u₂
−
u₁
)
2,000 = 20,000 × (0.12 − 0.02)
σ
=
f
A
20
=
2,000
100
Formulas are worked in MPa. The step-by-step shows the conversions.
Assumptions
•
Axial springs or bars, loaded only at the nodes; small displacements, linear elastic.
•
An element’s force is its stiffness times its stretch, k(uⱼ − uᵢ): tension is +.
•
One bar of one material and one area; the steps work in N, mm and MPa (N/mm²).
Step-by-step
WE KNOW
A = 100 mm², E = 200 GPa, L = 1,000 mm, u₁ = 0.02 mm, u₂ = 0.12 mm
FIND
element stiffness (k), element force (f), axial stress (σ)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to MPa).
E = 200 GPa = 200,000 MPa (1 GPa = 1000 MPa)
Step 2 · Find element stiffness (k)
Use k = A × E ÷ L
A
bar
under
axial
load
is
a
spring:
from
δ
=
PL
AE
,
its
stiffness
P
δ
is
AE
L
.
k
=
100
×
200,000
1,000
k
=
20,000,000
1,000
k
=
20,000
N/mm
Step 3 · Find element force (f)
Use f = k × (u₂ − u₁)
Row
2
of
the
element’s
stiffness
matrix
reads
f
=
−ku₁
+
ku₂:
k
times
the
stretch.
f
=
20,000
×
(0.12
−
0.02)
f
=
20,000
×
0.1
f
=
2,000
N
Step 4 · Find axial stress (σ)
Use σ = f ÷ A
Stress
is
the
element
force
over
the
area:
N
mm²
is
MPa.
σ
=
2,000
100
σ
=
20
MPa
Check
Put the numbers back in. Both sides should match.
20,000
=
100
×
200,000
1,000
✓
2,000 = 20,000 × (0.12 − 0.02) ✓
20
=
2,000
100
✓
Topic check
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