SME
Finite Element Analysis · Problem type
Nodes and degrees of freedom in a mesh
Use this for “A plate is meshed 10 × 5 with four-node quadrilaterals. How many nodes and degrees of freedom does the model have?”
Diagram
N = (nₓ + 1)(nᵧ + 1) = 11 × 6 = 66; DOF = 2 × 66 = 132 (u and v at each node).
This is an example. Type your own values to start a new problem.
N
=
(
nₓ
+
1)
×
(
nᵧ
+
1)
DOF
=
2
×
N
Formulas
N
=
(nₓ
+
1)
×
(nᵧ
+
1)
66 = (10 + 1) × (5 + 1)
DOF
=
2
×
N
132 = 2 × 66
Assumptions
•
A rectangular plate cut into nₓ × nᵧ four-node quadrilaterals.
•
Plane stress: two displacements, u and v, at each node.
•
Before any supports are applied: fixed nodes are still counted.
Step-by-step
WE KNOW
nₓ = 10, nᵧ = 5
FIND
number of nodes (N), degrees of freedom (DOF)
Step 1 · Find number of nodes (N)
Use N = (nₓ + 1) × (nᵧ + 1)
A row of n elements has n + 1 nodes, in each direction.
N
=
(10
+
1)
×
(5
+
1)
N
=
11
×
6
N
=
66
Step 2 · Find degrees of freedom (DOF)
Use DOF = 2 × N
Each node moves two ways, u and v.
DOF = 2 × 66
DOF = 132
Check
Put the numbers back in. Both sides should match.
66 = (10 + 1) × (5 + 1) ✓
132 = 2 × 66 ✓
Topic check
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