Calculator Input
Example Data Table
| Input Type | Example | Meaning |
|---|---|---|
| Node | 1,0,0 | Node 1 at x = 0 and y = 0. |
| Element | E1,1,2,200000000000,0.003 | Element E1 connects node 1 to node 2. |
| Support | 1x, 1y, 3y | These degrees of freedom are fixed. |
| Load | 2x=10000, 2y=-5000 | Loads are applied at node 2. |
Formula Used
The calculator uses the two dimensional truss element stiffness formula. Each node has two degrees of freedom: horizontal movement and vertical movement.
L = sqrt((xj - xi)^2 + (yj - yi)^2)
c = (xj - xi) / L
s = (yj - yi) / L
ke = (AE / L) *
[ c² cs -c² -cs
cs s² -cs -s²
-c² -cs c² cs
-cs -s² cs s² ]
Global assembly:
K[row, column] = K[row, column] + ke[local row, local column]
Reduced system:
Kff * uf = Ff
Support reaction:
R = K * u - F
How to Use This Calculator
- Enter every node on a separate line.
- Enter every element with start node, end node, modulus, and area.
- Add fixed degrees of freedom, such as 1x or 1y.
- Add loads by degree of freedom, such as 2y=-5000.
- Choose decimal places for the result display.
- Press the calculate button.
- Review the global matrix, reduced matrix, displacements, and reactions.
- Use CSV or PDF export to save the result.
Article: Understanding Global Stiffness Assembly
Understanding Global Stiffness Assembly
A global stiffness matrix links every free and restrained degree of freedom in a structure. It is central to the finite element method. Each member first has its own stiffness matrix. The calculator rotates that element matrix into global axes when the member is inclined. Then it places the four element degrees of freedom into the correct global rows and columns.
Why the Matrix Matters
The assembled matrix shows how the complete model resists movement. Large diagonal values mean strong resistance at that degree of freedom. Off diagonal values show coupling between two degrees of freedom. Engineers use this matrix before solving displacements, member forces, support reactions, and service checks. Students use it to confirm hand calculations and understand matrix placement.
Model Inputs
A clean model starts with node coordinates. Every node needs an identifier, an x value, and a y value. Elements connect two nodes. Each element also needs elastic modulus and cross sectional area. The calculator assumes a two dimensional truss member with axial stiffness only. Each node therefore has horizontal and vertical degrees of freedom.
Support and Load Handling
Supports remove selected degrees of freedom from the solvable system. The tool forms a reduced stiffness matrix for the remaining free degrees. Loads are entered by degree of freedom. After solving, the calculator rebuilds the full displacement vector. It also computes reactions from the complete matrix equation.
Practical Use
Use consistent units across the entire model. If modulus is in pascals, area should be in square meters, and coordinates should be in meters. The output force units will then match newtons. If using millimeters, keep modulus and area compatible. Wrong unit mixing can make results look precise but incorrect.
Checking Results
Always check element lengths and direction cosines. A zero length member cannot be assembled. A missing support can create a singular matrix. That means the structure can move as a mechanism. Add realistic restraints, then run the model again.
Exports and Records
CSV export helps you move matrix values into spreadsheets. The report button creates a simple printable record. Keep exported results with assumptions, units, and model sketches for review. This makes reviews easier when models are changed or expanded later by others.
FAQs
What does this calculator assemble?
It assembles the global stiffness matrix for a two dimensional truss model. It also forms a reduced matrix, solves free displacements when possible, and reports support reactions.
What element type is used?
The calculator uses axial truss elements. Each element carries force along its own length. Bending, shear deformation, and frame rotation degrees of freedom are not included.
How should I enter nodes?
Enter one node per line. Use the format node id, x coordinate, and y coordinate. Keep the same coordinate unit throughout the model.
How should I enter elements?
Enter one element per line. Use element id, start node, end node, elastic modulus, and cross sectional area. The start and end nodes must already exist.
What does a singular matrix mean?
A singular matrix usually means the structure is unstable or under restrained. Add realistic supports, check disconnected members, and confirm that all element lengths are valid.
Can I use different units?
Yes, but keep all units consistent. For example, use meters with square meters and pascals, or millimeters with compatible modulus and area units.
What are fixed degrees of freedom?
Fixed degrees of freedom are restrained movements. Use entries like 1x, 1y, or 3y. The calculator removes these from the reduced system.
What do the exports include?
The CSV export includes the global matrix, element summary, loads, displacements, and reactions. The PDF button creates a report from the displayed result tables.