Checking beam deflection with explicit section stiffness
Deflection depends on load, span, supports and flexural stiffness EI. A stress check alone does not establish acceptable displacement. Use section properties about the actual bending axis and an explicitly chosen deflection criterion.
Worked numbers are illustrative. Use your own operating conditions and documented material/process data.
Required inputs
- Support condition, span and load positions/distributions.
- Elastic modulus E at the material condition and second moment I about the bending axis.
- Deflection criterion and whether self-weight has been included as an explicit load.
Governing equations
EI y″(x) = M(x) under a consistent sign convention
Simply supported, central point: |δmax| = PL³/(48EI)
Simply supported, full UDL: |δmax| = 5wL⁴/(384EI)
Variables and units
- Using N and mm: P in N, w in N/mm, L in mm, E in N/mm², I in mm⁴ and δ in mm.
- I is second moment of area; Z is section modulus and cannot replace I.
Assumptions
- Prismatic slender beam, small deflection and linear-elastic bending.
- The example uses custom explicit properties, not an unsupported standard-section designation.
Calculation workflow
- Check force and moment equilibrium first.
- Use the closed form only when support and load conditions match exactly; otherwise use the supported beam model.
- Compare displacement with a stated project criterion and assess omitted stability/connection effects separately.
Worked example
- L = 4000 mm; P = 25000 N at centre; full-span w = 5 N/mm; E = 200000 N/mm²; I = 8330000 mm⁴.
- Point contribution = 20.008003 mm; UDL contribution = 10.004002 mm.
- By linear superposition, maximum central displacement magnitude = 30.012005 mm; reactions are 22.5 kN each.
Limitations
- Shear deformation, yielding, creep, buckling, connection flexibility and structural-code load combinations are not established by this model.
- Central-point formulas are not valid for arbitrary off-centre loads.
Common mistakes
- Using cm⁴ as mm⁴ without the factor 10000.
- Changing the section axis without changing I.
- Assuming a fixed end where the actual connection cannot restrain rotation.
Practical interpretation
These properties deliberately differ from the workspace’s standard ISMB 150 example. Doubling E or I halves displacement for the same load/support geometry. A commonly quoted L/250 limit is not applied automatically: the project and applicable design rules must define the criterion.
Related formulas
The governing equations, units and calculation steps above provide the formula basis for this guide. The related workspace also shows its calculation trace.
Elementary beam deflection equations →Related workspace
Open Beam Analysis →Sources and reference basis
Equations and worked examples are independently explained here. References reviewed 2026-10-17; no copyrighted standard tables are reproduced.