Problem statement and knowns/unknowns
This sheet combines first-pass structural checks that are often done in separate tools. You can estimate internal force effects and then evaluate stress-state implications in a single workflow.
- Compute support reactions and maximum bending moment for a simply supported beam.
- Estimate maximum normal stress from section geometry and load level.
- Calculate principal and maximum shear stresses with Mohr-circle equations.
Recommended workflow
Set beam variables first (`L`, `w`, `E`, `I`, `c`), then check `M_max`, `sigma_max`, and `delta_max`. After that, update the plane-stress terms (`sigma_x`, `sigma_y`, `tau_xy`) and review `sigma_1`, `sigma_2`, and `tau_max`.
Assumptions, validity limits, and unit discipline
- The beam formulas assume a simply supported, prismatic beam with uniform distributed load.
- `delta_max=5*w*L^4/(384*E*I)` is an Euler-Bernoulli small-deflection relation for linear-elastic behavior with constant `E` and `I`.
- Use consistent units across all rows, for example `L` in m, `w` in N/m, `E` in Pa, `I` in m^4, and `c` in m.
- For point loads, mixed loading, shear deformation, material nonlinearity, or large deflection, replace the default rows with an appropriate model.
Derivation and governing equations
R_A = R_B = wL/2
M_max = wL^2/8
sigma_max = M_max*c/I
delta_max = 5wL^4/(384EI)FAQ
Is this limited to uniform loads?
Yes for the default equations. Replace moment and deflection rows for point loads or mixed loading and keep the same sheet layout.
Can I adapt this for different units?
Yes. Keep one coherent unit system in every row so stress and deflection outputs remain physically valid.
Worked example (interpreted)
Step 1: With L=6 m and w=4500 N/m, reactions are R_A=R_B=13,500 N; the force-balance check row eq_force_error is zero.
Step 2: Maximum moment is 20,250 N*m, producing sigma_max=506 MPa with the current section assumptions.
Step 3: Deflection computes to 0.1187 m versus L/360=0.0167 m, so utilization is 7.12.
Interpretation: this starter geometry is intentionally overstressed for teaching. The worksheet should be used to iterate section stiffness and loading until utilization is below 1.0.
When this model is invalid
| Condition | Why invalid | Use instead |
|---|---|---|
| Deep beams or shear-dominated response | Euler-Bernoulli neglects shear deformation. | Timoshenko beam formulation or FEA with shear terms. |
| Large deflection / geometric nonlinearity | Small-deflection assumptions break as rotations grow. | Geometrically nonlinear beam analysis. |
| Point loads dominate | Uniform-load rows underpredict local effects. | Use the point-load variant rows (P, M_max_point, delta_max_point) and superposition as needed. |
How to validate your worksheet
eq_force_errorshould remain near zero; non-zero means load/support mismatch.delta_marginshould be positive for acceptable serviceability.invalidity_proxy = utilization_deflection - 1should be less than or equal to zero for a pass case.
Domain-specific variants in this template
The sheet includes a point-load branch (P) so you can compare uniform-load and midspan-point-load cases side-by-side without deleting baseline rows.
Practice problems (with answer outlines)
Practice 1
Recompute reactions and moment with a new uniform load while keeping span fixed.
Outline: Update w, then evaluate R_A, R_B, and M_max from equilibrium formulas.
Practice 2
Find minimum I to satisfy δ_{max}le L/360 at baseline load.
Outline: Rearrange deflection relation for I and verify utilization row falls below 1.
References and standards
- R.C. Hibbeler, Structural Analysis (textbook baseline). Edition reference.
- AISC 360 / Steel Construction Manual (design-code context for stress/serviceability checks). AISC resources.
- Roark's Formulas for Stress and Strain (practical closed-form beam cases). Reference page.