Reference worksheet · static beam model
Simply Supported Beam / Solid-Shaft Deflection Worksheet
Calculate the small-deflection midspan displacement of a uniform, prismatic, simply supported Euler–Bernoulli beam under a full-span uniform load and a point load located exactly at midspan.
Controlled inputs
Use the direct model when E, I and loading are already documented. Use the helper only for a uniform solid circular shaft whose self-weight is represented by a uniform line load. Changing a unit clears its paired numeric field and the previous result, so an existing number is never silently reinterpreted.
Reference result
Equations and dimensional boundary
For a constant-EI simply supported beam, a downward uniform line load q over the whole span and a downward center point load F are superposed:
q has dimension force/length, E has force/area, and I has length⁴. Each quotient therefore has dimension length. For nonnegative symmetric loads, the reported location is midspan and is the maximum downward deflection within this ideal model.
Optional uniform solid-shaft helper
The helper uses mass density, not weight density. For its self-weight term, substituting A and I gives δself = 5ρg₀L⁴/(24Ed²). Thus self-weight deflection scales as 1/d², while the center-force contribution scales as 1/d⁴. Increasing d from 50 mm to 60 mm reduces those terms by 30.56% and 51.77%, respectively—not one universal percentage.
What is excluded
Stepped or hollow shafts, overhangs, disks at arbitrary locations, multiple loads, nonuniform E/I, real bearing stiffness, shear deformation, local gravity, thermal bow, residual stress, assembly preload, contact, plasticity, large displacement, spin softening/stiffening, gyroscopic effects and rotor-dynamic response are outside this worksheet. Use a beam/FE/rotordynamic model that represents those features.
Independent published check
A Missouri University of Science and Technology instructional example reports a simply supported uniform-load case with q = 3 kip/ft, L = 10 ft, E = 29,000.02 ksi and I = 984 in⁴. Enter the equivalent direct inputs q = 250 lbf/in, L = 120 in, E = 29,000.02 ksi, I = 984 in⁴ and F = 0. The worksheet should reproduce approximately 0.02365 in (23.65 mil). This is a formula check, not an allowable-deflection statement.
Source classification
| Claim | Pengelasan | Sumber |
|---|---|---|
| 5qL⁴/(384EI), simply supported full-span uniform load | General Euler–Bernoulli beam solution; not an ISO formula | MIT OCW 2.082 notes; MIT OCW 1.050 formula sheet |
| FL³/(48EI), center point load | General Euler–Bernoulli beam solution; not an ISO formula | MIT OCW 2.72 lecture 3 / NPTEL vibration course |
| I = π(dₒ⁴−dᵢ⁴)/64 | Geometric second moment about a centroidal diameter | MIT OCW 2.72 lecture 3 |
| Linear superposition | Conditional model operation | MIT OCW 2.72 lecture 3 states linearity, load independence and small geometry change assumptions |
| g₀ and unit factors | Conventional metrology / exact definitions | BIPM CGPM Resolution 3-2; NIST SP 811 Appendix B |
- MIT OCW 2.082, Ship Structural Analysis & Design, buckling notes — page 1 shows the simply supported uniform-load maximum-deflection expression.
- MIT OCW 2.72, Elements of Mechanical Design, lecture 3 — center-load displacement, circular-section I and superposition assumptions.
- MIT OCW 1.050, Solid Mechanics, Problem Set 11 — beam displacement formula sheet.
- Missouri S&T, Computer Modeling lab — published numerical uniform-load check.
- BIPM, 3rd CGPM Resolution 2 — conventional standard gravity 9.80665 m/s² and mass/weight distinction.
- NIST SP 811 Appendix B.8 — inch, pound and pound-force conversion basis.
Accessed: 14 July 2026. The MIT PDF pages used above were rendered and visually checked during the audit.
Questions this worksheet does not answer
What shaft deflection is acceptable?
There is no universal 25–50 μm, 0.1 mm or “10% of bearing clearance” limit that this page can safely apply. Acceptance depends on the actual bearing, seal, coupling, rotor geometry, load case, operating state, measurement definition and controlled equipment/project requirements.
Is static sag the same as runout or 1× vibration?
No. Static elastic deflection, indicated runout and synchronous vibration are different quantities. A universal “TIR = 2 × sag” or “sag produces 1× vibration” rule is not used here; the measurement setup and rotating-system model must be defined.
Does increasing diameter always give a fourth-power reduction?
No. With fixed external q or F, I drives a fourth-power dependence for a solid circle. When q itself is the shaft’s own weight, q grows with d², so the resulting self-weight sag varies as 1/d². Mixed loads have no single diameter exponent.
