M025 · Structural mechanics

Beam Bending Laboratory

Change supports and solid or hollow rectangular sections. Read shear, bending moment, curvature and fiber stress; infer Young modulus from acquired deflections.

Model domain $\text{—}$

Left vertical reaction $\text{—}$

Right vertical reaction $\text{—}$

Largest displacement magnitude $\text{—}$

Probe vertical displacement $\text{—}$

Maximum axial stress magnitude $\text{—}$

Largest ideal utilization $\text{—}$

Elastic strain energy $\text{—}$

Cross-section area $\text{—}$

Section second moment $\text{—}$

Actual displacement display gain $\text{—}$

Acquired load stations $\text{—}$

Modulus fitted from raw data $\text{—}$

Fit residual RMS $\text{—}$

Clamp holding moment $\text{—}$

Moment at probe $\text{—}$

Right-hand shear at probe $\text{—}$

Curvature at probe $\text{—}$

Upper fiber stress at probe $\text{—}$

Lower fiber stress at probe $\text{—}$

Length / section depth $\text{—}$

DESIGN / LOAD / MEASURE

Can measured bending reveal material stiffness?

Applied loads and support reactionsFixed force scale per configurationBeam length

L

4.00\,m

Supports: simply supported / cantilever

s

0.00\,

Section: solid / rectangular tube

c

0.00\,

Section width

b

100.00\,mm

Section depth

h

180.00\,mm

Tube wall thickness

t

8.00\,mm

Young modulus

E

200.00\,GPa

Ideal yield strength

\sigma_y

250.00\,MPa

Point load

P

12.00\,kN

Load position / span

a/L

0.50\,

Displacement probe / span

x_p/L

0.50\,

Displacement noise bound

\epsilon

0.00\,mm

Requested displacement gain

G

100.00\,

Uniform distributed load

q

2.00\,kN/m

Uniform, isotropic Euler–Bernoulli beam with a point load and full-span distributed load. Upward displacement and sagging moment are positive. Solid and hollow rectangular sections use their actual dimensions. The cantilever clamp supplies a holding couple. Shear deformation, self-weight, plasticity, local wall buckling and lateral torsion are omitted.

A length-to-depth ratio below ten, a slope beyond one tenth, or a displacement above two percent of length triggers an approximation warning. Bending yield is screened separately. Computed curves remain visible, but out-of-domain acquisition is excluded. No valid-looking deformation is evidence of a safe real structure.

Raw points are simulated gauge readings with reproducible bounded station noise. Load sweeps retain only elastic stations. Least squares fits displacement against force, then known geometry converts compliance into modulus; the fit never reads the reference modulus. Different loads and display gain preserve observations. Geometry, material, probe or noise edits clear them. Repeating a station replaces it rather than creating an independent measurement. CSV excludes the reference modulus.

Linear elastic, quasi-static model. Displacements are magnified; readouts remain physical. Simulated sensor observations are separate from model references. Strength limits are ideal screening, not a certified design.

Start with the essentials

Focus question: Can measured bending reveal material stiffness?
One-sentence intuition: Geometry fixes the load paths; cross-section and modulus fix compliance.

Core mathematical model

Moment and curvature

EI\,v^{\prime\prime}(x)=M(x)

Upward displacement is positive. Positive sagging moment makes the upper fiber compressive and the lower fiber tensile.

Section and fiber stress

I=\frac{bh^3-b_i h_i^3}{12},\qquad\sigma_x=-\frac{My}{I}

The inner rectangle vanishes for a solid section. Height follows the vertical direction; width follows depth.

Distributed load and point jump

M^{\prime}=V,\quad V^{\prime}=-q,\quad V(a^+)-V(a^-)=-P

A downward point load gives a shear jump while moment, slope and displacement remain continuous.

Boundary conditions matter

v(0)=v(L)=0\quad\text{or}\quad v(0)=v^{\prime}(0)=0

The first pair is for a simply supported beam, the second for a left cantilever clamp. The free tip has zero moment and shear beyond applied tip loads.

Modulus from an acquired slope

v_p=\alpha P+\beta,\qquad\widehat E=\frac{C_{\mathrm{geometry}}}{\widehat\alpha}

Fit the actual simulated raw deflections. Known geometry gives unit-modulus compliance; a free intercept absorbs the fixed distributed-load offset.

Common difficulties

Magnification is not deformation

Typical misconceptionThe visibly bent apparatus gives the physical displacement directly.

Better mental modelUse the readout or quantitative plot; the 3D gain changes only presentation.

Limits do not predict collapse

Typical misconceptionA rod or beam beyond an ideal limit still obeys this linear model.

Better mental modelTreat the red solution as extrapolation; plastic flow and post-buckling are omitted.

Raw data come first

Typical misconceptionThe fitted modulus simply repeats the reference slider.

Better mental modelClear observations: the estimate disappears. Acquire several loads and change noise to inspect inference quality.

Run the experiment

01
Change the load path
Compare simply supported and cantilever presets.
What to observe: Reactions and internal forces change together.
02
Move the load and probe
Drag the two colored station handles below the apparatus.
What to observe: The probe follows physical deflection; moving it clears incompatible measurements.
03
Measure the material
Acquire nine load steps, add bounded sensor noise and export raw readings.
What to observe: Slope estimates modulus; residuals reveal sensor scatter.
04
Audit the boundary
Try a hollow section, then the short-beam warning.
What to observe: A low stress does not remove a shear-deformation warning.