M025 · Structural mechanics

Beam Bending Laboratory

Can measured bending reveal material stiffness? Yes, within the linear range. Deflection then depends on the supports, the cross-section, the load and Young's modulus. With the supports and cross-section known, plot deflection against load: the slope gives the modulus.

Change the supports and switch between solid and hollow rectangular sections. Read off the shear force, bending moment, curvature and fiber stress, then work out Young's modulus from the measured 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 a structure, then load and measure it

Can measured bending reveal material stiffness?

Applied loads and support reactionsEach configuration keeps one fixed force scale.Beam 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

The beam is a uniform, isotropic Euler–Bernoulli beam carrying a point load and a load spread over the whole span. Upward displacement and sagging moment count as positive. Solid and hollow rectangular sections use their real dimensions. The cantilever clamp supplies a holding couple. Shear deformation, self-weight, plasticity, local buckling of the walls and lateral torsion are left out.

A warning that the beam approximation is stretched appears when the length is less than ten times the depth, the slope exceeds one tenth or the deflection exceeds two percent of the length. Bending yield is checked separately. The computed curves stay on screen, but data outside the valid range cannot be acquired. However reasonable the deformation looks, it does not show that a real structure is safe.

The raw points are simulated gauge readings, each with bounded, reproducible noise. A load sweep keeps only the loads that stay elastic. A least-squares line of displacement against force gives the compliance, and the known geometry turns it into a modulus; the fit never reads the reference modulus. Changing the load or display gain keeps the readings, while changing the geometry, material, probe or noise clears them. Measuring the same station again replaces the old reading; it is not an independent measurement. The CSV leaves out the reference modulus.

The model is linear elastic and quasi-static. Displacements are drawn magnified, but the readouts show true physical values. Simulated sensor readings are kept separate from the model's reference values. The strength limits are an idealized first check, not a certified design.

Start with the essentials

Focus question: Can measured bending reveal material stiffness?
One-sentence intuition: Geometry decides how the load travels through the structure; the cross-section and modulus decide how much it deflects under that load.

Core mathematical model

Moment and curvature

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

Upward displacement is positive. A positive, sagging moment compresses the upper fiber and stretches the lower fiber.

Second moment of area and fiber stress

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

For a solid section the inner rectangle is zero. Height runs vertically and width runs in depth.

Distributed load and the jump at a point load

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

A downward point load makes the shear force jump, while moment, slope and displacement stay continuous.

Boundary conditions matter

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

The first pair describes a simply supported beam, the second a cantilever clamped at the left end. At the free tip, moment and shear are zero apart from any load applied there.

Modulus from the slope of measured deflection

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

Fit the raw simulated deflections. The known geometry gives the compliance for a unit modulus, and a free intercept absorbs the fixed offset from the distributed load.

Common difficulties

The drawing exaggerates the bend

Typical misconceptionYou can read the real displacement straight off the bent apparatus.

Better mental modelUse the readouts or the plotted curves. The 3D gain changes only how the bend is drawn.

A red limit is not a collapse prediction

Typical misconceptionA rod or beam past an ideal limit still follows this linear model.

Better mental modelRead the red solution as an extrapolation. Plastic flow and post-buckling behavior are not modeled.

The modulus comes from the data

Typical misconceptionThe fitted modulus just repeats the reference slider.

Better mental modelClear the observations and the estimate disappears. Record several loads and change the noise to see how good the estimate is.

Run the experiment

01
Change the load path
Compare the Simply supported and Cantilever tip load presets.
What to observe: The reactions and the 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 the real deflection. Moving it clears measurements that no longer match.
03
Measure the material
Press Acquire nine load steps, add bounded sensor noise and export the raw readings.
What to observe: The slope gives the modulus, and the residuals show the sensor scatter.
04
Test the model's limits
Try Rectangular tube, then Short-beam warning.
What to observe: A low stress does not clear the warning about shear deformation.