5. Non-Uniform (Cantilever) Bending — Young's Modulus¶
Module
Elasticity of Solids · Module 2
Try it live
Explore this on the interactive simulator with real-time sliders and graphs: 🔬 Launch in Simulator
Aim¶
To determine the Young's modulus of the material of a beam by the method of non-uniform (cantilever) bending.
Theoretical Background¶
When a beam is clamped rigidly at one end and loaded at the free end, it is called a cantilever, and the resulting bending is termed non-uniform bending because the bending moment varies continuously along the beam's length — it is greatest at the fixed end and zero at the free end where the load is applied.
Under a load, each thin longitudinal filament of the beam is either stretched (on the convex, outer side of the bend) or compressed (on the concave, inner side), while filaments along the central 'neutral axis' remain unchanged in length. The elastic restoring forces generated by this internal stretching and compression across the beam's cross-section balance the external bending moment produced by the load at every point along the beam.
Integrating this balance along the length of the cantilever gives the depression (vertical deflection) of the free end in terms of the applied load, the beam's length, its cross-sectional dimensions, and the material's Young's modulus — the deflection increases with the cube of the length, making even modest beams quite noticeably 'springy' when long.
This experiment gives one of the most direct and intuitive routes to Young's modulus, since the depression can be measured with a travelling microscope focused on a pin fixed to the loaded end, and its clean cubic dependence on length provides an excellent internal check of the theory.
Governing Formulas¶
Depression of the free end under a load W = Mg:
Young's modulus:
where \(l\) is the length of the cantilever (clamp to load point), \(b\) the breadth, \(d\) the thickness of the beam, and \(W\) the applied load.
Simulator Controls¶
| Control | Symbol | Range | Unit |
|---|---|---|---|
| Load | M | 50 – 500 | g |
| Cantilever length | l | 15 – 45 | cm |
| Beam breadth | b | 1.0 – 3.0 | cm |
| Beam thickness | d | 0.20 – 0.80 | cm |
Readouts:
| Quantity | Symbol | Unit |
|---|---|---|
| Depression of free end | y | mm |
Procedure¶
- Launch the Non-Uniform Bending panel; set the beam breadth b, thickness d and cantilever length l to their default values.
- Vary the load M in equal steps and record the depression y of the free end shown on the readout at each step.
- Plot y against M; the points should lie on a straight line through the origin (y ∝ M for fixed geometry).
- Reset the load to a middle value and instead vary the cantilever length l; tabulate y for each length and verify the expected y ∝ l³ dependence by plotting y against l³.
- Calculate Young's modulus Y from the slope of the y vs M graph, using the known l, b and d.
Observation Table¶
Table: Depression of the free end with increasing load (l, b, d held constant)
| S. No. | M (g) | l (cm) | b (cm) | d (cm) | Depression y (mm) |
|---|---|---|---|---|---|
| 1 | |||||
| 2 | |||||
| 3 | |||||
| 4 | |||||
| 5 | |||||
| 6 |
Graph¶
Plot Depression, y (y-axis) against Load, M (x-axis).
Expected trend: a straight line through the origin (y ∝ M).
y vs M, expected to be a straight line through the origin; slope = 4gl³/(Ybd³)
Calculation¶
- Compute the slope of the y vs M graph: slope = 4gl³/(Ybd³).
- Rearrange to find Y = 4gl³/(bd³ × slope), using the fixed l, b, d for that run.
- Separately verify the y ∝ l³ dependence from the length-sweep data and estimate Y from that graph's slope as a cross-check.
Result¶
Table: Young's modulus of the beam material
| Trial | l (cm) | b (cm) | d (cm) | Slope (mm/g) | Y (dyne/cm²) |
|---|---|---|---|---|---|
| 1 | |||||
| 2 | |||||
| 3 |
Maximum Permissible Error¶
Y = 4Wl³/(bd³y) depends on the cantilever length l cubed and on the thickness d cubed (inversely) — small errors in measuring l or, especially, the thin dimension d, are amplified threefold.
Precautions¶
- Load and unload in the same steps and average the depression for loading and unloading, to eliminate any small elastic-hysteresis or clamp-slip error.
- Wait for oscillations of the loaded beam to die out completely before taking a microscope reading.
- Measure the beam thickness d at several points with a screw gauge, since Y depends on d cubed and the beam may not be perfectly uniform.
Self-Check Quiz¶
1. For a fixed load, if the cantilever length l is doubled, the depression y becomes:
- A) Twice as large
- B) Four times as large
- C) Eight times as large
- D) Unchanged
Answer: C — y ∝ l³, so doubling l increases y by a factor of 2³ = 8.
2. If the beam thickness d is halved (all else fixed), the depression y:
- A) Halves
- B) Doubles
- C) Increases eightfold
- D) Decreases eightfold
Answer: C — y ∝ 1/d³, so halving d increases y by a factor of 2³ = 8 — thin beams are dramatically more flexible.
3. Along a loaded cantilever, the bending moment is:
- A) Zero everywhere
- B) Maximum at the free end, zero at the clamp
- C) Maximum at the clamped end, zero at the free end
- D) Constant along the length
Answer: C — The bending moment at any section equals the load times its distance from that section to the free end, so it is greatest at the fixed end and zero exactly at the point of loading.
Viva-Voce Questions¶
- Why is this bending called 'non-uniform', and how does that differ from the uniform bending experiment?
- Where along the neutral axis does a longitudinal filament experience zero strain, and why?
- Why does the depression depend on the cube of the length but only the cube of the thickness (inversely), rather than some other power?
- What experimental precautions minimise systematic error from the beam's own weight and any residual bending before loading?
- How would you distinguish, experimentally, whether a beam is behaving elastically (returns to zero on unloading) or has undergone some permanent set?
- Why is a travelling microscope, rather than a ruler, used to measure the depression?