Materials, stress, and stiffness

• Calculate axial stress, strain, and elastic extension for a uniform member.
• Explain why material, geometry, load case, and safety factor all affect a part choice.

• forces-free-body

A part can fail by breaking, yielding permanently, bending too far, buckling, wearing, or cracking after repeated loads. “Strong” therefore is incomplete. A plastic camera bracket may safely carry its load yet flex enough to blur an image; a thin printer guide may be stiff in one direction and flexible in another because its shape, not just its material, matters.

For a straight member under axial load F, normal stress is σ = F/A in pascals, and strain is ε = ΔL/L, a dimensionless ratio. In the linear elastic range, σ = Eε, where Young's modulus E measures stiffness. Combining them gives ΔL = FL/(AE). Yield strength marks the onset of substantial permanent deformation for a simple tensile test; an allowable stress might be σ_allow = σ_y/SF using safety factor SF. Bending stress and deflection depend strongly on cross-section shape; moving material away from the neutral axis increases second moment of area and can greatly improve bending stiffness without proportional mass.

The build decision is not “metal versus plastic” in the abstract. Identify the load path, environment, manufacturing direction, allowable motion, and credible peak load. A printed polymer part is anisotropic: layer adhesion, holes, notches, heat, and creep can dominate a handbook tensile number. Cardboard corrugation likewise gives direction-dependent stiffness.

A common misconception is that doubling thickness merely doubles stiffness. In many bending geometries, thickness enters the section property with a third power, although the exact relationship depends on cross-section and boundary conditions. Another failure is applying a safety factor to an uncertain internet strength value while ignoring stress concentration at a sharp corner. Treat simple equations as screening models, inspect where load actually enters, and test representative coupons when consequences matter.

Problem

A 200 mm long aluminum tie, cross-section 3.0 mm × 10 mm, carries 900 N axial tension. With E = 69 GPa and assumed yield strength 240 MPa, find stress, extension, and yield safety factor.

Steps

1. Area A = 30 mm² = 30 × 10⁻⁶ m².
2. Stress σ = 900 N ÷ (30 × 10⁻⁶ m²) = 30 MPa.
3. Strain ε = σ/E = 30 MPa ÷ 69,000 MPa = 0.000435.
4. Extension ΔL = εL = 0.000435 × 0.200 m = 0.000087 m = 0.087 mm; safety factor = 240/30 = 8.0.

Result

The simple axial model predicts 30 MPa stress, 0.087 mm extension, and safety factor 8 against the assumed yield value.

Sanity check

The strain is well below 0.1%, so a sub-millimetre extension over 200 mm is plausible; the result does not cover holes, bending, fatigue, or buckling.

Two rods of equal material and length carry the same axial tension. Rod B has twice Rod A's cross-sectional area. In the elastic range, what are B's stress and extension?

A. Both double B. Both halve C. Stress halves but extension stays equal D. Stress stays equal but extension halves

Your answer: ______________________________

Reasoning:

Goal

Compare material stiffness without confusing predicted deflection with measured deflection.

Equipment

• Equal-length strips of cardboard, wood, or printed plastic
• Small masses
• Ruler
• Two supports
• Clamps or tape

Procedure

1. Support each strip over the same span and mark its unloaded midpoint height.
2. Predict which orientation and material will deflect least under the same small mass, stating why.
3. Add masses incrementally, record actual midpoint deflection, and remove them to check whether the strip returns.
4. Deliberate fault: introduce a small notch in a sacrificial strip away from hands; identify localized bending or early damage rather than attributing it to bulk material alone.

What to observe

• Expected: edgewise or rib-aligned specimens are usually stiffer than flat or cross-grain specimens.
• Measure: load, span, orientation, deflection, and residual set; actual values depend on imperfect supports and material variation.

Transfer challenge

Redesign a sensor shelf by changing its cross-section before simply changing to a heavier material.

Cautions

• Wear eye protection around brittle samples and stop before sudden fracture; keep loads small.

Learner: ____________________ Date: ____________________

Evidence artifact or file: ____________________________________________

Notes: