Beams, Torsion & Deflection

A robot rail can remain below yield yet deflect enough to miss a hole. A shaft can survive torque yet twist enough to destabilize a control loop. Mechanical design therefore checks both strength and stiffness.

Read shear and moment before calculating stress

For distributed transverse load \(w(x)\), shear \(V(x)\), and bending moment \(M(x)\), the sign-convention-dependent relationships are

Point loads create jumps in shear; applied couples create jumps in moment. Extrema of moment occur where shear crosses zero or at boundaries. The bending stress estimate is

so material far from the neutral axis contributes disproportionately through the second moment of area \(I\). That is why tubes and I-sections can be stiff for their mass.

Worked cantilever

A 300 mm cantilever carries 100 N at its tip. Its rectangular cross-section is 30 mm wide and 10 mm deep in the bending direction.

• Root moment: \(M=FL=30{,}000\) N·mm.
• \(I=bh^3/12=30(10^3)/12=2500\) mm\(^4\).
• Extreme-fiber stress: \(\sigma=Mc/I=30{,}000(5)/2500=60\) MPa.
• For \(E=70{,}000\) MPa, end deflection \(\delta=FL^3/(3EI)=5.14\) mm.

Rotating the same bar so 30 mm is the depth raises \(I\) by a factor of nine and cuts deflection to about 0.57 mm. Orientation changed performance without adding material.

Torsion in a circular shaft

For a solid circular shaft under torque \(T\),

where \(J=\pi d^4/32\). The fourth-power dependence makes diameter powerful. Shoulders, keyways, splines, hollow sections, and noncircular shafts require refined models.

Deflection belongs in the error budget

Add predicted elastic motion to bearing clearance, manufacturing tolerance, thermal growth, foundation motion, sensor error, and control following error. If a pick must land within ±0.5 mm, a nominal 0.4 mm frame deflection is not “small”; it consumes most of the allowance.

Design exercise

Compare three cross-sections of equal area—a flat bar, a deep rectangular tube, and a square bar—for a horizontal sensor arm. Calculate or estimate \(I\), stress, and end deflection. Then record tradeoffs in access, joining, torsional stiffness, cable routing, and cost. The lightest beam equation answer is not automatically the best machine member.