Elastic Behaviour of Rigidly Constrained Structure

ME 370 · Machine Design · BYU–Idaho

Topic Overview

A fixed-fixed structural member — a hollow PVC window-blinds wand — loaded at an off-center point develops both bending stress and deflection. Because the supports prevent rotation at both ends, the system is statically indeterminate: equilibrium alone can't solve it, and the fixed-end moments must be found from the constraint that the slope is zero at each support. This study analyzes that behavior across three methods and validates them against a physical experiment.

Objective

Calculate the maximum bending stress and midspan deflection of a rigidly constrained structural member under a transverse point load, then verify those predictions against measured experimental data to confirm that the fixed-fixed beam model accurately describes real elastic behavior.

Method

The member was a hollow PVC tube (OD = 0.397 in, ID = 0.272 in, L = 32.25 in, I = 0.000951 in⁴). Material was identified as PVC through a float/burn test; the modulus of elasticity was initially assumed at 400 kpsi, then back-solved to E = 668,000 psi to match experimental data. Both ends were clamped with clamps and 3D-printed v-blocks; steel washer masses were applied at a point 20.125 in from end A (4 inches from midspan), and deflection was measured with a zeroed dial indicator at three load levels (F₁ = 0.298 lb, F₂ = 0.553 lb, F₃ = 0.861 lb). Deflection was calculated three ways: hand calculations using Euler-Bernoulli beam theory and fixed-end moment formulas (yielding δ = −0.2327F), singularity functions, and a SolidWorks FEA simulation. Results from all three methods were compared against measured experimental deflections.

Key Insights

All three analytical methods agreed with experiment to within 10% error across all load levels. At the highest load, hand calculations gave δ = 0.2004 in vs. measured 0.201 in (−0.30% error); FEA predicted 0.201 in (0.00% error). The back-solved modulus of E = 668,000 psi fell within the known PVC range (200–800 kpsi) and explained the small error at lower loads. The beam behaved elastically throughout, confirming that fixed-fixed beam theory and FEA together reliably predict deflection in rigidly constrained structural members — and that the indeterminate solution is necessary and sufficient to model real behavior.