Clamped-clamped beam — central point load

A prismatic slender beam of length \(L\), clamped rigidly at both ends, carrying a central transverse point load \(P\) deflects at its mid-span by

\[\delta_{\text{mid}} = \frac{P L^{3}}{192 E I}, \qquad I = \frac{b h^{3}}{12}.\]

One-quarter the simply-supported deflection: the fixed-end moments \(\pm P L / 8\) resist rotation at the supports and stiffen the beam considerably.

Problem

Parameter	Value
Length L	1.0 m
Cross-section	0.05 m × 0.05 m (square)
Young’s modulus E	200 GPa
Poisson’s ratio ν	0.30
Central load P	1 000 N (acts in -z)
Expected mid-span deflection	P L³ / (192 E I) = 5.00 × 10⁻⁵ m

Analytical reference

Integrating Euler–Bernoulli piecewise over [0, L/2] and [L/2, L] with clamped end BCs w(0)=w'(0)=w(L)=w'(L)=0 and the mid-span shear jump from \(P\) gives the closed form above (Timoshenko, SoM Part I, §5.7).

femorph-solver result

Ran by tests/validation/test_cc_beam_central_load.py on an SOLID185 enhanced-strain hex mesh. Both end faces carry an all-DOF Dirichlet clamp — no knife-edge idealisation needed since the fixed-fixed configuration removes every support-side freedom. The central load is lumped across the mid-span bottom- line nodes; mid-span deflection is extracted from the top-face centerline to sidestep the local 3D stress-concentration.

Discretisation	Mesh (nx × ny × nz)	δ_mid (m)	Error vs Euler–Bernoulli
Coarse	20 × 3 × 3	4.967 × 10⁻⁵	−0.7 %
Medium	40 × 3 × 3	5.050 × 10⁻⁵	+1.0 %
Refined	80 × 3 × 3	5.079 × 10⁻⁵	+1.6 %

The persistent ~2 % excess at fine-mesh convergence is the same 3D Poisson contribution noted on the simply-supported beam — a solid-mesh beam picks up curvature from transverse normal stresses that pure Euler–Bernoulli ignores. Well inside the 5 % tolerance.

Cross-references

Source	Reported δ_mid (m)	Problem ID / location
Closed form (Euler–Bernoulli)	5.000 × 10⁻⁵	Timoshenko SoM Part I §5.7
Gere & Goodno (2018) §10.3 Table 10-1	5.000 × 10⁻⁵	Fixed-fixed beam, central load
femorph-solver (refined mesh)	5.079 × 10⁻⁵	test_cc_beam_central_load.py
MAPDL Verification Manual	5.00 × 10⁻⁵	VM-2 Beam stresses and deflections (clamped variant)
Abaqus Verification Manual	5.00 × 10⁻⁵	AVM 1.5.x fixed-fixed beam family

Source

Problem class: femorph_solver.validation.problems.ClampedClampedBeamCentralLoad.

Backing regression test: tests/validation/test_cc_beam_central_load.py.