Propped cantilever — central point load

A propped cantilever is a beam clamped at one end (x = 0) and simply supported at the other (x = L). The extra support makes the problem statically indeterminate — the four reactions (R_A, M_A, R_B, plus the compatibility constraint on displacement at B) cannot be solved from statics alone; one integration step of the Euler–Bernoulli equation provides the missing compatibility equation.

Under a central point load \(P\) at \(x = L/2\):

\[\begin{split}\delta_{\text{mid}} &= \frac{7 P L^{3}}{768 E I}, \\ R_A &= \frac{11 P}{16} \text{ (fixed end)}, \\ R_B &= \frac{5 P}{16} \text{ (simple end)}, \\ M_A &= -\frac{3 P L}{16} \text{ (fixed-end moment)}.\end{split}\]

The 11/16 + 5/16 partition is the classic fingerprint of the propped cantilever — the fixed end carries more of the load than the simple end by a factor of 2.2.

Problem

Parameter	Value
Length L	1.0 m
Cross-section	0.05 m × 0.05 m (square)
Young’s modulus E	200 GPa
Central load P	1 000 N
Expected δ_mid	7 P L³ / (768 E I) = 8.75 × 10⁻⁵ m

femorph-solver result

Ran by tests/validation/test_propped_cantilever.py on an SOLID185 enhanced-strain hex mesh. BCs:

• x = 0: full all-DOF clamp on the left face.

• x = L: knife-edge roller at the bottom line (UZ = 0 along the full width), with UY = 0 pinned at one corner to kill the lateral rigid-body mode. UX left free so axial strain isn’t locked.

The central load is lumped across the mid-span bottom-line nodes; mid-span deflection is extracted from the top-face centerline (same as the static SS + CC beam problems, to avoid the local-load 3D stress-concentration indentation).

Refinement	Mesh (nx × ny × nz)	δ_mid (m)	Error vs Euler–Bernoulli
Coarse	20 × 3 × 3	8.713 × 10⁻⁵	−0.43 %
Medium	40 × 3 × 3	8.809 × 10⁻⁵	+0.68 %
Refined	80 × 3 × 3	8.843 × 10⁻⁵	+1.07 %

The ~1 % positive drift at fine-mesh convergence is the same 3D Poisson contribution the SS and CC beam problems exhibit — a solid-mesh beam picks up curvature from transverse normal stresses that pure Euler–Bernoulli ignores.

Cross-references

Source	Reported δ_mid (m)	Problem ID / location
Closed form (Gere & Goodno)	8.75 × 10⁻⁵	§10.3 Table 10-1 Case 6
Timoshenko (1955)	8.75 × 10⁻⁵	SoM Part I §5.8
femorph-solver (refined)	8.843 × 10⁻⁵	test_propped_cantilever.py
MAPDL Verification Manual	8.75 × 10⁻⁵	VM-41 family (propped cantilever)
Abaqus Verification Manual	8.75 × 10⁻⁵	AVM 1.5.x propped-cantilever family

Source

Problem class: femorph_solver.validation.problems.ProppedCantileverCentralLoad.

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