Cantilever Beam Bending (Concentrated Load)

Cantilever beam bending is a benchmark problem widely used in the industry as a starting point for FEA verification. Since there exists an exact solution for tip deflection $\delta = PL^3/(3EI)$ and maximum stress at the fixed end $\sigma_{max} = PLc/I$, it allows for quantitative checking of the implementation accuracy of numerical methods. This problem is also included in NAFEMS' introductory benchmark collection.

Exactly right. ASME V&V 10-2006 positions Code Verification as the stage to confirm the mathematical correctness of an analysis code. Demonstrating agreement between the numerical solution and problems with exact solutions is the first step. The cantilever beam is optimal as an entry point for this; within the range where Euler-Bernoulli beam theory assumptions hold, it matches exactly with 1D beam elements.

The deflection curve based on Euler-Bernoulli beam theory is as follows.

Here, $P$ is the tip load, $L$ is the beam length, $E$ is Young's modulus, and $I$ is the second moment of area. It is uniquely determined from the boundary conditions: $w=0$, $w'=0$ at the fixed end $x=0$, and moment and shear force are zero at the free end $x=L$.

Since the bending moment is maximum at the fixed end $M_{max} = PL$, the maximum bending stress occurs at the outermost fiber of the fixed end.

$c$ is the distance from the neutral axis to the outermost fiber, and $Z$ is the section modulus. For a rectangular cross-section, $I = bh^3/12$ and $c = h/2$.

Timoshenko beam theory considers shear deformation. An additional deflection increment due to shear deformation $\delta_s = \kappa PL/(GA)$ is added. $\kappa$ is the shear correction factor (5/6 for a rectangular cross-section). If the span/depth ratio $L/h$ is 10 or more, the contribution of shear deformation becomes less than 1%, so Euler-Bernoulli theory is sufficient. For deep beams with $L/h < 5$, Timoshenko theory or 3D solid elements become essential.

The standard settings are $L = 1$ m, $b = 0.1$ m, $h = 0.05$ m, $P = 1000$ N, $E = 200$ GPa, $\nu = 0.3$. For these, the theoretical values are $\delta_{tip} = 0.160$ mm, $\sigma_{max} = 240$ MPa.

HEX8 with full integration suffers from shear locking, causing the beam to behave stiffer than it actually is. Lower-order hexahedra are inherently disadvantaged in bending-dominated problems; convergence is slow unless reduced integration or the B-bar method is used. Quadratic elements can accurately represent bending deformation modes with their mid-side nodes, achieving high accuracy even with coarse meshes.

Yes. According to the FEM error estimation theorem (a priori error estimate derived from Céa's lemma), the energy norm error decreases as $O(h^p)$ for $p$-th order elements. That is, halving the element size reduces the error by about half for linear elements, and by about a quarter for quadratic elements. The essence of Code Verification is whether this theoretical convergence rate can be reproduced in actual mesh convergence studies.

A typical case is when there is a stress singularity. The fixed end of a cantilever beam is not actually a stress singularity, but depending on the implementation method of the fixed constraint, stress concentration can occur locally, degrading the convergence rate. Countermeasures include evaluating at a location sufficiently far from the constrained end, mindful of Saint-Venant's principle, or using distributed constraints.

It depends on the purpose. Beam elements directly discretize Euler-Bernoulli theory, so they reach the exact solution with the minimum degrees of freedom. When using shells or solids for verification purposes, discretization of the weak form via the Galerkin method is fundamental.

The element stiffness matrix is calculated by numerical integration.

$B$ is the strain-displacement matrix, $D$ is the material stiffness matrix, and $J$ is the Jacobian.

Exactly. For a scale like a cantilever beam, a direct method (Cholesky decomposition) is fine. When DOF exceeds tens of thousands, preconditioned CG methods are more memory efficient. The essence of this problem is confirming the accuracy of element formulation, so focus should be on element type and mesh density rather than solver selection.

In Nastran, the CBEAM element gives exact agreement for $w_{tip}$. In Abaqus, B31 (Timoshenko beam) is similar. For solid verification, Abaqus' C3D20R (quadratic hexahedron, reduced integration) is a standard. In Nastran, CHEXA (20-node).

HEX8 with full integration (2×2×2 Gauss points) locks in bending problems. Reduced integration (1×1×1) eliminates locking but risks zero-energy modes (hourglass modes). The B-bar method or EAS (Enhanced Assumed Strain) method avoids locking while also suppressing hourglassing. Abaqus' C3D8I (incompatible mode) is also an option.