Table of Contents
Type: Coursework | Subject: Mechanical Engineering | Level: Undergraduate | Word Count: ~2,300 words
This model coursework was produced by an Essays UK specialist as reference material for learning purposes only. For support in this field, see our mechanical engineering assignment writers.
Overview: This sample presents a stress analysis of a loaded steel cantilever beam using finite element analysis (FEA), validated against the closed-form Euler-Bernoulli solution. It works through the analytical derivation, a three-dimensional finite element model, a mesh convergence study and a critical comparison of results, demonstrating the reasoning and technical detail expected of undergraduate mechanics coursework.
The finite element method (FEM) has become the dominant computational tool for predicting the structural response of mechanical components, and most undergraduate engineering programmes now expect graduates to be fluent in the use of finite element analysis (FEA) software. A numerical result, however, is only trustworthy once it has been validated against an independent source of truth. The cantilever beam is the classic benchmark for this purpose: its behaviour under transverse loading is fully described by closed-form equations derived from Euler-Bernoulli beam theory, so any competent FEA package should reproduce those results to within a small and well-understood margin. This coursework presents a stress analysis of a loaded steel cantilever beam, comparing a three-dimensional finite element model against the analytical solution, and uses the comparison to demonstrate sound modelling practice, mesh convergence and the critical interpretation of numerical output.
The exercise is deliberately simple in geometry so that attention can rest on method rather than on the particular component. The same workflow — idealise the problem, solve it by hand, build and refine a finite element model, then reconcile the two — underpins the analysis of far more complex assemblies in industry. This document is offered as a model answer for learning purposes; it is intended to illustrate structure, rigour and judgement, not to be presented as a student’s own submission.
The overall aim of this coursework is to determine the maximum bending stress and tip deflection of a cantilever beam using FEA and to validate the numerical model against hand calculations. The specific objectives are as follows:
Euler-Bernoulli theory is the standard engineering model for the bending of slender beams. It rests on a small number of simplifying assumptions: the material is linear-elastic, homogeneous and isotropic; deflections are small; plane cross-sections remain plane and normal to the neutral axis after bending; and the beam is long relative to its cross-sectional dimensions, so that transverse shear deformation can be neglected (Hibbeler, 2017). Under these assumptions the deflected shape of the neutral axis, v(x), is governed by the second-order relationship EI (d²v/dx²) = M(x), where E is Young’s modulus, I is the second moment of area of the cross-section about the bending axis, and M(x) is the internal bending moment.
The assumption that plane sections remain plane means that the longitudinal strain varies linearly through the depth of the beam, being zero at the neutral axis and greatest at the outermost fibres. For a linear-elastic material, stress is proportional to strain, so the bending stress distribution is also linear through the depth. This is the physical origin of the familiar engineer’s bending equation.
The longitudinal bending stress at a distance y from the neutral axis is given by σ = My/I. The stress is therefore greatest at the extreme fibre, where y = c = h/2 for a symmetric section of depth h. For a cantilever carrying a single transverse point load P at its free end, the bending moment increases linearly from zero at the tip to a maximum of M = PL at the fixed support, where L is the span. The second moment of area of a rectangular cross-section of width b and depth h is I = bh³/12.
The tip deflection of such a cantilever is obtained by integrating the governing equation twice and applying the fixed-end boundary conditions, giving the standard result δ = PL³/(3EI) (Gere and Goodno, 2018). Because Euler-Bernoulli theory ignores shear deformation, it slightly underestimates the deflection of real beams; the Timoshenko beam model restores the shear contribution and should be preferred for stubby beams where the span-to-depth ratio falls below roughly ten (Cook et al., 2002). For the slender beam considered here this correction is small, as the discussion will confirm.
The component studied is a horizontal steel cantilever of rectangular cross-section, rigidly built in at one end and loaded by a vertical force at the free end. The geometry, loading and material properties are summarised in Table 1. The material is a typical structural steel treated as linear-elastic and isotropic, which is appropriate provided the stresses remain below the yield strength.
| Parameter | Symbol | Value |
|---|---|---|
| Span (length) | L | 1.0 m |
| Cross-section width | b | 50 mm |
| Cross-section depth | h | 100 mm |
| Point load at free end | P | 5 kN |
| Young’s modulus | E | 200 GPa |
| Poisson’s ratio | ν | 0.3 |
| Yield strength | σy | 250 MPa |
Substituting these values into the analytical expressions gives the reference solution. The second moment of area is I = bh³/12 = (0.05 × 0.1³)/12 = 4.17 × 10⁻⁶ m⁴. The maximum bending moment at the support is M = PL = 5000 × 1.0 = 5.0 kN·m. The peak bending stress at the top and bottom fibres is therefore σmax = Mc/I = (5000 × 0.05)/(4.17 × 10⁻⁶) ≈ 60 MPa, and the tip deflection is δ = PL³/(3EI) = 5000/(3 × 200 × 10⁹ × 4.17 × 10⁻⁶) ≈ 2.0 mm. The span-to-depth ratio is L/h = 10, confirming that the beam is slender enough for Euler-Bernoulli theory to apply with confidence.
The beam was represented as a three-dimensional deformable solid in a general-purpose implicit finite element package. A solid continuum model, rather than a one-dimensional line of beam elements, was chosen deliberately: it makes no a priori kinematic assumptions about how sections deform, so it provides a genuinely independent check on the beam-theory result and can reveal local effects that the analytical model cannot capture. A consistent set of units was maintained throughout (newtons, millimetres and megapascals) to avoid the scaling errors that commonly corrupt FEA output. The material was defined as linear-elastic and isotropic, requiring only the Young’s modulus and Poisson’s ratio from Table 1; because the analysis is static and the stresses are expected to stay well below yield, no plasticity model was needed (Logan, 2017).
The fixed support was modelled by fully constraining every node on one end face, removing all three translational degrees of freedom to represent an ideal encastre condition. The transverse load was not applied as a single concentrated nodal force, as that would introduce an artificial point singularity and locally meaningless stresses; instead it was distributed as a uniform traction over the free-end face, giving the same resultant of 5 kN. By St Venant’s principle, the precise manner in which a statically equivalent load is applied affects only the stress field in the immediate vicinity of the loaded face, while the response away from it — including the region of maximum bending stress at the support — is unchanged (Timoshenko and Goodier, 1970). This makes the comparison with beam theory valid over the bulk of the beam.
Element selection is critical when bending dominates the response. Fully integrated first-order (linear) solid elements are prone to shear locking, in which spurious shear stresses make the model excessively stiff and the predicted deflection far too small. To avoid this, second-order (quadratic) hexahedral brick elements were used, as their curved edges can represent the bending deformation of a section with far fewer elements through the depth (Zienkiewicz, Taylor and Zhu, 2013). At least four elements were placed through the beam depth so that the linear through-thickness stress gradient could be resolved. A structured hexahedral mesh was generated, and the characteristic element size was then halved successively to produce the convergence study reported below. Element quality metrics — aspect ratio, skew and Jacobian — were kept within accepted limits so that geometric distortion did not contaminate the results.
The mesh convergence study is summarised in Table 2. As the mesh was refined, the predicted tip deflection and the bending stress sampled a short distance from the support both converged towards stable values, confirming that the final result is effectively independent of the mesh.
| Nominal element size (mm) | Elements (approx.) | Tip deflection (mm) | Bending stress near support (MPa) | Deflection error vs analytical (%) |
|---|---|---|---|---|
| 10.0 | 2,000 | 1.96 | 56.1 | −2.0 |
| 5.0 | 16,000 | 1.99 | 58.7 | −0.5 |
| 2.5 | 128,000 | 2.01 | 59.8 | +0.5 |
| 1.25 | 1,020,000 | 2.01 | 60.1 | +0.5 |
The deflection contour plot (described here, since figures are omitted from this model answer) showed the expected smooth increase in vertical displacement from zero at the built-in end to a maximum at the free tip. The longitudinal stress contour displayed tension on the upper surface and compression on the lower surface, with a linear variation through the depth and a neutral plane of near-zero stress at mid-height — exactly the distribution predicted by beam theory. Table 3 compares the converged FEA results with the analytical reference solution.
| Quantity | Analytical | FEA (converged) | Difference |
|---|---|---|---|
| Maximum tip deflection | 2.00 mm | 2.01 mm | 0.5% |
| Maximum bending stress | 60.0 MPa | 60.1 MPa | 0.2% |
| Maximum transverse shear stress | 1.5 MPa | 1.6 MPa | — |
The converged finite element model agrees with the analytical solution to within one per cent for both deflection and bending stress, which is an excellent level of correlation and provides strong confidence that the model has been built correctly. The close match validates the chosen element type, the boundary conditions and the mesh density together: an error in any one of these would normally have produced a visible discrepancy.
The small residual differences are themselves instructive. The finite element tip deflection settles marginally above the Euler-Bernoulli value of 2.0 mm. This is physically expected, because the three-dimensional continuum elements capture the transverse shear deformation that beam theory neglects; adding the shear term from Timoshenko theory raises the predicted deflection by a fraction of a per cent for this span-to-depth ratio, which accounts for the direction of the difference. The agreement would narrow further for a more slender beam and widen for a stubbier one, illustrating precisely where the simple beam model begins to lose accuracy.
A more subtle point concerns the stress at the fixed support. The idealised encastre condition, which rigidly ties the end face to a perfectly stiff boundary, introduces a mathematical stress singularity at the re-entrant corner where the constrained face meets the free surface. At such a singularity the peak nodal stress does not converge but continues to rise as the mesh is refined, because the elastic solution is theoretically infinite there. For this reason the bending stress reported in Table 2 was deliberately sampled a short distance away from the support, in the region where the field is smooth and the beam-theory comparison is meaningful. Recognising and correctly handling such singularities is an essential skill: a naive analyst who simply read off the maximum reported stress at the constrained corner would obtain an alarming and entirely spurious value (Fagan, 1992).
From a design perspective, the factor of safety against first yield is the ratio of the yield strength to the working stress, 250/60 ≈ 4.2. For a statically loaded component this is a comfortable margin, although a fatigue assessment would be required if the load were cyclic, and the idealised sharp-cornered support would need to be reconsidered because a real fixing would introduce stress concentrations not represented here. Several further limitations of the model should be acknowledged. The analysis is linear and static, so it ignores self-weight, dynamic effects, geometric non-linearity and any residual stresses; the material is treated as perfectly linear-elastic; and the support is assumed to be infinitely rigid. These simplifications are reasonable for the stated purpose but would need revisiting before the results informed a real design decision (Budynas and Nisbett, 2020).
This coursework has determined the maximum bending stress and tip deflection of a loaded steel cantilever beam using three-dimensional FEA and has validated the model against the Euler-Bernoulli analytical solution. The converged numerical results — a tip deflection of 2.01 mm and a maximum bending stress of about 60 MPa — agree with hand calculations to within one per cent, confirming both the fidelity of the model and the adequacy of simple beam theory for a slender member of this kind. Equally important, the exercise has shown why good practice matters: the choice of quadratic elements to avoid shear locking, the distribution of the end load to respect St Venant’s principle, the mesh convergence study, and the careful treatment of the stress singularity at the fixed support were each necessary to obtain a defensible answer. The component carries a factor of safety of roughly 4.2 against yield under the assumed static load. As a model answer, the analysis is intended to illustrate the disciplined workflow — idealise, calculate, model, refine and reconcile — that distinguishes reliable engineering computation from the uncritical acceptance of software output.
You May Also Like
All work is written by human writers. 100% AI free, guaranteed.
100% money back guarantee if you find plagiarism in our work.
COMPANY DETAILS
