Assignment Sample: Stress Analysis of a Simply Supported Beam

Model Answer

Introduction

Beams are among the most common structural and mechanical elements, carrying loads primarily by bending rather than by direct tension or compression. Their design relies on Euler-Bernoulli beam theory, which assumes that plane cross-sections remain plane and perpendicular to the neutral axis after deformation, that the material behaves elastically, and that deflections are small relative to the span (Hibbeler, 2017). Under these assumptions, the internal bending moment and shear force at any section can be found from statics alone for a statically determinate beam such as the simply supported case considered here. This assignment analyses a 6 m simply supported steel beam that carries a uniformly distributed load representing the self-weight of a floor deck together with a point load representing a fixed item of plant. The support reactions, shear force and bending moment diagrams, maximum bending and shear stresses, and maximum deflection are calculated in turn, and the results are checked against the allowable stress and a serviceability deflection limit (Gere and Goodno, 2018). This type of analysis is routine in the early design stage of a structural or mechanical member, where an assumed trial section is checked against the governing loads before being refined or, where the margins are large, revised down to a lighter and more economical alternative; the calculations below therefore end with a brief comment on whether the trial section chosen here is well matched to the loading or unnecessarily conservative.

Approach/Method

The analysis follows the standard sequence used for statically determinate beams. First, the support reactions are found by applying the two equilibrium equations available for a coplanar force system, ΣF_y = 0 and ΣM = 0, treating the uniformly distributed load (UDL) as an equivalent point load acting at its centroid for the purpose of taking moments (Megson, 2019). Second, the beam is cut at a series of sections and the method of sections is used to derive expressions for shear force V(x) and bending moment M(x) as functions of distance x along the span, from which the shear force diagram (SFD) and bending moment diagram (BMD) are plotted and the position and magnitude of the maximum moment identified (where V(x) = 0). Third, the maximum bending stress is calculated from the engineer’s theory of bending, σ = My/I, evaluated at the extreme fibre (y = d/2) using the second moment of area I of the assumed equivalent rectangular cross-section, and compared against the allowable stress derived from the yield strength of the specified steel grade divided by a factor of safety (Hearn, 1997). Fourth, the maximum shear stress is estimated using the approximate formula for a rectangular section, τ_max = 1.5V/A, and compared against an allowable shear stress. Finally, the maximum deflection at midspan is found by superposition of the standard closed-form deflection formulae for a UDL and for a point load applied at a distance from one support, and compared against the common serviceability limit of span/360 (Young and Budynas, 2002). Throughout, sagging bending moments are taken as positive, and shear forces are taken as positive where the net upward force to the left of a section exceeds the net downward force, consistent with the standard beam sign convention used in most UK mechanics of materials textbooks (Hearn, 1997).

Worked Solution/Analysis

Beam Data and Support Reactions

The beam AB spans L = 6 m and is simply supported at A (x = 0) and B (x = 6 m). It carries a uniformly distributed load w = 12 kN/m over the full span and a point load P = 30 kN at x = 4 m from A. Taking moments about A:

R_B × 6 = (w × L × L/2) + (P × 4) = (12 × 6 × 3) + (30 × 4) = 216 + 120 = 336 kNm
R_B = 336 / 6 = 56 kN

From vertical equilibrium, R_A + R_B = wL + P = 72 + 30 = 102 kN, so R_A = 102 − 56 = 46 kN.

Shear Force and Bending Moment Diagrams

For 0 ≤ x < 4 m, V(x) = R_A − wx = 46 − 12x, and M(x) = R_Ax − wx²/2. At x = 4 m⁻, V = 46 − 48 = −2 kN. Immediately after the point load, V drops by 30 kN to −32 kN, and for 4 m < x ≤ 6 m the shear falls linearly to V(6⁻) = −32 − 12(2) = −56 kN, which correctly equals −R_B. The maximum moment occurs where V(x) = 0 in the first segment: 46 − 12x = 0 gives x = 3.83 m, at which M_max = 46(3.83) − 12(3.83)²/2 ≈ 88.2 kNm. As a check, at x = 6 m the moment correctly returns to zero: M(6) = 46(6) − 12(36)/2 − 30(2) = 276 − 216 − 60 = 0. An intermediate value at x = 2 m, well before the point load, confirms the expected parabolic shape of the moment diagram over the UDL-only region: M(2) = 46(2) − 12(2)²/2 = 92 − 24 = 68 kNm, consistent with a smoothly increasing moment between the left support and the maximum at 3.83 m.

Bending and Shear Stress

The beam is analysed using an equivalent solid rectangular cross-section of breadth b = 200 mm and depth d = 350 mm. The second moment of area is I = bd³/12 = (200 × 350³)/12 ≈ 714.6 × 10⁶ mm⁴, and the elastic section modulus is Z = I/(d/2) = 714.6 × 10⁶/175 ≈ 4.083 × 10⁶ mm³.

The maximum bending stress is σ_max = M_max/Z = (88.2 × 10⁶ Nmm)/(4.083 × 10⁶ mm³) ≈ 21.6 N/mm² (MPa). For Grade S275 structural steel, the yield strength is 275 MPa; applying a factor of safety of 1.5 gives an allowable stress of approximately 183 MPa (Benham, Crawford and Armstrong, 1996). Since 21.6 MPa is well below 183 MPa, the section passes the bending stress check with a large margin.

The maximum shear stress in a rectangular section occurs at the neutral axis and is approximated as τ_max = 1.5V_max/A, where A = bd = 200 × 350 = 70,000 mm². With V_max = 56 kN, τ_max = 1.5(56,000)/70,000 ≈ 1.2 N/mm², which is far below the allowable shear stress for this steel grade and is not a governing consideration for this section.

It is worth noting where each of these two stresses peaks within the cross-section, since this affects whether they need to be combined. Bending stress is greatest at the extreme fibres (y = ±d/2) and falls to zero at the neutral axis, whereas shear stress in a rectangular section is greatest at the neutral axis (y = 0) and falls to zero at the extreme fibres. The two peak stresses therefore occur at different points through the depth of the section and do not need to be combined via a von Mises or principal-stress check for a solid rectangular section of this kind; this simplification would not hold for a thin-walled I-section, where the web-flange junction can carry meaningful bending and shear stress simultaneously and a combined check is normally required (Ashby and Jones, 2012).

Deflection

Taking Young’s modulus for structural steel as E = 200,000 N/mm², the midspan deflection due to the UDL alone is found from δ_UDL = 5wL⁴/(384EI). Substituting w = 12 N/mm, L = 6000 mm, E = 200,000 N/mm² and I = 714.6 × 10⁶ mm⁴ gives a numerator of 5 × 12 × (6000)⁴ ≈ 7.776 × 10¹⁶ and a denominator of 384 × 200,000 × 714.6 × 10⁶ ≈ 5.488 × 10¹⁶, so δ_UDL ≈ 1.42 mm.

The midspan deflection due to the point load alone is found using the standard formula for a load P applied at distance a from the left support, evaluated at midspan x = L/2 = 3 m (which lies on the same side as the support, since x ≤ a here): δ_P(x) = Pbx(L² − b² − x²)/(6LEI), where b = L − a = 2 m. Substituting P = 30,000 N, b = 2000 mm, x = 3000 mm and L = 6000 mm gives L² − b² − x² = 36,000,000 − 4,000,000 − 9,000,000 = 23,000,000 mm², and evaluating the full expression gives δ_P ≈ 0.81 mm.

By superposition, the total midspan deflection is δ ≈ 1.42 + 0.81 = 2.22 mm. The common serviceability limit for floor beams of L/360 gives an allowable deflection of 6000/360 ≈ 16.7 mm (Ryder, 1969). The calculated deflection of 2.22 mm is comfortably within this limit.

Evaluation

The calculated results show that the assumed 200 mm × 350 mm equivalent rectangular section is adequate on every criterion checked, with the maximum bending stress reaching only around 12% of the allowable stress, the maximum shear stress a small fraction of one percent of the section’s shear capacity, and the maximum deflection around 13% of the serviceability limit. This large margin suggests the section is conservatively over-sized for this loading, and a more economical design — for example a standard rolled steel universal beam (UB) section chosen to give a similar second moment of area with substantially less material — could reduce both cost and self-weight while still satisfying both the strength and deflection checks (Ashby and Jones, 2012). It is also worth noting that bending governs the design comfortably before shear becomes significant, which is typical for beams whose span-to-depth ratio is relatively large, as is the case here (6000/350 ≈ 17); for much shorter, deeper beams shear can become the governing criterion instead. The analysis rests on several idealisations that should be stated explicitly: the point load from the plant item is treated as acting at a single point rather than being distributed over a bearing plate, the material is assumed to remain within the linear-elastic range with no local buckling or lateral-torsional instability considered, and the UDL is assumed uniform along the full span rather than varying with the actual floor loading pattern. A full design to a structural code such as Eurocode 3 would additionally require checks for lateral-torsional buckling of the compression flange, local web buckling under the point load, and a partial safety factor approach applied separately to loads and material strengths rather than the single lumped factor of safety used here for illustration (British Standards Institution, 2005). The utilisation ratios summarised above make the relative importance of each check easy to compare at a glance: bending stress, at roughly 12% utilisation, is the most heavily loaded criterion of the three, followed closely by deflection at roughly 13%, while shear stress is negligible at around 1%. This pattern is typical of moderately long, moderately loaded floor beams, where deflection and bending stress tend to govern jointly and shear rarely controls the design unless the beam is short and heavily loaded or has a thin web. Were the trial section to be optimised further, a beam with a second moment of area roughly one third of the value used here would still satisfy both the bending stress and deflection checks with some margin, indicating that the 200 mm × 350 mm equivalent section chosen for this exercise was a safe but not especially efficient starting point. Subject to these caveats, the beam as analysed is fit for purpose for the stated loading.

Conclusion

Applying standard statics and engineer’s theory of bending to the 6 m simply supported beam gave support reactions of R_A = 46 kN and R_B = 56 kN, a maximum bending moment of approximately 88.2 kNm at 3.83 m from the left support, a maximum bending stress of approximately 21.6 MPa against an allowable stress of approximately 183 MPa, a negligible maximum shear stress of approximately 1.2 MPa, and a maximum midspan deflection of approximately 2.22 mm against a serviceability limit of approximately 16.7 mm. On this basis, the equivalent 200 mm × 350 mm rectangular section comfortably satisfies both the strength and serviceability requirements for the combined uniformly distributed and point loading described in the brief, with scope identified for a lighter, more economical section should minimising material use become a design priority (Timoshenko and Gere, 1972). More broadly, the worked example demonstrates the standard sequence used throughout mechanical and structural design practice — equilibrium, internal force diagrams, stress recovery via the flexure and shear formulae, and a deflection check against a serviceability limit — and shows how each stage feeds directly into an engineering judgement about whether, and by how much, a trial section could safely be refined.

References

• Ashby, M.F. and Jones, D.R.H. (2012) Engineering Materials 1. 4th edn. Oxford: Butterworth-Heinemann.
• Benham, P.P., Crawford, R.J. and Armstrong, C.G. (1996) Mechanics of Engineering Materials. 2nd edn. Harlow: Longman.
• British Standards Institution (2005) BS EN 1993-1-1: Eurocode 3 — Design of Steel Structures. London: BSI.
• Gere, J.M. and Goodno, B.J. (2018) Mechanics of Materials. 9th edn. Boston: Cengage Learning.
• Hearn, E.J. (1997) Mechanics of Materials 1. 3rd edn. Oxford: Butterworth-Heinemann.
• Hibbeler, R.C. (2017) Mechanics of Materials. 10th edn. Harlow: Pearson.
• Megson, T.H.G. (2019) Structural and Stress Analysis. 4th edn. Oxford: Butterworth-Heinemann.
• Ryder, G.H. (1969) Strength of Materials. 3rd edn. London: Macmillan.
• Timoshenko, S.P. and Gere, J.M. (1972) Mechanics of Materials. New York: Van Nostrand Reinhold.
• Young, W.C. and Budynas, R.G. (2002) Roark’s Formulas for Stress and Strain. 7th edn. New York: McGraw-Hill.