Bending - Wikiparency

# Bending **Bending** is the behaviour of a slender structural element subjected to a load applied perpendicular to its long axis. An element loaded this way is a [[Beam_(structure)|beam]], and the load reaches its supports not by being carried along the member as a push or a pull, but by being converted into a [[Bending_moment|bending moment]] that the section resists with [[Stress_(mechanics)|stress]] varying across its depth: tension on the face that is stretched, compression on the face that is shortened, and a surface between them, the [[Neutral_axis|neutral axis]], where the stress is zero. Because the resisting stress grows with distance from the neutral axis, material far from that axis is worth far more than material near it, and the whole art of designing in bending is the art of putting material where the lever arm is long. The [[Second_moment_of_area|second moment of area]] I measures how well a section does this, and it rises with the cube of the depth — which is why floor joists stand on edge, why an [[I-beam|I-beam]] has flanges, and why a tube outperforms a solid rod of the same mass. The same reasoning, carried into two dimensions, gives the theory of plates, and carried into time, the theory of beam and plate vibration. The framework microsim *Bending: the shear, the moment and the sag of a loaded beam* draws the three diagrams a designer sketches before anything else — shear force `V(x)`, bending moment `M(x)` and deflection `y(x)` — for a beam whose supports, load and cross-section the reader chooses, so that it is clear at a glance which of the three moves when a choice changes: the moment sets the stress, and the section's I sets the sag. ## Quasi-static bending of beams In quasi-static bending the load is applied slowly enough that inertia plays no part, and the beam is in equilibrium at every instant. Three quantities describe what is happening inside it. The [[Shear_force|shear force]] `V(x)` is the transverse force transmitted across the section at x; the bending moment `M(x)` is the couple transmitted across it; and the [[Deflection_(engineering)|deflection]] `y(x)` is how far the axis has moved. They are not independent. For a distributed load `w(x)` acting downward, equilibrium of a slice gives `dV/dx = −w` and `dM/dx = V`, so the moment diagram is the integral of the shear diagram and the shear diagram is the integral of the load. A point load puts a step in the shear diagram and a kink in the moment diagram; the moment reaches an extremum where the shear passes through zero, which is the quickest way to find the worst section by hand. What the supports can resist decides the rest. A simple support carries force but no moment, a fixed or built-in support carries both, and a free end carries neither. A [[Cantilever|cantilever]] fixed at one end takes its whole load as a moment at the root; the same beam simply supported at both ends splits the load between them and carries a much smaller moment at midspan. That distinction is worth a factor of four in stress and forty in deflection for the same beam and load, and it is usually the cheapest change a designer can make. ### Euler–Bernoulli bending theory The classical theory rests on one kinematic assumption: plane sections perpendicular to the beam axis remain plane and perpendicular to it after bending. Shear deformation is therefore neglected, deflections are assumed small, and the material is assumed to obey [[Hooke's_law|Hooke's law]], `σ = E·ε` with E the [[Young's_modulus|Young's modulus]].[^up12-3] The assumption makes the axial strain vary linearly with distance from the neutral axis, so the stress does too, and integrating the stress over the section to recover the moment gives the two central results of [[Euler–Bernoulli_beam_theory|Euler–Bernoulli beam theory]]: `σ = M·c/I`, and `E·I·y'' = M(x)`, where c is the distance from the neutral axis to the point of interest, I the second moment of area, and `E·I` the flexural rigidity. The neutral axis passes through the [[Centroid|centroid]] of the section for a beam in pure bending of a material equally stiff in tension and compression. The ratio `S = I/c` is the [[Section_modulus|section modulus]], and writing the first result as `σ = M/S` reduces a strength check to one division. A worked case makes the scaling concrete. Take a steel cantilever 2 m long of rectangular section 50 mm wide and 100 mm deep, with `E` = 207 GPa, carrying a 5 kN load at its tip. Then `I = b·h³/12` = 4.17 × 10⁻⁶ m⁴, the root moment is 10 kN·m, the extreme-fibre stress is `M·c/I` = 120 MPa, and the tip deflection is `P·L³/(3·E·I)` = 15.5 mm, about one part in 129 of the span.[^johnson009] Lay the same section on its side, 100 mm wide and 50 mm deep, and nothing about the material or the load has changed, yet I falls to a quarter of its value: the stress doubles to 240 MPa and the deflection quadruples to 62 mm. Support the original beam at both ends instead, with the same load at midspan, and the moment falls to `P·L/4` = 2.5 kN·m, the stress to 30 MPa and the deflection to `P·L³/(48·E·I)` = 0.97 mm. Two of the microsim's conventions should be read as drawing rather than as data. The deflected shape is exaggerated, because a real beam's sag is invisible at the scale of the beam — the 15.5 mm above is a hundredth of the span — so the picture is ILLUSTRATIVE and the readout carries the number. The yield level the sim draws across the stress bar is 250 MPa, a nominal figure for mild steel and an ILLUSTRATIVE threshold rather than a property of any particular grade; a real check uses the specified minimum yield strength of the steel actually ordered.[^simbend] *Try: start with the cantilever and a tip point load and watch V(x) stay flat while M(x) ramps to a peak at the root; switch the supports to simply supported and see the moment peak drop to a quarter of it; then hold the load and the span and change the section from rectangle to I-beam to tube, and watch the deflection curve flatten while the shear diagram does not move at all.* ### Extensions of Euler-Bernoulli beam bending theory The classical theory assumes a linear material, small deflections, bending about a principal axis, and a section that does not change shape. Relaxing each assumption gives a recognised extension. **Plastic bending.** Past the [[Yield_(engineering)|yield strength]] the outer fibres stop following the straight line while the fibres nearer the neutral axis are still elastic, and the stress distribution changes from a triangle to a trapezoid and finally, when the whole section has yielded, to two rectangles. The moment at that limit is the plastic moment `Mp = σy·Z`, where Z is the plastic section modulus, and the ratio `Z/S` is the shape factor — exactly 1.5 for a rectangle, so the cantilever above would reach first yield at 20.8 kN·m and full plasticity at 31.3 kN·m on the sim's nominal 250 MPa. That reserve beyond first yield is the basis of plastic design in [[Structural_steel|steelwork]], and it does not exist in brittle materials, which is why cast iron and concrete are designed on the elastic distribution. [[Plasticity_(physics)|Plastic]] hinges forming at successive sections are also what turns a redundant frame into a mechanism, the collapse mode a structural engineer checks for. **Complex or asymmetrical bending.** When the moment does not act about a principal axis of the section, or when the section has no axis of symmetry, the neutral axis is no longer perpendicular to the plane of loading, and the beam deflects sideways as well as downward. The stress is then computed from the moments about both axes together with the product of inertia of the section, or, equivalently, by resolving the moment onto the principal axes and superposing. Angles, channels and unsymmetrical built-up sections all bend this way, and a channel loaded through its centroid rather than its shear centre twists as well as bends. **Large bending deformation.** If the deflection is not small compared with the span, the approximation of curvature by `y''` fails and the exact curvature must be used, giving the nonlinear problem of the [[Elastica_theory|elastica]] first solved for the buckled rod. Leaf springs, fishing rods, flexures in precision mechanisms and the compliant links of soft robots all work in this regime, where the geometry changes enough during loading to change the load path itself. ### Timoshenko bending theory Euler–Bernoulli theory discards shear deformation, and for a slender beam that is harmless: the shear contribution to the tip deflection of the cantilever above is well under one per cent. For a short deep beam it is not. Timoshenko bending theory keeps the assumption that sections remain plane but drops the assumption that they remain perpendicular to the axis, introducing a rotation of the section independent of the slope of the axis; the difference between the two is the shear strain. The result is a pair of coupled equations rather than one fourth-order equation, and a stiffness that includes an effective shear rigidity `κ·G·A`, with G the [[Shear_modulus|shear modulus]] and κ a shear correction factor depending on the shape of the section — 5/6 for a rectangle by the usual convention. The correction matters when the span-to-depth ratio falls below roughly ten, and it matters much sooner for [[Sandwich-structured_composite|sandwich panels]] and for [[Composite_material|composites]], whose cores and matrices are soft in shear compared with their faces and fibres. It matters again at high frequency, where even a slender beam's short-wavelength modes see it as deep, which is why the dynamic section below has a Timoshenko entry of its own. ## Beams on elastic foundations A beam does not have to rest on discrete supports. A [[Railway_track|railway rail]] rests on sleepers on ballast, a [[Pipeline|pipeline]] on soil, a [[Shallow_foundation|raft foundation]] on the ground beneath it, and in each case the support is distributed and yields in proportion to how far it is pushed down. The standard idealisation, due to Winkler, replaces the supporting medium by a bed of independent linear springs of modulus k, so that the reaction per unit length is `k·y` and the governing equation gains a term: `E·I·y'''' + k·y = q(x)`. Its solutions are exponentially decaying oscillations governed by the characteristic length `1/β` with `β = (k/(4·E·I))¹ᐟ⁴`. The practical consequence is localisation: a point load on such a beam is felt over a few multiples of `1/β` and essentially not at all beyond, so a long rail behaves as though only a short length of it were engaged by each wheel, and the moment under a wheel can be computed without knowing where the next one is. Stiffening the beam spreads the load, stiffening the foundation concentrates it, and the pattern of alternating positive and negative moment with distance from the load is what produces the characteristic uplift of a rail a metre or two ahead of a passing train. The model's weakness is the assumption that the springs are independent: real soil transmits shear between neighbouring points, so a real settlement trough is smoother and wider than Winkler predicts. Two-parameter foundation models and full elastic half-space solutions correct this at the cost of the closed forms, and [[Finite_element_method|finite element]] analysis is now usual where the answer matters. The Winkler form nonetheless remains the first estimate in track and pipeline work because it is a single differential equation with a single soil number.[^hetenyi] ## Dynamic bending of beams When a beam is loaded quickly, or is free to [[Vibration|vibrate]], the inertia of the beam itself enters the equations. Bending [[Wave|waves]] are the result, and they behave quite unlike the waves on a string: a beam is a dispersive medium, in which the wave speed depends on frequency, so a sharp impulse spreads as it travels and high-frequency components outrun low-frequency ones. The audible consequence is the ringing "chirp" of a struck rail or a long metal bar. ### Euler–Bernoulli theory Adding the inertia term to the static equation gives the classical beam equation for free vibration, `E·I·∂⁴w/∂x⁴ + ρ·A·∂²w/∂t² = 0`, with ρ the density and A the area. Substituting a travelling wave shows that the angular frequency is proportional to the square of the wavenumber, the dispersion relation that distinguishes bending waves from the non-dispersive waves of a taut string or a sound field. Free vibrations of a finite beam are the [[Normal_mode|normal modes]] that satisfy its end conditions, and for each of them the frequency follows from a dimensionless eigenvalue of the boundary problem. A simply supported beam has the exceptionally clean result `ωn = (n·π/L)²·√(E·I/(ρ·A))`; a cantilever's first mode has `β₁·L` = 1.875. For the steel cantilever of the worked example above — 2 m, 50 × 100 mm, so `ρ·A` = 39.25 kg/m at 7,850 kg/m³ and `E·I` = 862 kN·m² — the first [[Natural_frequency|natural frequency]] is 20.7 Hz, and the same beam simply supported at both ends comes out at 58.2 Hz. These numbers are the reason a designer who has just sized a beam for stress checks it again for [[Resonance|resonance]]: a floor beam with a fundamental near the 2 Hz of walking, or a machine bed near the running speed of what it carries, is a stiffness problem that no amount of strength will fix. ### Timoshenko–Rayleigh theory The classical dispersion relation cannot be right at high frequency, because it predicts a [[Phase_velocity|phase velocity]] that grows without limit as the wavelength shrinks. Two corrections fix it. Rayleigh's is the rotary inertia of the section, ignored in the classical theory, which resists the rapid rotation that short-wavelength bending demands; Timoshenko's is the shear flexibility described above. Including both gives a fourth-order system with two branches, of which the lower tends at high frequency to the speed of a shear wave in the material rather than to infinity, and the upper is a shear-dominated mode with a cut-off frequency below which it does not propagate. For free vibrations the practical effect is that the classical theory is accurate for the first few modes of a slender beam and progressively overestimates the frequencies of the higher ones. The error grows with mode number because each successive mode has a shorter half-wavelength, and it is the ratio of that half-wavelength to the section depth, rather than the slenderness of the whole beam, that decides whether the correction is needed. Structural dynamics and the analysis of impact, where high modes carry much of the response, use the corrected theory as a matter of course. ## Quasistatic bending of plates A plate is the two-dimensional counterpart of a beam: a flat body thin compared with its other dimensions, loaded transversely and resisting by bending, and the subject of [[Plate_theory|plate theory]]. The difference is not merely one of dimension. A plate can carry load by bending in two directions at once, and the two are coupled through [[Poisson's_ratio|Poisson's ratio]], so a plate strip is stiffer than a beam of the same section by a factor of `1/(1 − ν²)` — about 10 per cent for metals. The corresponding rigidity is `D = E·h³/(12·(1 − ν²))`, with h the thickness, and the cubic dependence on thickness is the same lever-arm effect that governs beams. ### Kirchhoff–Love theory of plates The thin-plate theory, or [[Kirchhoff–Love_plate_theory|Kirchhoff–Love plate theory]], makes the plate analogue of the Euler–Bernoulli assumption: straight lines normal to the mid-surface remain straight, unstretched and normal to it after deformation, so transverse shear strain is neglected. The deflection then satisfies the [[Biharmonic_equation|biharmonic equation]] `∇⁴w = q/D`, and the stresses vary linearly through the thickness exactly as in a beam. The theory's awkwardness is at its edges: the fourth-order equation admits only two boundary conditions per edge, so the three natural conditions — moment, twisting moment and shear — must be combined into an effective shear, which is the origin of the concentrated corner forces that appear at the corners of a simply supported rectangular plate and of the corner uplift that has to be held down in practice. Floor slabs, ship decks, aircraft skins and pressure-vessel heads are all designed on this theory, usually through tabulated coefficients rather than by solving the equation.[^theory-plates] ### Mindlin–Reissner theory of plates The thick-plate theories, of which [[Reissner–Mindlin_plate_theory|Mindlin–Reissner plate theory]] is the standard form, stand in the same relation to Kirchhoff–Love as Timoshenko does to Euler–Bernoulli: they allow the normals to rotate relative to the mid-surface, admitting transverse shear deformation and a shear correction factor. They matter for plates whose thickness exceeds roughly a tenth of the span, for sandwich and laminated construction where the core is soft in shear, and for higher vibration modes. They also have a practical advantage in computation: the governing equations are second order rather than fourth, so [[Finite_element_method|finite elements]] need only continuity of displacement across element boundaries instead of continuity of slope, which is why most general-purpose shell elements are built on a Mindlin–Reissner basis, with special measures to avoid the artificial stiffening known as shear locking in the thin limit. ## Dynamic bending of plates Adding inertia to the plate equation gives `D·∇⁴w + ρ·h·∂²w/∂t² = 0`, whose free solutions are the plate's normal modes. The modes are two-dimensional, indexed by two integers for a rectangle, and the frequencies do not fall into a harmonic series — which is why a struck plate or a cymbal gives a clang rather than a pitch, while a struck string gives a note. ### Dynamics of thin Kirchhoff plates For a simply supported rectangular plate of sides `Lx` and `Ly` the mode shapes are products of half sine waves and the frequencies follow `ωmn = π²·((m/Lx)² + (n/Ly)²)·√(D/(ρ·h))`, the plate counterpart of the beam result above and, in form, the same two-index [[Standing_wave|standing wave]] that governs a rectangular membrane. Each mode has nodal lines along which the plate does not move, and sand sprinkled on a driven plate collects along them — the figures [[Ernst_Chladni|Ernst Chladni]] drew in the eighteenth century, and still the quickest way to see a plate's mode shape without instruments. Bending waves in plates are dispersive for the same reason they are in beams, and their speed at audio frequencies is comparable with the [[Speed_of_sound|speed of sound]] in air. The frequency at which the two coincide is the critical frequency, above which a panel radiates sound efficiently and below which it does not, and it is the single most useful number in predicting how much noise a partition, a floor or a machine casing will let through. Thin-plate dynamics is therefore as much a subject of [[Soundproofing|soundproofing]] and [[Noise_control|noise control]] as of structural analysis. ## Minnesota *This section is specific to Wikitube.* The [[I-35W_Mississippi_River_bridge|Interstate 35W Mississippi River bridge]] in Minneapolis, [[Minnesota]], collapsed during the evening rush on August 1, 2007, killing 13 people and injuring 145. The [[National_Transportation_Safety_Board|National Transportation Safety Board]] concluded that the probable cause was the inadequate load capacity of the gusset plates at the U10 nodes of the deck truss — plates about half an inch (13 mm) thick where the design required twice that — which had been undersized by an error in the original design and which finally failed under the weight of the structure, its accumulated deck resurfacings and the construction loads stockpiled over those nodes that day.[^ntsb] The bridge is a lesson about where bending goes in a truss. A deck truss does not carry the deck's bending in its members' sections; it converts that bending into axial tension and compression in the chords and diagonals, and the [[Gusset_plate|gusset plates]] are where those forces meet and are redirected. A plate that is too thin buckles or tears in shear long before the members it joins reach their own limits, so the connection, not the member, sets the capacity — the reason a modern check sizes gussets explicitly rather than assuming them stronger than what they join. Its replacement, the [[I-35W_Saint_Anthony_Falls_Bridge|I-35W Saint Anthony Falls Bridge]], opened on September 18, 2008 and takes the opposite approach: a [[Prestressed_concrete|post-tensioned]] concrete [[Box_girder_bridge|box girder]], in which the deck itself is the bending member and the box's depth — varying along the span, deepest over the piers where the moment is greatest — is the second moment of area doing exactly what this article's first section describes. ## See also - [[Beam_(structure)]] - [[Euler–Bernoulli_beam_theory]] - [[Bending_moment]] - [[Second_moment_of_area]] - [[Deflection_(engineering)]] - [[Section_modulus]] - [[Neutral_axis]] - [[Stress_(mechanics)]] - [[Buckling]] - [[Plate_theory]] - [[Shear_force]] - [[Structural_engineering]] ## References [^johnson009]: Johnson, Eric R. *Aerospace Structures*. Portal Books 009, pp. 417–421: the landing-strut sizing worked through the bending moment, the Castigliano deflection `Δ = R·l³/(3·E·I·sin θ)`, the rectangular section's `I = b·h³/12`, and the margin of safety against an allowable stress. The cantilever form `P·L³/(3·E·I)` used above is that deflection with the strut normal to the load. [^up12-3]: OpenStax (2016). *University Physics Volume 1*. Ling, S. J.; Sanny, J.; Moebs, W. Rice University. Chapter 12 "Static Equilibrium and Elasticity," §12.3 "Stress, Strain, and Elastic Modulus," pp. 587–600 (stress, strain and Young's modulus). https://openstax.org/details/books/university-physics-volume-1 (Portal Books 077). [^simbend]: Portal engineering pack, `design.beam.solve` for the shear, moment and deflection — the standard beam-table closed forms for a cantilever and a fixed-fixed beam under a tip or midspan point load or a uniform load, and a simply supported beam under a midspan point load or a uniform load, in the forms tabulated in Hibbeler, *Mechanics of Materials*, Appendix C — with `design.sections.rect`, `ibeam` and `tube` for the section properties; sim spec `specs/sims/Bending.json`. Two ILLUSTRATIVE items, both labelled on the sim itself: the deflected shape is drawn exaggerated, and the 250 MPa yield level is a mild-steel nominal threshold, not a graded property. Appendix letter as recalled; the closed forms themselves are standard beam-table results. [^hetenyi]: Hetényi, Miklós (1946). *Beams on Elastic Foundation: Theory with Applications in the Fields of Civil and Mechanical Engineering*. Ann Arbor: University of Michigan Press. The standard treatment of the Winkler model, the characteristic length and the decaying-oscillation solutions. (Pages not re-checked for this article.) [^theory-plates]: Timoshenko, Stephen P.; Woinowsky-Krieger, S. (1959). *Theory of Plates and Shells*, 2nd edition. New York: McGraw-Hill. The source of the tabulated plate coefficients used in design practice, and of the corner-force result quoted above. (Chapter and pages not re-checked for this article.) [^ntsb]: National Transportation Safety Board (2008). *Collapse of I-35W Highway Bridge, Minneapolis, Minnesota, August 1, 2007*. Highway Accident Report NTSB/HAR-08/03, adopted November 14, 2008. (Report number and adoption date as recalled; the report is published by the NTSB at https://www.ntsb.gov/ and the finding on the undersized U10 gusset plates is its central conclusion. The casualty figures of 13 killed and 145 injured are those the report and contemporaneous reporting carry.)  **Microsim — three.js (Wikitube framework):** *Bending* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/engineering/Bending.html" data-title="Bending"></div> *Built from `MICROSIM_GUIDE/specs/sims/Bending.json`; part of the [[PORTAL_Engineering|Engineering portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Bending) : [Wikitube](https://en.wikitube.io/wiki/Bending) - skeleton pinned to revision 1351473367 (2026-09-18).