Stress_(mechanics) - Wikiparency

# Stress (mechanics) In [[Continuum_mechanics|continuum mechanics]], **stress** is the measure of the internal forces that neighbouring particles of a deformable body exert on one another, reckoned as force per unit of the area across which it acts. It is what a cut reveals: imagine slicing a loaded bar and holding the two halves apart, and the distributed force that must be applied to the exposed face to keep the half in equilibrium is the stress on that face. Because the exposed face can be cut at any angle, stress at a point is not one number but a [[Cauchy_stress_tensor|tensor]] — a rule that turns the direction of a cut into the force per unit area transmitted across it. Stress has the dimensions of [[Pressure|pressure]], and its SI unit is the [[Pascal_(unit)|pascal]]; engineering practice works in megapascals, one of which is exactly one newton per square millimetre. The component of the traction along the normal to the cut is the normal stress, tensile when it pulls and compressive when it pushes; the component lying in the cut face is the [[Shear_stress|shear stress]]. Stress is the quantity every later calculation in [[Solid_mechanics|solid mechanics]] produces and every material property is measured against: a [[Bending|bent beam]], a [[Torsion_(mechanics)|twisted shaft]] and a slender [[Buckling|column]] differ only in the rule that turns their load into a stress field, and each is safe while that field stays clear of [[Yield_(engineering)|yield]] and [[Fracture|fracture]]. The framework microsim *Stress: one state at a point, and what every cut through it sees* puts one stress state on a small element and lets the reader turn a cut through it: the normal and shear stress on the cut face are read off as arrows and as numbers, the same point rides around [[Mohr's_circle|Mohr's circle]] beside the element at twice the cut angle, and the principal planes and the plane of maximum shear announce themselves as the cut sweeps past them. ## History The idea that a material fails when something inside it reaches a limit is older than the quantity that measures it. [[Galileo_Galilei|Galileo Galilei]] opened the subject in 1638 with the problem of the breaking cantilever, asking how the strength of a beam scales with its dimensions and getting the geometry right while the distribution of force across the broken section eluded him.[^galileo1638] [[Robert_Hooke|Robert Hooke]] supplied the deformation half of the story in 1678, stating as *ut tensio, sic vis* the proportionality between extension and force that is now [[Hooke's_law|Hooke's law]].[^hooke1678] The modern quantity is Augustin-Louis Cauchy's. In the 1820s [[Augustin-Louis_Cauchy|Cauchy]] replaced the idea of a force acting on a section by the idea of a traction acting at a point on a plane of a given orientation, showed that the traction depends linearly on the plane's normal, and so produced the nine-component object that carries his name.[^cauchy1823] Claude-Louis Navier, Siméon Denis Poisson and Adhémar Jean Claude Barré de Saint-Venant built the theory of [[Elasticity_(physics)|elasticity]] on it over the following decades, Saint-Venant adding the principle that the details of how a load is applied stop mattering a short distance away, which is what licenses the simple formulas of engineering practice.[^stvenant] Otto Mohr drew the construction that made the tensor visible to engineers, publishing in 1882 the circle on which every plane through a stress state appears as a single point.[^mohr1882] Drawn by hand for a century, it is the same circle the section's microsim redraws as the cut turns. ## Definition Consider a small area ΔA on a plane through a point inside a loaded body, with unit normal n, and let ΔF be the force that the material on the positive side of the plane exerts on the material on the negative side. The traction, or stress vector, on that plane is the limit `t(n) = lim ΔF/ΔA` as the area shrinks to the point. The limit exists and is finite for a [[Continuum_mechanics|continuum]]; it is the idealisation that replaces the discrete forces between atoms by a smooth field, and it fails only where the continuum picture itself fails, at a crack tip or across a few atomic spacings. The traction is a vector with the units of pressure, but it is not a property of the point alone: cut the same point on a different plane and a different traction appears. What belongs to the point is the map from n to t, and Cauchy's theorem states that this map is linear, `t(n) = σ·n`, so that three tractions — on three mutually perpendicular planes — determine the traction on every other plane through that point. ### Normal and shear Any traction splits into two parts relative to the face it acts on. The part along the normal is the **normal stress** `σ_n = t·n`, positive in tension by the usual engineering convention and negative in compression; what remains lies in the plane of the cut and is the **shear stress** τ. A hydraulic ram, a cable and a column carry almost pure normal stress; a rivet, a bolt in a lap joint and the glue line of a bonded lap carry almost pure shear. The two components are not separate stresses but two readings of one state, and they trade against each other as the cut turns. In the microsim's default state — `σx` = 100 MPa, `σy` = 0 and `τxy` = 50 MPa — a cut at 50° from the x axis carries a normal stress of 90.6 MPa and a shear stress of −57.9 MPa, while the same point cut at 22.5° carries the largest normal stress the state can produce, 120.7 MPa, and no shear at all.[^simstress] Nothing about the material changed between those two readings; only the question did. That is the sense in which stress is a tensor, and it is the one fact the section's sim exists to make unavoidable. Every number the sim prints comes from the transformation equations below rather than from a fit or a stand-in, so nothing in this sim is ILLUSTRATIVE; what the sim does take liberties with is drawing, and it says so on its own sheet — arrow lengths are scaled to the stresses, and the axes of the circle rescale to keep it in frame.[^simstress] *Try: set sigma_x, sigma_y and tau_xy on the element, then drag the cut angle theta through a full half-turn and watch the orange normal and shear arrows on the cut face swap size while the blue face arrows stay put; stop where the shear arrow vanishes and read the principal stress off the readout.* ## Units Stress is force divided by area, so its SI unit is the pascal, `1 Pa = 1 N/m²`. The pascal is inconveniently small for solids: atmospheric pressure is about 101 kPa, while the stresses that matter in a steel part run to hundreds of millions of pascals. Engineering therefore works in megapascals, and the convenient identity is that `1 MPa = 1 N/mm²`, which lets a load in newtons be divided by an area in square millimetres to give megapascals directly. A tensile bar of 50 mm² cross-section carrying 5 kN is under `5000/50 = 100 MPa`. Gigapascals appear for the [[Elastic_modulus|elastic moduli]] rather than for working stresses: steel's [[Young's_modulus|Young's modulus]] is about 207 GPa, some two thousand times a typical design stress, which is why elastic strains in metals are measured in parts per ten thousand.[^up12-3] United States practice still quotes pounds per square inch and the kip per square inch, with `1 ksi = 6.895 MPa`; the 100 MPa bar above is a 14.5 ksi bar. Because stress and pressure share a unit, tabulated values must say which convention signs them: pressure is positive in compression, engineering normal stress positive in tension. ## Causes and effects Stress arises whenever a body is prevented from taking the shape it would otherwise take. Applied loads are the obvious cause — the weight a bridge carries, the pressure inside a vessel, the torque in a shaft — and self-weight is a load like any other, dominant in dams and in long-span structures. Less obvious causes matter as much in practice. A change of temperature makes a body try to change size, and a body restrained against that change carries a stress `σ = E·α·ΔT` with no external load at all: a steel member held rigidly while it cools by 40 K picks up about 96 MPa, taking `E` = 200 GPa and a [[Thermal_expansion|coefficient of thermal expansion]] of 12 × 10⁻⁶ K⁻¹. [[Residual_stress|Residual stresses]] left by [[Welding|welding]], casting, quenching and cold forming are locked-in fields that satisfy equilibrium with no external load and can be a large fraction of yield before the part is ever used. Phase changes, moisture swelling and differential settlement act the same way. The effects run in a sequence. A small stress produces a proportional [[Strain_(mechanics)|strain]] and nothing else, and the body returns to its shape when unloaded. Past the elastic limit, metals flow plastically and keep part of the deformation; the whole story is the [[Stress–strain_curve|stress–strain curve]]. Held long enough at high temperature, materials creep; cycled below yield for long enough, they crack by [[Fatigue_(material)|fatigue]]; loaded with a flaw present, they may fracture before they yield at all. Stress is the common input to every one of these limits, which is why it is computed first and compared afterwards. ## Simple types Most engineering calculations do not need the full tensor, because most parts are designed so that one simple state dominates. Four such states recur, and each is a special case of the general one that the microsim can be set to show. ### Uniaxial normal A straight bar pulled along its axis carries a normal stress `σ = F/A` uniform over any cross-section perpendicular to the axis, and nothing else — on that plane. The state is called uniaxial because only one principal stress is non-zero; it is the state of a tie rod, a suspension cable and the tensile test specimen from which a material's properties are read.[^up12-3] Cut the same bar obliquely, however, and shear appears: the uniaxial state has a maximum shear stress of `σ/2` on planes at 45° to the axis, which is why ductile metals pulled in tension slip and neck on inclined planes rather than parting straight across. ### Shear Pure shear is the state in which the normal stresses vanish on a pair of perpendicular planes and only `τ` remains. It is what a bolt in single shear, a shaft in [[Torsion_(mechanics)|torsion]] at its surface, and a thin web between flanges see. Pure shear is also the clearest demonstration that a stress state has no single value: a state of 60 MPa pure shear has principal stresses of exactly +60 MPa and −60 MPa on planes at 45°, so the material is in tension and compression of equal size at the same point, and Mohr's circle for it is centred on the origin.[^simstress] Brittle materials in torsion accordingly break on a 45° helix, along the plane of maximum tension. *Try: open the Shear stress variant on its pure-shear preset (0, 0, 60 MPa) with the cut on the x face, then turn the cut to 45 degrees and watch the shear arrow fall to zero while two equal and opposite normal arrows appear.* ### Isotropic If all three principal stresses are equal, the state is isotropic, or hydrostatic: every plane through the point carries the same normal stress and no shear whatever. A body immersed in a still fluid is in this state, with `σ = −p` on every face. Mohr's circle for it degenerates to a single point, radius zero, which the microsim shows directly when the state is set to equal normal stresses with no shear.[^simstress] The distinction matters because the hydrostatic part of a stress state changes a metal's volume but not its shape, and, to a good approximation, does not cause yielding — the reason yield criteria are written on the deviatoric part alone. ### Cylinder A thin-walled cylinder under internal pressure is the classic biaxial state, and the one most engineers meet first outside the tensile bar. Equilibrium of a half-cylinder gives a circumferential, or hoop, stress `σθ = p·r/t`, and equilibrium of an end cap gives an axial stress `σz = p·r/(2t)` — exactly half as much.[^up12-3] A pipe of 0.5 m radius and 10 mm wall at 2 MPa internal pressure carries 100 MPa hoop and 50 MPa axial stress, and the in-plane maximum shear is 25 MPa, half their difference. The two-to-one ratio explains why pressurised cylinders burst along a line parallel to the axis rather than around it, and it is the starting point for [[Cylinder_stress|cylinder stress]] design in [[Pressure_vessel|pressure vessels]]. ## General types When no single simple state dominates, the full description is needed. A general three-dimensional stress state at a point has nine components, `σij`, giving the force per unit area in direction *i* on the face whose normal is *j*; balance of angular momentum on a vanishing element makes the tensor symmetric, `σij = σji`, so six independent numbers remain — three normal and three shear. Combined loading is the usual reason. A crankshaft journal carries bending and torsion at once, a bolt carries tension and shear, an aircraft wing spar carries bending about two axes plus torsion plus the shear that comes with them, and the stress state at any point in the section is the superposition of all of them. Superposition is exact as long as the material stays linear and deflections stay small, which is the working assumption of [[Linear_elasticity|linear elasticity]]. In general the six components vary from point to point, so the object of an analysis is a stress *field* rather than a stress, and the goal is usually the worst point in it rather than a typical one. ## Cauchy tensor The Cauchy stress tensor σ is the linear map that answers, for a single point, the question every cut asks. Its components in a chosen coordinate frame are the tractions on the three coordinate faces: the column *j* holds the traction on the face whose outward normal is the *j* axis. Cauchy's theorem, `t(n) = σ·n`, then delivers the traction on every other plane through the point, which is why three faces suffice. The tensor is symmetric, and — being a true tensor — its components change when the frame turns, while the physical state does not. ### Change of coordinates Turning the frame by an angle θ in the plane transforms the components by `σ_n = (σx + σy)/2 + (σx − σy)/2 · cos 2θ + τxy · sin 2θ`, with the companion `τ_n = −(σx − σy)/2 · sin 2θ + τxy · cos 2θ`. Both are the parametric equations of a circle in the (σ, τ) plane: the centre sits at `C = (σx + σy)/2` on the normal-stress axis and the radius is `R = √(((σx − σy)/2)² + τxy²)`. For the microsim's default state the centre is at 50 MPa and the radius is 70.7 MPa, so the principal stresses — the extreme normal stresses, where the circle crosses the axis and the shear vanishes — are `C + R` = 120.7 MPa and `C − R` = −20.7 MPa, on planes at `θp = ½·arctan(2τxy/(σx − σy))` = 22.5°.[^simstress] The maximum shear stress is the radius itself, 70.7 MPa, on planes 45° from the principal ones. The factor of two in the angle is the construction's one trap and its one insight: a cut turned through 45° in the material moves a full 90° around the circle. A quantity that does not change under rotation is an invariant, and the three invariants of the tensor — in the plane case the trace `σx + σy` and the determinant — are what let a criterion of failure be written once instead of once per coordinate choice. *Try: open the Mohr's circle variant, which starts with the cut already on the principal plane at 22.5 degrees, then step theta and watch the point travel twice as fast around the circle as the cut turns through the element.* ### Tensor field In a body, the tensor is a field: σ(x) varies from point to point, and the variation is not free. Balance of linear momentum on an infinitesimal element requires `∂σij/∂xj + fi = 0` in statics, with `fi` the body force per unit volume, and the same equation with `ρ·ai` on the right in dynamics. These three partial differential equations, together with the strain–displacement relations and a constitutive law such as Hooke's law, close the problem of [[Linear_elasticity|elasticity]]; every closed-form result in strength of materials is a solution of them under simplifying assumptions, and every [[Finite_element_method|finite element]] run is a numerical solution of them. ### Thin plates A plate is thin enough that the stress through its thickness may be neglected, leaving a two-dimensional state — the condition called [[Plane_stress|plane stress]]. It is convenient to integrate the in-plane stresses through the thickness into resultants: membrane forces per unit length, which act at the mid-surface, and bending moments per unit length, which vary linearly through the thickness and vanish at the mid-surface. Aircraft skins, pressure-vessel walls and floor slabs are designed on these resultants rather than on point stresses, and the local stress is recovered at the end by dividing back by the thickness or by the section modulus. The plate equations themselves belong to [[Bending|bending]]. ### Thin beams A slender beam reduces the tensor further still. Under the assumption that plane sections remain plane, the axial stress varies linearly across the depth, `σ = M·y/I`, where M is the [[Bending_moment|bending moment]], y the distance from the neutral axis and I the [[Second_moment_of_area|second moment of area]]; the transverse shear stress is distributed parabolically across a rectangular section with a peak of `1.5·V/A` at the neutral axis. The two are at their maxima at different points — the axial stress at the extreme fibre where the shear is zero, the shear at the mid-depth where the axial stress is zero — which is why short deep beams and long slender ones fail in different ways, and why [[Euler–Bernoulli_beam_theory|Euler–Bernoulli beam theory]] is enough for the second but not for the first. ## Analysis ### Goals and assumptions A stress analysis answers one of two questions: whether a given part under a given load stays below its limits, or what dimensions make it do so. Either way the analysis produces a field and then reduces it to a single comparison at the worst point. Nearly every practical calculation assumes a continuum, small strains, and either [[Plane_stress|plane stress]] or plane strain to cut three dimensions to two; most assume the material is linear, isotropic and homogeneous. These assumptions are not innocent — they are exactly what fails near a crack tip, a sharp re-entrant corner or a bond line between dissimilar materials, where the idealised solution predicts infinite stress and the real answer belongs to [[Fracture_mechanics|fracture mechanics]]. ### Methods Three families of method are in use. Closed-form solutions from [[Strength_of_materials|strength of materials]] — the axial bar, the bent beam, the twisted shaft, the thin cylinder — remain the first resort because they are auditable and fast, and because a designer who cannot estimate an answer cannot check a computed one. Numerical methods, above all the [[Finite_element_method|finite element method]], solve the equilibrium equations on a mesh for geometry and loading that no closed form covers. Experimental methods measure what actually happens: bonded [[Strain_gauge|strain gauges]] read strain at a point and give stress through the material law, [[Photoelasticity|photoelasticity]] shows the difference of principal stresses as coloured fringes in a transparent model, and full-field optical correlation maps strain across a surface. The comparison at the end needs a single number, because material strength is measured in a uniaxial test while the part is in a multiaxial state. For ductile metals the usual reduction is the [[Von_Mises_yield_criterion|von Mises]] equivalent stress, `σ_e = √(σx² − σx·σy + σy² + 3τxy²)` in plane stress, which for the microsim's default state is 132 MPa against principal stresses of 120.7 and −20.7 MPa.[^simstress] The part is judged by comparing that equivalent stress with the material's [[Yield_(engineering)|yield strength]], and the ratio of the two is the factor of safety. ## Measures The Cauchy stress is the true stress: force per unit of *current*, deformed area. While strains stay small the distinction is immaterial, since the area barely changes, and all of the formulas above are written for it. Once deformation is large the reference area and the current area differ enough to matter, and several distinct stress measures appear, each pairing with a particular strain measure so that their product gives work per unit volume correctly. The first Piola–Kirchhoff stress, the engineering or nominal stress, reports the current force per unit of *original* area; it is what a tensile testing machine reports without correction, and its divergence from the true stress after necking is why an engineering [[Stress–strain_curve|stress–strain curve]] turns down past the ultimate tensile strength while the true curve continues to rise. It is not in general symmetric. The second Piola–Kirchhoff stress pulls both the force and the area back to the reference configuration, is symmetric, and is the measure used in the constitutive laws of [[Finite_strain_theory|finite strain theory]] and in most nonlinear finite element codes; the Kirchhoff stress, the Cauchy stress scaled by the volume ratio, is convenient in plasticity. These are alternative accounts of one physical state, and quoting a large-deformation stress without naming its measure leaves the number ambiguous.[^stressmeasures] ## Minnesota *This section is specific to Wikitube.* Stress cannot be measured; strain can, and the two are connected by the material law. That is the working principle of the instrumentation built into the [[I-35W_Saint_Anthony_Falls_Bridge|I-35W Saint Anthony Falls Bridge]] in Minneapolis, [[Minnesota]], the concrete box-girder bridge opened on September 18, 2008 to replace the [[I-35W_Mississippi_River_bridge|Interstate 35W Mississippi River bridge]] that collapsed the previous year. Gauges cast into the girders and mounted on the structure were designed to report strain, temperature and movement continuously from the day it opened, giving the Minnesota Department of Transportation and the University of Minnesota a record of how the bridge actually responds to traffic and to the seasons rather than only how it was calculated to respond.[^mnshm] The temperature channels are not an afterthought in a state whose recorded extreme is −51 °C (−60 °F), measured at Tower on February 2, 1996.[^mndnr] A member restrained against a swing of that size would carry a thermal stress of the same order as its working stress, by the `σ = E·α·ΔT` estimate above, so the gauges must separate the strain that comes from load from the strain that comes from the thermometer before either can be read as stress. Continuous monitoring of this kind is the field application of the section's subject: a tensor that cannot be seen, inferred point by point from a deformation that can. ## See also - [[Mohr's_circle]] - [[Shear_stress]] - [[Cauchy_stress_tensor]] - [[Strain_(mechanics)]] - [[Stress–strain_curve]] - [[Hooke's_law]] - [[Bending]] - [[Buckling]] - [[Torsion_(mechanics)]] - [[Von_Mises_yield_criterion]] - [[Linear_elasticity]] - [[Strength_of_materials]] ## References [^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. https://openstax.org/details/books/university-physics-volume-1 (Portal Books 077). [^galileo1638]: Galilei, Galileo (1638). *Discorsi e dimostrazioni matematiche intorno a due nuove scienze*. Leiden: Elzevir. Second Day (the strength of beams). https://archive.org/details/discorsiedimost00galigoog [^hooke1678]: Hooke, Robert (1678). *Lectures de Potentia Restitutiva, or of Spring, Explaining the Power of Springing Bodies*. London: John Martyn. (Publisher as recalled; the title, the year and the phrase *ut tensio, sic vis* are standard in histories of mechanics.) [^cauchy1823]: Cauchy, Augustin-Louis (1823). "Recherches sur l'équilibre et le mouvement intérieur des corps solides ou fluides, élastiques ou non élastiques." *Bulletin de la Société Philomathique de Paris*. (Volume and pages as recalled; the 1823 announcement and its elaboration in the *Exercices de mathématiques* of 1827–1828 are the standard citation for the stress tensor.) [^stvenant]: Saint-Venant's principle is stated in Barré de Saint-Venant's memoirs on torsion and flexure presented to the Académie des Sciences in the 1850s; it is given in modern form in every strength-of-materials text (Hibbeler, *Mechanics of Materials*, chapter 4). Specific memoir and pages not re-checked for this article. [^mohr1882]: Mohr, Otto (1882). "Über die Darstellung des Spannungszustandes und des Deformationszustandes eines Körperelementes." *Der Civilingenieur* 28. (Volume year as recalled; pages not re-checked. Karl Culmann's graphical statics of 1866 is the acknowledged predecessor of the construction.) [^simstress]: Portal engineering pack, `design.stress` — `principal(σx, σy, τxy)` for the centre, radius, σ₁, σ₂ and the principal angle, `mohrPoint(σx, σy, τxy, θ)` for the stresses on the cut face, and `vonMises` for the equivalent stress — with the sim spec `specs/sims/Stress_(mechanics).json`; hand-checked against `tools/test_design.mjs` in the run report of 2026-09-18. Every number the sim prints is computed from these functions: nothing in this sim is ILLUSTRATIVE. Two drawing conventions are stated on the sim's own sheet rather than implied — arrow length scales with stress, and the Mohr axes rescale to keep the circle in frame — and the circle is drawn in the Hibbeler convention, with τ positive downward. [^stressmeasures]: Standard treatment: Ogden, R. W. (1984). *Non-Linear Elastic Deformations*. Chichester: Ellis Horwood (Dover reprint 1997), chapter 3 (stress tensors and their conjugate strain measures). Chapter as recalled; consult also Hibbeler, *Mechanics of Materials*, for the engineering-versus-true stress distinction. [^mnshm]: Minnesota Department of Transportation and University of Minnesota, structural-health-monitoring programme for the I-35W Saint Anthony Falls Bridge (bridge 27409), instrumented at construction and reporting since the bridge opened on September 18, 2008. *Citation needed*: the qualitative description here is well established in MnDOT and university reporting on the bridge, but the sensor inventory, the channel list and the commissioning date of the monitoring system were not re-verified for this article; the MnDOT research report on the bridge's instrumentation would settle them, and no count of sensors is quoted above for that reason. [^mndnr]: Minnesota Department of Natural Resources, State Climatology Office. "Minnesota's record low temperature": −60 °F at Tower, February 2, 1996. https://www.dnr.state.mn.us/climate/index.html  **Microsim — three.js (Wikitube framework):** *Stress (mechanics)* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/engineering/Stress_(mechanics).html" data-title="Stress (mechanics)"></div> *Built from `MICROSIM_GUIDE/specs/sims/Stress_(mechanics).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/Stress_(mechanics)) : [Wikitube](https://en.wikitube.io/wiki/Stress_(mechanics)) - skeleton pinned to revision 1370311866 (2026-09-18).