Stress (mechanics)Stressis theforceper unitareaon a body that tends to cause it tochange shape.[2]Stress is a measure of the internal forces in a body between itsparticles.[2]These internal forces are a reaction to the external forces applied on the body that cause it to separate, compress or slide.[2]External forces are eithersurface forcesorbody forces. Stress is the average force per unit area that a particle of a body exerts on an adjacent particle, across an imaginary surface that separates them.The formula for uniaxial normal stress is:

where is the stress, F is the force and A is the surface area.InSIunits, force is measured innewtonsand area insquare metres. This means stress is newtons per square meter, or N/m2. However, stress has its own SI unit, called thepascal. 1 pascal (symbol Pa) is equal to 1 N/m2. InImperial units, stress is measured inpound-forceper squareinch, which is often shortened to "psi". The dimension of stress is the same as that ofpressure.In continuum mechanics, the loaded deformable body behaves as acontinuum. So, these internal forces are distributed continually within the volume of the material body. (This means that the stress distribution in the body is expressed as apiecewisecontinuous functionof space and time.) The forces causedeformationof the body's shape. The deformation can lead to a permanent shape change or structural failure if thematerial is not strong enough.Some models of continuum mechanics treat force as something that can change. Other models look at the deformation of matter and solid bodies, because the characteristics of matter and solids are three dimensional. Each approach can give different results. Classical models of continuum mechanics assume an average force and do not properly include "geometrical factors". (The geometry of the body can be important to how stress is shared out and how energy builds up during the application of the external force.)Contents[hide] 1Shear stress 2Simple stresses 2.1Uniaxial normal stress 3Stress in one-dimensional bodies 4Related pages 5References 6Bibliography 7Other websitesShear stress[change|change source]For more details, seeShear stressSimple stresses[change|change source]In some situations, the stress within an object can be described by a single number, or by a singlevector(a number and a direction). Three suchsimple stresssituations are theuniaxial normal stress, thesimple shear stress, and theisotropic normal stress.[3]Uniaxial normal stress[change|change source]Tensile stress(ortension) is the stress state leading toexpansion; that is, the length of a material tends to increase in the tensile direction. The volume of the material stays constant. When equal and opposite forces are applied on a body, then the stress due to this force is called tensile stress.Therefore in a uniaxial material the length increases in the tensile stress direction and the other two directions will decrease in size. In the uniaxial manner oftension, tensile stress is induced by pulling forces. Tensile stress is the opposite ofcompressive stress.Structural members in direct tension areropes,soilanchorsandnails,bolts, etc.Beamssubjected to bendingmomentsmay include tensile stress as well as compressive stress and/orshear stress.Tensile stress may be increased until the reach oftensile strength, namely thelimit stateof stress.Stress in one-dimensional bodies[change|change source]All real objects occupy three-dimensional space. However, if two dimensions are very large or very small compared to the others, the object may be modelled as one-dimensional. This simplifies themathematical modellingof the object. One-dimensional objects include a piece of wire loaded at the ends and viewed from the side, and a metal sheet loaded on the face and viewed up close and through the cross section.

Incontinuum mechanics,stressis aphysical quantitythat expresses the internalforcesthat neighboringparticlesof acontinuous materialexert on each other, while strain is the measure of the deformation of the material. For example, when asolidvertical bar is supporting aweight, each particle in the bar pushes on the particles immediately below it. When aliquidis in a closed container underpressure, each particle gets pushed against by all the surrounding particles. The container walls and thepressureinducing surface (such as a piston), inreaction, push against them. These macroscopic forces are actually the average of a very large number ofintermolecular forcesandcollisionsbetween the particles in thosemolecules.Strain inside a material may arise by various mechanisms, such asstressas applied by external forces to the bulk material (likegravity) or to its surface (likecontact forces, external pressure, orfriction). Anystrain (deformation)of a solid material generates an internalelastic stress, analogous to the reaction force of aspring, that tends to restore the material to its original non-deformed state. In liquids andgases, only deformations that change the volume generate persistent elastic stress. However, if the deformation is gradually changing with time, even in fluids there will usually be someviscous stress, opposing that change. Elastic and viscous stresses are usually combined under the namemechanical stress.Significant stress may exist even when deformation is negligible or non-existent (a common assumption when modeling the flow of water). Stress may exist in the absence of external forces; suchbuilt-in stressis important, for example, inprestressed concreteandtempered glass. Stress may also be imposed on a material without the application of net forces, for example bychanges in temperatureorchemicalcomposition, or by externalelectromagnetic fields(as inpiezoelectricandmagnetostrictivematerials).The relation between mechanical stress, deformation, and therate of change of deformationcan be quite complicated, although alinear approximationmay be adequate in practice if the quantities are small enough. Stress that exceeds certainstrength limitsof the material will result in permanent deformation (such asplastic flow,fracture,cavitation) or even change itscrystal structureandchemical composition.In some branches ofengineering, the termstressis occasionally used in a looser sense as a synonym of "internal force". For example, in the analysis oftrusses, it may refer to the total traction or compression force acting on a beam, rather than the force divided by the area of itscross-section.Contents[hide] 1History 2Overview 3Simple stress 4General stress 5Stress analysis 6Alternative measures of stress 7See also 8Further reading 9ReferencesHistory[edit]

Roman-era bridge inSwitzerland

Incabridge on theApurimac RiverSince ancient times humans have been consciously aware of stress inside materials. Until the 17th century the understanding of stress was largely intuitive and empirical; and yet it resulted in some surprisingly sophisticated technology, like thecomposite bowandglass blowing.[1]Over several millennia, architects and builders, in particular, learned how to put together carefully shaped wood beams and stone blocks to withstand, transmit, and distribute stress in the most effective manner, with ingenious devices such as thecapitals,arches,cupolas,trussesand theflying buttressesofGothic cathedrals.Ancient and medieval architects did develop some geometrical methods and simple formulas to compute the proper sizes of pillars and beams, but the scientific understanding of stress became possible only after the necessary tools were invented in the 17th and 18th centuries:Galileo's rigorousexperimental method,Descartes'scoordinatesandanalytic geometry, andNewton'slaws of motion and equilibriumandcalculus of infinitesimals.[2]With those tools,Cauchywas able to give the first rigorous and general mathematical model for stress in a homogeneous medium.[citation needed]Cauchy observed that the force across an imaginary surface was a linear function of its normal vector; and, moreover, that it must be a symmetric function (with zero total momentum).[citation needed]The understanding of stress in liquids started with Newton, who provided a differential formula for friction forces (shear stress) in parallellaminar flow.Overview[edit]Definition[edit]Stress is defined as the average force per unit area that some particle of a body exerts on an adjacent particle, across an imaginary surface that separates them.[3]Being derived from a fundamental physical quantity (force) and a purely geometrical quantity (area), stress is also a fundamental quantity, like velocity,torqueorenergy, that can be quantified and analyzed without explicit consideration of the nature of the material or of its physical causes.Following the basic premises of continuum mechanics, stress is amacroscopicconcept. Namely, the particles considered in its definition and analysis should be just small enough to be treated as homogeneous in composition and state, but still large enough to ignorequantumeffects and the detailed motions of molecules. Thus, the force between two particles is actually the average of a very large number of atomic forces between their molecules; and physical quantities like mass, velocity, and forces that act through the bulk of three-dimensional bodies, like gravity, are assumed to be smoothly distributed over them.[4]:p.90106Depending on the context, one may also assume that the particles are large enough to allow the averaging out of other microscopic features, like the grains of ametalrod or thefibersof a piece ofwood.

The stress across a surface element (yellow disk) is the force that the material on one side (top ball) exerts on the material on the other side (bottom ball), divided by the area of the surface.Quantitatively, the stress is expressed by theCauchy traction vectorTdefined as the traction forceFbetween adjacent parts of the material across an imaginary separating surfaceS, divided by the area ofS.[5]:p.4150In afluidat rest the force is perpendicular to the surface, and is the familiarpressure. In asolid, or in aflowof viscousliquid, the forceFmay not be perpendicular toS; hence the stress across a surface must be regarded a vector quantity, not a scalar. Moreover, the direction and magnitude generally depend on the orientation ofS. Thus the stress state of the material must be described by atensor, called the(Cauchy) stress tensor; which is alinear functionthat relates thenormal vectornof a surfaceSto the stressTacrossS. With respect to any chosencoordinate system, the Cauchy stress tensor can be represented as asymmetricmatrixof 33 real numbers. Even within ahomogeneousbody, the stress tensor may vary from place to place, and may change over time; therefore, the stress within a material is, in general, a time-varyingtensor field.Normal and shear stress[edit]Further information:compression (physical)andShear stressIn general, the stressTthat a particlePapplies on another particleQacross a surfaceScan have any direction relative toS. The vectorTmay be regarded as the sum of two components: thenormal stress(compressionortension) perpendicular to the surface, and theshear stressthat is parallel to the surface.If the normal unit vectornof the surface (pointing fromQtowardsP) is assumed fixed, the normal component can be expressed by a single number, thedot productTn. This number will be positive ifPis "pulling" onQ(tensile stress), and negative ifPis "pushing" againstQ(compressive stress) The shear component is then the vectorT (Tn)n.Units[edit]The dimension of stress is that ofpressure, and therefore its coordinates are commonly measured in the same units as pressure: namely,pascals(Pa, that is,newtonspersquare metre) in theInternational System, orpoundspersquare inch(psi) in theImperial system.Causes and effects[edit]

Glass vase with thecraqueleffect. The cracks are the result of brief but intense stress created when the semi-molten piece is briefly dipped in water.[6]Stress in a material body may be due to multiple physical causes, including external influences and internal physical processes. Some of these agents (like gravity, changes intemperatureandphase, and electromagnetic fields) act on the bulk of the material, varying continuously with position and time. Other agents (like external loads and friction, ambient pressure, and contact forces) may create stresses and forces that are concentrated on certain surfaces, lines, or points; and possibly also on very short time intervals (as in theimpulsesdue to collisions). In general, the stress distribution in the body is expressed as apiecewisecontinuous functionof space and time.Conversely, stress is usually correlated with various effects on the material, possibly including changes in physical properties likebirefringence,polarization, andpermeability. The imposition of stress by an external agent usually creates somestrain (deformation)in the material, even if it is too small to be detected. In a solid material, such strain will in turn generate an internal elastic stress, analogous to the reaction force of a stretchedspring, tending to restore the material to its original undeformed state. Fluid materials (liquids,gasesandplasmas) by definition can only oppose deformations that would change their volume. However, if the deformation is changing with time, even in fluids there will usually be some viscous stress, opposing that change.The relation between stress and its effects and causes, including deformation and rate of change of deformation, can be quite complicated (although alinear approximationmay be adequate in practice if the quantities are small enough). Stress that exceeds certainstrength limitsof the material will result in permanent deformation (such asplastic flow,fracture,cavitation) or even change itscrystal structureandchemical composition.Simple stress[edit]In some situations, the stress within a body may adequately be described by a single number, or by a single vector (a number and a direction). Three suchsimple stresssituations, that are often encountered in engineering design, are theuniaxial normal stress, thesimple shear stress, and theisotropic normal stress.[7]Uniaxial normal stress[edit]

Idealized stress in a straight bar with uniform cross-section.A common situation with a simple stress pattern is when a straight rod, with uniform material and cross section, is subjected totensionby opposite forces of magnitudealong its axis. If the system is inequilibriumand not changing with time, and the weight of the bar can be neglected, then through each transversal section of the bar the top part must pull on the bottom part with the same forceF. Therefore the stress throughout the bar, across anyhorizontalsurface, can be described by the number=F/A, whereAis the area of the cross-section.On the other hand, if one imagines the bar being cut along its length, parallel to the axis, there will be no force (hence no stress) between the two halves across the cut.This type of stress may be called (simple)normal stressoruniaxial stress; specifically, (uniaxial,simple, etc.)tensile stress.[7]If the load iscompressionon the bar, rather than stretching it, the analysis is the same except that the forceFand the stresschange sign, and the stress is calledcompressive stress.

The ratiomay be only an average stress. The stress may be unevenly distributed over the cross section (mm), especially near the attachment points (nn).This analysis assumes the stress is evenly distributed over the entire cross-section. In practice, depending on how the bar is attached at the ends and how it was manufactured, this assumption may not be valid. In that case, the value=F/Awill be only the average stress, calledengineering stressornominal stress. However, if the bar's lengthLis many times its diameterD, and it has no gross defects orbuilt-in stress, then the stress can be assumed to be uniformly distributed over any cross-section that is more than a few timesDfrom both ends. (This observation is known as theSaint-Venant's principle).Normal stress occurs in many other situations besides axial tension and compression. If an elastic bar with uniform and symmetric cross-section is bent in one of its planes of symmetry, the resultingbending stresswill still be normal (perpendicular to the cross-section), but will vary over the cross section: the outer part will be under tensile stress, while the inner part will be compressed. Another variant of normal stress is thehoop stressthat occurs on the walls of a cylindricalpipeorvesselfilled with pressurized fluid.Simple shear stress[edit]

Shear stress in a horizontal bar loaded by two offset blocks.Another simple type of stress occurs when a uniformly thick layer of elastic material like glue or rubber is firmly attached to two stiff bodies that are pulled in opposite directions by forces parallel to the layer; or a section of a soft metal bar that is being cut by the jaws of ascissors-like tool. LetFbe the magnitude of those forces, andMbe the midplane of that layer. Just as in the normal stress case, the part of the layer on one side ofMmust pull the other part with the same forceF. Assuming that the direction of the forces is known, the stress acrossMcan be expressed by the single number=F/A, whereFis the magnitude of those forces andAis the area of the layer.However, unlike normal stress, thissimple shear stressis directed parallel to the cross-section considered, rather than perpendicular to it.[7]For any planeSthat is perpendicular to the layer, the net internal force acrossS, and hence the stress, will be zero.As in the case of an axially loaded bar, in practice the shear stress may not be uniformly distributed over the layer; so, as before, the ratioF/Awill only be an average ("nominal", "engineering") stress. However, that average is often sufficient for practical purposes.[8]:p.292Shear stress is observed also when a cylindrical bar such as ashaftis subjected to opposite torques at its ends. In that case, the shear stress on each cross-section is parallel to the cross-section, but oriented tangentially relative to the axis, and increases with distance from the axis. Significant shear stress occurs in the middle plate (the "web") ofI-beamsunder bending loads, due to the web constraining the end plates ("flanges").Isotropic stress[edit]

Isotropic tensile stress. Top left: Each face of a cube of homogeneous material is pulled by a force with magnitudeF, applied evenly over the entire face whose area isA. The force across any sectionSof the cube must balance the forces applied below the section. In the three sections shown, the forces areF(top right),F(bottom left), andF(bottom right); and the area ofSisA,AandA, respectively. So the stress acrossSisF/Ain all three cases.Another simple type of stress occurs when the material body is under equal compression or tension in all directions. This is the case, for example, in a portion of liquid or gas at rest, whether enclosed in some container or as part of a larger mass of fluid; or inside a cube of elastic material that is being pressed or pulled on all six faces by equal perpendicular forces provided, in both cases, that the material is homogeneous, without built-in stress, and that the effect of gravity and other external forces can be neglected.In these situations, the stress across any imaginary internal surface turns out to be equal in magnitude and always directed perpendicularly to the surface independently of the surface's orientation. This type of stress may be calledisotropic normalor justisotropic; if it is compressive, it is calledhydrostatic pressureor justpressure. Gases by definition cannot withstand tensile stresses, but liquids may withstand very small amounts of isotropic tensile stress.Cylinder stresses[edit]Parts withrotational symmetry, such as wheels, axles, pipes, and pillars, are very common in engineering. Often the stress patterns that occur in such parts have rotational or evencylindrical symmetry. The analysis of suchcylinder stressescan take advantage of the symmetry to reduce the dimension of the domain and/or of the stress tensor.General stress[edit]Often, mechanical bodies experience more than one type of stress at the same time; this is calledcombined stress. In normal and shear stress, the magnitude of the stress is maximum for surfaces that are perpendicular to a certain direction, and zero across any surfaces that are parallel to. When the stress is zero only across surfaces that are perpendicular to one particular direction, the stress is calledbiaxial, and can be viewed as the sum of two normal or shear stresses. In the most general case, calledtriaxial stress, the stress is nonzero across every surface element.The Cauchy stress tensor[edit]Main article:Cauchy stress tensor

Components of stress in three dimensions

Illustration of typical stresses (arrows) across various surface elements on the boundary of a particle (sphere), in a homogeneous material under uniform (but not isotropic) triaxial stress. The normal stresses on the principal axes are +5, +2, and 3 units.Combined stresses cannot be described by a single vector. Even if the material is stressed in the same way throughout the volume of the body, the stress across any imaginary surface will depend on the orientation of that surface, in a non-trivial way.However, Cauchy observed that the stress vectoracross a surface will always be alinear functionof the surface'snormal vector, the unit-length vector that is perpendicular to it. That is,, where the functionsatisfies

for any vectorsand any real numbers. The function, now called the(Cauchy) stress tensor, completely describes the stress state of a uniformly stressed body. (Today, any linear connection between two physical vector quantities is called atensor, reflecting Cauchy's original use to describe the "tensions" (stresses) in a material.) Intensor calculus,is classified as second-order tensor oftype (0,2).Like any linear map between vectors, the stress tensor can be represented in any chosenCartesian coordinate systemby a 33 matrix of real numbers. Depending on whether the coordinates are numberedor named, the matrix may be written asorThe stress vectoracross a surface with normal vectorwith coordinatesis then a matrix product(where T in upper index is transposition) (look onCauchy stress tensor), that is

The linear relation betweenandfollows from the fundamental laws ofconservation of linear momentumandstatic equilibriumof forces, and is therefore mathematically exact, for any material and any stress situation. The components of the Cauchy stress tensor at every point in a material satisfy the equilibrium equations (Cauchys equations of motionfor zero acceleration). Moreover, the principle ofconservation of angular momentumimplies that the stress tensor issymmetric, that is,, and. Therefore, the stress state of the medium at any point and instant can be specified by only six independent parameters, rather than nine. These may be written

where the elementsare called theorthogonal normal stresses(relative to the chosen coordinate system), andtheorthogonal shear stresses.Change of coordinates[edit]The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. A graphical representation of this transformation law is theMohr's circleof stress distribution.As a symmetric 33 real matrix, the stress tensorhas three mutually orthogonal unit-lengtheigenvectorsand three realeigenvalues, such that. Therefore, in a coordinate system with axes, the stress tensor is a diagonal matrix, and has only the three normal componentstheprincipal stresses. If the three eigenvalues are equal, the stress is anisotropiccompression or tension, always perpendicular to any surface; if there is no shear stress, the tensor is a diagonal matrix in any coordinate frame.Stress as a tensor field[edit]In general, stress is not uniformly distributed over a material body, and may vary with time. Therefore the stress tensor must be defined for each point and each moment, by considering aninfinitesimalparticle of the medium surrounding that point, and taking the average stresses in that particle as being the stresses at the point.Stress in thin plates[edit]

Atank carmade from bent and welded steel plates.Man-made objects are often made from stock plates of various materials by operations that do not change their essentially two-dimensional character, like cutting, drilling, gentle bending and welding along the edges. The description of stress in such bodies can be simplified by modeling those parts as two-dimensional surfaces rather than three-dimensional bodies.In that view, one redefines a "particle" as being an infinitesimal patch of the plate's surface, so that the boundary between adjacent particles becomes an infinitesimal line element; both are implicitly extended in the third dimension, straight through the plate. "Stress" is then redefined as being a measure of the internal forces between two adjacent "particles" across their common line element, divided by the length of that line. Some components of the stress tensor can be ignored, but since particles are not infinitesimal in the third dimension one can no longer ignore the torque that a particle applies on its neighbors. That torque is modeled as abending stressthat tends to change thecurvatureof the plate. However, these simplifications may not hold at welds, at sharp bends and creases (where theradius of curvatureis comparable to the thickness of the plate).Stress in thin beams[edit]

For stress modeling, afishing polemay be considered one-dimensional.The analysis of stress can be considerably simplified also for thin bars,beamsor wires of uniform (or smoothly varying) composition and cross-section that are subjected to moderate bending and twisting. For those bodies may consider only cross-sections that are perpendicular to the bar's axis, and redefine a "particle" as being a piece of wire with infinitesimal length between two such cross sections. The ordinary stress is then reduced to a scalar (tension or compression of the bar), but one must take into account also abending stress(that tries to change the bar's curvature, in some direction perpendicular to the axis) and atorsional stress(that tries to twist or un-twist it about its axis).Other descriptions of stress[edit]The Cauchy stress tensor is used for stress analysis of material bodies experiencingsmall deformationswhere the differences in stress distribution in most cases can be neglected. For large deformations, also calledfinite deformations, other measures of stress, such as thefirst and second PiolaKirchhoff stress tensors, theBiot stress tensor, and theKirchhoff stress tensor, are required.Solids, liquids, and gases havestress fields. Static fluids support normal stress but will flow undershear stress. Movingviscous fluidscan support shear stress (dynamic pressure). Solids can support both shear and normal stress, withductilematerials failing under shear andbrittlematerials failing under normal stress. All materials have temperature dependent variations in stress-related properties, andnon-Newtonianmaterials have rate-dependent variations.Stress analysis[edit]Stress analysisis a branch ofapplied physicsthat covers the determination of the internal distribution of stresses in solid objects. It is an essential tool inengineeringfor the study and design of structures such astunnels,dams, mechanical parts, and structural frames, under prescribed or expected loads. It is also important in many other disciplines; for example, ingeology, to study phenomena likeplate tectonics,vulcanismandavalanches; and inbiology, to understand theanatomyof living beings.Goals and assumptions[edit]Stress analysis is generally concerned with objects and structures that can be assumed to be in macroscopicstatic equilibrium. ByNewton's laws of motion, any external forces are being applied to such a system must be balanced by internal reaction forces,[9]:p.97which are almost always surface contact forces between adjacent particles that is, as stress.[5]Since every particle needs to be in equilibrium, this reaction stress will generally propagate from particle, creating a stress distribution throughout the body.The typical problem in stress analysis is to determine these internal stresses, given the external forces that are acting on the system. The latter may bebody forces(such as gravity or magnetic attraction), that act throughout the volume of a material;[10]:p.4281or concentrated loads (such as friction between an axle and abearing, or the weight of a train wheel on a rail), that are imagined to act over a two-dimensional area, or along a line, or at single point.In stress analysis one normally disregards the physical causes of the forces or the precise nature of the materials. Instead, one assumes that the stresses are related to deformation (and, in non-static problems, to the rate of deformation) of the material by knownconstitutive equations.[11]Methods[edit]Stress analysis may be carried out experimentally, by applying loads to the actual artifact or to scale model, and measuring the resulting stresses, by any of several available methods. This approach is often used for safety certification and monitoring. However, most stress analysis is done by mathematical methods, especially during design.The basic stress analysis problem can be formulated byEuler's equations of motionfor continuous bodies (which are consequences ofNewton's lawsfor conservation oflinear momentumandangular momentum) and theEuler-Cauchy stress principle, together with the appropriate constitutive equations. Thus one obtains a system ofpartial differential equationsinvolving the stress tensor field and thestrain tensorfield, as unknown functions to be determined. The external body forces appear as the independent ("right-hand side") term in the differential equations, while the concentrated forces appear as boundary conditions. The basic stress analysis problem is therefore aboundary-value problem.Stress analysis forelasticstructures is based on thetheory of elasticityandinfinitesimal strain theory. When the applied loads cause permanent deformation, one must use more complicated constitutive equations, that can account for the physical processes involved (plastic flow,fracture,phase change, etc.).However, engineered structures are usually designed so that the maximum expected stresses are well within the range oflinear elasticity(the generalization ofHookes lawfor continuous media); that is, the deformations caused by internal stresses are linearly related to them. In this case the differential equations that define the stress tensor are linear, and the problem becomes much easier. For one thing, the stress at any point will be a linear function of the loads, too. For small enough stresses, even non-linear systems can usually be assumed to be linear.

Simplified model of a truss for stress analysis, assuming unidimensional elements under uniform axial tension or compression.Stress analysis is simplified when the physical dimensions and the distribution of loads allow the structure to be treated as one- or two-dimensional. In the analysis of trusses, for example, the stress field may be assumed to be uniform and uniaxial over each member. Then the differential equations reduce to a finite set of equations (usually linear) with finitely many unknowns. In other contexts one may be able to reduce the three-dimensional problem to a two-dimensional one, and/or replace the general stress and strain tensors by simpler models like uniaxial tension/compression, simple shear, etc.Still, for two- or three-dimensional cases one must solve a partial differential equation problem. Analytical or closed-form solutions to the differential equations can be obtained when the geometry, constitutive relations, and boundary conditions are simple enough. Otherwise one must generally resort to numerical approximations such as thefinite element method, thefinite difference method, and theboundary element method.Alternative measures of stress[edit]Main article:Stress measuresOther useful stress measures include the first and secondPiolaKirchhoff stress tensors, theBiot stress tensor, and theKirchhoff stress tensor.PiolaKirchhoff stress tensor[edit]In the case offinite deformations, thePiolaKirchhoff stress tensorsexpress the stress relative to the reference configuration. This is in contrast to theCauchy stress tensorwhich expresses the stress relative to the present configuration. For infinitesimal deformations and rotations, the Cauchy and PiolaKirchhoff tensors are identical.Whereas the Cauchy stress tensorrelates stresses in the current configuration, the deformationgradientand strain tensors are described by relating the motion to the reference configuration; thus not all tensors describing the state of the material are in either the reference or current configuration. Describing the stress, strain and deformation either in the reference or current configuration would make it easier to define constitutive models (for example, the Cauchy Stress tensor is variant to a pure rotation, while the deformation strain tensor is invariant; thus creating problems in defining a constitutive model that relates a varying tensor, in terms of an invariant one during pure rotation; as by definition constitutive models have to be invariant to pure rotations). The 1st PiolaKirchhoff stress tensor,is one possible solution to this problem. It defines a family of tensors, which describe the configuration of the body in either the current or the reference state.The 1st PiolaKirchhoff stress tensor,relates forces in thepresentconfiguration with areas in thereference("material") configuration.

whereis thedeformation gradientandis theJacobiandeterminant.In terms of components with respect to anorthonormal basis, the first PiolaKirchhoff stress is given by

Because it relates different coordinate systems, the 1st PiolaKirchhoff stress is atwo-point tensor. In general, it is not symmetric. The 1st PiolaKirchhoff stress is the 3D generalization of the 1D concept ofengineering stress.If the material rotates without a change in stress state (rigid rotation), the components of the 1st PiolaKirchhoff stress tensor will vary with material orientation.The 1st PiolaKirchhoff stress is energy conjugate to the deformation gradient.2nd PiolaKirchhoff stress tensor[edit]Whereas the 1st PiolaKirchhoff stress relates forces in the current configuration to areas in the reference configuration, the 2nd PiolaKirchhoff stress tensorrelates forces in the reference configuration to areas in the reference configuration. The force in the reference configuration is obtained via a mapping that preserves the relative relationship between the force direction and the area normal in the reference configuration.

Inindex notationwith respect to an orthonormal basis,

This tensor, a one-point tensor, is symmetric.If the material rotates without a change in stress state (rigid rotation), the components of the 2nd PiolaKirchhoff stress tensor remain constant, irrespective of material orientation.The 2nd PiolaKirchhoff stress tensor is energy conjugate to theGreenLagrange finite strain tensor.

STRESS is the intensity of force inside a solid. The basic unit of stress is the Pascal (Pa) which is Newton per square metre.In engineering it is more convenient to measured as the force (N) per square mm. This gives the common engineering unit of stress, MPa.PropertyFormulaUnitsExample

DENSITY: Mass per unit volume= Mass (kg) / Volume (m3)kg / m3Steel = 7800

STRENGTH: How much Stress it can 'take'Ultimate Strength (max stress before breaking)Yield Strength (max stress before plastic)Stresses: Tensile & Compression (Axial), ShearFatigue: Max stress under millions of repsWorking/Allowable; 'Safe' stress, design value= Force (N) / Area (mm2)MPa1020 Steel UTS = 400MPa1020 Steel YS = 200MPa1020 Steel SS = MPaSteel Grade 250 FS = 207MPa1020 ATS = 120MPa

HARDNESS: Resistance to indentation or abrasionSize or depth of indentvariesHRC55 (Rockwell) etc

STIFFNESS: How much Stress for a certain StrainYoung's Modulus, Elastic Modulus= Stress (MPa) / StrainMPa1020 Steel E = 205GPa

TOUGHNESS: Energy to break=Area under Stress-Strain curveJ / m2Charpy Test (Joules)

ELASTICITY: Ability to Stretch with plasticity= Strain at yield%1020 Steel: 0.01% @ yield

PLASTICITY: Permanent deformation:Ductility = tensile plasticityMalleablility = compressive plasticity= (L2 - L1) / L1%1020 Steel: 25%

POISSON'S RATIO: side strain to axial strainv = ex / ey-1020 Steel v = 0.29

DEFINEFormulaUnitsDiagram

Axial Stress (Tension or Compression)Stress = Force / AreaMPa

Axial Strain (Tension or Compression)Strain = extension / original Length-

Shear StressStress = Force / AreaMPa

Modulus of Elasticity (Young's Mod)E = Stress / StrainGPaSlope of Stress:Strain diagram

Modulus of Rigidity (Shear Mod.) =~ 0.4EG = S. Stress / S. StrainGPaSlope of S.Stress:S.Strain diagram

Shear StrainStrain = movement / original Depth-

Shear in Detail:Shear Strain is usually small enough to ignore the changes in L with angle.Angle is in radians.Area is the zone that would slide apart assuming it broke in shear.

What is a Stress?STRESS is the intensity of force inside a solid.It has the same units as Pressure (Pa, kPa, MPa, etc), so you could think of stress as pressure in a solid. The difference is, pressure acts equally in every direction, but stress has a certain direction.Stress = Force/AreaThe base unit for pressure and stress is the Pascal (Pa), but this is way too small for engineering use - except perhaps when measuring the pressure of air conditioning ducts or something. Certainly nothing compared to the stress required to break steel. In most engineering situations, the strength of a material is measured in MPa (MegaPascals)Stress (MPa) = Force (N) / Area (mm2)

COMMON MISTAKE: (FORCE DOUBLING).When drawing a Free Body Diagram of a component under stress, you will always end up with a pair of forces (e.g. 1 up, 1 down). This is thedefinitionof stress - that the cross-sectional area has to sustain the 2 forces trying to tear it apart. If you add the 2 forces together you are probably making a mistake! (Besides, if you did try to add them they would cancel each other out anyway, since they are in opposite directions.)Worked Example 1: Tensile force of 5kN acting on a 6mm diameter rod. What is the stress?

Worked Example 2: A block made of 40MPa concrete with dimensions as shown. What is the maximum load (mass) it can support?

Worked Example 3: Tensile force of 1kN, with steel of UTS=750MPa and Factor of Safety of 2.5. What is maximum force?

DIFFERENT SYMBOLS:Watch out for different symbols for stress. Ivanoff (and some TAFE publications) usefbut the rest of the world (internet and other textbooks) use the Greek symbol sigma.

Tensile, Compressive and Shear stressThere are 3 types of stress in the world; Tensile = pulling apart Compressive = squashing together Shear = sliding apartAny of these 3 types of stress are calculated the same way, with the same units - it the area that is different. Always think of what area must be broken when the component fails (the broken area).

Strength Of Materials. Part I. Simple StressI. Stress. When forces are applied to a body they tend in a greater or less degree to break it. Preventing or tending to prevent the rupture, there arise, generally, forces between every two adjacent parts of the body. Thus, when a load is suspended by means of anironrod, the rod is subjected to a downward pull at its lower end and to an upward pull at itsupperend, and these two forces tend to pull it apart. At any cross-section of the rod the iron on either side "holds fast" to that on the other, and these forces which the parts of the rod exert upon each other prevent the tearing of the rod. Forexample, in Fig. 1, let a represent the rod and its suspended load, 1,000pounds; then the pull on the lower end equals 1,000 pounds. If we neglect theweightof the rod, the pull on the upper end is also 1,000 pounds, as shown in Fig. 1 (b); and the upper part A exerts on the lower part B an upward pull Q equal to 1,000 pounds, while the lower part exerts on the upper a force P also equal to 1,000 pounds. These two forces, P and Q, prevent rupture of the rod at the "section" C; at any other section there are two forces like P and Q preventing rupture at that section.By stress at a section of a body is meant the force which the part of the body on either side of the section exerts on the other. Thus, the stress at the section C (Fig. 1) is P (or Q), and it equals 1,000 pounds.a. Stresses are usually expressed (in America) in pounds, sometimes in tons. Thus the stress P in the preceding article is

Fig. 1.1.000 pounds, or ton. Notice that thisvaluehas nothing to do with thesizeof the cross-section on which the stress acts.3. Kinds of Stress, (a) When the forces acting on a body (as a rope or rod) are such that they tend to tear it, the stress at any cross-section is called a tension or a tensile stress. The stresses P and Q, of Fig. 1, are tensile stresses. Stretched ropes, loaded "tie rods" ofroofsand bridges, etc., are under tensile stress. (b.) "When the forces acting on a body (as a shortpost,brick, etc.) are such that they tend to crush it, the stress at any section at rightanglesto the direction of the crushing forces is called a pressure or a compres-sive stress. Fig. 2 (a) represents a loaded post, and Fig. 2 (b) the upper and lower parts. The upper part presses down on B, and the lower part presses up on A, as shown. P or Q is the compressive stress in the post at section C. Loaded posts, or struts, piers, etc., are under compressive stress.(c.) When the forces acting on a body (as a rivet in a bridgejoint) are such that they tend to cut or "shear" it across, the stress at a section along which there is a tendency to cut is called a shear or a shearing stress. This kind of stress takes its name from the act of cutting with a pair of shears. In amaterialwhich is being cut in this way, the stresses that are being "overcome" are shearing stresses. Fig. 3 (a) represents ariveted joint, and Fig. 3 (b) two parts of the rivet. The forces applied to the joint are such that A tends to slide to the left, and B to the right; then B exerts on A a force P toward the right, and A on B a force Q toward the left as shown. P or Q is the shearing stress in the rivet.Tensions, Compressions and Shears are called simple stresses. "Forces may act upon a body so as to produce a combination of simple stresses on some section; such a combination is called a complex stress. The stresses inbeamsare usually complex. There are other terms used to describe stress; they will be defined farther on.

Fig. 2.4. Unit=Stress. It is often necessary to specify not merely the amount of the entire stress which acts on an area, but also the amount which acts on each unit of area (square inch for example). By unit-stress is meant stress per unit area.To find the value of a unit-stress: Divide the whole stress by the whole area of the section on which it acts, or over which it is distributed. Thus, letP denote the value of the whole stress,A the area on which it acts, andS the value of the unit-stress; thenS = P/A, also P = AS.(I)Strictly these formulas apply only when the stress P is uniform, that is, when it is uniformly distributed over the area, each square inch for example sustaining the same amount of stress. When the stress is not uniform, that is, when the stresses on different square inches are not equal, then PA equals the average value of the unit-stress.

Fig. 3.5. Unit-stresses are usually expressed (in America) in pounds per square inch, sometimes in tons per square inch. If P and A in equation 1 are expressed in pounds and square inches respectively, then S will be in pounds per square inch; and if P and A are expressed in tons and square inches, S will be in tons per square inch.Examples. 1. Suppose that the rod sustaining the load in Fig. 1 is 2 square inches in cross-section, and that the load weighs l000 pounds. What is the value of the unit-stress ?Here P = 1,000 pounds, A= 2 square inches; hence.S = 1,000/ 2 = 500 pounds per square inch.2. Suppose that the rod is one-half square inch in cross-section. What is the value of the unit-stress?A = square inch, and, as before, P = 1,000 pounds; hence S = 1,000 = 2,000 pounds per square inch.Notice that one must always divide the whole stress by the area to get the unit-stress, whether the area is greater or less than one.6. Deformation. "Whenever forces are applied to a body itchangesin size, and usually in shape also. This change ofsizeand shape is called deformation. Deformations are usually measured in inches; thus, if a rod is stretched 2 inches, the "elongation" = 2 inches.7. Unit-Deformation. It is sometimes necessary to specify not merely the value of a total deformation but its amount per unit length of the deformed body. Deformation per unit length of the deformed body is called unit-deformation.

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Normal Stress/Shear StressClick here for discussion of Shear StressKey Concepts:Normal stress can be viewed as force per unit area acting normal to aninternal section of a structural element, typically called a bar or an axial member.

In a Nut Shell:Definition of an Axial MemberA structure that is generally long in one direction (perhaps in the x-direction), straight,and has a constant (or mildly tapered) cross-section is generally termed an axialmember.The cross-section of the axial member will have acentroid.The x-axis of theaxial member is assumed to lie along thecentroidof each cross-section.Click here toview a typical axial member.

Definition of Normal Stress in an Axial MemberThe average normal stress. , in an axial member is the force, P, in the member dividedby its cross-sectional area, A.=P / ACommon units for stress arepsi,ksi,MPa, N/mm2(English/Metric)

Definition of Normal Stress at a Point, P, in an axial memberIts possible that the axial force might vary over the cross-section of a structuralmember.In that case let the element of force beFover an element of areaAfor the cross-section.Then the normal stress,p, at pointPin the cross-section of the axial member isp=limF / AA 0assuming the limit exists.If the cross-section lies in the y-z plane then the axial stress may bea function of both y and z so that=(y,z)and the total axial force, F, acts through thecentroidat any given cross-section is simply the integral ofover the cross-section.Clickhere for a review ofcentroids.F= (y,z)dA

Normal StressExample:LinkAChas a uniform cross section in wide and 1/16 in thick.A smooth pinconnectsthe link to the rectangular plate at A.Smooth pins also are at C and B.A cord rapsaroundthe smooth pulleys as shown.The tension in the cord, P, is 300 lb.Find the normalstressmidway along the link AC.Strategy:Construct a free body diagrams of the plate and of link, AC.Use equilibrium tofindthe force in the link.Then the normal stress in the link is just the force it carries dividedbythe cross-sectional area of the link.The equations of equilibrium are:Fx= 0,Fy=0,and MB= 0.Click here to continue with this example.

Normal StressExample:(continued)For equilibrium of the plate:CCWMB= 0- Ax(12) + 300(6)=0Ax= 150.0 lbNote from statics that the link is a two-force member.So that it is either in tension orincompression.In this case link AC carries a tensile force, R, directed along the link.For equilibrium oflink AC:Fx=0-Rcos(30)+ 150.0=0R=173.2 lbCross-sectional area of link=Alink= (1/4)(1/16)=1/64 in2So the normal stress in link AC=link= 173.2/(1/64)=11085psi =11.09ksi(result)

Normal Stress*Example:While stopping a car the driver exerts a force, P, of 10 lb on the brake pedal at Casshown below.The brake rod is pinned normal to the brake pedal ABC at B.d = 2 in. ande = 10 in.The diameter of the brake rod is 3/16 in.Find the normal stress in brake rod BD.Strategy:Construct a free body diagram of the brake pedal.Use equilibrium to find theforceexerted by the brake rod on the brake pedal.Then the normal stress in the brakerodis just the force it carries divided by the cross-sectional area of the brake rod.The free body diagram of the brake pedal, ABC, is:Apply the equation ofequilibriumMA= 0.R(d) P(d+e)=0,R=P(d+e)/dR=10(2+10)/2=60 lbx-sectional area of brake rod =(3/16)2/4 = 0.0276 in2The force in the brake rod is equal and opposite.Sothe normal stress in the brake rod is: =-60/0.0276=- 2170 psiNote:The brake rod is in compression.(result)

Normal Strain/Shear StrainKey Concepts:Normal strain is the change in length of a structural element divided by theoriginal length of the structural element.Shear strain is the change in angle (distortion)between any two lines at a section of the structural element resulting from applied loads.

In a Nut Shell:Imagine an arbitrary structural member say in the shape of a flat ovular plate.See the figure below.Beforeloading let an arbitrary material segment, AB, in the plate be oflength L.After loading thematerial segment may rotate and change length.Let the length ofthe same material segment become A*B*,after loading be L*.Then the normal strain, , isdefined as=( L*-L)/L=normal strainCommon units of strain are ininches/inch,millimeters/millimeter, ormicrostrain(dimensionless).Normal strain is considered to be positive when the material segment becomeslonger and negative if it shortens (sign convention).Usually normal strains arevery small usually much less than 1.| |