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.
Read
more:http://chestofbooks.com/architecture/Cyclopedia-Carpentry-Building-1-3/Strength-Of-Materials-Part-I-Simple-Stress.html#.VMeeu_6UfpU#ixzz3Q22lH4WC
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.| |
