Stress–strain analysis (or stress analysis) is an
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
discipline that uses many methods to determine the
stresses and
strains in materials and structures subjected to
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s. In
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, stress is a
physical quantity
A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For examp ...
that expresses the internal
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s that neighboring
particle
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object which can be described by several physical property, physical or chemical property, chemical ...
s of a
continuous material exert on each other, while strain is the measure of the deformation of the material.
In simple terms we can define stress as the force of resistance per unit area, offered by a body against deformation. Stress is the ratio of force over area (S =R/A, where S is the stress, R is the internal resisting force and A is the cross-sectional area). Strain is the ratio of change in length to the original length, when a given body is subjected to some external force (Strain= change in length÷the original length).
Stress analysis is a primary task for
civil
Civil may refer to:
*Civic virtue, or civility
*Civil action, or lawsuit
* Civil affairs
*Civil and political rights
*Civil disobedience
*Civil engineering
*Civil (journalism), a platform for independent journalism
*Civilian, someone not a membe ...
,
mechanical
Mechanical may refer to:
Machine
* Machine (mechanical), a system of mechanisms that shape the actuator input to achieve a specific application of output forces and movement
* Mechanical calculator, a device used to perform the basic operations of ...
and
aerospace engineers
Aerospace engineering is the primary field of engineering concerned with the development of aircraft and spacecraft. It has two major and overlapping branches: aeronautical engineering and astronautical engineering. Avionics engineering is si ...
involved in the design of structures of all sizes, such as
tunnel
A tunnel is an underground passageway, dug through surrounding soil, earth or rock, and enclosed except for the entrance and exit, commonly at each end. A pipeline is not a tunnel, though some recent tunnels have used immersed tube cons ...
s,
bridge
A bridge is a structure built to span a physical obstacle (such as a body of water, valley, road, or rail) without blocking the way underneath. It is constructed for the purpose of providing passage over the obstacle, which is usually somethi ...
s and
dam
A dam is a barrier that stops or restricts the flow of surface water or underground streams. Reservoirs created by dams not only suppress floods but also provide water for activities such as irrigation, human consumption, industrial use, a ...
s,
aircraft
An aircraft is a vehicle that is able to fly by gaining support from the air. It counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines ...
and
rocket
A rocket (from it, rocchetto, , bobbin/spool) is a vehicle that uses jet propulsion to accelerate without using the surrounding air. A rocket engine produces thrust by reaction to exhaust expelled at high speed. Rocket engines work entirely fr ...
bodies, mechanical parts, and even
plastic cutlery
Cutlery (also referred to as silverware, flatware, or tableware), includes any hand implement used in preparing, serving, and especially eating food in Western culture. A person who makes or sells cutlery is called a cutler. The city of Sheffie ...
and
staples. Stress analysis is also used in the maintenance of such structures, and to investigate the causes of structural failures.
Typically, the starting point for stress analysis are a
geometrical
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
description of the structure, the properties of the materials used for its parts, how the parts are joined, and the maximum or typical forces that are expected to be applied to the structure. The output data is typically a quantitative description of how the applied forces spread throughout the structure, resulting in stresses, strains and the deflections of the entire structure and each component of that structure. The analysis may consider forces that vary with time, such as
engine
An engine or motor is a machine designed to convert one or more forms of energy into mechanical energy.
Available energy sources include potential energy (e.g. energy of the Earth's gravitational field as exploited in hydroelectric power gen ...
vibrations or the load of moving vehicles. In that case, the stresses and deformations will also be functions of time and space.
In engineering, stress analysis is often a tool rather than a goal in itself; the ultimate goal being the design of structures and artifacts that can withstand a specified load, using the minimum amount of material or that satisfies some other optimality criterion.
Stress analysis may be performed through classical mathematical techniques, analytic mathematical modelling or computational simulation, experimental testing, or a combination of methods.
The term stress analysis is used throughout this article for the sake of brevity, but it should be understood that the strains, and deflections of structures are of equal importance and in fact, an analysis of a structure may begin with the calculation of deflections or strains and end with calculation of the stresses.
Scope
General principles
Stress analysis is specifically concerned with solid objects. The study of stresses in liquids and gases is the subject of
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
.
Stress analysis adopts the macroscopic view of materials characteristic of
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, namely that all properties of materials are homogeneous at small enough scales. Thus, even the smallest
particle
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object which can be described by several physical property, physical or chemical property, chemical ...
considered in stress analysis still contains an enormous number of atoms, and its properties are averages of the properties of those atoms.
In stress analysis one normally disregards the physical causes of forces or the precise nature of the materials. Instead, one assumes that the stresses are related to
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
of the material by known
constitutive equations
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approx ...
.
By
Newton's laws of motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in moti ...
, any external forces that act on a system must be balanced by internal reaction forces,
[ or cause the particles in the affected part to accelerate. In a solid object, all particles must move substantially in concert in order to maintain the object's overall shape. It follows that any force applied to one part of a solid object must give rise to internal reaction forces that propagate from particle to particle throughout an extended part of the system. With very rare exceptions (such as ]ferromagnetic
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
materials or planet-scale bodies), internal forces are due to very short range intermolecular interactions, and are therefore manifested as surface contact forces between adjacent particles — that is, as stress.[
]
Fundamental problem
The fundamental problem in stress analysis is to determine the distribution of internal stresses throughout the system, given the external forces that are acting on it. In principle, that means determining, implicitly or explicitly, the Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that complete ...
at every point.[
The external forces may be ]body force
In physics, a body force is a force that acts throughout the volume of a body.
Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref>
Forces due to gravity, electric fields and magnetic fields are examples of body forces. Bo ...
s (such as gravity or magnetic attraction), that act throughout the volume of a material;[ or concentrated loads (such as friction between an axle and a bearing, 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. The same net external force will have a different effect on the local stress depending on whether it is concentrated or spread out.
]
Types of structures
In civil engineering applications, one typically considers structures to be in static equilibrium
In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero. By extension, a physical system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is zero ...
: that is, are either unchanging with time, or are changing slowly enough for viscous stresses to be unimportant (quasi-static). In mechanical and aerospace engineering, however, stress analysis must often be performed on parts that are far from equilibrium, such as vibrating plates or rapidly spinning wheels and axles. In those cases, the equations of motion must include terms that account for the acceleration of the particles. In structural design applications, one usually tries to ensure the stresses are everywhere well below the yield strength
In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and wi ...
of the material. In the case of dynamic loads, the material fatigue
In materials science, fatigue is the initiation and propagation of cracks in a material due to cyclic loading. Once a fatigue crack has initiated, it grows a small amount with each loading cycle, typically producing striations on some parts of ...
must also be taken into account. However, these concerns lie outside the scope of stress analysis proper, being covered in materials science under the names strength of materials
The field of strength of materials, also called mechanics of materials, typically refers to various methods of calculating the stresses and strains in structural members, such as beams, columns, and shafts. The methods employed to predict the re ...
, fatigue
Fatigue describes a state of tiredness that does not resolve with rest or sleep. In general usage, fatigue is synonymous with extreme tiredness or exhaustion that normally follows prolonged physical or mental activity. When it does not resolve ...
analysis, stress corrosion, creep modeling, and other.
Experimental methods
Stress analysis can be performed experimentally by applying forces to a test element or structure and then determining the resulting stress using sensor
A sensor is a device that produces an output signal for the purpose of sensing a physical phenomenon.
In the broadest definition, a sensor is a device, module, machine, or subsystem that detects events or changes in its environment and sends ...
s. In this case the process would more properly be known as ''testing'' ( destructive or non-destructive). Experimental methods may be used in cases where mathematical approaches are cumbersome or inaccurate. Special equipment appropriate to the experimental method is used to apply the static or dynamic loading.
There are a number of experimental methods which may be used:
*Tensile testing
Tensile testing, also known as tension testing, is a fundamental materials science and engineering test in which a sample is subjected to a controlled tension until failure. Properties that are directly measured via a tensile test are ultimate ...
is a fundamental materials science test in which a sample is subjected to uniaxial tension until failure. The results from the test are commonly used to select a material for an application, for quality control
Quality control (QC) is a process by which entities review the quality of all factors involved in production. ISO 9000 defines quality control as "a part of quality management focused on fulfilling quality requirements".
This approach places ...
, or to predict how a material will react under other types of forces. Properties that are directly measured via a tensile test are the ultimate tensile strength
Ultimate tensile strength (UTS), often shortened to tensile strength (TS), ultimate strength, or F_\text within equations, is the maximum stress that a material can withstand while being stretched or pulled before breaking. In brittle materials t ...
, maximum elongation and reduction in cross-section
Cross section may refer to:
* Cross section (geometry)
** Cross-sectional views in architecture & engineering 3D
*Cross section (geology)
* Cross section (electronics)
* Radar cross section, measure of detectability
* Cross section (physics)
**Ab ...
area. From these measurements, properties such as Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
, Poisson's ratio
In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Pois ...
, yield strength
In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and wi ...
, and the strain-hardening characteristics of the sample can be determined.
*Strain gauge
A strain gauge (also spelled strain gage) is a device used to measure strain on an object. Invented by Edward E. Simmons and Arthur C. Ruge in 1938, the most common type of strain gauge consists of an insulating flexible backing which supports ...
s can be used to experimentally determine the deformation of a physical part. A commonly used type of strain gauge is a thin flat resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
that is affixed to the surface of a part, and which measures the strain in a given direction. From the measurement of strain on a surface in three directions the stress state that developed in the part can be calculated.
* Neutron diffraction is a technique that can be used to determine the subsurface strain in a part.
*The photoelastic method relies on the fact that some materials exhibit birefringence
Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefring ...
on the application of stress, and the magnitude of the refractive indices at each point in the material is directly related to the state of stress at that point. The stresses in a structure can be determined by making a model of the structure from such a photoelastic material.
*Dynamic mechanical analysis
Dynamic mechanical analysis (abbreviated DMA) is a technique used to study and characterize materials. It is most useful for studying the viscoelastic behavior of polymers
A polymer (; Greek '' poly-'', "many" + ''-mer'', "part")
is a subst ...
(DMA) is a technique used to study and characterize viscoelastic
In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly wi ...
materials, particularly polymers. The viscoelastic property of a polymer is studied by dynamic mechanical analysis where a sinusoidal force (stress) is applied to a material and the resulting displacement (strain) is measured. For a perfectly elastic solid, the resulting strains and the stresses will be perfectly in phase. For a purely viscous fluid, there will be a 90 degree phase lag of strain with respect to stress. Viscoelastic polymers have the characteristics in between where some phase lag will occur during DMA tests.
Mathematical methods
While experimental techniques are widely used, most stress analysis is done by mathematical methods, especially during design.
Differential formulation
The basic stress analysis problem can be formulated by Euler's equations of motion for continuous bodies (which are consequences of Newton's laws
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
for conservation of linear momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
and angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
) and the Euler-Cauchy stress principle
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that complete ...
, together with the appropriate constitutive equations.
These laws yield a system of partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
that relate the stress tensor field to the strain tensor
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally ...
field as unknown functions to be determined. Solving for either then allows one to solve for the other through another set of equations called constitutive equations. Both the stress and strain tensor fields will normally be continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
within each part of the system and that part can be regarded as a continuous medium with smoothly varying constitutive equations.
The external body forces will appear as the independent ("right-hand side") term in the differential equations, while the concentrated forces appear as boundary conditions. An external (applied) surface force, such as ambient pressure or friction, can be incorporated as an imposed value of the stress tensor across that surface. External forces that are specified as line loads (such as traction) or point loads (such as the weight of a person standing on a roof) introduce singularities in the stress field, and may be introduced by assuming that they are spread over small volume or surface area. The basic stress analysis problem is therefore a boundary-value problem.
Elastic and linear cases
A system is said to be elastic
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rubber used to hold objects togeth ...
if any deformations caused by applied forces will spontaneously and completely disappear once the applied forces are removed. The calculation of the stresses (stress analysis) that develop within such systems is based on the theory of elasticity
Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and ot ...
and infinitesimal strain theory
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally ...
. When the applied loads cause permanent deformation, one must use more complicated constitutive equations, that can account for the physical processes involved (plastic flow
In engineering, deformation refers to the change in size or shape of an object. ''Displacements'' are the ''absolute'' change in position of a point on the object. Deflection is the relative change in external displacements on an object. Strain ...
, fracture
Fracture is the separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually occurs due to the development of certain displacement discontinuity surfaces within the solid. If a displa ...
, phase change
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic State of ...
, etc.)
Engineered structures are usually designed so that the maximum expected stresses are well within the realm of linear elastic (the generalization of Hooke’s law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of th ...
for continuous media) behavior for the material from which the structure will be built. That is, the deformations caused by internal stresses are linearly related to the applied loads. In this case the differential equations that define the stress tensor are also linear. Linear equations are much better understood than non-linear ones; for one thing, their solution (the calculation of stress at any desired point within the structure) will also be a linear function of the applied forces. For small enough applied loads, even non-linear systems can usually be assumed to be linear.
Built-in stress (preloaded)
A preloaded structure is one that has internal forces, stresses, and strains imposed within it by various means prior to application of externally applied forces. For example, a structure may have cables that are tightened, causing forces to develop in the structure, before any other loads are applied. Tempered glass is a commonly found example of a preloaded structure that has tensile forces and stresses that act on the plane of the glass and in the central plane of glass that causes compression forces to act on the external surfaces of that glass.
The mathematical problem represented is typically ill-posed
The mathematical term well-posed problem stems from a definition given by 20th-century French mathematician Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that:
# a solution exists,
# the sol ...
because it has an infinitude of solutions. In fact, in any three-dimensional solid body one may have infinitely many (and infinitely complicated) non-zero stress tensor fields that are in stable equilibrium even in the absence of external forces. These stress fields are often termed hyperstatic stress fields and they co-exist with the stress fields that balance the external forces. In linear elasticity, their presence is required to satisfy the strain/displacement compatibility requirements and in limit analysis their presence is required to maximise the load carrying capacity of the structure or component.
Such built-in stress may occur due to many physical causes, either during manufacture (in processes like extrusion
Extrusion is a process used to create objects of a fixed cross-sectional profile by pushing material through a die of the desired cross-section. Its two main advantages over other manufacturing processes are its ability to create very complex c ...
, casting
Casting is a manufacturing process in which a liquid material is usually poured into a mold, which contains a hollow cavity of the desired shape, and then allowed to solidify. The solidified part is also known as a ''casting'', which is ejected ...
or cold working
In metallurgy, cold forming or cold working is any metalworking process in which metal is shaped below its recrystallization temperature, usually at the ambient temperature. Such processes are contrasted with hot working techniques like hot roll ...
), or after the fact (for example because of uneven heating, or changes in moisture content or chemical composition). However, if the system can be assumed to behave in a linear fashion with respect to the loading and response of the system, then effect of preload can be accounted for by adding the results of a preloaded structure and the same non-preloaded structure.
If linearity cannot be assumed, however, any built-in stress may affect the distribution of internal forces induced by applied loads (for example, by changing the effective stiffness of the material) or even cause an unexpected material failure. For these reasons, a number of techniques have been developed to avoid or reduce built-in stress, such as annealing of cold-worked glass and metal parts, expansion joint
An expansion joint, or movement joint, is an assembly designed to hold parts together while safely absorbing temperature-induced expansion and contraction of building materials, and vibration, or to allow movement due to ground settlement or seis ...
s in buildings, and roller joints for bridges.
Simplifications
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 a bridge, its three dimensional structure may be idealized as a single planar structure, if all forces are acting in the plane of the trusses of the bridge. Further, each member of the truss structure might then be treated a uni-dimensional members with the forces acting along the axis of each member. In which case, the differential equations reduce to a finite set of equations with finitely many unknowns.
If the stress distribution can be assumed to be uniform (or predictable, or unimportant) in one direction, then one may use the assumption of plane stress
In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular plane. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analysi ...
and plane strain
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally ...
behavior and the equations that describe the stress field are then a function of two coordinates only, instead of three.
Even under the assumption of linear elastic behavior of the material, the relation between the stress and strain tensors is generally expressed by a fourth-order stiffness tensor
Stiffness is the extent to which an object resists deformation in response to an applied force.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
Calculations
The stiffness, k, of a bo ...
with 21 independent coefficients (a symmetric 6 × 6 stiffness matrix). This complexity may be required for general anisotropic materials, but for many common materials it can be simplified. For orthotropic material
In material science and solid mechanics, orthotropic materials have material properties at a particular point which differ along three orthogonal axes, where each axis has twofold rotational symmetry. These directional differences in strength can b ...
s such as wood, whose stiffness is symmetric with respect to each of three orthogonal planes, nine coefficients suffice to express the stress–strain relationship. For isotropic materials, these coefficients reduce to only two.
One may be able to determine ''a priori
("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ex ...
'' that, in some parts of the system, the stress will be of a certain type, such as uniaxial tension
Tension may refer to:
Science
* Psychological stress
* Tension (physics), a force related to the stretching of an object (the opposite of compression)
* Tension (geology), a stress which stretches rocks in two opposite directions
* Voltage or el ...
or compression
Compression may refer to:
Physical science
*Compression (physics), size reduction due to forces
*Compression member, a structural element such as a column
*Compressibility, susceptibility to compression
*Gas compression
*Compression ratio, of a c ...
, simple shear
Shear may refer to:
Textile production
*Animal shearing, the collection of wool from various species
**Sheep shearing
*The removal of nap during wool cloth production
Science and technology Engineering
*Shear strength (soil), the shear strength ...
, isotropic compression or tension, torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
, bending
In applied mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element.
The structural element is assumed to ...
, etc. In those parts, the stress field may then be represented by fewer than six numbers, and possibly just one.
Solving the equations
In any case, for two- or three-dimensional domains one must solve a system of partial differential equations with specified boundary conditions. Analytical (closed-form) solutions to the differential equations can be obtained when the geometry, constitutive relations, and boundary conditions are simple enough. For more complicated problems one must generally resort to numerical approximations such as the finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
, the finite difference method
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are di ...
, and the boundary element method
The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in ''boundary integral'' form), including fluid mechanics, acoustics, ele ...
.
Factor of safety
The ultimate purpose of any analysis is to allow the comparison of the developed stresses, strains, and deflections with those that are allowed by the design criteria. All structures, and components thereof, must obviously be designed to have a capacity greater than what is expected to develop during the structure's use to obviate failure. The stress that is calculated to develop in a member is compared to the strength of the material from which the member is made by calculating the ratio of the strength of the material to the calculated stress. The ratio must obviously be greater than 1.0 if the member is to not fail. However, the ratio of the allowable stress to the developed stress must be greater than 1.0 as a factor of safety (design factor) will be specified in the design requirement for the structure. All structures are designed to exceed the load those structures are expected to experience during their use. The design factor (a number greater than 1.0) represents the degree of uncertainty in the value of the loads, material strength, and consequences of failure. The stress (or load, or deflection) the structure is expected to experience are known as the working, the design or limit stress. The limit stress, for example, is chosen to be some fraction of the yield strength
In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and wi ...
of the material from which the structure is made. The ratio of the ultimate strength of the material to the allowable stress is defined as the factor of safety against ultimate failure.
Laboratory tests are usually performed on material samples in order to determine the yield and ultimate strengths of those materials. A statistical analysis of the strength of many samples of a material is performed to calculate the particular material strength of that material. The analysis allows for a rational method of defining the material strength and results in a value less than, for example, 99.99% of the values from samples tested. By that method, in a sense, a separate factor of safety has been applied over and above the design factor of safety applied to a particular design that uses said material.
The purpose of maintaining a factor of safety on yield strength is to prevent detrimental deformations that would impair the use of the structure. An aircraft with a permanently bent wing might not be able to move its control surfaces, and hence, is inoperable. While yielding of material of structure could render the structure unusable it would not necessarily lead to the collapse of the structure. The factor of safety on ultimate tensile strength is to prevent sudden fracture and collapse, which would result in greater economic loss and possible loss of life.
An aircraft wing might be designed with a factor of safety of 1.25 on the yield strength of the wing and a factor of safety of 1.5 on its ultimate strength. The test fixtures that apply those loads to the wing during the test might be designed with a factor of safety of 3.0 on ultimate strength, while the structure that shelters the test fixture might have an ultimate factor of safety of ten. These values reflect the degree of confidence the responsible authorities have in their understanding of the load environment, their certainty of the material strengths, the accuracy of the analytical techniques used in the analysis, the value of the structures, the value of the lives of those flying, those near the test fixtures, and those within the building.
The factor of safety is used to calculate a maximum allowable stress:
:
Load transfer
The evaluation of loads and stresses within structures is directed to finding the load transfer path. Loads will be transferred by physical contact between the various component parts and within structures. The load transfer may be identified visually or by simple logic for simple structures. For more complex structures more complex methods, such as theoretical solid mechanics
Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and ot ...
or numerical methods may be required. Numerical methods include direct stiffness method As one of the methods of structural analysis, the direct stiffness method, also known as the matrix stiffness method, is particularly suited for computer-automated analysis of complex structures including the statically indeterminate type. It is a ...
which is also referred to as the finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
.
The object is to determine the critical stresses in each part, and compare them to the strength of the material (see strength of materials
The field of strength of materials, also called mechanics of materials, typically refers to various methods of calculating the stresses and strains in structural members, such as beams, columns, and shafts. The methods employed to predict the re ...
).
For parts that have broken in service, a forensic engineering
Forensic engineering has been defined as ''"the investigation of failures - ranging from serviceability to catastrophic - which may lead to legal activity, including both civil and criminal".'' It includes the investigation of materials, product ...
or failure analysis Failure analysis is the process of collecting and analyzing data to determine the cause of a failure, often with the goal of determining corrective actions or liability.
According to Bloch and Geitner, ”machinery failures reveal a reaction chain o ...
is performed to identify weakness, where broken parts are analysed for the cause or causes of failure. The method seeks to identify the weakest component in the load path. If this is the part which actually failed, then it may corroborate independent evidence of the failure. If not, then another explanation has to be sought, such as a defective part with a lower tensile strength
Ultimate tensile strength (UTS), often shortened to tensile strength (TS), ultimate strength, or F_\text within equations, is the maximum stress that a material can withstand while being stretched or pulled before breaking. In brittle materials t ...
than it should for example.
Uniaxial stress
A linear element of a structure is one that is essentially one dimensional and is often subject to axial loading only. When a structural element is subjected to tension or compression its length will tend to elongate or shorten, and its cross-sectional area changes by an amount that depends on the Poisson's ratio
In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Pois ...
of the material. In engineering applications, structural members experience small deformations and the reduction in cross-sectional area is very small and can be neglected, i.e., the cross-sectional area is assumed constant during deformation. For this case, the stress is called ''engineering stress'' or ''nominal stress'' and is calculated using the original cross section.
:
where P is the applied load, and Ao is the original cross-sectional area.
In some other cases, e.g., elastomer
An elastomer is a polymer with viscoelasticity (i.e. both viscosity and elasticity) and with weak intermolecular forces, generally low Young's modulus and high failure strain compared with other materials. The term, a portmanteau of ''elastic p ...
s and plastic
Plastics are a wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adaptab ...
materials, the change in cross-sectional area is significant. For the case of materials where the volume is conserved (i.e. Poisson's ratio
In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Pois ...
= 0.5), if the ''true stress'' is desired, it must be calculated using the true cross-sectional area instead of the initial cross-sectional area, as:
:,
where
: is the nominal (engineering) strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
, and
: is nominal (engineering) stress.
The relationship between true strain and engineering strain is given by
:.
In uniaxial tension, true stress is then greater than nominal stress. The converse holds in compression.
Graphical representation of stress at a point
''Mohr's circle'', ''Lame's stress ellipsoid'' (together with the ''stress director surface''), and ''Cauchy's stress quadric'' are two-dimensional graphical representations of the state of stress at a point. They allow for the graphical determination of the magnitude of the stress tensor at a given point for all planes passing through that point. Mohr's circle is the most common graphical method.
''Mohr's circle'', named after Christian Otto Mohr
Christian Otto Mohr (8 October 1835 – 2 October 1918) was a German civil engineer. He is renowned for his contributions to the field of structural engineering, such as Mohr's circle, and for his study of stress.
Biography
He was born on 8 Octo ...
, is the locus of points that represent the state of stress on individual planes at all their orientations. The abscissa
In common usage, the abscissa refers to the (''x'') coordinate and the ordinate refers to the (''y'') coordinate of a standard two-dimensional graph.
The distance of a point from the y-axis, scaled with the x-axis, is called abscissa or x coo ...
, , and ordinate
In common usage, the abscissa refers to the (''x'') coordinate and the ordinate refers to the (''y'') coordinate of a standard two-dimensional graph.
The distance of a point from the y-axis, scaled with the x-axis, is called abscissa or x coo ...
, , of each point on the circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
are the normal stress and shear stress components, respectively, acting on a particular cut plane with a unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
with components .
Lame's stress ellipsoid
The surface of the ellipsoid represents the locus of the endpoints of all stress vectors acting on all planes passing through a given point in the continuum body. In other words, the endpoints of all stress vectors at a given point in the continuum body lie on the stress ellipsoid surface, i.e., the radius-vector from the center of the ellipsoid, located at the material point in consideration, to a point on the surface of the ellipsoid is equal to the stress vector on some plane passing through the point. In two dimensions, the surface is represented by an ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
(Figure coming).
Cauchy's stress quadric
The Cauchy's stress quadric, also called the ''stress surface'', is a surface of the second order that traces the variation of the normal stress vector as the orientation of the planes passing through a given point is changed.
The complete state of stress in a body at a particular deformed configuration, i.e., at a particular time during the motion of the body, implies knowing the six independent components of the stress tensor , or the three principal stresses , at each material point in the body at that time. However, numerical analysis and analytical methods allow only for the calculation of the stress tensor at a certain number of discrete material points. To graphically represent in two dimensions this partial picture of the stress field different sets of contour lines
A contour line (also isoline, isopleth, or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensional graph ...
can be used:[
* ''Isobars'' are curves along which the principal stress, e.g., is constant.
* ''Isochromatics'' are curves along which the maximum shear stress is constant. These curves are directly determined using photoelasticity methods.
* ''Isopachs'' are curves along which the mean normal stress is constant.
* ''Isostatics'' or ''stress trajectories'' are a system of curves which are at each material point tangent to the principal axes of stress - see figure ]
* ''Isoclinics'' are curves on which the principal axes make a constant angle with a given fixed reference direction. These curves can also be obtained directly by photoelasticity methods.
* '' Slip lines'' are curves on which the shear stress is a maximum.
See also
*Forensic engineering
Forensic engineering has been defined as ''"the investigation of failures - ranging from serviceability to catastrophic - which may lead to legal activity, including both civil and criminal".'' It includes the investigation of materials, product ...
*Piping
Within industry, piping is a system of pipes used to convey fluids (liquids and gases) from one location to another. The engineering discipline of piping design studies the efficient transport of fluid.
Industrial process piping (and accompan ...
*Rockwell scale
The Rockwell scale is a hardness scale based on indentation hardness of a material. The Rockwell test measures the depth of penetration of an indenter under a large load (major load) compared to the penetration made by a preload (minor load). The ...
*Structural analysis
Structural analysis is a branch of Solid Mechanics which uses simplified models for solids like bars, beams and shells for engineering decision making. Its main objective is to determine the effect of loads on the physical structures and thei ...
*Stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
* Worst case circuit analysis
*List of finite element software packages
This is a list of notable software packages that implement the finite element method for solving partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various ...
*Stress–strain curve
In engineering and materials science, a stress–strain curve for a material gives the relationship between stress and strain. It is obtained by gradually applying load to a test coupon and measuring the deformation, from which the stress and ...
References
[
Donald Ray Smith and Clifford Truesdell (1993]
"An Introduction to Continuum Mechanics after Truesdell and Noll". Springer.
[
John Conrad Jaeger, N. G. W. Cook, and R. W. Zimmerman (2007)]
"Fundamentals of Rock Mechanics"
(4th edition) Wiley-Blackwell.
[
Fridtjov Irgens (2008)]
"Continuum Mechanics"
Springer.
Dr. R.K Bansal (2009) "Strength in Materials" 4th edition
[ Michael J. Fagan and Michiel Postema (2007)]
"Introduction to stress and strain analysis"
Kingston upon Hull: The University of Hull
[
I-Shih Liu (2002)]
"Continuum Mechanics"
Springer
{{DEFAULTSORT:Stress Analysis
Structural analysis