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104-430: The Osterøy Bridge is a 1,065-metre (3,494 ft) long suspension bridge that has a main span of 595 metres (1,952 ft). There are 8 spans, and none of the piers are in the water, just on land. There is 53 metres (174 ft) of clearance below the bridge. The two suspension towers are each 121.5 metres (399 ft) high. The bridge was completed on 3 October 1997 and cost about 308 million  kr . The bridge

208-410: A cable-stayed bridge in which the deck is in compression. Cable-stayed bridges and suspension bridges may appear to be similar, but are quite different in principle and in their construction. In suspension bridges, large main cables (normally two) hang between the towers and are anchored at each end to the ground. The main cables, which are free to move on bearings in the towers, bear the load of

312-420: A flow of viscous liquid , the force F may not be perpendicular to S ; 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 of S . Thus the stress state of the material must be described by a tensor , called the (Cauchy) stress tensor ; which is a linear function that relates the normal vector n of

416-426: A linear approximation may be adequate in practice if the quantities are sufficiently small. Stress that exceeds certain strength limits of the material will result in permanent deformation (such as plastic flow , fracture , cavitation ) or even change its crystal structure and chemical composition . Humans have known about stress inside materials since ancient times. Until the 17th century, this understanding

520-439: 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, normal to (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

624-460: A 200 feet span (also termed Beose Bridge) was constructed near Sagar, India during 1828–1830 by Duncan Presgrave, Mint and Assay Master. The Clifton Suspension Bridge (designed in 1831, completed in 1864 with a 214 m central span), is similar to the Sagar bridge. It is one of the longest of the parabolic arc chain type. The current Marlow suspension bridge was designed by William Tierney Clark and

728-492: A bridge has a tendency to collapse simply because of the gravitational forces acting on the materials of which the bridge is made. Live load refers to traffic that moves across the bridge as well as normal environmental factors such as changes in temperature, precipitation, and winds. Dynamic load refers to environmental factors that go beyond normal weather conditions, factors such as sudden gusts of wind and earthquakes. All three factors must be taken into consideration when building

832-468: A bridge. Silver Bridge (USA) was an eyebar chain highway bridge, built in 1928, that collapsed in late 1967, killing forty-six people. The bridge had a low-redundancy design that was difficult to inspect. The collapse inspired legislation to ensure that older bridges were regularly inspected and maintained. Following the collapse a bridge of similar design was immediately closed and eventually demolished. A second similarly-designed bridge had been built with

936-788: A bridge. The principles of suspension used on a large scale also appear in contexts less dramatic than road or rail bridges. Light cable suspension may prove less expensive and seem more elegant for a cycle or footbridge than strong girder supports. An example of this is the Nescio Bridge in the Netherlands, and the Roebling designed 1904 Riegelsville suspension pedestrian bridge across the Delaware River in Pennsylvania. The longest pedestrian suspension bridge, which spans

1040-392: A coordinate system with axes e 1 , e 2 , e 3 {\displaystyle e_{1},e_{2},e_{3}} , the stress tensor is a diagonal matrix, and has only the three normal components λ 1 , λ 2 , λ 3 {\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}

1144-444: A crumpled sponge, is subject to compressive stress and may undergo shortening. The greater the force and the smaller the cross-sectional area of the body on which it acts, the greater the stress. Stress has dimension of force per area, with SI units of newtons per square meter (N/m ) or pascal (Pa). Stress expresses the internal forces that neighbouring particles of a continuous material exert on each other, while strain

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1248-457: A cylindrical bar such as a shaft is 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") of I-beams under bending loads, due to the web constraining the end plates ("flanges"). Another simple type of stress occurs when

1352-454: A differential formula for friction forces (shear stress) in parallel laminar flow . Stress is defined as the force across a small boundary per unit area of that boundary, for all orientations of the boundary. Derived from a fundamental physical quantity (force) and a purely geometrical quantity (area), stress is also a fundamental quantity, like velocity, torque or energy , that can be quantified and analyzed without explicit consideration of

1456-403: A higher margin of safety and remained in service until 1991. The Tacoma Narrows Bridge , (USA), 1940, was vulnerable to structural vibration in sustained and moderately strong winds due to its plate-girder deck structure. Wind caused a phenomenon called aeroelastic fluttering that led to its collapse only months after completion. The collapse was captured on film. There were no human deaths in

1560-538: A pedestrian suspension bridge over the Machchhu River in the city of Morbi, Gujarat, India collapsed, leading to the deaths of at least 141 people. Stress (mechanics) In continuum mechanics , stress is a physical quantity that describes forces present during deformation . For example, an object being pulled apart, such as a stretched elastic band, is subject to tensile stress and may undergo elongation . An object being pushed together, such as

1664-403: A piston) push against them in (Newtonian) reaction . These macroscopic forces are actually the net result of a very large number of intermolecular forces and collisions between the particles in those molecules . Stress is frequently represented by a lowercase Greek letter sigma ( σ ). Strain inside a material may arise by various mechanisms, such as stress as applied by external forces to

1768-647: A proposal by Robert Stevenson for a bridge over the River Almond near Edinburgh . Roebling's Delaware Aqueduct (begun 1847) consists of three sections supported by cables. The timber structure essentially hides the cables; and from a quick view, it is not immediately apparent that it is even a suspension bridge. The main suspension cables in older bridges were often made from a chain or linked bars, but modern bridge cables are made from multiple strands of wire. This not only adds strength but improves reliability (often called redundancy in engineering terms) because

1872-541: A sequence generally described as follows. Depending on length and size, construction may take anywhere between a year and a half (construction on the original Tacoma Narrows Bridge took only 19 months) up to as long as a decade (the Akashi-Kaikyō Bridge's construction began in May 1986 and was opened in May 1998 – a total of twelve years). Suspension bridges are typically ranked by the length of their main span. These are

1976-452: A surface S to the traction vector T across S . With respect to any chosen coordinate system , the Cauchy stress tensor can be represented as a symmetric matrix of 3×3 real numbers. Even within a homogeneous body, 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-varying tensor field . In general,

2080-1007: A surface will always be a linear function of the surface's normal vector n {\displaystyle n} , the unit-length vector that is perpendicular to it. That is, T = σ ( n ) {\displaystyle T={\boldsymbol {\sigma }}(n)} , where the function σ {\displaystyle {\boldsymbol {\sigma }}} satisfies σ ( α u + β v ) = α σ ( u ) + β σ ( v ) {\displaystyle {\boldsymbol {\sigma }}(\alpha u+\beta v)=\alpha {\boldsymbol {\sigma }}(u)+\beta {\boldsymbol {\sigma }}(v)} for any vectors u , v {\displaystyle u,v} and any real numbers α , β {\displaystyle \alpha ,\beta } . The function σ {\displaystyle {\boldsymbol {\sigma }}} , now called

2184-434: A surface with normal vector n {\displaystyle n} (which is covariant - "row; horizontal" - vector) with coordinates n 1 , n 2 , n 3 {\displaystyle n_{1},n_{2},n_{3}} is then a matrix product T = n ⋅ σ {\displaystyle T=n\cdot {\boldsymbol {\sigma }}} (where T in upper index

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2288-413: A system must be balanced by internal reaction forces, which are almost always surface contact forces between adjacent particles — that is, as stress. Since every particle needs to be in equilibrium, this reaction stress will generally propagate from particle to particle, creating a stress distribution throughout the body. The typical problem in stress analysis is to determine these internal stresses, given

2392-434: A system of partial differential equations involving the stress tensor field and the strain tensor field, 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 a boundary-value problem . Stress analysis for elastic structures

2496-499: A temporary walkway. Poured sockets are used to make a high strength, permanent cable termination. They are created by inserting the suspender wire rope (at the bridge deck supports) into the narrow end of a conical cavity which is oriented in-line with the intended direction of strain. The individual wires are splayed out inside the cone or 'capel', and the cone is then filled with molten lead-antimony-tin (Pb80Sb15Sn5) solder. Most suspension bridges have open truss structures to support

2600-489: 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

2704-1092: Is transposition , and as a result we get covariant (row) vector) (look on Cauchy stress tensor ), that is [ T 1 T 2 T 3 ] = [ n 1 n 2 n 3 ] ⋅ [ σ 11 σ 21 σ 31 σ 12 σ 22 σ 32 σ 13 σ 23 σ 33 ] {\displaystyle {\begin{bmatrix}T_{1}&T_{2}&T_{3}\end{bmatrix}}={\begin{bmatrix}n_{1}&n_{2}&n_{3}\end{bmatrix}}\cdot {\begin{bmatrix}\sigma _{11}&\sigma _{21}&\sigma _{31}\\\sigma _{12}&\sigma _{22}&\sigma _{32}\\\sigma _{13}&\sigma _{23}&\sigma _{33}\end{bmatrix}}} The linear relation between T {\displaystyle T} and n {\displaystyle n} follows from

2808-410: 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. Depending on the context, one may also assume that the particles are large enough to allow the averaging out of other microscopic features, like

2912-583: Is an essential tool in engineering for the study and design of structures such as tunnels, dams, mechanical parts, and structural frames, under prescribed or expected loads. It is also important in many other disciplines; for example, in geology, to study phenomena like plate tectonics , vulcanism and avalanches ; and in biology, to understand the anatomy of living beings. Stress analysis is generally concerned with objects and structures that can be assumed to be in macroscopic static equilibrium . By Newton's laws of motion , any external forces being applied to such

3016-406: Is assumed fixed, the normal component can be expressed by a single number, the dot product T · n . This number will be positive if P is "pulling" on Q (tensile stress), and negative if P is "pushing" against Q (compressive stress). The shear component is then the vector T − ( T · n ) n . The dimension of stress is that of pressure , and therefore its coordinates are measured in

3120-478: Is based on the theory of elasticity and infinitesimal 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.). Engineered structures are usually designed so the maximum expected stresses are well within the range of linear elasticity (the generalization of Hooke's law for continuous media); that is,

3224-645: Is considered the last remaining Inca rope bridge and is rebuilt annually. The first iron chain suspension bridge in the Western world was the Jacob's Creek Bridge (1801) in Westmoreland County, Pennsylvania , designed by inventor James Finley . Finley's bridge was the first to incorporate all of the necessary components of a modern suspension bridge, including a suspended deck which hung by trusses. Finley patented his design in 1808, and published it in

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3328-641: Is given in the article on viscosity . The same for normal viscous stresses can be found in Sharma (2019). The relation between stress and its effects and causes, including deformation and rate of change of deformation, can be quite complicated (although a linear approximation may be adequate in practice if the quantities are small enough). Stress that exceeds certain strength limits of the material will result in permanent deformation (such as plastic flow , fracture , cavitation ) or even change its crystal structure and chemical composition . In some situations,

3432-428: Is important, for example, in prestressed concrete and tempered glass . Stress may also be imposed on a material without the application of net forces , for example by changes in temperature or chemical composition, or by external electromagnetic fields (as in piezoelectric and magnetostrictive materials). The relation between mechanical stress, strain, and the strain rate can be quite complicated, although

3536-513: Is often used for safety certification and monitoring. Most stress is analysed by mathematical methods, especially during design. The basic stress analysis problem can be formulated by Euler's equations of motion for continuous bodies (which are consequences of Newton's laws for conservation of linear momentum and angular momentum ) and the Euler-Cauchy stress principle , together with the appropriate constitutive equations. Thus one obtains

3640-408: Is perpendicular to the layer, the net internal force across S , 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 ratio F / A will only be an average ("nominal", "engineering") stress. That average is often sufficient for practical purposes. Shear stress is observed also when

3744-412: Is subjected to tension by opposite forces of magnitude F {\displaystyle F} along its axis. If the system is in equilibrium and 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 force, F with continuity through the full cross-sectional area , A . Therefore,

3848-403: Is the measure of the relative deformation of the material. For example, when a solid vertical bar is supporting an overhead weight , each particle in the bar pushes on the particles immediately below it. When a liquid is in a closed container under pressure , each particle gets pushed against by all the surrounding particles. The container walls and the pressure -inducing surface (such as

3952-437: Is then reduced to a scalar (tension or compression of the bar), but one must take into account also a bending stress (that tries to change the bar's curvature, in some direction perpendicular to the axis) and a torsional stress (that tries to twist or un-twist it about its axis). Stress analysis is a branch of applied physics that covers the determination of the internal distribution of internal forces in solid objects. It

4056-576: 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 stretched spring , tending to restore the material to its original undeformed state. Fluid materials (liquids, gases and plasmas ) by definition can only oppose deformations that would change their volume. If the deformation changes with time, even in fluids there will usually be some viscous stress, opposing that change. Such stresses can be either shear or normal in nature. Molecular origin of shear stresses in fluids

4160-463: Is tuned so that its greatest oscillation occurs when the wind is about 10 metres per second (22 mph) such as a light breeze. This Vestland location article is a stub . You can help Misplaced Pages by expanding it . This article about a bridge in Norway is a stub . You can help Misplaced Pages by expanding it . Suspension bridge A suspension bridge is a type of bridge in which

4264-505: 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 a tensor , reflecting Cauchy's original use to describe the "tensions" (stresses) in a material.) In tensor calculus , σ {\displaystyle {\boldsymbol {\sigma }}} is classified as a second-order tensor of type (0,2) or (1,1) depending on convention. Like any linear map between vectors,

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4368-836: The Mahakam River , located in Kutai Kartanegara Regency , East Kalimantan district on the Indonesia island of Borneo , was built in 1995, completed in 2001 and collapsed in 2011. Dozens of vehicles on the bridge fell into the Mahakam River . As a result of this incident, 24 people died and dozens of others were injured and were treated at the Aji Muhammad Parikesit Regional Hospital. Meanwhile, 12 people were reported missing, 31 people were seriously injured, and 8 people had minor injuries. Research findings indicate that

4472-610: The capitals , arches , cupolas , trusses and the flying buttresses of Gothic 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 Galilei 's rigorous experimental method , René Descartes 's coordinates and analytic geometry , and Newton 's laws of motion and equilibrium and calculus of infinitesimals . With those tools, Augustin-Louis Cauchy

4576-519: The deck is hung below suspension cables on vertical suspenders. The first modern examples of this type of bridge were built in the early 1800s. Simple suspension bridges , which lack vertical suspenders, have a long history in many mountainous parts of the world. Besides the bridge type most commonly called suspension bridges, covered in this article, there are other types of suspension bridges . The type covered here has cables suspended between towers , with vertical suspender cables that transfer

4680-448: The live and dead loads of the deck below, upon which traffic crosses. This arrangement allows the deck to be level or to arc upward for additional clearance. Like other suspension bridge types, this type often is constructed without the use of falsework . The suspension cables must be anchored at each end of the bridge, since any load applied to the bridge is transformed into tension in these main cables. The main cables continue beyond

4784-993: The orthogonal shear stresses . 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 the Mohr's circle of stress distribution. As a symmetric 3×3 real matrix, the stress tensor σ {\displaystyle {\boldsymbol {\sigma }}} has three mutually orthogonal unit-length eigenvectors e 1 , e 2 , e 3 {\displaystyle e_{1},e_{2},e_{3}} and three real eigenvalues λ 1 , λ 2 , λ 3 {\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}} , such that σ e i = λ i e i {\displaystyle {\boldsymbol {\sigma }}e_{i}=\lambda _{i}e_{i}} . Therefore, in

4888-457: The principal stresses . If the three eigenvalues are equal, the stress is an isotropic compression or tension, always perpendicular to any surface, there is no shear stress, and the tensor is a diagonal matrix in any coordinate frame. 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 an infinitesimal particle of

4992-777: The Philadelphia journal, The Port Folio , in 1810. Early British chain bridges included the Dryburgh Abbey Bridge (1817) and 137 m Union Bridge (1820), with spans rapidly increasing to 176 m with the Menai Bridge (1826), "the first important modern suspension bridge". The first chain bridge on the German speaking territories was the Chain Bridge in Nuremberg . The Sagar Iron Suspension Bridge with

5096-552: The River Paiva, Arouca Geopark , Portugal, opened in April 2021. The 516 metres bridge hangs 175 meters above the river. Where such a bridge spans a gap between two buildings, there is no need to construct towers, as the buildings can anchor the cables. Cable suspension may also be augmented by the inherent stiffness of a structure that has much in common with a tubular bridge . Typical suspension bridges are constructed using

5200-432: The bridge deck. Before the deck is installed, the cables are under tension from their own weight. Along the main cables smaller cables or rods connect to the bridge deck, which is lifted in sections. As this is done, the tension in the cables increases, as it does with the live load of traffic crossing the bridge. The tension on the main cables is transferred to the ground at the anchorages and by downwards compression on

5304-422: The bulk material (like gravity ) or to its surface (like contact forces , external pressure, or friction ). Any strain (deformation) of a solid material generates an internal elastic stress , analogous to the reaction force of a spring , that tends to restore the material to its original non-deformed state. In liquids and gases , only deformations that change the volume generate persistent elastic stress. If

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5408-449: 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 the impulses due to collisions). In active matter , self-propulsion of microscopic particles generates macroscopic stress profiles. In general,

5512-491: The chains are not attached to abutments as is usual, but instead are attached to the main girders, which are thus in compression. Here, the chains are made from flat wrought iron plates, eight inches (203 mm) wide by an inch and a half (38 mm) thick, rivetted together. The first wire-cable suspension bridge was the Spider Bridge at Falls of Schuylkill (1816), a modest and temporary footbridge built following

5616-631: The collapse of James Finley's nearby Chain Bridge at Falls of Schuylkill (1808). The footbridge's span was 124 m, although its deck was only 0.45 m wide. Development of wire-cable suspension bridges dates to the temporary simple suspension bridge at Annonay built by Marc Seguin and his brothers in 1822. It spanned only 18 m. The first permanent wire cable suspension bridge was Guillaume Henri Dufour 's Saint Antoine Bridge in Geneva of 1823, with two 40 m spans. The first with cables assembled in mid-air in

5720-529: The collapse was largely caused by the construction failure of the vertical hanging clamp. It was also found that poor maintenance, fatigue in the cable hanger construction materials, material quality, and bridge loads that exceed vehicle capacity, can also have an impact on bridge collapse. In 2013 the Kutai Kartanegara Bridge rebuilt the same location and completed in 2015 with a Through arch bridge design. On 30 October 2022, Jhulto Pul ,

5824-522: The collapse; several drivers escaped their cars on foot and reached the anchorages before the span dropped. Yarmouth suspension bridge (England) was built in 1829 and collapsed in 1845, killing 79 people. Peace River Suspension Bridge (Canada), which was completed in 1943, collapsed when the north anchor's soil support for the suspension bridge failed in October 1957. The entire bridge subsequently collapsed. Kutai Kartanegara Bridge (Indonesia) over

5928-498: 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 the hoop stress that occurs on the walls of a cylindrical pipe or vessel filled with pressurized fluid. 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

6032-425: The deformation changes gradually with time, even in fluids there will usually be some viscous stress , opposing that change. Elastic and viscous stresses are usually combined under the name mechanical 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; such built-in stress

6136-402: 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. Stress analysis is simplified when

6240-709: 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 called isotropic normal or just isotropic ; if it is compressive, it is called hydrostatic pressure or just pressure . Gases by definition cannot withstand tensile stresses, but some liquids may withstand very large amounts of isotropic tensile stress under some circumstances. see Z-tube . Parts with rotational symmetry , such as wheels, axles, pipes, and pillars, are very common in engineering. Often

6344-434: The elements σ x , σ y , σ z {\displaystyle \sigma _{x},\sigma _{y},\sigma _{z}} are called the orthogonal normal stresses (relative to the chosen coordinate system), and τ x y , τ x z , τ y z {\displaystyle \tau _{xy},\tau _{xz},\tau _{yz}}

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6448-424: The external forces that are acting on the system. The latter may be body forces (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. In stress analysis one normally disregards

6552-584: The failure of a few flawed strands in the hundreds used pose very little threat of failure, whereas a single bad link or eyebar can cause failure of an entire bridge. (The failure of a single eyebar was found to be the cause of the collapse of the Silver Bridge over the Ohio River .) Another reason is that as spans increased, engineers were unable to lift larger chains into position, whereas wire strand cables can be formulated one by one in mid-air from

6656-412: The fundamental laws of conservation of linear momentum and static equilibrium of 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 ( Cauchy's equations of motion for zero acceleration). Moreover, the principle of conservation of angular momentum implies that

6760-461: The grains of a metal rod or the fibers of a piece of wood . Quantitatively, the stress is expressed by the Cauchy traction vector T defined as the traction force F between adjacent parts of the material across an imaginary separating surface S , divided by the area of S . In a fluid at rest the force is perpendicular to the surface, and is the familiar pressure . In a solid , or in

6864-432: The highway, which may be supported by suspender cables or their own trusswork . In cases where trusswork supports the spans, there will be very little arc in the outboard main cables. The earliest suspension bridges were ropes slung across a chasm, with a deck possibly at the same level or hung below the ropes such that the rope had a catenary shape. The Tibetan siddha and bridge-builder Thangtong Gyalpo originated

6968-408: The layer; or a section of a soft metal bar that is being cut by the jaws of a scissors-like tool . Let F be the magnitude of those forces, and M be the midplane of that layer. Just as in the normal stress case, the part of the layer on one side of M must pull the other part with the same force F . Assuming that the direction of the forces is known, the stress across M can be expressed simply by

7072-534: The live loads. In an underspanned suspension bridge, also called under-deck cable-stayed bridge, the main cables hang entirely below the bridge deck, but are still anchored into the ground in a similar way to the conventional type. Very few bridges of this nature have been built, as the deck is inherently less stable than when suspended below the cables. Examples include the Pont des Bergues of 1834 designed by Guillaume Henri Dufour ; James Smith's Micklewood Bridge; and

7176-429: 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

7280-519: The medium surrounding that point, and taking the average stresses in that particle as being the stresses at the point. Human-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

7384-702: The modern method was Joseph Chaley 's Grand Pont Suspendu in Fribourg , in 1834. In the United States, the first major wire-cable suspension bridge was the Wire Bridge at Fairmount in Philadelphia, Pennsylvania. Designed by Charles Ellet Jr. and completed in 1842, it had a span of 109 m. Ellet's Niagara Falls suspension bridge (1847–48) was abandoned before completion. It was used as scaffolding for John A. Roebling 's double decker railroad and carriage bridge (1855). The Otto Beit Bridge (1938–1939)

7488-448: The most general case, called triaxial stress , the stress is nonzero across every surface element. 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. Cauchy observed that the stress vector T {\displaystyle T} across

7592-420: The nature of the material or of its physical causes. Following the basic premises of continuum mechanics, stress is a macroscopic concept. 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 ignore quantum effects and the detailed motions of molecules. Thus, the force between two particles

7696-452: 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 known constitutive equations . 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

7800-424: 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

7904-406: The pillars to deck-level supports, and further continue to connections with anchors in the ground. The roadway is supported by vertical suspender cables or rods, called hangers. In some circumstances, the towers may sit on a bluff or canyon edge where the road may proceed directly to the main span. Otherwise, the bridge will typically have two smaller spans, running between either pair of pillars and

8008-445: The plate). The analysis of stress can be considerably simplified also for thin bars, beams or wires of uniform (or smoothly varying) composition and cross-section that are subjected to moderate bending and twisting. For those bodies, one 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

8112-413: The railing and the walking layer of Gyalpo's bridges used wires. The stress points that carried the screed were reinforced by the iron chains. Before the use of iron chains it is thought that Gyalpo used ropes from twisted willows or yak skins. He may have also used tightly bound cloth. The Inca used rope bridges , documented as early as 1615. It is not known when they were first made. Queshuachaca

8216-612: The roadbed, particularly owing to the unfavorable effects of using plate girders, discovered from the Tacoma Narrows Bridge (1940) bridge collapse. In the 1960s, developments in bridge aerodynamics allowed the re-introduction of plate structures as shallow box girders , first seen on the Severn bridge , built 1961–1966. In the picture of the Yichang Bridge , note the very sharp entry edge and sloping undergirders in

8320-671: The same units as pressure: namely, pascals (Pa, that is, newtons per square metre ) in the International System , or pounds per square inch (psi) in the Imperial system . Because mechanical stresses easily exceed a million Pascals, MPa, which stands for megapascal, is a common unit of stress. 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 in temperature and phase , and electromagnetic fields) act on

8424-424: The single number τ {\displaystyle \tau } , calculated simply with the magnitude of those forces, F and the cross sectional area, A . τ = F A {\displaystyle \tau ={\frac {F}{A}}} Unlike normal stress, this simple shear stress is directed parallel to the cross-section considered, rather than perpendicular to it. For any plane S that

8528-407: The stress T that a particle P applies on another particle Q across a surface S can have any direction relative to S . The vector T may be regarded as the sum of two components: the normal stress ( compression or tension ) perpendicular to the surface, and the shear stress that is parallel to the surface. If the normal unit vector n of the surface (pointing from Q towards P )

8632-507: The stress can be assumed to be uniformly distributed over any cross-section that is more than a few times D from both ends. (This observation is known as the Saint-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 resulting bending stress will still be normal (perpendicular to

8736-411: The stress distribution in a body is expressed as a piecewise continuous function of space and time. Conversely, stress is usually correlated with various effects on the material, possibly including changes in physical properties like birefringence , polarization , and permeability . The imposition of stress by an external agent usually creates some strain (deformation) in the material, even if it

8840-475: 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 σ {\displaystyle \sigma } = F / A will be only the average stress, called engineering stress or nominal stress . If the bar's length L is many times its diameter D , and it has no gross defects or built-in stress , then

8944-424: The stress is maximum for surfaces that are perpendicular to a certain direction d {\displaystyle d} , and zero across any surfaces that are parallel to d {\displaystyle d} . When the shear stress is zero only across surfaces that are perpendicular to one particular direction, the stress is called biaxial , and can be viewed as the sum of two normal or shear stresses. In

9048-399: The stress patterns that occur in such parts have rotational or even cylindrical symmetry . The analysis of such cylinder stresses can take advantage of the symmetry to reduce the dimension of the domain and/or of the stress tensor. Often, mechanical bodies experience more than one type of stress at the same time; this is called combined stress . In normal and shear stress, the magnitude of

9152-684: 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 [ σ x τ x y τ x z τ x y σ y τ y z τ x z τ y z σ z ] {\displaystyle {\begin{bmatrix}\sigma _{x}&\tau _{xy}&\tau _{xz}\\\tau _{xy}&\sigma _{y}&\tau _{yz}\\\tau _{xz}&\tau _{yz}&\sigma _{z}\end{bmatrix}}} where

9256-411: 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 a bending stress that tends to change the curvature of the plate. These simplifications may not hold at welds, at sharp bends and creases (where the radius of curvature is comparable to the thickness of

9360-1620: The stress tensor can be represented in any chosen Cartesian coordinate system by a 3×3 matrix of real numbers. Depending on whether the coordinates are numbered x 1 , x 2 , x 3 {\displaystyle x_{1},x_{2},x_{3}} or named x , y , z {\displaystyle x,y,z} , the matrix may be written as [ σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 ] {\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{bmatrix}}} or [ σ x x σ x y σ x z σ y x σ y y σ y z σ z x σ z y σ z z ] {\displaystyle {\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\\\end{bmatrix}}} The stress vector T = σ ( n ) {\displaystyle T={\boldsymbol {\sigma }}(n)} across

9464-431: The stress tensor is symmetric , that is σ 12 = σ 21 {\displaystyle \sigma _{12}=\sigma _{21}} , σ 13 = σ 31 {\displaystyle \sigma _{13}=\sigma _{31}} , and σ 23 = σ 32 {\displaystyle \sigma _{23}=\sigma _{32}} . Therefore,

9568-423: The stress within a body may adequately be described by a single number, or by a single vector (a number and a direction). Three such simple stress situations, that are often encountered in engineering design, are the uniaxial normal stress , the simple shear stress , and the isotropic normal stress . A common situation with a simple stress pattern is when a straight rod, with uniform material and cross section,

9672-440: The stress σ throughout the bar, across any horizontal surface, can be expressed simply by the single number σ, calculated simply with the magnitude of those forces, F , and cross sectional area, A . σ = F A {\displaystyle \sigma ={\frac {F}{A}}} 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

9776-401: The suspension bridge shown. This enables this type of construction to be used without the danger of vortex shedding and consequent aeroelastic effects, such as those that destroyed the original Tacoma Narrows bridge. Three kinds of forces operate on any bridge: the dead load, the live load, and the dynamic load. Dead load refers to the weight of the bridge itself. Like any other structure,

9880-514: The ten bridges with the longest spans, followed by the length of the span and the year the bridge opened for traffic: (Chronological) Broughton Suspension Bridge (England) was an iron chain bridge built in 1826. One of Europe's first suspension bridges, it collapsed in 1831 due to mechanical resonance induced by troops marching in step. As a result of the incident, the British Army issued an order that troops should "break step" when crossing

9984-489: The towers. In cable-stayed bridges, the towers are the primary load-bearing structures that transmit the bridge loads to the ground. A cantilever approach is often used to support the bridge deck near the towers, but lengths further from them are supported by cables running directly to the towers. By design, all static horizontal forces of the cable-stayed bridge are balanced so that the supporting towers do not tend to tilt or slide and so must only resist horizontal forces from

10088-440: The two halves across the cut. This type of stress may be called (simple) normal stress or uniaxial stress; specifically, (uniaxial, simple, etc.) tensile stress. If the load is compression on the bar, rather than stretching it, the analysis is the same except that the force F and the stress σ {\displaystyle \sigma } change sign, and the stress is called compressive stress. This analysis assumes

10192-568: The use of iron chains in his version of simple suspension bridges . In 1433, Gyalpo built eight bridges in eastern Bhutan . The last surviving chain-linked bridge of Gyalpo's was the Thangtong Gyalpo Bridge in Duksum en route to Trashi Yangtse , which was finally washed away in 2004. Gyalpo's iron chain bridges did not include a suspended-deck bridge , which is the standard on all modern suspension bridges today. Instead, both

10296-423: The weight of the cables is small compared to the weight of the deck. One can see the shape from the constant increase of the gradient of the cable with linear (deck) distance, this increase in gradient at each connection with the deck providing a net upward support force. Combined with the relatively simple constraints placed upon the actual deck, that makes the suspension bridge much simpler to design and analyze than

10400-400: Was able to give the first rigorous and general mathematical model of a deformed elastic body by introducing the notions of stress and strain. 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). The understanding of stress in liquids started with Newton, who provided

10504-661: Was built between 1829 and 1832, replacing a wooden bridge further downstream which collapsed in 1828. It is the only suspension bridge across the non-tidal Thames. The Széchenyi Chain Bridge , (designed in 1840, opened in 1849), spanning the River Danube in Budapest, was also designed by William Clark and it is a larger-scale version of Marlow Bridge. An interesting variation is Thornewill and Warham 's Ferry Bridge in Burton-on-Trent , Staffordshire (1889), where

10608-506: Was designed by the structural engineering firm Aas-Jakobsen . It was put into service 28 years after the first plans for a connection between Osterøy and Bergen were prepared. It was opened for traffic by Sissel Rønbeck , the Norwegian Minister of Transport and Communications . The bridge was built to withstand quite strong winds. Experts have indicated that the bridge should be capable of surviving an extreme storm. The bridge

10712-408: Was largely intuitive and empirical, though this did not prevent the development of relatively advanced technologies like the composite bow and glass blowing . 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

10816-404: Was the first modern suspension bridge outside the United States built with parallel wire cables. Two towers/pillars, two suspension cables, four suspension cable anchors, multiple suspender cables, the bridge deck. The main cables of a suspension bridge will form a catenary when hanging under their own weight only. When supporting the deck, the cables will instead form a parabola , assuming

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