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From Wikipedia
In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or Î¼, is defined as the ratio of shear stress to the shear strain:
 G \ \stackrel{\mathrm{def}}{=}\ \frac {\tau_{xy}} {\gamma_{xy}} = \frac{F/A}{\Delta x/l} = \frac{F l}{A \Delta x}
where
 \tau_{xy} = F/A \, = shear stress;
 F is the force which acts
 A is the area on which the force acts
 \gamma_{xy} = \Delta x/l = \tan \theta \, = shear strain;
 \Delta x is the transverse displacement
 l is the initial length
Shear modulus is usually expressed in gigapascals (GPa) or thousands of pounds per square inch (ksi).
Explanation
The shear modulus is one of several quantities for measuring the stiffness of materials. All of them arise in the generalized Hooke's law:
 Young's modulus describes the material's response to linear strain (like pulling on the ends of a wire),
 the bulk modulus describes the material's response to uniform pressure, and
 the shear modulus describes the material's response to shearing strains.
The shear modulus is concerned with the deformation of a solid when it experiences a force parallel to one of its surfaces while its opposite face experiences an opposing force (such as friction). In the case of an object that's shaped like a rectangular prism, it will deform into a parallelepiped. Anisotropic materials such as wood and paper exhibit differing material response to stress or strain when tested in different directions. In this case, when the deformation is small enough so that the deformation is linear, the elastic moduli, including the shear modulus, will then be a tensor, rather than a single scalar value.
Waves
In homogeneous and isotropic solids, there are two kinds of waves, pressure waves and shear waves. The velocity of a shear wave, (v_s) is controlled by the shear modulus,
 v_s = \sqrt{\frac {G} {\rho} }
where
 G is the shear modulus
 \rho is the solid's density.
Shear modulus of metals
The shear modulus of metals measures the resistance to glide over atomic planes in crystals of the metal. In polycrystalline metals there are also grain boundary factors that have to be considered. In metal alloys, the shear modulus is observed to be higher than in pure metals due to the presence of additional sources of resistance to glide.
The shear modulus of metals is usually observed to decrease with increasing temperature. At high pressures, the shear modulus also appears to increase with the applied pressure. Correlations between the melting temperature, vacancy formation energy, and the shear modulus have been observed in many metals.
Several models exist that attempt to predict the shear modulus of metals (and possibly that of alloys). Shear modulus models that have been used in plastic flow computations include:
 the MTS shear modulus model developed by and used in conjunction with the Mechanical Threshold Stress (MTS) plastic flow stress model.
 the SteinbergCochranGuinan (SCG) shear modulus model developed by and used in conjunction with the SteinbergCochranGuinanLund (SCGL) flow stress model.
 the Nadal and LePoac (NP) shear modulus model that uses Lindemann theory to determine the temperature dependence and the SCG model for pressure dependence of the shear modulus.
MTS shear modulus model
The MTS shear modulus model has the form:
\mu(T) = \mu_0  \frac{D}{\exp(T_0/T)  1}
where Âµ_{0} is the shear modulus at 0 K, and D and T_{0}are material constants.
SCG shear modulus model
The SteinbergCochranGuinan (SCG) shear modulus model is pressure dependent and has the form
\mu(p,T) = \mu_0 + \frac{\partial \mu}{\partial p} \frac{p}{\eta^{1/3}} + \frac{\partial \mu}{\partial T}(T  300) ; \quad \eta := \rho/\rho_0 where, Âµ_{0} is the shear modulus at the reference state (T = 300 K, p = 0, Î· = 1), p is the pressure, and T is the temperature.
NP shear modulus model
The NadalLe Poac (NP) shear modulus model is a modified version of the SCG model. The empirical temperature dependence of the shear modulus in the SCG model is replaced with an equation based on Lindemann melting theory. The NP shear modulus model has the form:
\mu(p,T) = \frac{1}{\mathcal{J}(\hat{T})} \left[ \left(\mu_0 + \frac{\partial \mu}{\partial p} \cfrac{p}{\eta^{1/3}} \right) (1  \hat{T}) + \frac{\rho}{Cm}~k_b~T\right]; \quad C := \cfrac{(6\pi^2)^{2/3}}{3} f^2
where
\mathcal{J}(\hat{T}) := 1 + \exp\left[\cfrac{1+1/\zeta} {1+\zeta/(1\hat{T})}\right] \quad \text{for} \quad \hat{T}:=\frac{T}{T_m}\in[0,1+\zeta],
and Âµ_{0} is the shear modulus at 0 K and ambient pressure, Î¶ is a material parameter, k_{b}is theBoltzmann constant, m is the atomic mass, and f is the Lindemann constant.
Stiffness is the resistance of an elastic body to deformation by an applied force along a given degree of freedom (DOF) when a set of loading points and boundary conditions are prescribed on the elastic body. It is an extensive material property.
Calculations
The stiffness, k, of a body is a measure of the resistance offered by an elastic body to deformation. For an elastic body with a single Degree of Freedom (for example, stretching or compression of a rod), the stiffness is defined as
 k=\frac {F} {\delta}
where
 F is the force applied on the body
 δ is the displacement produced by the force along the same degree of freedom (for instance, the change in length of a stretched spring)
In the International System of Units, stiffness is typically measured in newtons per metre. In English Units, stiffness is typically measured in pound force (lbf) per inch.
Generally speaking, deflections (or motions) of an infinitesimal element (which is viewed as a point) in an elastic body can occur along multiple Degrees of Freedom (maximum of six Degrees of Freedom at a point). For example, a point on a horizontal beam can undergo both a vertical displacement and a rotation relative to its undeformed axis. When the Degrees of Freedom is M, for example, a M x M matrix must be used to describe the stiffness at the point. The diagonal terms in the matrix are the directrelated stiffnesses (or simply stiffnesses) along the same degree of freedom and the offdiagonal terms are the coupling stiffnesses between two different degrees of freedom (either at the same or different points) or the same degree of freedom at two different points. In industry, the term influence coefficient is sometimes used to refer to the coupling stiffness.
It is noted that for a body with multiple Degrees of Freedom, Equation (1) generally does not apply since the applied force generates not only the deflection along its own direction (or degree of freedom), but also those along other directions (or Degrees of Freedom). For example, for a cantilevered beam, the stiffness at its free end is 12*E*I/L^3 rather than 3*E*I/L^3 if calculated with Equation (1).
For a body with multiple Degrees of Freedom, to calculate a particular directrelated stiffness (the diagonal terms), the corresponding Degree of Freedom is left free while the remaining Degrees of Freedom should be constrained. Under such a condition, Equation (1) can be used to obtain the directrelated stiffness for the degree of freedom which is unconstrained. The ratios between the reaction forces (or moments) and the produced deflection are the coupling stiffnesses.
The inverse of stiffness is compliance, typically measured in units of metres per newton. In rheology it may be defined as the ratio of strain to stress , and so take the units of reciprocal stress, e.g. 1/Pa.
Rotational stiffness
A body may also have a rotational stiffness, k, given by
 k=\frac {M} {\theta}
where
 M is the applied moment
 θ is the rotation
In the SI system, rotational stiffness is typically measured in newtonmetres per radian.
In the SAE system, rotational stiffness is typically measured in inchpounds per degree.
Further measures of stiffness are derived on a similar basis, including:
 shear stiffness  ratio of applied shear force to shear deformation
 torsional stiffness  ratio of applied torsion moment to angle of twist
Relationship to elasticity
In general, elastic modulus is not the same as stiffness. Elastic modulus is a property of the constituent material; stiffness is a property of a structure. That is, the modulus is an intensive property of the material; stiffness, on the other hand, is an extensive property of the solid body dependent on the material and the shape and boundary conditions. For example, for an element in tension or compression, the axial stiffness is
 k=\frac {AE} {L}
where
 A is the crosssectional area,
 E is the (tensile) elastic modulus (or Young's modulus),
 L is the length of the element.
Similarly, the rotational stiffness is
 k=\frac {nEI} {L}
where
 "I" is the moment of inertia,
 "n" is an integer depending on the boundary condition (=4 for fixed ends)
For the special case of unconstrained uniaxial tension or compression, Young's moduluscan be thought of as a measure of the stiffness of a material.
Use in engineering
The stiffness of a structure is of principal importance in many engineering applications, so the modulus of elasticity is often one of the primary properties considered when selectin
A column in structural engineering is a vertical structural element that transmits, through compression, the weight of the structure above to other structural elements below. For the purpose of wind or earthquake engineering, columns may be designed to resist lateral forces. Other compression members are often termed "columns" because of the similar stress conditions. Columns are frequently used to support beams or arches on which the upper parts of walls or ceilings rest. In architecture "column" refers to such a structural element that also has certain proportional and decorative features. A column might also be a decorative or triumphant feature but need not be supporting any structure e.g. a statue on top.
History
In the architecture of ancient Egypt as early as 2600 BC the architect Imhotep made use of stone columns whose surface was carved to reflect the organic form of bundled reeds; in later Egyptian architecture faceted cylinders were also common.
Some of the most elaborate columns in the ancient world were those of Persia especially the massive stone columns erected in Persepolis. They included doublebull structures in their capitals. The Hall of Hundred Columns at Persepolis, measuring 70 Ã— 70 meters was built by the Achaemenid king Darius I (524â€“486 BC). Many of the ancient Persian columns are standing, some being more than 30 meters tall.
The Greeks pioneered the use of the classical orders (Doric, Ionic, Corinthian) which was expanded by the Romans to include the Tuscan and Composite styles.
The impost (or pier) is the topmost member of a column. The bottommost part of the arch, called the springing, rests on the impost.
Structure
Early columns were constructed of stone, some out of a single piece of stone, usually by turning on a lathelike apparatus. Singlepiece columns are among the heaviest stones used in architecture. Other stone columns are created out of multiple sections of stone, mortared or dryfit together. In many classical sites, sectioned columns were carved with a center hole or depression so that they could be pegged together, using stone or metal pins. The design of most classical columns incorporates entasis (the inclusion of a slight outward curve in the sides) plus a reduction in diameter along the height of the column, so that the top is as little as 83% of the bottom diameter. This reduction mimics the parallax effects which the eye expects to see, and tends to make columns look taller and straighter than they are while entasis adds to that effect.
Modern columns are constructed out of steel, poured or precast concrete, or brick. They may then be clad in an architectural covering (or veneer), or left bare.
Equilibrium, instability, and loads
As the axial load on a perfectly straight slender column with elastic material properties is increased in magnitude, this ideal column passes through three states: stable equilibrium, neutral equilibrium, and instability. The straight column under load is in stable equilibrium if a lateral force, applied between the two ends of the column, produces a small lateral deflection which disappears and the column returns to its straight form when the lateral force is removed. If the column load is gradually increased, a condition is reached in which the straight form of equilibrium becomes socalled neutral equilibrium, and a small lateral force will produce a deflection that does not disappear and the column remains in this slightly bent form when the lateral force is removed. The load at which neutral equilibrium of a column is reached is called the critical or buckling load. The state of instability is reached when a slight increase of the column load causes uncontrollably growing lateral deflections leading to complete collapse.
For an axially loaded straight column with any end support conditions, the equation of static equilibrium, in the form of a differential equation, can be solved for the deflected shape and critical load of the column. With hinged, fixed or free end support conditions the deflected shape in neutral equilibrium of an initially straight column with uniform cross section throughout its length always follows a partial or composite sinusoidal curve shape, and the critical load is given by
f_{cr}\equiv\frac{\pi^2\textit{E}I_{min}}^2}\qquad (1)
where E = elastic modulus of the material, I_{min}= the minimal moment of inertia of the cross section, and L = actual length of the column between its two end supports. A variant of (1) is given by
f_{cr}\equiv\frac{\pi^{2}E_T}{(\frac{KL}{r})^{2}}\qquad (2)
where r = radius of gyration of [column]crosssection which is equal to the square root of (I/A), K = ratio of the longest half sine wave to the actual column length, and KL = effective length (length of an equivalent hingedhinged column). From Equation (2) it can be noted that the buckling strength of a column is inversely proportional to the square of its length.
When the critical stress, F_{cr} (F_{cr} =P_{cr}/A, where A = crosssectional area of the column), is greater than the proportional limit of the material, the column is experiencing inelastic buckling. Since at this stress the slope of the material's stressstrain curve, E_{t} (called the tangent modulus), is smaller than that below the proportional limit, the critical load at inelastic buckling is reduced. More complex formulas and procedures apply for such cases, but in its simplest form the critical buckling load formula is given as Equation (3),
f_{cr}\equiv{F_y}\frac{F^{2}_{y}}{4\pi^{2}E}\left(\frac{KL}{r^2}\right)\qquad (3)
where E_{t} = tangent modulus at the stress F_{cr}
A column with a cross section that lacks symmetry may suffer torsional buckling (sudden twisting) before, or in combination with, lateral buckling. The presence of the twisting deformations renders both theoretical analyses and practical designs rather complex.
Eccentricity of the load, or imperfections such as initial crookedness, decreases column strength. If the axial load on the column is not concentric, that is, its line of action is not precisely coincident with the centroidal axi
Longitudinal waves are waves that have the same direction of vibration as their direction of travel, which means that the movement of the medium is in the same direction as or the opposite direction to the motion of the wave. Mechanical longitudinal waves have been also referred to as compressional waves or compression waves.
Nonelectromagnetic
Longitudinal waves include sound waves (alternation in pressure, particle displacement, or particle velocity propagated in an elastic material) and seismic Pwaves (created by earthquakes and explosions).
Sound waves
In the case of longitudinal harmonic sound waves, the frequency and wavelength can be described with the formula
 y(x,t) = y_0 \sin\Bigg( \omega \left(t\frac{x}{c} \right) \Bigg)
where:
 y is the displacement of the point on the traveling sound wave;
 x is the distance the point has traveled from the wave's source;
 t is the time elapsed;
 y_{0} is the amplitude of the oscillations,
 c is the speed of the wave; and
 Ï‰ is the angular frequency of the wave.
The quantity x/c is the time that the wave takes to travel the distance x.
The ordinary frequency (f) of the wave is given by
 f = \frac{\omega}{2 \pi}.
For sound waves, the amplitude of the wave is the difference between the pressure of the undisturbed air and the maximum pressure caused by the wave.
Sound's propagation speed depends on the type, temperature and pressure of the medium through which it propagates.
Pressure waves
In an elastic medium with rigidity, a harmonic pressure wave oscillation has the form,
 y(x,t)\, = y_0 \cos(k x  \omega t +\varphi)
where:
 y_{0} is the amplitude of displacement,
 k is the wavenumber,
 x is distance along the axis of propagation,
 Ï‰ is angular frequency,
 t is time, and
 Ï† is phase difference.
The force acting to return the medium to its original position is provided by the medium's bulk modulus.
Electromagnetic
Maxwell's equations lead to the prediction of electromagnetic waves in a vacuum, which are transverse (in that the electric fields and magnetic fields vary perpendicularly to the direction of propagation). However, waves can exist in plasma or confined spaces. These are called plasma waves and can be longitudinal, transverse, or a mixture of both. Plasma waves can also occur in forcefree magnetic fields.
In the early development of electromagnetism there was some suggesting that longitudinal electromagnetic waves existed in a vacuum. After Heaviside's attempts to generalize Maxwell's equations, Heaviside came to the conclusion that electromagnetic waves were not to be found as longitudinal waves in "free space" or homogeneous media. But it should be stated that Maxwell's equations do lead to the appearance of longitudinal waves under some circumstances in either plasma waves or guided waves. Basically distinct from the "freespace" waves, such as those studied by Hertz in his UHF experiments, areZenneck waves. The longitudinal mode of a resonant cavity is a particular standing wave pattern formed by waves confined in a cavity. The longitudinal modes correspond to the wavelengths of the wave which are reinforced by constructive interference after many reflections from the cavity's reflecting surfaces. Recently, Haifeng Wang et al. proposed a method that can generate a longitudinal electromagnetic (light) wave in free space, and this wave can propagate without divergence for a few wavelengths.
From Yahoo Answers
Answers:For a perfectly rigid body, the strain produced is zero, no matter how much the stress, so the Young's modulus of such a body is infinite. THe dimensions should be that of distance as this is the equation for the distance covered in the nth second. In isothermal process, the temperature should remain constant. In adiabatic proces, there should be no exchange of temperature between the system and the surroundings. The temperature of the system might increase but it should not be distributed to the surroundings. SO the enclosure of such a system should be perfect insulators of heat.
Answers:Firstly we need to find the Pot. Energy(U) stored in the wire due to stretching. We apply the formula: U=(AY/2L)*l*l, where A=Area of crosssection of wire, Y=Young's Mod., L=original length, l=change in length. We can see that the formula demands value of area. As density(D) of wire is given, we can find area by: D=Mass/Vol, So Vol=Mass/D. Also Vol=Area*L. So Area=Vol/L. So pot. energy of wire(U) is found out. Assuming there is no heat loss, we can say that U=Heat required to raise temp. of water. So U=mcT, where m=mass of water c=specific heat capacity of water = 4200 J per kg K T=rise in temp. of water in degree Celsius or Kelvin. Solve the above equation for T. Note: If mass of water and wire is NOT same, then their respective masses have to be given. Otherwise this question will have insufficient data and cannot be solved for an exact value for rise in temperature of water.
Answers:You can get beam properties here: http://www.efunda.com/math/areas/RolledSteelBeamsW.cfm Here's a calculator for beams: http://www.engineeringtoolbox.com/beamstressdeflectiond_1312.html As the worst case is when your crane is in the middle of the bridge, your case is somewhat like a simply supported beam with a single center load. Thus, scoll down the link to the "Beam Supported at Both Ends, Load at Center" section. Standard beams are generally A36, which is 36,000 psi (I think yield) strength, and 18,000 (2x design factor) is probably a little too high. Stronger steels would improve the design factor at little additional cost, but the steel is less weldable, and likely not in stock. Running this through the beam calculator, you need at least a W12x14 beam. W or wide beams are more efficient, as the flanges are bigger. This is 14#/ft, or 400 lbs total, with over 1" deflection in the middle when loaded (just on the bridge beam). Going substantially larger and/or heavier will make it more rigid, say W14x2, which deflects less than 1/2 inch. The loads transfered from the bridge beam will tend to deflect the legs. The bridge will have additional deflection from this deflection. In addition, you are likely to see side loads on the trolleys. You need to have an engineer analyze this to determine a safe design.
Answers:WOW that's so impressive, I don't think I would be able to handle it. Maybe if I was as smart as you....
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