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# elastic section modulus equation

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Shear modulus

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:

1. the MTS shear modulus model developed by and used in conjunction with the Mechanical Threshold Stress (MTS) plastic flow stress model.
2. the Steinberg-Cochran-Guinan (SCG) shear modulus model developed by and used in conjunction with the Steinberg-Cochran-Guinan-Lund (SCGL) flow stress model.
3. 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 Âµ0 is the shear modulus at 0 K, and D and T0are material constants.

### SCG shear modulus model

The Steinberg-Cochran-Guinan (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, Âµ0 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 Nadal-Le 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 Âµ0 is the shear modulus at 0 K and ambient pressure, Î¶ is a material parameter, kbis theBoltzmann constant, m is the atomic mass, and f is the Lindemann constant.

Stiffness

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
&delta; 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 direct-related stiffnesses (or simply stiffnesses) along the same degree of freedom and the off-diagonal 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 direct-related 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 direct-related 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
&theta; is the rotation

In the SI system, rotational stiffness is typically measured in newton-metres per radian.

In the SAE system, rotational stiffness is typically measured in inch-pounds 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 cross-sectional 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

Column

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 double-bull structures in their capitals. The Hall of Hundred Columns at Persepolis, measuring 70&nbsp;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 bottom-most 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 lathe-like apparatus. Single-piece columns are among the heaviest stones used in architecture. Other stone columns are created out of multiple sections of stone, mortared or dry-fit 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 so-called 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

where E = elastic modulus of the material, Imin= 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

where r = radius of gyration of [column]cross-section 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 hinged-hinged column). From Equation&nbsp;(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, Fcr (Fcr =Pcr/A, where A&nbsp;= cross-sectional 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 stress-strain curve, Et (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&nbsp;(3),

where Et = tangent modulus at the stress Fcr

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 wave

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.

## Non-electromagnetic

Longitudinal waves include sound waves (alternation in pressure, particle displacement, or particle velocity propagated in an elastic material) and seismic P-waves (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;
• y0 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:

• y0 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 force-free 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 "free-space" 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.

Question:what are the dimensions of Xnth in the standard equation Xnth = u + a/2 (2n-1) where u is intial velocity ,a is uniform accleration and n is time? what are the 2 essential conditions in each isothermal and adiabatic process to take place? thank you

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.

Question:calculate the rise in temperature of wire if Young's modulus and density of wire is known The rise in temperature of wire and not water

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 cross-section 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.

Question:OK i want to build a bridge crane in my shop. i will need to span 29 feet (roughly) and the ceiling hieght is 11 feet 4 inches. i plan on using heavy wall pipe for the legs, anchored to concrete pilings in the floor. i plan on supporting the length wise I beam evry 10 feet, ( or what ever distance the material i use needs for proper support. i want to be able to pick up and stripped car body IE 55 chevy station wagon) and move it side to side, forward and back and up and down. i just need to know what size of I beam i would need to span that 29 feet to pick up that type of load. for math purposes lets quantify the desired left capacity to be 3000 lbs. any help would be great thanks Pablo I Kinda figured i was not providing a clear picture of my question. The unsupported portion of the bridge crane will be 29 feet, this will be the portion from which the trolly hangs. The support for the two rails, from which the bridge protion will ride is open to what ever the material requires IE Every 10 feet every 8 feet Etc. I guess what i need to know is how large a Jr I beam i need to span that distance and still be able to lift 3000 lbs. I have no real workiing knowlege of Modulus of Elasticity, or the Moment of Inertia, I am not an engineer but I am smart enough to ask one before i build something LOL i hope this helps you to help me Pablo