C1B Stress Analysis Lecture 1: Minimum Energy Principles in Mechanics Prof Alexander M. Korsunsky Hilary Term (January 08)
|
|
- Camilla Cole
- 5 years ago
- Views:
Transcription
1 CB Stress Anaysis ecture : Minimum Energy Principes in Mechanics Prof Aexander M. Korsunsky Hiary Term (January 8)
2 This course introduces seected chapters in Stress Anaysis. Energy principes in deformation theories. Variationa cacuus. Euer equation and boundary conditions. Appication to beam bending. The Rayeigh-Ritz method.. Further Euer, Rayeigh-Ritz, and Gaerkin. Generaisation to higher dimensions. Piecewise approximation, and the connection with the FEM. 3. Fundamentas of anisotropic easticity: Stress, strain, eastic constants. The system of equations of easticity. Anaytica soution of eastic probems. Pane stress and pane strain. The wedge probem. Fundamenta singuar soution for probems invoving pane contacts. Connection with the Boundary Eement Method (BEM). Soution of contact probems using BEM. 4. Fundamenta soutions in easticity theory. The Wiiams wedge anaysis. The Westergaard stress function.
3 Energy minimisation A simpe spring mode shows how probems can be soved using the energy minimisation approach, rather than the system of equiibrium equations. The eastic energy stored in the spring is found as the work done by the interna forces over the dispacements, i.e. du Fdx kxdx, U ε Fxdx During uniaxia deformation of ties, coumns, beams etc. the strain energy density (i.e. the eastic stored energy per unit voume) is given by: du σdε σdε Eεdε, U σdε ε kx Eε Key steps of soution by energy minimisation: Identify key parameters that describe deformation Express tota energy in terms of unknown parameters Minimise energy with respect to these parameters To deveop understanding of soving probems by energy minimisation, we wi foow this strategy: ). Discuss fundamentas of minimising functionas (e.g. integras), known as the cacuus of variations. ). earn about the derivation and use of Euer s equation, and the attendant boundary conditions. We wi appy Euer s equation to the probem of bending a simpy supported beam and a buit-in cantiever. 3). earn how fast and error-free derivations can be carried out using a symboic agebra package, such as Mathematica. 3). Introduce the direct parametrisation and minimisation technique known as the Rayeigh-Ritz method. We wi then assess its accuracy by comparison with exact soutions. 4). Introduce another method for soving minimisation probems, known as the Gaerkin method. 3
4 Energy Functionas The energy to be minimised is often expressed as an integra. As exampes we can consider: eastic bending of a beam axia tension or compression in ties and coumns torsion of shafts et v(x) denote the defection. Strain energy density for a bent beam : du σdε Eεdε, U σdε Eε W Eε dadx E y da v'' dx V For a tie/coumn: W EA For a shaft: W GJ θ ( u' ) dx ( ') dx A ε ( ) EI ( v'' ) dx Note that in a cases the energy is expressed as an integra of some derivative of the deformed shape. In order to find a soution, the integra must be minimised with respect to the choice of a function, e.g., v(x), u(x) or θ (x). Additionay, the function must satisfy some boundary conditions at the ends of the interva (or possiby within). For exampe, at the ends x, of a bent beam free, simpy supported, or buit-in conditions may be prescribed. Note that, in genera, the energy integra contains a combination of the function and its derivatives of different orders. Minimisation of a functiona is the probem of the cacuus of variations 4
5 Variationa cacuus Probem: Find a function y(x) that makes the foowing integra W[y] take a minimum vaue: W b [ y] F( x, y, y', y' ')dx a Boundary conditions on the unknown function y(x) may be prescribed: y(a)y, y(b)y, y (a)y, y (b)y, etc. If y(x) is the soution, how do we seek it? et s try to adjust the methods known from minimisation of functions. et y(x) be a soution. Construct comparison or tria soutions When α, y becomes the soution. Require y y( x) + αη( x) η( a), η( b) so that y sti satisfies a boundary conditions., etc., Now minimise W as a function of α. How do we determine the conditions on the soution y(x)? We know that at a minimum of a function f(x), its derivative f x vanishes. Another way of saying the same thing is that the variation δff x δxvanishes. We now deveop the same approach to finding the minimum of functiona W[y]. Consider an increment of the introduced parameter α, and find the expression for the variation δw. We identify the conditions this imposes on the functions invoved, which are ikey to emerge as differentia equations, and aso some conditions at the region boundary. 5
6 Euer equation We are seeking the variation of W[y] with y. Consider δww[y+δy]-w[y]: b [ y] F( x, y + δy, y' + δy', y' ' + y' ')dx F( x, y, y', y' ')dx W[ y + δy] W δ δw a b [ ' y] ( Fyδy + Fy ' δy' + Fy ' δy' ') dx a By carrying out integration by parts as many times as necessary, a derivatives of y can be eiminated from under the integra sign, giving b δw[ y] [ F ( F ')' + ( F '')''] δydx a y y W [ y] F( x, y, y', y' ')dx y b a Now in order to ensure that W is at a minimum, this variation must be zero for any δy. This can ony be guaranteed if the expression in brackets is zero. b a (.) The differentia equation for the soution y(x) is the Euer equation: F y ( Fy ' ')' + ( Fy ' )'' Probem: find the shape of a string of ength suspended in tension T due to distributed oad w. A ine eement of string dx becomes after oading. The strain is ½ (y ) and strain energy is ½ T(y ), whie the work done by the oad is wy per unit ength. The tota energy is given by (.), where F( x, y, y', y' ') To write the Euer equation using (.), we need to differentiate F with respect to y and y as if they were two independent variabes. Hence Appy boundary conditions y()y(). The soution: 6 dx + ( y' ) dx( + T ( y' ) Ty' ' +w wy (.) ( y' ) w y( x) x( x). T )
7 Euer: Bent Beam Probem: Minimise the potentia energy of a beam with defection v under genera appied force f(x) (may be a combination of distributed and point oads). Tota energy of the system: The second integra term describes the work done by the force f over dispacement v. The first variation of W is found by considering increments δv and δv δw and using integration by parts δw The Euer equation for a bent beam is found as: ( v'' ) dx We now discuss the boundary conditions. Some boundary conditions arise from the variation of δw (see above). They are not a consequence of any additiona constraints imposed on the system, and for this reason are caed natura boundary conditions. Other boundary conditions appear because of additiona constraints. For a bent beam, the foowing options can be encountered: W EI fvd EI v'' δv'' dx fδvdx EI[ v' ' δv' ] EI v'' ' δv' dx fδv d x EI[ v' ' δv' ] EI[ v' '' δv] + [ EIv' ''' f ] δvdx (a)if beam ends are free, then δv and δv are arbitrary, and their coefficients in the expression for δw must vanish (these are the natura b.c.). Hence at a free end v and v. (b)if an end is simpy supported, then the defection and the bending moment M-Eiv must vanish, eading to the requirements v and v. (c)if an end is buit-in, then the defection and sope must vanish, eading to the requirements v and v. EIv '''' f (.3) x 7
8 Euer: Bent Beam EIv' ''' f Exampe probem: a simpy supported beam of ength and bending stiffness EI is oaded by a combination of UD ω and a point oad P at mid-span. Use Euer equation to sove the probem. The oad is given by f ( x) w + Pδ ( x / ) Here δ(x) is Dirac s deta-function. It is often used in physics and engineering to represent point oads, concentrated charges, etc. Its most important property is expressed in an integra sense: when the deta function is integrated over an interva which contains the zero of its argument, the resut is unity; otherwise it is zero. Note that this is precisey what we mean by a unit point oad! To find v(x), we must integrate (four times) Euer equation in the form v '''' [ w + Pδ ( x / )] / EI Integrating the deta-function is easy: the first integra produces a unit-step function, which is changes from zero to unity as we go through zero of the argument. If we denote this first integra of the deta-function by H(x), then subsequent integrations produce functions of the form H n (x)h(x) x n- /(n-)! y(x) Integrating δ-function δ(x) H(x) H (x) H 3 (x) H 4 (x) You must note here the simiarity between the deta-function and its integras, and the Macauay brackets. It is aso important to note that constants of integration must be introduced when soving differentia equations x
9 Using Mathematica In order to find the soution of the exampe probem, integration must be performed four times. Four constants wi be introduced in the process, which wi need to be found from the four boundary conditions. You may want to practice doing that for the exam. For our purposes we need an error-free and fast way to perform these anaytica manipuations. We can use one of the symboic agebra packages, such as Mathematica. euerbeam.nb v, v, v3, c, c, c3, c4d H Euer equation: EI v''''w + P DiracDeta@x dd v3 Integrate@Hw + P DiracDeta@x dd ê EI, xd + c v Integrate@v3, xd + c v Integrate@v, xd + c3 v Integrate@v, xd + c4 wx + PUnitStepHx- d c + EI wx PHx- d UnitStepHx - d + c x + c + EI EI wx 3 c x dphx- d UnitStepHx - d + + c x + c3-6ei EI wx 4 c x3 c x c3 x+ c4 + d PHx - d UnitStepHx- d 4 EI 6 EI v ê.x ê. UnitStep@ dd ê. UnitStep@ dd v ê.x ê. UnitStep@ dd ê. UnitStep@ dd + PHx - d UnitStepHx - d EI - dphx - d UnitStepHx - d EI + PHx3 - d 3 UnitStepHx - d 6EI c c4 v v ê.c ê.c4 ; v v ê.c ê.c4 ; eq v ê.x ê. UnitStep@ dd ê. UnitStep@ dd v v ê. Sove@eq, cd@@dd v v ê. Sove@eq, cd@@dd w EI + c + H - d P EI wx 4 4 EI - Hw + P - dp x 3 + c3 x + d PHx - d UnitStepHx- d EI EI wx EI - Hw + P- dp x PHx- d UnitStepHx - d + EI EI - dphx - d UnitStepHx - d EI eq Simpify@v ê.x ê. UnitStep@ dd ê. UnitStep@ dd ê.c ê.c4 D v v ê. Sove@eq, c3d@@dd v v ê. Sove@eq, c3d@@dd + PHx3 - d 3 UnitStepHx - d 6EI - w4 + 8dP - 4 c3 EI - d P + 4d 3 P 4EI wx 4 4EI - Hw + P - dp x 3 - H-w4-8dP + d P - 4d 3 P x + d PHx - d UnitStepHx- d - dphx - d UnitStepHx- d + PHx3 - d 3 UnitStepHx - d EI 4 EI EI EI 6EI wx EI - Hw + P- dp x + EI PHx- d UnitStepHx - d EI 9
10 Using Mathematica In[6]: vauep 8EI,, d.5, P, w <; vauew 8EI,, d.5, P, w <; nvp v ê. vauep nvp v ê. vauep nvw v ê. vauew nvw v ê. vauew Pot@8nvp, nvw<, 8x,, <, PotStye 8Hue@.D, Hue@D<D 8nvp ê.x.5, nvw ê.x.5, nvp ê.x.5, nvw ê.x.5< Pot@8 nvp, nvw<, 8x,, <, PotStye 8Hue@.D, Hue@D<D 8 nvp ê.x.5, nvw ê.x.5, nvp ê.x.5, nvw ê.x.5< Out[8] x x +.5 Hx -.5 UnitStepHx Ix -.5M UnitStepHx Ix3 -.5M UnitStepHx -.5 Out[9] Hx-.5 UnitStepHx x Out[3] x x3 + x 4 Out[3] x - x Out[3] Ü Graphics Ü Out[33] ,.38,.439,.97734< Out[34] Ü Graphics Ü Out[35] 8.5,.5,.5,.9375< Summary EIv(/) EIv(/4) -EIv (/) -EIv (/4) P at /.8P 3.43P 3.5P.5P UD ω.3ω 4.93ω 4.5ω.94ω
11 Rayeigh-Ritz Method This approach is a powerfu aternative to finding and soving Euer equation. It is a direct method, in the sense that the soution is assumed to be unknown but of a certain form containing unknowns, which are then found by minimisation. Key steps: (a) Express the unknown function (e.g. defected shape) as a inear combination of known functions with unknown coefficients: (b) Express the tota energy of the system W as a function of constants A i. (c) Find the minimum by requiring v ( 3 x) A v ( x) + Av ( x) + A3v ( x) +... grad A i W i.e. W / A The Rayeigh-Ritz method transforms a probem with an infinite number of degrees of freedom into one with a finite number (as many as A i s used). Rayeigh-Ritz: Bent Beam We seek a soution to the bent beam probem by seeking the defected shape as a series of the type (.4), but containing ony one term: v( x) πx Asin We find right away that: The function v(x) is chosen to satisfy the defection boundary conditions v()v() at simpy supported ends. Note that it aso satisfies the moment boundary conditions, since for it v ()v (). Tota energy of the system can be evauated (e.g. using Mathematica) W PA 4 π EIA EI( v' ') dx wvdx Pv( / ) 3 4 π v''( x) A Since in this case (.5) reduces to a singe equation, the immediate resut is i πx sin W 4 3 π EIA w w P, A P A 4 π π EI π wa π (.4) (.5)
12 Rayeigh-Ritz Method In Mathematica, the soution is found by foowing the three steps we have indentified (ritzbeam.nb): Out[6] - Out[7] Out[8] H Bent beam by Rayeigh Ritz, one term Off@Genera::spe, Genera::speD H Step : Define shape in terms of A v ASin@Pi x ê D;v D@v, 8x, <D H Step : Cacuate W in terms of A W Integrate@EI v^ ê wv,8x,, <D P H v ê.x ê H Step 3: Minimise dwêda to find soution vr v ê. Sove@D@W, AD, AD@@DD Ap sinj p x N A EI p 5-8 A 4 w AP p 3 HpP+ w sinj p x EI p 5 N It is now possibe to compare the resuts with the exact soutions (by Euer equation), by giving the foowing commands: In[]: vr D@vr, 8x, <D nvrp vr ê. vauep nvrp vr ê. vauep nvrw vr ê. vauew nvrw vr ê. vauew Pot@8nvrp, nvp<, 8x,, <, PotStye 8Grayeve@.6D, Grayeve@D<D Pot@8nvrw, nvw<, 8x,, <, PotStye 8Grayeve@.6D, Grayeve@D<D Pot@8 nvrp, nvp<, 8x,, <, PotStye 8Grayeve@.6D, Grayeve@D<D Pot@8 nvrw, nvw<, 8x,, <, PotStye 8Grayeve@.6D, Grayeve@D<D We can now compare the soutions for defected shapes and bending moment profies
13 Rayeigh-Ritz Method 3 We compare the exact soution with the R-R approximation: In[59]: H Comparison for defection H Exact 8nvp ê.x.5, nvw ê.x.5, nvp ê.x.5, nvw ê.x.5< H R R 8nvrp ê.x.5, nvrw ê.x.5, nvrp ê.x.5, nvrw ê.x.5< Out[59] ,.38,.439,.97734< Out[6] 8.53,.37,.4583,.9463< H Comparison for bending moment H Exact 8 nvp ê.x.5, nvw ê.x.5, nvp ê.x.5, nvw ê.x.5< H R R 8 nvrp ê.x.5, nvrw ê.x.5, nvrp ê.x.5, nvrw ê.x.5< Out[57] 8.5,.5,.5,.9375< Out[58] 8.64,.96,.439,.9< R-R captures the defected shape better than the bending moment profie (i.e. stress). This is because stresses and strains depend on the derivatives of defection, and require higher order approximation for good agreement. The quaity of approximation can ony improve with extra terms in (.4). Assume v( x) i.. N iπx Ai sin An efficient soution is obtained if, after writing down the integra expression for W, differentiation with respect to A i is carried out under the integra sign: W A i EI v' ' v v' ' dx P A A i i ( / ) The resuting integras are famiiar from Fourier anaysis. In particuar from it foows that, and You can inspect a possibe Mathematica impementation of this sine series soution in the fie ritzbeamn.nb, or construct your own. v w A i dx iπx jπx sin sin dx δij 4 iπ iπ w EI Ai Psin (cosiπ ) iπ 3 iπ w A i Psin + (cosiπ ) 4 ( iπ ) EI iπ 3
14 Mathematica tutoria H Introduction to using Mathematica for 4 ME6 Stress Anaysis H In the derivations I use a very sma part of what Mathematica can do. H You wi have guessed that parenthesis asterisk is used for comments H Mathematica understands about most functions. To run a CE, you need to press Shitf+Enter top status bar shoud fash ' RUNNING', and come up with a pot beow Pot@8Sin@xD, Cos@xD, Exp@xD, Cosh@xD<, 8x,, Pi<D Ü Graphics Ü H You may have noticed that the functions are grouped together between cury brackets, and separated by commas they form a IST. The potting argument and range aso form a ist 8x,, Pi<. The pot above ooks difficut to read, since no coour was used. Instruction can be given to the pot command, using PotStye 8Hue...< to choose a HUE for each function respectivey. HUE argument can go from to. To find what' s what, change the numbers beow, and re run Pot@8Sin@xD, Cos@xD, Exp@xD, Cosh@xD<, 8x,, Pi<, PotStye 8Hue@.D, Hue@.3D, Hue@.5D, Hue@.7D<D Ü Graphics Ü H Mathematica can aso buid ists using Tabe: Tabe@Sin@nPiê 6D, 8n,, 6<D Pot@8Sin@xD, Sin@ xd, Sin@3 xd, Sin@4 xd<, 8x,, Pi<, PotStye Tabe@Hue@.5 jd, 8j,, 4<DD :,, è 3,, è 3,,> Ü Graphics Ü 4
15 Mathematica tutoria H It can aso buid arrays: consts Array@c, 5D 8cH, ch, ch3, ch4, ch5< H It' s very good at producing pretty 3 D pots u Sin@rhoD H3+ Cos@thD; v Cos@rhoDH3+ Sin@thD; z Sin@thD; ParametricPot3D@8u, v, z<, 8rho,, Pi<, 8th,, Pi<D Ü Graphics3D Ü H Mathematica can differentiate using D@D Heven severa times and find indefinite and definite integras using Integrate@D D@Sin@xD,xD D@Sin@xD, 8x, <D Integrate@Sin@xD,xD Integrate@Sin@xD, 8x,, Pi<D coshx -sinhx -coshx H It can do series! Series@Tan@xD, 8x,, 5<D x+ x3 3 + x x x x x x OIx6 M H Mathematica can sove equations note the use of Sove@x^ + 5 x 6, xd 88x Ø-6<, 8x Ø << H It can aso sove differentia equations! DSove@u''@xD 9 u@xd, u@xd,xd 99uHx Ø -3 x c + 3 x c DSove@u''@xD 4 u'@xd + 4 u@xd, u@xd,xd 99uHx Ø x c + x xc 5
16 Mathematica tutoria 3 H Mathematica is very good at soving systems of equations. Command SOVE needs a ist of equations, foowed by a ist of unknowns to find. SOUTION Sove@8x+ 3 y + z, x y + z 5, 3 x+ 7 y + z 3<, 8x, y, z<d ::x Ø- 3, y Ø, z Ø 5 >> H We ca the resut ' SOUTION'. It contains RUES. Each RUE describes a substitution: it shows, using an arrow, which vaues shoud be given to each unknown. Our SOUTION contains the rues encosed in doube cury brackets. To extract the rues, we needto use doube square brackets, and specify we need the first eement SOUTION@@DD :x Ø- 3, y Ø, z Ø 5 > H To use the soution, we need to perform the substitution. This is done by using the command ê. The resut is a ist of vaues of x,y,z that we found 8x, y, z<ê. SOUTION@@DD :- 3,, 5 > H Finay, procedures can be defined using MODUE. This one cacuates the sum of odd integers up to n. It uses OddQ a query function, which is true of the integer is odd, and fase otherwise. Note the C stye cacuation of the tota OddSum@n_D : Modue@8i, tota <, Do@If@OddQ@iD, tota + i, D, 8i, n<d; tota D H We can now use OddSum as a function OddSum@D 5 In order to appreciate the capabiities of Mathematica, you must try running it yoursef. You can use the program to study the exampes from in the ECS ab (the Soarium) on any of the machines booth36.ecs to booth5.ecs (incusive). To run Mathematica, type start_mathematica. This wi start a window tited "Mathematica4. Execution Window". Run Mathematica in this window by typing mathematica. 6
VTU-NPTEL-NMEICT Project
MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 VTU-NPTE-NMEICT Project Progress Report The Project on Deveopment of Remaining Three Quadrants to NPTE Phase-I under grant in aid
More informationWork and energy method. Exercise 1 : Beam with a couple. Exercise 1 : Non-linear loaddisplacement. Exercise 2 : Horizontally loaded frame
Work and energy method EI EI T x-axis Exercise 1 : Beam with a coupe Determine the rotation at the right support of the construction dispayed on the right, caused by the coupe T using Castigiano s nd theorem.
More informationLecture 6: Moderately Large Deflection Theory of Beams
Structura Mechanics 2.8 Lecture 6 Semester Yr Lecture 6: Moderatey Large Defection Theory of Beams 6.1 Genera Formuation Compare to the cassica theory of beams with infinitesima deformation, the moderatey
More informationStrain Energy in Linear Elastic Solids
Strain Energ in Linear Eastic Soids CEE L. Uncertaint, Design, and Optimiation Department of Civi and Environmenta Engineering Duke Universit Henri P. Gavin Spring, 5 Consider a force, F i, appied gradua
More informationNumerical methods for PDEs FEM - abstract formulation, the Galerkin method
Patzhater für Bid, Bid auf Titefoie hinter das Logo einsetzen Numerica methods for PDEs FEM - abstract formuation, the Gaerkin method Dr. Noemi Friedman Contents of the course Fundamentas of functiona
More informationSEMINAR 2. PENDULUMS. V = mgl cos θ. (2) L = T V = 1 2 ml2 θ2 + mgl cos θ, (3) d dt ml2 θ2 + mgl sin θ = 0, (4) θ + g l
Probem 7. Simpe Penduum SEMINAR. PENDULUMS A simpe penduum means a mass m suspended by a string weightess rigid rod of ength so that it can swing in a pane. The y-axis is directed down, x-axis is directed
More informationCE601-Structura Anaysis I UNIT-IV SOPE-DEFECTION METHOD 1. What are the assumptions made in sope-defection method? (i) Between each pair of the supports the beam section is constant. (ii) The joint in
More informationLecture 9. Stability of Elastic Structures. Lecture 10. Advanced Topic in Column Buckling
Lecture 9 Stabiity of Eastic Structures Lecture 1 Advanced Topic in Coumn Bucking robem 9-1: A camped-free coumn is oaded at its tip by a oad. The issue here is to find the itica bucking oad. a) Suggest
More informationSECTION A. Question 1
SECTION A Question 1 (a) In the usua notation derive the governing differentia equation of motion in free vibration for the singe degree of freedom system shown in Figure Q1(a) by using Newton's second
More information4 1-D Boundary Value Problems Heat Equation
4 -D Boundary Vaue Probems Heat Equation The main purpose of this chapter is to study boundary vaue probems for the heat equation on a finite rod a x b. u t (x, t = ku xx (x, t, a < x < b, t > u(x, = ϕ(x
More informationFOURIER SERIES ON ANY INTERVAL
FOURIER SERIES ON ANY INTERVAL Overview We have spent considerabe time earning how to compute Fourier series for functions that have a period of 2p on the interva (-p,p). We have aso seen how Fourier series
More informationTorsion and shear stresses due to shear centre eccentricity in SCIA Engineer Delft University of Technology. Marijn Drillenburg
Torsion and shear stresses due to shear centre eccentricity in SCIA Engineer Deft University of Technoogy Marijn Drienburg October 2017 Contents 1 Introduction 2 1.1 Hand Cacuation....................................
More information14 Separation of Variables Method
14 Separation of Variabes Method Consider, for exampe, the Dirichet probem u t = Du xx < x u(x, ) = f(x) < x < u(, t) = = u(, t) t > Let u(x, t) = T (t)φ(x); now substitute into the equation: dt
More information1D Heat Propagation Problems
Chapter 1 1D Heat Propagation Probems If the ambient space of the heat conduction has ony one dimension, the Fourier equation reduces to the foowing for an homogeneous body cρ T t = T λ 2 + Q, 1.1) x2
More informationC. Fourier Sine Series Overview
12 PHILIP D. LOEWEN C. Fourier Sine Series Overview Let some constant > be given. The symboic form of the FSS Eigenvaue probem combines an ordinary differentia equation (ODE) on the interva (, ) with a
More informationNonlinear Analysis of Spatial Trusses
Noninear Anaysis of Spatia Trusses João Barrigó October 14 Abstract The present work addresses the noninear behavior of space trusses A formuation for geometrica noninear anaysis is presented, which incudes
More informationSeparation of Variables and a Spherical Shell with Surface Charge
Separation of Variabes and a Spherica She with Surface Charge In cass we worked out the eectrostatic potentia due to a spherica she of radius R with a surface charge density σθ = σ cos θ. This cacuation
More information3.10 Implications of Redundancy
118 IB Structures 2008-9 3.10 Impications of Redundancy An important aspect of redundant structures is that it is possibe to have interna forces within the structure, with no externa oading being appied.
More informationMath 124B January 17, 2012
Math 124B January 17, 212 Viktor Grigoryan 3 Fu Fourier series We saw in previous ectures how the Dirichet and Neumann boundary conditions ead to respectivey sine and cosine Fourier series of the initia
More informationModel Solutions (week 4)
CIV-E16 (17) Engineering Computation and Simuation 1 Home Exercise 6.3 Mode Soutions (week 4) Construct the inear Lagrange basis functions (noda vaues as degrees of freedom) of the ine segment reference
More informationUnit 48: Structural Behaviour and Detailing for Construction. Deflection of Beams
Unit 48: Structura Behaviour and Detaiing for Construction 4.1 Introduction Defection of Beams This topic investigates the deformation of beams as the direct effect of that bending tendency, which affects
More information2.1. Cantilever The Hooke's law
.1. Cantiever.1.1 The Hooke's aw The cantiever is the most common sensor of the force interaction in atomic force microscopy. The atomic force microscope acquires any information about a surface because
More informationMath 124B January 31, 2012
Math 124B January 31, 212 Viktor Grigoryan 7 Inhomogeneous boundary vaue probems Having studied the theory of Fourier series, with which we successfuy soved boundary vaue probems for the homogeneous heat
More informationAPPENDIX C FLEXING OF LENGTH BARS
Fexing of ength bars 83 APPENDIX C FLEXING OF LENGTH BARS C.1 FLEXING OF A LENGTH BAR DUE TO ITS OWN WEIGHT Any object ying in a horizonta pane wi sag under its own weight uness it is infinitey stiff or
More informationMA 201: Partial Differential Equations Lecture - 10
MA 201: Partia Differentia Equations Lecture - 10 Separation of Variabes, One dimensiona Wave Equation Initia Boundary Vaue Probem (IBVP) Reca: A physica probem governed by a PDE may contain both boundary
More informationCABLE SUPPORTED STRUCTURES
CABLE SUPPORTED STRUCTURES STATIC AND DYNAMIC ANALYSIS OF CABLES 3/22/2005 Prof. dr Stanko Brcic 1 Cabe Supported Structures Suspension bridges Cabe-Stayed Bridges Masts Roof structures etc 3/22/2005 Prof.
More informationUI FORMULATION FOR CABLE STATE OF EXISTING CABLE-STAYED BRIDGE
UI FORMULATION FOR CABLE STATE OF EXISTING CABLE-STAYED BRIDGE Juan Huang, Ronghui Wang and Tao Tang Coege of Traffic and Communications, South China University of Technoogy, Guangzhou, Guangdong 51641,
More information4 Separation of Variables
4 Separation of Variabes In this chapter we describe a cassica technique for constructing forma soutions to inear boundary vaue probems. The soution of three cassica (paraboic, hyperboic and eiptic) PDE
More information1 Equations of Motion 3: Equivalent System Method
8 Mechanica Vibrations Equations of Motion : Equivaent System Method In systems in which masses are joined by rigid ins, evers, or gears and in some distributed systems, various springs, dampers, and masses
More informationPhysicsAndMathsTutor.com
. Two points A and B ie on a smooth horizonta tabe with AB = a. One end of a ight eastic spring, of natura ength a and moduus of easticity mg, is attached to A. The other end of the spring is attached
More information1 Heat Equation Dirichlet Boundary Conditions
Chapter 3 Heat Exampes in Rectanges Heat Equation Dirichet Boundary Conditions u t (x, t) = ku xx (x, t), < x (.) u(, t) =, u(, t) = u(x, ) = f(x). Separate Variabes Look for simpe soutions in the
More informationCourse 2BA1, Section 11: Periodic Functions and Fourier Series
Course BA, 8 9 Section : Periodic Functions and Fourier Series David R. Wikins Copyright c David R. Wikins 9 Contents Periodic Functions and Fourier Series 74. Fourier Series of Even and Odd Functions...........
More informationGauss Law. 2. Gauss s Law: connects charge and field 3. Applications of Gauss s Law
Gauss Law 1. Review on 1) Couomb s Law (charge and force) 2) Eectric Fied (fied and force) 2. Gauss s Law: connects charge and fied 3. Appications of Gauss s Law Couomb s Law and Eectric Fied Couomb s
More informationHYDROGEN ATOM SELECTION RULES TRANSITION RATES
DOING PHYSICS WITH MATLAB QUANTUM PHYSICS Ian Cooper Schoo of Physics, University of Sydney ian.cooper@sydney.edu.au HYDROGEN ATOM SELECTION RULES TRANSITION RATES DOWNLOAD DIRECTORY FOR MATLAB SCRIPTS
More informationLecture Notes for Math 251: ODE and PDE. Lecture 32: 10.2 Fourier Series
Lecture Notes for Math 251: ODE and PDE. Lecture 32: 1.2 Fourier Series Shawn D. Ryan Spring 212 Last Time: We studied the heat equation and the method of Separation of Variabes. We then used Separation
More informationIn-plane shear stiffness of bare steel deck through shell finite element models. G. Bian, B.W. Schafer. June 2017
In-pane shear stiffness of bare stee deck through she finite eement modes G. Bian, B.W. Schafer June 7 COLD-FORMED STEEL RESEARCH CONSORTIUM REPORT SERIES CFSRC R-7- SDII Stee Diaphragm Innovation Initiative
More informationLobontiu: System Dynamics for Engineering Students Website Chapter 3 1. z b z
Chapter W3 Mechanica Systems II Introduction This companion website chapter anayzes the foowing topics in connection to the printed-book Chapter 3: Lumped-parameter inertia fractions of basic compiant
More informationStrauss PDEs 2e: Section Exercise 2 Page 1 of 12. For problem (1), complete the calculation of the series in case j(t) = 0 and h(t) = e t.
Strauss PDEs e: Section 5.6 - Exercise Page 1 of 1 Exercise For probem (1, compete the cacuation of the series in case j(t = and h(t = e t. Soution With j(t = and h(t = e t, probem (1 on page 147 becomes
More informationAssignment 7 Due Tuessday, March 29, 2016
Math 45 / AMCS 55 Dr. DeTurck Assignment 7 Due Tuessday, March 9, 6 Topics for this week Convergence of Fourier series; Lapace s equation and harmonic functions: basic properties, compuations on rectanges
More informationHigher dimensional PDEs and multidimensional eigenvalue problems
Higher dimensiona PEs and mutidimensiona eigenvaue probems 1 Probems with three independent variabes Consider the prototypica equations u t = u (iffusion) u tt = u (W ave) u zz = u (Lapace) where u = u
More informationTHE OUT-OF-PLANE BEHAVIOUR OF SPREAD-TOW FABRICS
ECCM6-6 TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS, Sevie, Spain, -6 June 04 THE OUT-OF-PLANE BEHAVIOUR OF SPREAD-TOW FABRICS M. Wysocki a,b*, M. Szpieg a, P. Heström a and F. Ohsson c a Swerea SICOMP
More informationMATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES
MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES Separation of variabes is a method to sove certain PDEs which have a warped product structure. First, on R n, a inear PDE of order m is
More informationCS229 Lecture notes. Andrew Ng
CS229 Lecture notes Andrew Ng Part IX The EM agorithm In the previous set of notes, we taked about the EM agorithm as appied to fitting a mixture of Gaussians. In this set of notes, we give a broader view
More informationLecture Notes for Math 251: ODE and PDE. Lecture 34: 10.7 Wave Equation and Vibrations of an Elastic String
ecture Notes for Math 251: ODE and PDE. ecture 3: 1.7 Wave Equation and Vibrations of an Eastic String Shawn D. Ryan Spring 212 ast Time: We studied other Heat Equation probems with various other boundary
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
odue 2 naysis of Staticay ndeterminate Structures by the atri Force ethod Version 2 E T, Kharagpur esson 12 The Three-oment Equations- Version 2 E T, Kharagpur nstructiona Objectives fter reading this
More information1 Equivalent SDOF Approach. Sri Tudjono 1,*, and Patria Kusumaningrum 2
MATEC Web of Conferences 159, 01005 (018) IJCAET & ISAMPE 017 https://doi.org/10.1051/matecconf/01815901005 Dynamic Response of RC Cantiever Beam by Equivaent Singe Degree of Freedom Method on Eastic Anaysis
More informationLecture Notes 4: Fourier Series and PDE s
Lecture Notes 4: Fourier Series and PDE s 1. Periodic Functions A function fx defined on R is caed a periodic function if there exists a number T > such that fx + T = fx, x R. 1.1 The smaest number T for
More informationVibrations of Structures
Vibrations of Structures Modue I: Vibrations of Strings and Bars Lesson : The Initia Vaue Probem Contents:. Introduction. Moda Expansion Theorem 3. Initia Vaue Probem: Exampes 4. Lapace Transform Method
More information$, (2.1) n="# #. (2.2)
Chapter. Eectrostatic II Notes: Most of the materia presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartoo, Chap... Mathematica Considerations.. The Fourier series and the Fourier
More informationСРАВНИТЕЛЕН АНАЛИЗ НА МОДЕЛИ НА ГРЕДИ НА ЕЛАСТИЧНА ОСНОВА COMPARATIVE ANALYSIS OF ELASTIC FOUNDATION MODELS FOR BEAMS
СРАВНИТЕЛЕН АНАЛИЗ НА МОДЕЛИ НА ГРЕДИ НА ЕЛАСТИЧНА ОСНОВА Милко Стоянов Милошев 1, Константин Савков Казаков 2 Висше Строително Училище Л. Каравелов - София COMPARATIVE ANALYSIS OF ELASTIC FOUNDATION MODELS
More informationc 2007 Society for Industrial and Applied Mathematics
SIAM REVIEW Vo. 49,No. 1,pp. 111 1 c 7 Society for Industria and Appied Mathematics Domino Waves C. J. Efthimiou M. D. Johnson Abstract. Motivated by a proposa of Daykin [Probem 71-19*, SIAM Rev., 13 (1971),
More informationPhysics 235 Chapter 8. Chapter 8 Central-Force Motion
Physics 35 Chapter 8 Chapter 8 Centra-Force Motion In this Chapter we wi use the theory we have discussed in Chapter 6 and 7 and appy it to very important probems in physics, in which we study the motion
More information221B Lecture Notes Notes on Spherical Bessel Functions
Definitions B Lecture Notes Notes on Spherica Besse Functions We woud ike to sove the free Schrödinger equation [ h d r R(r) = h k R(r). () m r dr r m R(r) is the radia wave function ψ( x) = R(r)Y m (θ,
More informationWeek 6 Lectures, Math 6451, Tanveer
Fourier Series Week 6 Lectures, Math 645, Tanveer In the context of separation of variabe to find soutions of PDEs, we encountered or and in other cases f(x = f(x = a 0 + f(x = a 0 + b n sin nπx { a n
More informationFinite element method for structural dynamic and stability analyses
Finite eement method for structura dynamic and stabiity anayses Modue-9 Structura stabiity anaysis Lecture-33 Dynamic anaysis of stabiity and anaysis of time varying systems Prof C S Manohar Department
More informationModule 22: Simple Harmonic Oscillation and Torque
Modue : Simpe Harmonic Osciation and Torque.1 Introduction We have aready used Newton s Second Law or Conservation of Energy to anayze systems ike the boc-spring system that osciate. We sha now use torque
More informationELASTICITY PREVIOUS EAMCET QUESTIONS ENGINEERING
ELASTICITY PREVIOUS EAMCET QUESTIONS ENGINEERING. If the ratio of engths, radii and young s modui of stee and brass wires shown in the figure are a, b and c respectivey, the ratio between the increase
More informationSection 6: Magnetostatics
agnetic fieds in matter Section 6: agnetostatics In the previous sections we assumed that the current density J is a known function of coordinates. In the presence of matter this is not aways true. The
More informationApplication of the Finite Fourier Sine Transform Method for the Flexural-Torsional Buckling Analysis of Thin-Walled Columns
IOSR Journa of Mechanica and Civi Engineering (IOSR-JMCE) e-issn: 78-1684,p-ISSN: 3-334X, Voume 14, Issue Ver. I (Mar. - Apr. 17), PP 51-6 www.iosrjournas.org Appication of the Finite Fourier Sine Transform
More informationLECTURE 10. The world of pendula
LECTURE 0 The word of pendua For the next few ectures we are going to ook at the word of the pane penduum (Figure 0.). In a previous probem set we showed that we coud use the Euer- Lagrange method to derive
More informationRadiation Fields. Lecture 12
Radiation Fieds Lecture 12 1 Mutipoe expansion Separate Maxwe s equations into two sets of equations, each set separatey invoving either the eectric or the magnetic fied. After remova of the time dependence
More informationTechnical Data for Profiles. Groove position, external dimensions and modular dimensions
Technica Data for Profies Extruded Profie Symbo A Mg Si 0.5 F 25 Materia number.206.72 Status: artificiay aged Mechanica vaues (appy ony in pressing direction) Tensie strength Rm min. 245 N/mm 2 Yied point
More informationWave Equation Dirichlet Boundary Conditions
Wave Equation Dirichet Boundary Conditions u tt x, t = c u xx x, t, < x 1 u, t =, u, t = ux, = fx u t x, = gx Look for simpe soutions in the form ux, t = XxT t Substituting into 13 and dividing
More informationNotes: Most of the material presented in this chapter is taken from Jackson, Chap. 2, 3, and 4, and Di Bartolo, Chap. 2. 2π nx i a. ( ) = G n.
Chapter. Eectrostatic II Notes: Most of the materia presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartoo, Chap... Mathematica Considerations.. The Fourier series and the Fourier
More informationCryptanalysis of PKP: A New Approach
Cryptanaysis of PKP: A New Approach Éiane Jaumes and Antoine Joux DCSSI 18, rue du Dr. Zamenhoff F-92131 Issy-es-Mx Cedex France eiane.jaumes@wanadoo.fr Antoine.Joux@ens.fr Abstract. Quite recenty, in
More informationFirst-Order Corrections to Gutzwiller s Trace Formula for Systems with Discrete Symmetries
c 26 Noninear Phenomena in Compex Systems First-Order Corrections to Gutzwier s Trace Formua for Systems with Discrete Symmetries Hoger Cartarius, Jörg Main, and Günter Wunner Institut für Theoretische
More informationMA 201: Partial Differential Equations Lecture - 11
MA 201: Partia Differentia Equations Lecture - 11 Heat Equation Heat conduction in a thin rod The IBVP under consideration consists of: The governing equation: u t = αu xx, (1) where α is the therma diffusivity.
More informationUNCOMPLICATED TORSION AND BENDING THEORIES FOR MICROPOLAR ELASTIC BEAMS
11th Word Congress on Computationa Mechanics WCCM XI 5th European Conference on Computationa Mechanics ECCM V 6th European Conference on Computationa Fuid Dynamics ECFD VI E. Oñate J. Oiver and. Huerta
More informationSchool of Electrical Engineering, University of Bath, Claverton Down, Bath BA2 7AY
The ogic of Booean matrices C. R. Edwards Schoo of Eectrica Engineering, Universit of Bath, Caverton Down, Bath BA2 7AY A Booean matrix agebra is described which enabes man ogica functions to be manipuated
More informationSchool of Electrical Engineering, University of Bath, Claverton Down, Bath BA2 7AY
The ogic of Booean matrices C. R. Edwards Schoo of Eectrica Engineering, Universit of Bath, Caverton Down, Bath BA2 7AY A Booean matrix agebra is described which enabes man ogica functions to be manipuated
More informationRelated Topics Maxwell s equations, electrical eddy field, magnetic field of coils, coil, magnetic flux, induced voltage
Magnetic induction TEP Reated Topics Maxwe s equations, eectrica eddy fied, magnetic fied of cois, coi, magnetic fux, induced votage Principe A magnetic fied of variabe frequency and varying strength is
More information22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 2: The Moment Equations
.615, MHD Theory of Fusion ystems Prof. Freidberg Lecture : The Moment Equations Botzmann-Maxwe Equations 1. Reca that the genera couped Botzmann-Maxwe equations can be written as f q + v + E + v B f =
More informationStrauss PDEs 2e: Section Exercise 1 Page 1 of 7
Strauss PDEs 2e: Section 4.3 - Exercise 1 Page 1 of 7 Exercise 1 Find the eigenvaues graphicay for the boundary conditions X(0) = 0, X () + ax() = 0. Assume that a 0. Soution The aim here is to determine
More informationInstructional Objectives:
Instructiona Objectives: At te end of tis esson, te students soud be abe to understand: Ways in wic eccentric oads appear in a weded joint. Genera procedure of designing a weded joint for eccentric oading.
More informationCombining reaction kinetics to the multi-phase Gibbs energy calculation
7 th European Symposium on Computer Aided Process Engineering ESCAPE7 V. Pesu and P.S. Agachi (Editors) 2007 Esevier B.V. A rights reserved. Combining reaction inetics to the muti-phase Gibbs energy cacuation
More information9. EXERCISES ON THE FINITE-ELEMENT METHOD
9. EXERCISES O THE FIITE-ELEMET METHOD Exercise Thickness: t=; Pane strain proem (ν 0): Surface oad Voume oad; 4 p f ( x, ) ( x ) 0 E D 0 0 0 ( ) 4 p F( xy, ) Interna constrain: rigid rod etween D and
More informationLecture Note 3: Stationary Iterative Methods
MATH 5330: Computationa Methods of Linear Agebra Lecture Note 3: Stationary Iterative Methods Xianyi Zeng Department of Mathematica Sciences, UTEP Stationary Iterative Methods The Gaussian eimination (or
More information2M2. Fourier Series Prof Bill Lionheart
M. Fourier Series Prof Bi Lionheart 1. The Fourier series of the periodic function f(x) with period has the form f(x) = a 0 + ( a n cos πnx + b n sin πnx ). Here the rea numbers a n, b n are caed the Fourier
More informationSlender Structures Load carrying principles
Sender Structures Load carrying principes Cabes and arches v018-1 ans Weeman 1 Content (preiminary schedue) Basic cases Extension, shear, torsion, cabe Bending (Euer-Bernoui) Combined systems - Parae systems
More informationPhysics 505 Fall 2007 Homework Assignment #5 Solutions. Textbook problems: Ch. 3: 3.13, 3.17, 3.26, 3.27
Physics 55 Fa 7 Homework Assignment #5 Soutions Textook proems: Ch. 3: 3.3, 3.7, 3.6, 3.7 3.3 Sove for the potentia in Proem 3., using the appropriate Green function otained in the text, and verify that
More information12.2. Maxima and Minima. Introduction. Prerequisites. Learning Outcomes
Maima and Minima 1. Introduction In this Section we anayse curves in the oca neighbourhood of a stationary point and, from this anaysis, deduce necessary conditions satisfied by oca maima and oca minima.
More informationVolume 13, MAIN ARTICLES
Voume 13, 2009 1 MAIN ARTICLES THE BASIC BVPs OF THE THEORY OF ELASTIC BINARY MIXTURES FOR A HALF-PLANE WITH CURVILINEAR CUTS Bitsadze L. I. Vekua Institute of Appied Mathematics of Iv. Javakhishvii Tbiisi
More informationCOUPLED FLEXURAL TORSIONAL VIBRATION AND STABILITY ANALYSIS OF PRE-LOADED BEAMS USING CONVENTIONAL AND DYNAMIC FINITE ELEMENT METHODS
COUPLED FLEXURAL TORSIONAL VIBRATION AND STABILITY ANALYSIS OF PRE-LOADED BEAMS USING CONVENTIONAL AND DYNAMIC FINITE ELEMENT METHODS by Heenkenda Jayasinghe, B. Eng Aeronautica Engineering City University
More informationThe Bending of Rectangular Deep Beams with Fixed at Both Ends under Uniform Load
Engineering,,, 8-9 doi:.6/eng..7 Pubised Onine December (ttp://.scirp.org/journa/eng) Te Bending of Rectanguar Deep Beams it Fied at Bot Ends under Uniform Load Abstract Ying-Jie Cen, Bao-Lian Fu, Gang
More information6 Wave Equation on an Interval: Separation of Variables
6 Wave Equation on an Interva: Separation of Variabes 6.1 Dirichet Boundary Conditions Ref: Strauss, Chapter 4 We now use the separation of variabes technique to study the wave equation on a finite interva.
More informationModal analysis of a multi-blade system undergoing rotational motion
Journa of Mechanica Science and Technoogy 3 (9) 5~58 Journa of Mechanica Science and Technoogy www.springerin.com/content/738-494x DOI.7/s6-9-43-3 Moda anaysis of a muti-bade system undergoing rotationa
More informationMechanics 3. Elastic strings and springs
Chapter assessment Mechanics 3 Eastic strings and springs. Two identica ight springs have natura ength m and stiffness 4 Nm -. One is suspended verticay with its upper end fixed to a ceiing and a partice
More informationFourier series. Part - A
Fourier series Part - A 1.Define Dirichet s conditions Ans: A function defined in c x c + can be expanded as an infinite trigonometric series of the form a + a n cos nx n 1 + b n sin nx, provided i) f
More informationOn a geometrical approach in contact mechanics
Institut für Mechanik On a geometrica approach in contact mechanics Aexander Konyukhov, Kar Schweizerhof Universität Karsruhe, Institut für Mechanik Institut für Mechanik Kaiserstr. 12, Geb. 20.30 76128
More informationHomework #04 Answers and Hints (MATH4052 Partial Differential Equations)
Homework #4 Answers and Hints (MATH452 Partia Differentia Equations) Probem 1 (Page 89, Q2) Consider a meta rod ( < x < ), insuated aong its sides but not at its ends, which is initiay at temperature =
More informationMalaysian Journal of Civil Engineering 30(2): (2018)
Maaysian Journa of Ci Engineering 3():331-346 (18) BUBNOV-GALERKIN METHOD FOR THE ELASTIC BUCKLING OF EULER COLUMNS Ofondu I.O. 1, Ikwueze E. U. & Ike C. C. * 1 Dept. of Mechanica and Production Engineering,
More informationMat 1501 lecture notes, penultimate installment
Mat 1501 ecture notes, penutimate instament 1. bounded variation: functions of a singe variabe optiona) I beieve that we wi not actuay use the materia in this section the point is mainy to motivate the
More informationSolution of Wave Equation by the Method of Separation of Variables Using the Foss Tools Maxima
Internationa Journa of Pure and Appied Mathematics Voume 117 No. 14 2017, 167-174 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-ine version) ur: http://www.ijpam.eu Specia Issue ijpam.eu Soution
More informationKINGS COLLEGE OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING. Question Bank. Sub. Code/Name: CE1303 Structural Analysis-I
KINGS COLLEGE OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING Question Bank Sub. Code/Name: CE1303 Structura Anaysis-I Year: III Sem:V UNIT-I DEFLECTION OF DETERMINATE STRUCTURES 1.Why is it necessary to
More informationPHYS 110B - HW #1 Fall 2005, Solutions by David Pace Equations referenced as Eq. # are from Griffiths Problem statements are paraphrased
PHYS 110B - HW #1 Fa 2005, Soutions by David Pace Equations referenced as Eq. # are from Griffiths Probem statements are paraphrased [1.] Probem 6.8 from Griffiths A ong cyinder has radius R and a magnetization
More informationInternational Journal of Advance Engineering and Research Development
Scientific Journa of Impact Factor (SJIF): 4.4 Internationa Journa of Advance Engineering and Research Deveopment Voume 3, Issue 3, March -206 e-issn (O): 2348-4470 p-issn (P): 2348-6406 Study and comparison
More informationExplicit overall risk minimization transductive bound
1 Expicit overa risk minimization transductive bound Sergio Decherchi, Paoo Gastado, Sandro Ridea, Rodofo Zunino Dept. of Biophysica and Eectronic Engineering (DIBE), Genoa University Via Opera Pia 11a,
More informationIntroduction. Figure 1 W8LC Line Array, box and horn element. Highlighted section modelled.
imuation of the acoustic fied produced by cavities using the Boundary Eement Rayeigh Integra Method () and its appication to a horn oudspeaer. tephen Kirup East Lancashire Institute, Due treet, Bacburn,
More informationDYNAMIC RESPONSE OF CIRCULAR FOOTINGS ON SATURATED POROELASTIC HALFSPACE
3 th Word Conference on Earthquake Engineering Vancouver, B.C., Canada August -6, 4 Paper No. 38 DYNAMIC RESPONSE OF CIRCULAR FOOTINGS ON SATURATED POROELASTIC HALFSPACE Bo JIN SUMMARY The dynamic responses
More informationSome Measures for Asymmetry of Distributions
Some Measures for Asymmetry of Distributions Georgi N. Boshnakov First version: 31 January 2006 Research Report No. 5, 2006, Probabiity and Statistics Group Schoo of Mathematics, The University of Manchester
More information