Advanced Mathematics and Mechanics Applications Using MATLAB phần 9 pps

Let us solve for the ßow around a circular cylinder in the region|ζ| ≥ 1, ζ = ξ+iη with the requirement that the velocity components at inÞnity have constant values where φ is a harmonic

86: print(’Input data are incorrect The ’);

87: print(’following r values lie outside the ’);

88: print(’unit circle:’); disp(rvec(kout)’);

89: return

90: end

91:

92: if bvtyp==1 % Solve a Dirichlet problem

93: % Check for points on the boundary where

94: % function values are known Interpolate

122: disp(’CORRECT THE INPUT DATA AND RERUN.’);

131: surf(r.*cos(th),r.*sin(th),u);

132: xlabel(’x axis’); ylabel(’y axis’);

133: zlabel(’function u’); title(titl);

134: colormap(’default’);

135: grid on; figure(gcf);

136: % print -deps dirich

6: % This function solves a mixed boundary

7: % value problem for the interior of a circle

8: % by numerically evaluating a Cauchy integral

9: %

12: % nquad - order of Gauss quadrature used to

28: fb=cos(th)./fb; fb(1)=0; fb(end)=0;

29: F=cauchint(fb,zb,z,nquad);

30: F=F.*sqrt(z-i).*sqrt(z+i); u=2*real(F);

31:

32: surf(real(z),imag(z),u); xlabel(’x axis’);

33: ylabel(’y axis’); zlabel(’Solution Value’)

34: title([’Approximate Solution to ’,

36: grid on; figure(gcf); %grặ4);

37: fprintf(’\nPress [Enter] to solution error\n’);

46: grid on; figure(gcf); %grặ4)

47: %print -deps caucher2

56: % This function determines a function which is

57: % harmonic for abs(z)<1 and satisfies at r=1,

58: % u=cos(theta), -pi/2<theta<pi/2

59: % du/dr=0, pi/2<theta<3*pi/2

60: % The solution only applies for points inside

61: % or on the unit circlẹ

62: %

65: % noplot - option set to one if no plot is

68: %

71: %

72: % User m functions called: none

88: title([’Mixed Boundary Value Problem ’,

90: grid; figure(gcf); %grặ4), pause

91: %print -deps mbvtest

101: % This function numerically evaluates a Cauchy

102: % integral of the form:

103: %

104: % F(z)=1/(2*pi*i)*Integral(f(t)/(t-z)*dt)

105: %

106: % where t denotes points on a curve in the

107: % complex planẹ The boundary curve is defined

108: % by spline interpolation through data points

109: % zb lying on the curvẹ The values of f(t)

110: % are also specified by spline interpolation

111: % through values fb corresponding to the

112: % points zb Numerical evaluation of the

113: % integral is performed using a composite

114: % Gauss formula of arbitrary order

115: %

116: % fb - values of density function f

118: % zb - points where fb is given The

128: % nquad - the order of Gauss quadrature

131: %

153: disp([’WARNING! SOME DATA VALUES ARE ’,

155: disp([’THE BOUNDARY COMPUTED RESULTS ’,

12.12 Inviscid Fluid Flow around an Elliptic Cylinder

This section analyzes inviscid ßow around an elliptic cylinder in an inÞnite Þeld Flow around a circular cylinder is treated Þrst Then the function conformally map- ping the exterior of a circle onto the exterior of an ellipse is used in conjunction with the invariance of harmonic functions under a conformal transformation Results de- scribing the elliptic cylinder ßow Þeld for uniform velocity components at inÞnity are presented.

Let us solve for the ßow around a circular cylinder in the region|ζ| ≥ 1, ζ = ξ+iη

with the requirement that the velocity components at inÞnity have constant values

where φ is a harmonic function The velocity normal to the cylinder boundary must

be zero This requires that the function ψ, the harmonic conjugate of φ, must be

con-stant on the boundary The concon-stant can be taken as zero without loss of generality.

In terms of the complex velocity potential

A Laurent series can be used to represent f(ζ) in the form

Now consider ßow about an elliptic cylinder lying in the z-plane If the velocity

at inÞnity has components (U, V ) then we need a velocity potential F (z) such that

where ω(ζ) is the mapping function

To get values for a particular choice of z we can use the inverse mapping function

ζ = z +

√

z2− 4mR2

2R

to eliminate ζ or we can compute results in terms of ζ.

To complete our discussion of this ßow problem we will graph the lines

charac-terizing the directions of ßow The velocity potential F = φ + iψ satisÞes

Consequently, the ßow lines are the contours of function ψ, which is called the

stream function The function we want to contour does not exist inside the ellipse,

but we can circumvent this problem by computing ψ in the ellipse exterior and then

setting ψ to zero inside the ellipse The function elipcyl analyzes the cylinder ßow

and produces the accompanying contour plot shown in Figure 12.9

x axis

Figure 12.9: Elliptic Cylinder Flow Field for Angle = 30◦

12.12.1 Program Output and Code

6: % This function computes the flow field around

7: % an elliptic cylinder The velocity direction

8: % at infinity is arbitrary

9: %

10: % a - defines the region -a<x<a, -a<y<a

15: % rx,ry - major and minor semi-diameters af the

17: % respectively

18: % ang - the angle in degrees which the

21: %

22: % x,y - matrices of points where the velocity

58: hold on; fill(xb,yb,[127/255 1 212/255]);

59: xlabel(’x axis’); ylabel(’y axis’);

60: title([’Elliptic Cylinder Flow Field for ’,

62: colormap hsv; figure(gcf); hold off;

63: %print -deps elipcyl

12.13 Torsional Stresses in a Beam Mapped onto a Unit Disk

Torsional stresses in a cylindrical beam can be computed from an integral formula

when the function z = ω(ζ) mapping the unit disk, |ζ| ≤ 1, onto the beam cross

section is known [90] The complex stress function

where γ denotes the unit circle, can be evaluated exactly by contour integration in

some cases However, an approach employing series methods is easy to implement

and gives satisfactory results if enough series terms are taken When ω(ζ) is a nomial, f(ζ) is a polynomial of the same order as ω(ζ) Furthermore, when ω(ζ)

poly-is a rational function, residue calculus can be employed to compute f(ζ) exactly, provided the poles of ω(1/ζ) can be found A much simpler approach is to use the FFT to expand ω(σ)ω(σ) in a complex Fourier series and write

where µ is the shear modulus and ε is the angle of twist per unit length The capital

Z subscript on shear stresses refers to the direction of the beam axis normal to the

xy plane rather than the complex variable z = x + ıy The series expansion gives

y distance along the side

Max Shear Stress = 0.67727

Number of Series Terms = 800

Maximum Stress Error = 0.44194%

Stiffness Factor Error = 0.052216%

Radial shear stress Tangential shear stress

Figure 12.10: Torsional Shear Stresses on a Square Cross Section

rational function mapping|ζ| < 1 onto a square deÞned by |x| ≤ 1 and |y| ≤ 1 was

employed Function mapsqr which computes z(ζ) and z  (ζ)is used by function

torstres to evaluate stresses in terms of ζ A short driver program runtors evaluates

stresses on the boundary for x = 1, 0 ≤ y ≤ 1 Stresses divided by the side length of

2 are plotted and results produced from a highly accurate solution [90] are compared

with values produced using 800 terms in f(ζ) Results depicted in Figure 12.10 show that the error in maximum shear stress was only 0.44% and the torsional stiff- ness was accurate within 0.05% The numerical solution gives a nonzero stress value for y = 1, which disagree with the exact solution This error is probably due more to

the mapping function giving slightly rounded corners than to slow convergence of the series solution Even though the differentiated series converges slowly, computation time is still small The reader can verify that using 1500 terms reduces the bound- ary stress oscillations to negligible magnitude and produces a maximum stress error

of 0.03% Although taking 1500 terms to achieve accurate results seems excessive,

less than 400 nonzero terms are actually involved because geometrical symmetry plies a series increasing in powers of four For simplicity and generality, no attempt was made to account for geometrical symmetry exhibited by a particular mapping function It appears that a series solution employing a mapping function is a viable computational tool to deal with torsion problems.

im-12.13.1 Program Output and Code

5: % Example showing torsional stress computation

6: % for a beam of square cross section using

7: % conformal mapping and a complex stress

19: % Compute stresses using an approximate rational

20: % function mapping function for the square

37: xlabel(’y distance along the side’);

38: ylabel(’shear stresses at the boundary’);

39: title([’Torsional Shear Stresses on a ’,

46: [’Maximum Stress Error = ’,num2str(err),’%’]);

47: text(.05,.22,[’Stiffness Factor Error = ’,

48: num2str(ster),’%’]);

49: legend(’Radial shear stress’,

50: ’Tangential shear stress’);

51: figure(gcf);

52: %disp(’Use mouse to locate legend block’);

53: %disp(’Press [Enter] when finished’);

54: %print -deps torsion

65: % This function computes torsional stresses in

66: % a beam such that abs(zeta)<=1 is mapped onto

67: % the beam cross section by a function named

68: % mapfun

69: %

70: % mapfun - a character string giving the name

72: % zeta - values in the zeta plane

75: % ntrms - the number of terms used in the

78: % nft - the number of function values

82: %

83: % trho - torsional stresses in directions

86: % should be zero at the boundary

88: % talpha - torsional stresses in directions

108: % Compute boundary values of the mapping

109: % function needed to construct the complex

118: % Compute Fourier coefficients for the complex

119: % stress function and its derivative

128: % trho is the radial shear stress that should

129: % vanish at the boundary

130: trho=real(tcplx);

132: % talpha is the circumferential stress which

133: % gives the maximum stress of interest at the

144: % This function maps the interior of a circle

145: % onto the interior of a square using a rational

146: % function of the approximate form:

155: % z,zp - matrices of values of the mapping

172: % Exit if the derivative of z is not needed

173: if nargout==1, return, end

174:

175: % evaluate z’(zeta)

176: na=length(a); nb=length(b);

177: pp=polyval(flipud((4*(1:na)’-3).*a),zeta4);

178: qp=4*zeta.^3.*polyval(flipud((1:nb)’.*b),zeta4);

179: zp=(q.*pp-p.*qp)./q.^2;

12.14 Stress Analysis by the Kolosov-Muskhelishvili Method

Two-dimensional problems in linear elastostatics of homogeneous bodies can be analyzed with the use of analytic functions The primary quantities of interest are

cartesian stress components τ xx , τ yy , and τ xy and displacement components u and

v These can be expressed as

τ xx + τ yy = 2[Φ(z) + Φ(z)]

−τ xx + τ yy + 2iτ xy= 2[¯zΦ  (z) + Ψ(z)]

2µ(u + iv) = κφ(z) − zΦ(z) − ψ(z) φ(z) =



Φ(z) dz , ψ(z) =



Ψ(z) dz where µ is the shear modulus and κ depends on Poisson’s ratio ν according to

κ = 3 − 4ν for plane strain or κ = (3 − ν)/(1 + ν) for plane stress The above

relations are known as the Kolosov-Muskhelishvili formulas [73] and they have been used to solve many practical problems employing series or integral methods Bod- ies such as a circular disk, a plate with a circular hole, and a circular annulus can

be handled for quite general boundary conditions Solutions can also be developed for geometries where a rational function is known that maps the interior of a circle onto the desired geometry Futhermore, complex variable methods provide the most general techniques available for solving a meaningful class of mixed boundary value problems such as contact problems typiÞed by pressing a rigid punch into a half plane.

Fully understanding all of the analyses presented in [72, 73] requires familiarity with contour integration, conformal mapping, and multivalued functions However, some of the closed form solutions given in these texts can be used without extensive background in complex variable methods or the physical concepts of elasticity the- ory With that perspective let us examine the problem of computing stresses in an

inÞnite plate uniformly stressed at inÞnity and having a general normal stress N(θ) and tangential shear T (θ) applied to the hole We will use the general solution of

Muskhelishvili 1 [72] to evaluate stresses anywhere in the plate with particular est on stress concentrations occurring around the hole The stress functions Ψ and Φ

can be represented as follows

where γ denotes counterclockwise contour integration around the boundary of the

hole and the other constants are given by

Parameters α and δ depend only on the components of stress at inÞnity, while β is

determined by the force resultant on the hole caused by the applied loading The

quantity N + ıT is the boundary value of radial stress τ rr and shear stress τ rθ in polar coordinates Hence

N + ıT = τ rr + iτ rθ , |z| = 1

The transformation formulas relating Cartesian stresses τ xx , τ yy , τ xy and polar

co-ordinate stresses τ rr , τ θθ , τ rθare

or we can compute the approximate coefÞcients more readily using the FFT.

The stress function Ψ(z) is related to Φ(z) according to

Ψ = 1

z2Φ

 1

¯



− d dz

1

which has the form

where the coefÞcients b nare obtainable by comparing coefÞcients of corresponding

powers in the two series Hence, the series expansions of functions Φ(z) and Ψ(z) can be generated in terms of the coefÞcients c n and the stress components at inÞn- ity The stresses can be evaluated by using the stress functions Displacements can also be obtained by integrating Φ and Ψ, but this straightforward calculation is not discussed here.

The program runplate was written to evaluate the above formulas by expanding

N + iT using the FFT Truncating the series for harmonics above some speciÞed

order, say np, gives approximations for Φ(z) and Ψ(z), which exactly represent

the solution corresponding to the boundary loading deÞned by the truncated Fourier

series Using the same approach employed in Chapter 6 we can deÞne N and T as piecewise linear functions of the polar angle θ.

The program utilizes several routines described in the table below.

runplate deÞne N, T , stresses at inÞnity, z-points

where results are requested, and the number

of series terms used.

platecrc computes series coefÞcients deÞning the

stress functions.

strfun evaluates Φ, Ψ, and Φ.

cartstrs evaluates Cartesian stresses for given values

of z and the stress functions.

rec2polr transforms from Cartesian stresses to polar

coordinate stresses.

polßip simpliÞed interface to function polyval.

The program solves two sample problems The Þrst one analyzes a plate having

no loading on the hole, and stresses at inÞnity given by τ ∞

Figure 12.11: Stress Concentration around a Circular Hole in a Plate

3, producing a stress concentration factor of three due to the presence of the hole The second problem applies a sinusoidally varying normal stress on the hole while the stresses at inÞnity are zero Taking

inves-−2 −1.5 −1 −0.5

0 0.5 1 1.5 2

−2

−1 0 1 2

12.14.1 Program Output and Code

5: % Example to compute stresses around a

6: % circular hole in a plate using the

7: % Kolosov-Muskhelishvili method

8: %

9: % User m functions required:

12:

13: if nargin==0

14: titl=[’Stress Concentration Around a ’,

35: fprintf(’\n\nStress components at infinity ’)

36: fprintf(’are: ’); fprintf(’%g ’,ti);

37: fprintf(’\nNormal stresses on the hole are ’)

38: fprintf([’defined by ’,Nn]);

39: fprintf(’\nTangential stresses on the hole ’)

40: fprintf([’are defined by ’,Tt])

41: fprintf(’\nElastic constant kappa equals: ’)

49: fprintf(’\nThe Kolosov-Muskhelishvili stress ’);

50: fprintf(’functions have\nthe series forms:’);

51: fprintf(’\nPhi=sum(a(k)*z^(-k+1), k=1:np+1)’);

52: fprintf(’\nPsi=sum(b(k)*z^(-k+1), k=1:np+3)’);

53: fprintf(’\n’);

54: fprintf(’\nCoefficients defining stress ’);

55: fprintf(’function Phi are:\n’);

56: disp(a(:));

57: fprintf(’Coefficients defining stress ’);

58: fprintf(’function Psi are:\n’);

75: [’Maximum Principal Stress = ’,num2str(pmax)]);

76: fprintf(’\nPress [Enter] for a surface ’);

77: fprintf(’plot of the\ncircumferential stress ’);

78: fprintf(’in the plate\n’); input(’’,’s’); clf;

79: close; colormap(’hsv’);

80: surf(x,y,tt); xlabel(’x axis’); ylabel(’y axis’);

81: zlabel(’Circumferential Stress’);

82: title(titl); grid on; view(viewpnt); figure(gcf);

83: %if nargin==0, print -deps strconc1

84: %else, print -deps strconc2; end

85: fprintf(’All Done\n’);

94: % This function computes coefficients in the

95: % series expansions that define the

Kolosov-96: % Muskhelishvili stress functions for a plate

97: % having a circular hole of unit radius The

98: % plate is uniformly stressed at infinity On

99: % the surface of the hole, normal and tangential

100: % stress distributions N and T defined as

101: % piecewise linear functions are applied

102: %

103: % N - a two column matrix with each row

117: % T - a two column matrix defining values of

125: % ti - vector of Cartesian stress components

127: % kapa - a constant depending on Poisson’s ratio

128: % nu

131: % When the resultant force on the hole

134: % np - the highest power of exp(i*theta) used

137: %

138: % a - coefficients in the series expansion

141: % b - coefficients in the series expansion

168: % Generate a and b coefficients using the

169: % Fourier coefficients of N+i*T

192: % This function evaluates the complex

193: % stress functions Phi(z) and Psi(z)

194: % as well as the derivative function Phi’(z)

195: % using series coefficients determined from

196: % function platecrc The calculation also

197: % uses a function polflip defined such that

204: % Phi,Psi - complex stress function values

205: % Phip - derivative Phi’(z)

221: %

222: % This function uses values of the complex

223: % stress functions to evaluate Cartesian stress

224: % components relative to the x,y axes

225: %

228: % Phi,Psi - matrices containing complex stress

231: %

232: % tx,ty,txy - values of the Cartesian stress

234: % tp1,tp2 - values of maximum and minimum

252: % This function transforms Cartesian stress

253: % components tx,ty,txy to polar coordinate

254: % stresses tr,tt,trt

255: %

256: % tx,ty,txy - matrices of Cartesian stress

263: %

264: % tr,tt,trt - matrices of polar coordinate

283: % This function evaluates polyval(a,x) with

284: % the order of the elements reversed

12.14.2 Stressed Plate with an Elliptic Hole

This chapter is concluded with an example using conformal mapping in elasticity theory We discussed earlier the useful property that harmonic functions remain harmonic under a conformal transformation However, linear elasticity leads to the biharmonic Airy stress function which satisÞes

Conse-elasticity This does not preclude use of conformal mapping in elasticity, but we encounter equations of very different structure in the mapped variables We will examine that problem enough to illustrate the kind of differences involved Let a

mapping function z = ω(ζ) deÞne curvilinear coordinate lines in the z-plane A lar coordinate grid corresponding to arg(ζ) = constant and |ζ| = constant maps into

po-curves we term ρ lines and α lines, respectively Plotting of such lines was

demon-strated previously with function gridview (mapping the exterior of a circle onto the

exterior of an ellipse) It can be shown that curvilinear coordinate stresses τ ρρ , τ αα,

τ ραare related to cartesian stresses according to

τ ρρ + τ αα = τ xx + τ yy , −τ ρρ + τ αα + 2iτ ρα = h( −τ xx + τ yy + 2iτ xy) where

z = ω(ζ) = R



ζ + m ζ

When ζ is selected as the primary reference variable, we have to perform chain rule

differentiation and write

in order to compute stresses in terms of the ζ-variable Readers unaccustomed to

using conformal mapping in this context should remember that there is no stress

state in the ζ-plane comparable to the analogous velocity components which can

be envisioned in a potential ßow problem We are simply using ζ as a convenient

reference variable to analyze physical stress and displacement quantities existing

only in the z-plane.

Suppose the inÞnite plate has an elliptic hole deÞned by



x

r x

2+

and the hole is free of applied tractions The stress state at inÞnity consists of a

tension p inclined at angle λ with the x-axis The stress functions relating to that

problem are found to be ([72], page 338)

for a plate with a circular hole The function eliphole computes curvilinear

coordi-nate stresses in the z-plane expressed in terms of the ζ-variable When λ = π/2, the plate tension acts along the y-axis and the maximum circumferential stress occurs at

z = r x corresponding to ζ = 1 A surface plot produced by eliphole for the default

data case using r x= 2and r y= 1 is shown in Figure 12.13 It is also interesting to

graph τmax

yy as a function of r x /r y The program elpmaxst produces the plot in

Figure 12.14 showing that the circumferential stress concentration increases linearly according to

−2 0 2 4

−3

−2

−1 0 1 2 3

Stress Concentration Around an Elliptical Hole

ratio ( max diameter ) / ( min diameter )

Figure 12.14: Stress Concentration around an Elliptical Hole

12.14.3 Program Output and Code

5: % MATLAB example to plot the stress

6: % concentration around an elliptic hole

7: % as a function of the semi-diameter ratio

25: ylabel([’( max circumferential stress ) / ’,

27: grid on; figure(gcf);

28: %print -deps elpmaxst

39: % This function determines curvilinear

40: % coordinate stresses around an elliptic hole

41: % in a plate uniformly stressed at infinity.

51: % ifplot - optional parameter that is given

54: %

75: % The complex stress functions and mapping

76: % function have the form

97 : % having a circular hole of unit radius The

98 : % plate is uniformly stressed at infinity On

99 : % the surface of the hole, normal and. .. class="page_container" data-page="26">

192 : % This function evaluates the complex

193 : % stress functions Phi(z) and Psi(z)

194 : % as well as the derivative... function Phi’(z)

195 : % using series coefficients determined from

196 : % function platecrc The calculation also

197 : % uses a function