Description

EX_AXISTRESSSTRAIN2 Example for a pressurized hollow sphere axisymmetric stress-strain.

[ FEA, OUT ] = EX_AXISTRESSSTRAIN2( VARARGIN ) Example to calculate displacements and stresses in a pressurized hollow sphere in axisymmetric/cylindrical coordinates.

Ref. 4.1.4 Pressurized hollow sphere. [1] Applied Mechanics of Solids, Allan F. Bower, 2012 (http://solidmechanics.org/).

Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
a           scalar {1}             Cylinder inner radius
b           scalar {2}             Cylinder outer radius
p           scalar {20e4}          Load force
E           scalar {200e9}         Modulus of elasticity
nu          scalar {0.3}           Poissons ratio
igrid       scalar 0/{<0}          Cell type (0=quadrilaterals, <0=triangles)
sfun        string {sflag2}        Shape function for displacements
iplot       scalar 0/{1}           Plot solution (=1)
                                                                                  .
Output      Value/(Size)           Description
-----------------------------------------------------------------------------------
fea         struct                 Problem definition struct
out         struct                 Output struct

Code listing

 cOptDef = { 'a',        1;
             'b',        2;
             'p',        20e4;
             'E',        200e9;
             'nu',       0.3;
             'igrid',    0;
             'sfun',     'sflag2';
             'iplot',    1;
             'tol',      5e-3;
             'fid',      1 };
 [got,opt] = parseopt(cOptDef,varargin{:});
 fid       = opt.fid;


% Geometry and grid.
 a = opt.a;
 b = opt.b;
 if ( opt.igrid==1 )
   error('ex_axistressstrain2: unstructured grid not supported.')
 else
   fea.grid = ringgrid( 12, 72, a, b );
   fea.grid = delcells( fea.grid, selcells( fea.grid, '(x<=eps) | (y<=eps)') );
   if( opt.igrid<0 )
     fea.grid = quad2tri( fea.grid );
   end
 end
 n_bdr = max(fea.grid.b(3,:));   % Number of boundaries.


% Axisymmetric stress-strain equation definitions.
 fea.sdim = { 'r', 'z' };
 fea = addphys( fea, @axistressstrain );
 fea.phys.css.eqn.coef{1,end} = { opt.nu };
 fea.phys.css.eqn.coef{2,end} = { opt.E  };
 fea.phys.css.sfun            = { opt.sfun opt.sfun };   % Set shape functions.


% Boundary conditions.
 bctype = mat2cell( zeros(2,n_bdr), [1 1], ones(1,n_bdr) );
 bctype{1,4} = 1;
 bctype{2,3} = 1;
 fea.phys.css.bdr.coef{1,5} = bctype;

 bccoef = mat2cell( zeros(2,n_bdr), [1 1], ones(1,n_bdr) );
 bccoef{1,1} = ['-r*nr*',num2str(opt.p),];
 bccoef{2,1} = ['-r*nz*',num2str(opt.p),];
 fea.phys.css.bdr.coef{1,end} = bccoef;


% Solve.
 fea       = parsephys( fea );
 fea       = parseprob( fea );
 fea.sol.u = solvestat( fea, 'icub', 1+str2num(strrep(opt.sfun,'sflag','')), 'fid', fid );


% Postprocessing.
 n = 20;
 r = linspace(a,b,n);
 z = zeros(1,n);
 u_ref = 1./(2*opt.E*(b^3-a^3)*r'.^2) .* (2*(opt.p*a^3)*(1-2*opt.nu)*r'.^3+opt.p*(1+opt.nu)*b^3*a^3);
 u = evalexpr( 'r*u', [r;z], fea );
 if( opt.iplot>0 )
   subplot(1,2,1)
   postplot( fea, 'surfexpr', 'sqrt((r*u)^2+w^2)', 'arrowexpr', {'r*u' 'w'} )
   title('computed displacement')
   subplot(1,2,2), hold on
   plot(u_ref,r,'r-')
   plot(u,r,'b.')
   title( 'radial displacement')
   legend('exact solution','computed solution')
   xlabel('r')
   grid on
 end


% Error checking.
 out.err  = norm( u_ref - u )/norm( u_ref );
 out.pass = out.err < opt.tol;


 if( nargout==0 )
   clear fea out
 end