Description

EX_AXISTRESSSTRAIN1 Example for hollow cylider axisymmetric stress-strain.

[ FEA, OUT ] = EX_AXISTRESSSTRAIN1( VARARGIN ) Example to calculate displacements and stresses in a hollow cylinder in axisymmetric/cylindrical coordinates.

Ref. 4.1.9 Long (generalized plane strain) cylinder subjected to internal and external pressure. [1] Applied Mechanics of Solids, Allan F. Bower, 2012 (http://solidmechanics.org/).

Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
a           scalar {1.5}           Cylinder inner radius
b           scalar {2}             Cylinder outer radius
l           scalar {3}             Cylinder length
pa          scalar {5e3}           Inner load force
pb          scalar {20e4}          Outer load force
E           scalar {200e9}         Modulus of elasticity
nu          scalar {0.3}           Poissons ratio
igrid       scalar 0/{1}           Cell type (0=quadrilaterals, 1=triangles)
hmax        scalar {0.1}           Max grid cell size
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',        0.9;
             'b',        2;
             'l',        3;
             'pa',       5e3;
             'pb',       20e4;
             'E',        200e9;
             'nu',       0.3;
             'igrid',    0;
             'hmax',     0.01;
             'sfun',     'sflag2';
             'iplot',    1;
             'tol',      5e-2;
             'fid',      1 };
 [got,opt] = parseopt(cOptDef,varargin{:});
 fid       = opt.fid;


% Geometry and grid.
 a = opt.a;
 b = opt.b;
 l = opt.l;
 fea.geom.objects = { gobj_rectangle( a, b, 0, l, 'R1' ) };
 if ( opt.igrid==1 )
   fea.grid = gridgen( fea, 'hmax', opt.hmax, 'fid', fid );
 else
   fea.grid = rectgrid( ceil((b-a)/opt.hmax), ceil(l/opt.hmax), [a b;0 l] );
   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{2,:}] = deal( 1 );
 fea.phys.css.bdr.coef{1,5} = bctype;

 bccoef = mat2cell( zeros(2,n_bdr), [1 1], ones(1,n_bdr) );
 bccoef{1,2} = opt.pb*b;
 bccoef{1,4} = opt.pa*a;
 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 = l/2*ones(1,n);
 pa = opt.pa; pb = -opt.pb; E = opt.E; nu = opt.nu;
% 4.1.9 http://solidmechanics.org/Text/Chapter4_1/Chapter4_1.php#Sect4_1_9
 u_ref  = (1+nu)*a^2*b^2/E/(b^2-a^2)*( (pa-pb)./r' + (1-2*nu)*(pa*a^2 - pb*b^2)/a^2/b^2*r' );
 sr_ref = (pa*a^2-pb*b^2)/(b^2-a^2) - a^2*b^2/(b^2-a^2)./(r').^2*(pa-pb);
 st_ref = (pa*a^2-pb*b^2)/(b^2-a^2) + a^2*b^2/(b^2-a^2)./(r').^2*(pa-pb);
 sz_ref = 2*nu*(pa*a^2-pb*b^2)/(b^2-a^2);
 u  = evalexpr( fea.phys.css.eqn.vars{3,2}, [r;z], fea );
 w  = evalexpr( fea.phys.css.eqn.vars{4,2}, [r;z], fea );
 sr = evalexpr( fea.phys.css.eqn.vars{5,2}, [r;z], fea );
 st = evalexpr( fea.phys.css.eqn.vars{6,2}, [r;z], fea );
 sz = evalexpr( fea.phys.css.eqn.vars{7,2}, [r;z], fea );
 if( opt.iplot>0 )
   subplot(1,2,1)
   postplot( fea, 'surfexpr', fea.phys.css.eqn.vars{3,2} )
   title('r-displacement')
   subplot(1,2,2), hold on
   plot(u_ref,r,'r-')
   plot(u,r,'b.')
   legend('exact solution','computed solution')
   xlabel('r')
   grid on
 end


% Error checking.
 out.erru  = norm( u_ref - u )/norm( u_ref );
 out.errw  = norm( w );
 out.errsr = norm( sr_ref - sr )/norm( sr_ref );
 out.errst = norm( st_ref - st )/norm( st_ref );
 out.errsz = norm( sz_ref - sz )/norm( sz_ref );
 out.err   = [ out.erru, out.errw, out.errsr, out.errst, out.errsz ];
 out.pass  = all(out.err < opt.tol);


 if( nargout==0 )
   clear fea out
 end