/*  *  ========================================================================= *  Force-displacement calculation for mass-spring system having springs  *  with bi-linear force-displacement characteristics. * *  We will compute : force displacement curve. *                    system-level energy versus time. *                    element-level energy versus time. * *  Written By: Wane-Jang Lin and Mark Austin       October 1997 - March 1998 *  ========================================================================= */ /* Define bi-linear spring behavior */   ks1 = 2.0 N/cm;   ks2 = 1.5 N/cm;   kt1 = 0.8 N/cm;   kt2 = 0.5 N/cm;   fy1 = 18 N;       fy2 = 15 N;   Ks = [ks1; ks2];   Kt = [kt1; kt2];   Fy = [fy1; fy2];/* Compute and store displacements for incipient yielding of springs */   ey = Matrix([2,1]);   for( ele=1; ele<=2; ele=ele+1 ) {      ey[ele][1] = Fy[ele][1]/Ks[ele][1];   }/* Initial condition, unstressed */   P   = [0 N;   0 N];       /* structure force           */   p   = [0 cm; 0 cm];       /* structure displacement    */   PR  = [0 N;   0 N];       /* structure resisting force */   PRe = [0 N;   0 N];       /* element resisting force   */   Q_saved  = [0  N; 0  N];  /* matrix of element forces        */   q_saved  = [0 cm; 0 cm];  /* matrix of element displacements *//* initialize flags, stresses and strains before applying any load step */   index = [0;0];  /* index for loading flag   */   flag1 = [0;0];  /* yielding at load step k  */   flag2 = [0;0];  /* pre_range at load step k */   flag3 = [0;0];  /* pre_load at load step k  */   yielding  = [0;0];   pre_range = [0;0];   pre_load  = [0;0];   loading   = [0;0];   sr = [0 N;   0 N];   er = [0 cm; 0 cm];   s0 = [0 N;   0 N];   e0 = [0 cm; 0 cm];   sr_saved = [0 N;   0 N];   er_saved = [0 cm; 0 cm];   s0_saved = [0 N;   0 N];   e0_saved = [0 cm; 0 cm];   sx_saved = [0 N;   0 N];   ex_saved = [0 cm; 0 cm];/* transformation matrix Lele, d_q = Lele*d_p[ele] for each element    *//* transform structural displacements d_p to element deformations d_q  */   Rigid = [-1,1];   Transform = [1,0;0,1];   L = Rigid*Transform;/* temporary matrix to store element tangent stiffness Ks at each step */   tangent = [Ks[1][1] ; Ks[2][1]];/* force interpolation matrices b(x), D(x) = b(x)*Q *//* relate section force D(x) with element force Q */   bx = 1;/* assemble initial structure tangent stiffness matrix BigK */   BigK = [ Ks[1][1]+Ks[2][1], -Ks[2][1];                    -Ks[2][1],  Ks[2][1] ];   total_step = 400;/* Allocate storage matrix for force-displacement history *//* column[1] : external applied load at end node (2)      *//* column[2] : total elogation at end node (2) = [3]+[4]  *//* column[3] : element elogation 1, node 1                *//* column[4] : element elogation 2, node 2                */   result = ColumnUnits( Matrix([total_step+1,4]), [N,cm,cm,cm] );   result[1][1] = P[2][1];   result[1][2] = p[2][1];   result[1][3] = p[1][1];   result[1][4] = p[2][1] - p[1][1];/* Allocate storage matrix system/element energy calculations   *//* column[1] : total external work for whole system             *//* column[2] : total internal energy = [3]+[4] for whole system *//* column[3] : elastic strain energy for whole system           *//* column[4] : plastic strain energy for whole system           */   system_energy = ColumnUnits( Matrix([total_step+1,4]), [Jou] );   system_energy[1][1] = 0 Jou;   system_energy[1][2] = 0 Jou;   system_energy[1][3] = 0 Jou;   system_energy[1][4] = 0 Jou;/* column[1] : internal strain energy for element 1 *//* column[2] : elastic strain energy for element 1  *//* column[3] : internal strain energy for element 2 *//* column[4] : elastic strain energy for element 2  */   element_energy = ColumnUnits( Matrix([total_step+1,4]), [Jou] );   element_energy[1][1] = 0 Jou;   element_energy[1][2] = 0 Jou;   element_energy[1][3] = 0 Jou;   element_energy[1][4] = 0 Jou;/* Increase external load, structure determination */tol = 0.00001;for ( k=1; k <= total_step ; k = k+1 ) {   print "start step no = ", k, "\n";   /* Define incremental loading */   if( k<=20 )           { d_P =  [0 N; 1 N]; }   if( k>20  && k<=60  ) { d_P = -[0 N; 1 N]; }   if( k>60  && k<=105 ) { d_P =  [0 N; 1 N]; }   if( k>105 && k<=155 ) { d_P = -[0 N; 1 N]; }   if( k>155 && k<=190 ) { d_P =  [0 N; 1 N]; }   if( k>190 && k<=200 ) { d_P = -[0 N; 1 N]; }   if( k>200 )           { d_P =  [0 N; 0 N]; }   P = P + d_P;   element_energy[k+1][1] = element_energy[k][1];   element_energy[k+1][3] = element_energy[k][3];   index[1][1] = 1;   index[2][1] = 1;   err = tol+1;   while( err > tol )  { /* i-th Newton-Raphson iteration */      /* compute displacement increment */      d_p = Solve(BigK,d_P);      p   = p + d_p;      /* Element level state determination */      for( ele = 1; ele <= 2; ele = ele + 1 ) {         /* Retrieve element level displacements   */         if( ele==1 ) {             d_pe = [       0 m; d_p[1][1] ];         }         if( ele==2 ) {             d_pe = [ d_p[1][1]; d_p[2][1] ];         }         /* Retrieve values from previous iteration */         Q  = Q_saved[ele][1];      /* element-level forces           */         q  = q_saved[ele][1];      /* element-level displacements    */         Dx = bx*Q;                 /* section forces                 */         dx = bx*q;                 /* section displacements          */         K  = tangent[ele][1];      /* tangent stiffness of element   */         kx = tangent[ele][1];      /* tangent stiffness of element   */         fx = 1/kx;                 /* tangent flexibility of element */         rx  = 0 cm;         DUx = 1E+7 N;         /* Increment element deformation */         d_q = QuanCast(L*d_pe);         q   = q + d_q;         /* iterative loop for convergence of element forces */         while( abs(DUx) > 0.00001 N ) {            d_Q = K*d_q;          /* element force increment */            Q   = Q + d_Q;        /* update element force    */            /* determine the section force increments */            /* repeat for all integration points of the element */            d_Dx = bx*d_Q;        /* section force increment           */            d_dx = rx + fx*d_Dx;  /* section deformation increment     */            Dx = Dx + d_Dx;       /* update section force              */            dx = dx + d_dx;       /* update section deformation vector */            /* get new section tangent flexibility f(x) from new d(x)     */            /* and section resisting force DR(x) from material properties */            /* set flags and stresses and strains for each i-th iteration */            if( index[ele][1] == 1 ) {               if( d_dx >  0 cm ) { loading[ele][1] =  1; }               if( d_dx <  0 cm ) { loading[ele][1] = -1; }               if( d_dx == 0 cm ) { loading[ele][1] = pre_load[ele][1]; }               index[ele][1] = index[ele][1] + 1;            }            yielding[ele][1]  = flag1[ele][1];            pre_range[ele][1] = flag2[ele][1];            pre_load[ele][1]  = flag3[ele][1];            sr[ele][1] = sr_saved[ele][1];            er[ele][1] = er_saved[ele][1];            s0[ele][1] = s0_saved[ele][1];            e0[ele][1] = e0_saved[ele][1];            /* material is still in elastic range, and therefore does not have */            /* any plastic residual                                            */            if( yielding[ele][1] == 0 ) then {                if( abs(dx) <= ey[ele][1] ) then {                    kx = Ks[ele][1];                    sx = 0 N;                    ex = 0 cm;                    yielding[ele][1]  = 0;                    pre_range[ele][1] = 0;                } else {                    kx = Kt[ele][1];                    s0[ele][1] = Fy[ele][1]*loading[ele][1];                    e0[ele][1] = ey[ele][1]*loading[ele][1];                    sx = s0[ele][1];                    ex = e0[ele][1];                    yielding[ele][1]  = 1;                    pre_range[ele][1] = 1;                }            } else {   /* plastic residual occurs , yielding[ele][1] = 1 */              if( pre_load[ele][1] != loading[ele][1] ) then {              if( pre_range[ele][1] == 1 ) then {                  sr[ele][1] = sx_saved[ele][1];                  er[ele][1] = ex_saved[ele][1];                  kx = Ks[ele][1];                  sx = sr[ele][1];                  ex = er[ele][1];                  pre_range[ele][1] = 0;              } else {   /* pre_range[ele][1] = 0 */                 if( loading[ele][1]*dx <= loading[ele][1]*er[ele][1] ) then {                     kx = Ks[ele][1];                     sx = sr[ele][1];                     ex = er[ele][1];                     pre_range[ele][1] = 0;                 } else {                     if( loading[ele][1]*ex_saved[ele][1] >= loading[ele][1]*er[ele][1] ) then {                         kx = Ks[ele][1];                         sx = sr[ele][1];                         ex = er[ele][1];                         pre_range[ele][1] = 0;                     } else {                         kx = Kt[ele][1];                         sx = sr[ele][1];                         ex = er[ele][1];                         pre_range[ele][1] = 1;                     }                }              }              } else {   /* pre_load[ele][1] == loading[ele][1] */              if( pre_range[ele][1] == 1 ) then {                  kx = Kt[ele][1];                  sx = s0[ele][1];                  ex = e0[ele][1];                  pre_range[ele][1] = 1;              } else {  /* pre_range[ele][1] = 0 */                  if( loading[ele][1]*ex_saved[ele][1] <= loading[ele][1]*er[ele][1] ) then {                  if( loading[ele][1]*dx <= loading[ele][1]*er[ele][1] ) then {                      kx = Ks[ele][1];                      sx = sr[ele][1];                      ex = er[ele][1];                      pre_range[ele][1] = 0;                  } else { /* line 2 -> line 1 */                      kx = Kt[ele][1];                      sx = sr[ele][1];                      ex = er[ele][1];                      pre_range[ele][1] = 1;                  }                  } else { /* line 4 */                  if( abs(dx-er[ele][1]) <= 2*ey[ele][1] ) then {                      kx = Ks[ele][1];                      sx = sr[ele][1];                      ex = er[ele][1];                      pre_range[ele][1] = 0;                  } else { /* line 4 -> line 1 */                      s0[ele][1] = sr[ele][1] + loading[ele][1]*2*Fy[ele][1];                      e0[ele][1] = er[ele][1] + loading[ele][1]*2*ey[ele][1];                      kx = Kt[ele][1];                      sx = s0[ele][1];                      ex = e0[ele][1];                      pre_range[ele][1] = 1;                  }                  } /* end of line 4 */              }              }            }            /* calculate stress and flexibility */            fx  = 1/kx;             /*                              */            DRx = sx + kx*(dx-ex);  /*                              */            DUx = Dx - DRx;         /* section unbalanced force     */            rx  = fx*DUx;           /* section residual deformation */            /* finish for all integration points of the element      */            /* update the element flexibility and stiffness matrices */            F = bx*fx*bx;            K = 1/F;            /* check for element convergence */            s = bx*rx;   /* element residual deformation */            d_q = -s;         }  /* j, while loop to check element convergence */         /* energy calculations for each element ele */         dWork    = 0.5*(Q_saved[ele][1]+Q)*(q-q_saved[ele][1]);         element_energy[k+1][2*ele-1] = element_energy[k+1][2*ele-1] + dWork;         dElastic = 0.5*Q*Q/Ks[ele][1];         element_energy[k+1][2*ele]   = dElastic;         Q_saved[ele][1] = Q;         q_saved[ele][1] = q;         tangent[ele][1] = K; /* element stiffness Ke =LT*K*L=[K,-K;-K,K]    */         PRe[ele][1] = Q;     /* element resistant force PRe = LT*Q = [-Q;Q] */      }       /* assemble structure resistant force */      PR[1][1] = PRe[1][1] - PRe[2][1];      PR[2][1] = PRe[2][1];      /* assemble new structure stiffness */      BigK[1][1] = tangent[1][1] + tangent[2][1];      BigK[1][2] = -tangent[2][1];      BigK[2][1] = -tangent[2][1];      BigK[2][2] = tangent[2][1];      d_P = P - PR;      err = L2Norm(d_P);   }  /* i-th iteration in Newton-Raphson while loop */   /* Updating history: save flags and reversal points for each load step k */   for( ele=1 ; ele<=2 ; ele=ele+1 ) {      pre_load[ele][1] = loading[ele][1];      flag1[ele][1]    = yielding[ele][1];      flag2[ele][1]    = pre_range[ele][1];      flag3[ele][1]    = pre_load[ele][1];      sr_saved[ele][1] = sr[ele][1];      er_saved[ele][1] = er[ele][1];      s0_saved[ele][1] = s0[ele][1];      e0_saved[ele][1] = e0[ele][1];      sx_saved[ele][1] = Q_saved[ele][1];      ex_saved[ele][1] = q_saved[ele][1];   }   /* store analysis results */   result[k+1][1] = P[2][1];   result[k+1][2] = p[2][1];   result[k+1][3] = p[1][1];   result[k+1][4] = p[2][1] - p[1][1];   /* reassemble the system energy */   dWork = 0.5*(result[k+1][1]+result[k][1])*(result[k+1][2]-result[k][2]);   system_energy[k+1][1] =  system_energy[k][1]   + dWork;   system_energy[k+1][2] = element_energy[k+1][1] + element_energy[k+1][3];   system_energy[k+1][3] = element_energy[k+1][2] + element_energy[k+1][4];   system_energy[k+1][4] =  system_energy[k+1][2] - system_energy[k+1][3];} PrintMatrix(result);PrintMatrix(element_energy);PrintMatrix(system_energy);quit;