
>  # Torsionsmoment mit der Torsionsfunktion (7.63c)
> Phi[3]:=(1-xi^2)*(1-eta^2)*(1.2201+1.083*xi^2*eta^2-0.2457*xi^4*eta^4)
> ;
> 

                   2          2
  Phi[3] := (1 - xi ) (1 - eta )

                          2    2            4    4
        (1.2201 + 1.083 xi  eta  - 0.2457 xi  eta )

> M:=2*G*D*Int(Int(Phi,xi=-1..1),eta=-1..1);

                                1    1
                               /    /
                              |    |
                  M := 2 G D  |    |   Phi dxi deta
                              |    |
                             /    /
                               -1   -1

> M[3]:=2*Int(Int(Phi_[3],xi=-1..1),eta=-1..1)=      
> evalf(2*int(int(Phi[3],xi=-1..1),eta=-1..1),15);

                    1    1
                   /    /
                  |    |
       M[3] := 2  |    |   Phi_[3] dxi deta = 4.48574171428572
                  |    |
                 /    /
                   -1   -1

# Werte des Integranden in de GAUSS-Punkten:
> a:=sqrt((3+2*sqrt(6/5))/7);   b:=sqrt((3-2*sqrt(6/5))/7);

                                        1/2 1/2
                            (525 + 70 30   )
                       a := -------------------
                                    35


                                        1/2 1/2
                            (525 - 70 30   )
                       b := -------------------
                                    35

> Digits:=15:
> F[1]:=subs({xi=-a,eta=-a},Phi[3]);

          /          1/2\2
          |      2 30   |
  F[1] := |4/7 - -------|  (1.2201
          \        35   /

                               -6             1/2 2
         + 0.721699291961683 10   (525 + 70 30   )

                               -12             1/2 4
         - 0.109109056843419 10    (525 + 70 30   ) )

> F[1]:=simplify(%);

                      F[1] := 0.116310640889380

> F[2]:=subs({xi=a,eta=-a},Phi[3]);

          /          1/2\2
          |      2 30   |
  F[2] := |4/7 - -------|  (1.2201
          \        35   /

                               -6             1/2 2
         + 0.721699291961683 10   (525 + 70 30   )

                               -12             1/2 4
         - 0.109109056843419 10    (525 + 70 30   ) )

> F[2]:=simplify(%);

                      F[2] := 0.116310640889380

> F[3]:=subs({xi=a,eta=a},Phi[3]);

          /          1/2\2
          |      2 30   |
  F[3] := |4/7 - -------|  (1.2201
          \        35   /

                               -6             1/2 2
         + 0.721699291961683 10   (525 + 70 30   )

                               -12             1/2 4
         - 0.109109056843419 10    (525 + 70 30   ) )

> F[3]:=simplify(%);

                      F[3] := 0.116310640889380

> F[4]:=subs({xi=-a,eta=a},Phi[3]);

          /          1/2\2
          |      2 30   |
  F[4] := |4/7 - -------|  (1.2201
          \        35   /

                               -6             1/2 2
         + 0.721699291961683 10   (525 + 70 30   )

                               -12             1/2 4
         - 0.109109056843419 10    (525 + 70 30   ) )

> F[4]:=simplify(%);

                      F[4] := 0.116310640889380

> F[5]:=subs({xi=-b,eta=-b},Phi[3]);

          /          1/2\2
          |      2 30   |
  F[5] := |4/7 + -------|  (1.2201
          \        35   /

                               -6             1/2 2
         + 0.721699291961683 10   (525 - 70 30   )

                               -12             1/2 4
         - 0.109109056843419 10    (525 - 70 30   ) )

> F[5]:=simplify(%);

                      F[5] := 0.965628708762553

> F[6]:=subs({xi=b,eta=-b},Phi[3]);

          /          1/2\2
          |      2 30   |
  F[6] := |4/7 + -------|  (1.2201
          \        35   /

                               -6             1/2 2
         + 0.721699291961683 10   (525 - 70 30   )

                               -12             1/2 4
         - 0.109109056843419 10    (525 - 70 30   ) )

> F[6]:=simplify(%);

                      F[6] := 0.965628708762553

> F[7]:=subs({xi=b,eta=b},Phi[3]);

          /          1/2\2
          |      2 30   |
  F[7] := |4/7 + -------|  (1.2201
          \        35   /

                               -6             1/2 2
         + 0.721699291961683 10   (525 - 70 30   )

                               -12             1/2 4
         - 0.109109056843419 10    (525 - 70 30   ) )

> F[7]:=simplify(%);

                      F[7] := 0.965628708762553

> F[8]:=subs({xi=-b,eta=b},Phi[3]);

          /          1/2\2
          |      2 30   |
  F[8] := |4/7 + -------|  (1.2201
          \        35   /

                               -6             1/2 2
         + 0.721699291961683 10   (525 - 70 30   )

                               -12             1/2 4
         - 0.109109056843419 10    (525 - 70 30   ) )

> F[8]:=simplify(%);

                      F[8] := 0.965628708762553

> F[9]:=subs({xi=-a,eta=-b},Phi[3]);

          /          1/2\ /          1/2\
          |      2 30   | |      2 30   |
  F[9] := |4/7 - -------| |4/7 + -------| (1.2201
          \        35   / \        35   /

                               -6             1/2              1/2
         + 0.721699291961683 10   (525 + 70 30   ) (525 - 70 30   )

         -

                            -12             1/2 2             1/2 2
        0.109109056843419 10    (525 + 70 30   )  (525 - 70 30   ) )

> F[9]:=simplify(%);

                      F[9] := 0.299685355102041

> F[10]:=subs({xi=a,eta=-b},Phi[3]);

           /          1/2\ /          1/2\
           |      2 30   | |      2 30   |
  F[10] := |4/7 - -------| |4/7 + -------| (1.2201
           \        35   / \        35   /

                               -6             1/2              1/2
         + 0.721699291961683 10   (525 + 70 30   ) (525 - 70 30   )

         -

                            -12             1/2 2             1/2 2
        0.109109056843419 10    (525 + 70 30   )  (525 - 70 30   ) )

> F[10]:=simplify(%);

                      F[10] := 0.299685355102041

> F[11]:=subs({xi=a,eta=b},Phi[3]);

           /          1/2\ /          1/2\
           |      2 30   | |      2 30   |
  F[11] := |4/7 - -------| |4/7 + -------| (1.2201
           \        35   / \        35   /

                               -6             1/2              1/2
         + 0.721699291961683 10   (525 + 70 30   ) (525 - 70 30   )

         -

                            -12             1/2 2             1/2 2
        0.109109056843419 10    (525 + 70 30   )  (525 - 70 30   ) )

> F[11]:=simplify(%);

                      F[11] := 0.299685355102041

> F[12]:=subs({xi=-a,eta=b},Phi[3]);

           /          1/2\ /          1/2\
           |      2 30   | |      2 30   |
  F[12] := |4/7 - -------| |4/7 + -------| (1.2201
           \        35   / \        35   /

                               -6             1/2              1/2
         + 0.721699291961683 10   (525 + 70 30   ) (525 - 70 30   )

         -

                            -12             1/2 2             1/2 2
        0.109109056843419 10    (525 + 70 30   )  (525 - 70 30   ) )

> F[12]:=simplify(%);

                      F[12] := 0.299685355102041

> F[13]:=subs({xi=-b,eta=-a},Phi[3]);

           /          1/2\ /          1/2\
           |      2 30   | |      2 30   |
  F[13] := |4/7 - -------| |4/7 + -------| (1.2201
           \        35   / \        35   /

                               -6             1/2              1/2
         + 0.721699291961683 10   (525 + 70 30   ) (525 - 70 30   )

         -

                            -12             1/2 2             1/2 2
        0.109109056843419 10    (525 + 70 30   )  (525 - 70 30   ) )

> F[13]:=simplify(%);

                      F[13] := 0.299685355102041

> F[14]:=subs({xi=b,eta=-a},Phi[3]);

           /          1/2\ /          1/2\
           |      2 30   | |      2 30   |
  F[14] := |4/7 - -------| |4/7 + -------| (1.2201
           \        35   / \        35   /

                               -6             1/2              1/2
         + 0.721699291961683 10   (525 + 70 30   ) (525 - 70 30   )

         -

                            -12             1/2 2             1/2 2
        0.109109056843419 10    (525 + 70 30   )  (525 - 70 30   ) )

> F[14]:=simplify(%);

                      F[14] := 0.299685355102041

> F[15]:=subs({xi=b,eta=a},Phi[3]);

           /          1/2\ /          1/2\
           |      2 30   | |      2 30   |
  F[15] := |4/7 - -------| |4/7 + -------| (1.2201
           \        35   / \        35   /

                               -6             1/2              1/2
         + 0.721699291961683 10   (525 + 70 30   ) (525 - 70 30   )

         -

                            -12             1/2 2             1/2 2
        0.109109056843419 10    (525 + 70 30   )  (525 - 70 30   ) )

> F[15]:=simplify(%);

                      F[15] := 0.299685355102041

> F[16]:=subs({xi=-b,eta=a},Phi[3]);

           /          1/2\ /          1/2\
           |      2 30   | |      2 30   |
  F[16] := |4/7 - -------| |4/7 + -------| (1.2201
           \        35   / \        35   /

                               -6             1/2              1/2
         + 0.721699291961683 10   (525 + 70 30   ) (525 - 70 30   )

         -

                            -12             1/2 2             1/2 2
        0.109109056843419 10    (525 + 70 30   )  (525 - 70 30   ) )

> F[16]:=simplify(%);

                      F[16] := 0.299685355102041

# Wichtungsfaktoren:
> w[1..4]:=59/216-(1/6)*sqrt(5/6);

                                            1/2
                                    59    30
                       w[1 .. 4] := --- - -----
                                    216    36

> w[5..8]:=59/216+(1/6)*sqrt(5/6);

                                            1/2
                                    59    30
                       w[5 .. 8] := --- + -----
                                    216    36

> w[9..16]:=49/216;

                                        49
                          w[9 .. 16] := ---
                                        216

> GAUSS_Quadratur:=2*w[1..4]*sum(F[i],i=1..4)+
> 2*w[5..8]*sum(F[j],j=5..8)+2*w[9..16]*sum(F[k],k=9..16);

                                                              1/2
    GAUSS_Quadratur := 3.45198468257273 + 0.188737348416261 30

> GAUSS_Quadratur:=evalf(%,15);

                 GAUSS_Quadratur := 4.48574171428571

# Dieser Wert stimmt mit dem Integralwert fr das Torsionsmoment
# berein.
> 
> 
> 
> 
> 
> 
> 
