exs_beam2¶

Purpose:

Analysis of a plane frame.

Description:

A frame consists of one horizontal and two vertical beams according to the figure.

Material and geometric properties:

\(E\)	\(=\)	\(200\) GPa
\(A_1\)	\(=\)	\(2.0 \times 10^{-3}\) m²
\(I_1\)	\(=\)	\(1.6 \times 10^{-5}\) m⁴
\(A_2\)	\(=\)	\(6.0 \times 10^{-3}\) m²
\(I_2\)	\(=\)	\(5.4 \times 10^{-5}\) m⁴
\(P\)	\(=\)	\(2.0\) kN
\(q_0\)	\(=\)	\(10.0\) kN/m

The corresponding finite element model consists of three beam elements and twelve degrees of freedom.

Example:

A topology matrix Edof, a global stiffness matrix K and load vector f are defined. The element matrices Ke and fe are computed by the function beam2e. These matrices are then assembled in the global matrices using the function assem:

>>  Edof=[1  4  5  6  1  2  3;
          2  7  8  9 10 11 12;
          3  4  5  6  7  8  9];

>> K=zeros(12);  f=zeros(12,1);  f(4)=2e+3;

>> E=200e9;
>> A1=2e-3;    A2=6e-3;
>> I1=1.6e-5;  I2=5.4e-5;
>> ep1=[E A1 I1];  ep3=[E A2 I2];

>> ex1=[0 0];  ex2=[6 6];  ex3=[0 6];
>> ey1=[0 4];  ey2=[0 4];  ey3=[4 4];
>> eq1=[0 0];  eq2=[0 0];  eq3=[0 -10e+3];

>> Ke1=beam2e(ex1,ey1,ep1);
>> Ke2=beam2e(ex2,ey2,ep1);
>> [Ke3,fe3]=beam2e(ex3,ey3,ep3,eq3);

>> K=assem(Edof(1,:),K,Ke1);
>> K=assem(Edof(2,:),K,Ke2);
>> [K,f]=assem(Edof(3,:),K,Ke3,f,fe3);

The system of equations are solved considering the boundary conditions in bc:

>> bc=[1 0; 2 0; 3 0; 10 0; 11 0];
>> [a,r]=solveq(K,f,bc)

a =                         r =

         0                    1.0e+004 *
         0
         0                      0.1927
    0.0075                      2.8741
   -0.0003                      0.0445
   -0.0054                           0
    0.0075                      0.0000
   -0.0003                     -0.0000
    0.0047                     -0.0000
         0                           0
         0                      0.0000
   -0.0052                     -0.3927
                                3.1259
                                     0

The element displacements are obtained from the function extract, and the function beam2s computes the section forces and the displacements along the element:

>> Ed=extract_ed(Edof,a);
>> [es1,edi1]=beam2s(ex1,ey1,ep1,Ed(1,:),eq1,21)

es1 =                           edi1 =

  1.0e+004 *                        0.0003    0.0075
                                    0.0003    0.0065
   -2.8741    0.1927    0.8152        .         .
   -2.8741    0.1927    0.7767      0.0000    0.0000
      .         .         .
   -2.8741    0.1927    0.0445

>> [es2,edi2]=beam2s(ex2,ey2,ep1,Ed(2,:),eq2,21)

es2 =                           edi2 =

  1.0e+004 *                        0.0003    0.0075
                                    0.0003    0.0084
   -3.1259   -0.3927   -1.5707        .         .
   -3.1259   -0.3927   -1.4922      0.0000    0.0000
      .         .         .
   -3.1259   -0.3927   -0.0000

>> [es3,edi3]=beam2s(ex3,ey3,ep3,Ed(3,:),eq3,21)

es3 =                           edi3 =

  1.0e+004 *                        0.0075   -0.0003
                                    0.0075   -0.0019
   -0.3927   -2.8741   -0.8152        .         .
   -0.3927   -2.5741    0.0020      0.0075   -0.0003
      .         .         .
   -0.3927    3.1259   -1.5707

A displacement diagram is displayed using the function dispbeam2 and section force diagrams using the function secforce2:

>> figure(1)
>> plotpar=[2 1 0];
>> eldraw2(ex1,ey1,plotpar);
>> eldraw2(ex2,ey2,plotpar);
>> eldraw2(ex3,ey3,plotpar);
>> sfac=scalfact2(ex3,ey3,Ed(3,:),0.1);
>> plotpar=[1 2 1];
>> dispbeam2(ex1,ey1,edi1,plotpar,sfac);
>> dispbeam2(ex2,ey2,edi2,plotpar,sfac);
>> dispbeam2(ex3,ey3,edi3,plotpar,sfac);
>> axis([-1.5 7.5 -0.5 5.5]);
>> scalgraph2(sfac,[1e-2 0.5 0]);
>> title('Displacements')

>> figure(2)
>> plotpar=[2 1];
>> sfac=scalfact2(ex1,ey1,es1(:,1),0.2);
>> secforce2(ex1,ey1,es1(:,1),plotpar,sfac);
>> secforce2(ex2,ey2,es2(:,1),plotpar,sfac);
>> secforce2(ex3,ey3,es3(:,1),plotpar,sfac);
>> axis([-1.5 7.5 -0.5 5.5]);
>> scalgraph2(sfac,[3e4 1.5 0]);
>> title('Normal force')

>> figure(3)
>> plotpar=[2 1];
>> sfac=scalfact2(ex3,ey3,es3(:,2),0.2);
>> secforce2(ex1,ey1,es1(:,2),plotpar,sfac);
>> secforce2(ex2,ey2,es2(:,2),plotpar,sfac);
>> secforce2(ex3,ey3,es3(:,2),plotpar,sfac);
>> axis([-1.5 7.5 -0.5 5.5]);
>> scalgraph2(sfac,[3e4 0.5 0]);
>> title('Shear force')

>> figure(4)
>> plotpar=[2 1];
>> sfac=scalfact2(ex3,ey3,es3(:,3),0.2);
>> secforce2(ex1,ey1,es1(:,3),plotpar,sfac);
>> secforce2(ex2,ey2,es2(:,3),plotpar,sfac);
>> secforce2(ex3,ey3,es3(:,3),plotpar,sfac);
>> axis([-1.5 7.5 -0.5 5.5]);
>> scalgraph2(sfac,[3e4 0.5 0]);
>> title('Moment')