Calculation of Stiffness in Structural elements

    1)

                                               

        The given structure can be analysed by any of the standard methods, including the moment distribution method. Here,          the definition of stiffness influence coefficients is used to solve the problem. The system has the three DOFs as shown.

• To obtain the first column of the 3×3 stiffness matrix, it is necessary to impose unit displacement in DOF u1, with
  u2 = u3 = 0. The forces ki1 required to maintain this deflected shape are shown.
• The elements ki2 in the second column of the stiffness matrix are determined by imposing u2 = 1 with u1 = u3 = 0.
  
  
• Similarly, the elements ki3 in the third column of the stiffness matrix can be determined by imposing displacements
  u3 = 1 with u1 = u2 = 0.

                             

       2:a) 

                                                     

                                                            

  2:b)                                                                                                                                                                                                                                                                                                                                                      

         we know that, no.of degrees of freedoms for dynamic analysis is 1 and the equation of motion is given by

                                                                   `mddotu+ku`                                                                      

          we have to know the stiffness of the three columns.

          stiffness of first column k1=`(12EI)/h^3`

          stiffness of secod column k2=`(12EI)/h^3`

          stiffness of third column k3=`(3EI)/h^3`

          hence,total stiffness=k1+k2+k3=`(27EI)/h^3`

          Now,on sustituting the value of 'k' in the equation of motion we get,

               `mddotu+(27EI)/h^3(u)=0`

       we have natural frequency (fn)=`(1/(Tn))`and wn=`sqrt(k/m)`,where m=mass.

      On substituting the value of 'k' in wn ,we will get,wn=`sqrt(((27EI)/h^3)/m)`

               and we know that Tn=`(2pi)/(wn)`.

              On substituting wn value in Tn equation,we get, Tn=`2pi*sqrt(mh^3)/(sqrt(27EI))`

              On substituting the value of 'Tn' in the equation for natural frequency ,we get  

                                     fn= `sqrt(27EI)/(sqrt(mh^3)`  

              3)

                                      

                                    

              4)

                  a. An equation for the effective stiffness of fig.(a) can be given as,K=`(3EI)/L^3`

                 b.   An equation for the effective stiffness of fig.(b) can be given as,K=`(3EI)/L^3+(3EI)/L^3=(6EI)/L^3`

                 c.  An equation for the effective stiffness of fig.(c) can be given as,K=`(12EI)/L^3+(12EI)/L^3=(24EI)/L^3`