Calculation of Stiffness in Structural elements

1 solution:-  lateral stiffness of one storey frame-

            -  the structure can be analyzed by any of the standard methods including moment distribution method ,

             here we use the definition of stiffness influence coefiicients to solve 

             -  thus the 3*3 stiffnss matrix of the structure is known and the equilibrium equations can be written for a frame 

               ib=ic subjected to lateral force fs ,and DOF is 3    

              step1 

              U1=1,U2=0,U3=0

              the stiffness matrix is calculated 

              K11=24EIc/h^3

              K12=6EIc/h^2

              K13=6EIc/h^2

              step2 

              U1=0,U2=1,U3=0

              the stiffness matrix is calculated 

              K21=6EIc/h^2

              K22=4EIc/h+4EIb/L

              K23=2EIb/L

              step3

              U1=0,U2=0,U3=1

              the stiffness matrix is calculated 

              K31=6EIc/h^2

              K32=2EIB/L

              K33=4EIc/L+4EIb/L

             hence, K=matrix of [K11 K12 K13

                       K21 K22 K23

                       K31 K32 K33]

             K=EIc/h^3 matrix of [24 6h 6h {u1={fs

             6h 6h^2 h^2 u2 0

             6h h^2 6h^22]u3}0}

             the equation now can be written as 

             24u1+6hu2+6hu3=fs.........1

             6hu1+h^2u2+h^2u3=0.......2

             6hu1+h^2u2+6h^2u3=0.......3

             the equation can be solved by elimination or static condensation method

              after solving equation is 

              u1=96/7 *EIc/h^3  *u1

              stiffness equation K=96/7 * EIC/h^3

2 solution - a) aim- to determine DOF for dynamic analysis 

                 * establish the equation of motion 

                    calculate their natural frequencies 

                    DOF =1

                * equation of motion is   mu+ku=0  ( mu is mass , ku is stiffness of frame )

                   ku =3EI/h^3

                * natural frequency wn=k/m

                  m is mass (M)

                   sub the value of m and k in wn formula 

                  therefore natural frequency of wn = root 3EI/h^3/m

                b) DOF=1

                   * equation of motion =mu+ku=0

                   * stiffness of frame K=12EI/h^3+12EI/h^3+3EI/h^3

                                                  =27EI/h^3

                   *for natural frequency wn=k/m

                     sub the value of k in wn formula

                      therefore, natural frequency of wn =root (27EI/h^3/m)

3 solution - The lateral stiffness we need to add the both stiffness of beam therefore we known,

                 ,the simple supported beam lateral stiffness is =48EI/L^3

                   for cantilever supported beam lateral stiffness is =3EI/L^3

                  add both beam stiffness 

                  K=(48EI/L^3+3EI/L^3)

                  K=51(EI/L^3)

                  the natural frequency wn =root (k/m)

                 wn =root (51(EI/L^3)/m

4solution - Expression for the effective stiffness

                1. the effective stiffness of cantilever beam is K=48EI/L^3

                2. the effective stiffness of simply supported beam is K=3EI/L^3

                3. the effective stiffness of 2 simply supported beam is 

                                                          K=3EI/L^3+3EI/L^3

                                                            =6EI/L^3