Calculation of Stiffness in Structural elements

1.

The elements Ki2 in the first column, we determine the stiffness matrix by imposing u1=1 with u2=u3=0.

The elements Ki2 in the secon column,we determine the stiffness matrix by imposing u2=1 with u1=u3=0.

The elements Ki3 in the third column, we determine the stiffness matrix by imposing u3=1 with u1=u2=0.

`EIc-:h^3``[[24,6h,6h],[6h,6h^2,h^2],[6h,h^2,6h]][(u1),(u2),(u3)]=[(fs),(0),(0)]`

From the 2nd and third equations, the joint rotations can be expressed interms of lateral displacements as follows.

`[(u2),(u3)]=-[[6h^2,h^2],[h^2,6h^2]]^-1[[6h,6h]u1= -6-:7h[[1,1]]u1`

Now sustituting the above equation in the first equation,as shown below.

`fs=(24EIc-:h^3-EIc-:h^3 6-:7h(6h 6h)[[1,1]]u1=96-:7 EIc-:h^3u1`

finally the lateral stiffness of frame is `96-:7 EIc-:h^3`

`k=96-:7EIc-:h^3`

2.

3.

4.

a)Expression for the effective stiffness,

->>`k=3EI-:L^3`

b)Expression for effective stiffness,

->>`k=3EI-:L^3+3EI-:L^3=6EI-:L^3`

c)Expression for effective stress,

->>`k=12EI-:L^3+12EI-:L^3=24EI-:L^3`