Calculation of Stiffness in Structural elements

1.] Compute lateral stiffness of the one story frame with an intermediate realistic stiffness of the beam. The system has 3 DOFs as shown. Assume L = 2h and El_b = El_c

Ans. This structure can be analysed by any of the standard methods including moment distribution.

Given that the degree of freedom is '3'.

`underset(11)(K)`= 2(12)EIc/h^3.

`underset(21)(K)`= 6EIc/h^2.

`underset(31)(K)`= 6EIc/h^1.

`underset(12)(K)`= 6EIc/h^2.

`underset(22)(K)`= 4EIc/h + 4EIb/L

`underset(32)(K)`= 2EIb/L.

The elements Ki1 in the first column, we determine the stiffness matrix by imposing u1=1, with u2=u3=0.

The elements Ki2 in the second 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^2]]``((u1),(u2),(u3))`= `((fs),(0),(0))`

From the 2nd and 3rd equations, the jointrotations can be expressed interms of lateral displacements as follows.

`[[u2],[u3]]`= -`[[6h^2,h^2],[h^2,6h^2]]``[[6h],[6h]]u1`= -`6/7h[[1],[1]]u1`

Now by substituting 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^3 u1`

Finally the lateral stiffness of frame is `96/7 EIc/h^3`

K = `96/7 EIc/h^3`

2.] For the following structures:

• Determine the number of degrees-of-freedom for dynamic analysis
• Establish the equation of motion
• Calculate their natural frequencies

a)

Ans.

The number of degree of freedom for dynamic analysis is 1.

The equation of motion is ,

`mddot(u)+ku=0` -------------equation 1.

Here, we consider stiffness of frame ,

`K=(3EI)/h^3`

Now we have to substitute the value of K in equation 1,

Equation of motion = `mddot(u)+ku=0`

= `mddot(u)+(3EI)/h^3.u=0`

= `mddot(u)+(3EI)/h^3.(1)=0`

Natural frequency Fn = 1/Tn ------------equation 2.

Now `underset(n)(w) =sqrt(K/M)` -----------------equation 3.

Here M = mass

Next substitute the K value in equation 3

`underset(n)(w)=sqrt((3EI)/h^3)/sqrt(M)` -------------------equation 4.

Now , Tn = `(2pi)/underset(n)(w)` ---------------------equation 5.

Substitute the equation 4 in equation 5

Tn = `(2pi)/sqrt((3EI)/h^3)/sqrt(M)` --------------------equation 6.

Now substitute equation 6 in equation 2.

Fn = `sqrt(EI)/(2pisqrt(Mh^3)`

b)

Ans.

The number of degree of freedom for dynamic analysis is 1.

The equation of motion is ,

`mddot(u)+ku=0`

First we need to know the stiffness of the three columns,

Stiffness of 1st column K1 = 12EI/h^3

Stiffness of 2nd column K2 = 12EI/h^3

Stiffness of 3rd column K3 = 3EI/h^3

K = 12EI/h^3 + 12EI/h^3 + 3EI/h^3 = 27EI/h^3

Next we need to substitute K value in the equation of motion ,

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

Natural frequency Fn = 1/Tn ---------------equation 1.

Now `underset(n)(w) =sqrt(K/M)` --------------equation 2.

Now , Tn = `(2pi)/underset(n)(w)` -----------equation 3.

Substitute equation 2 in equation 3

Tn = `(2pi)/sqrt((27EI)/h^3)/sqrt(M)`

Now substitute Tn in equation 1

Fn = `(1)/(2pi)/sqrt((27EI)/h^3)/sqrt(M)`

3.] Consider the propped cantilever shown in the figure below. The beams are made from the same steel section and have lengths as shown on the diagram. Determine the natural period of this system if a large mass, M, is placed at the intersection of the beams at point A. The weight of the beams in comparison with the mass M is very small.

Ans.

The two beams contribute to the overall stiffness of the system in a parallel manner.

This implies that the system stiffness can be found from the direct addition of the stiffnesses of the individual beams.

Lateral stiffness of cantilever beam (Kc) = `3EI/underset(c)(L)^3`

Lateral stiffness of simply supported beam (Kb) = `48EI/underset(b)(L)^3`

Now we need to add the lateral stiffness of both cantilever beam and simply supported beam.

• K = Kc + Kb
• K = `3EI/underset(c)(L)^3`+`48EI/underset(b)(L)^3`
• K = `3EI(1/underset(c)(L)^3+16/underset(b)(L)^3)`

Natural Frequency (fn) = `1/Tunderset(n)`

• Wn = `sqrt (k/M)`

Now substitute the k value in the above equation.

• Wn = `sqrt(3EI(1/underset (c)(L)^3+16/underset(b)(L)^3))/sqrtM`

The natural period of the system can be found from this natural frequency using the expression, below.

• Tn = `2pi/underset(n)(W)`
• Tn = `(2pitimessqrtM)/(sqrt3EI(1/underset(c)(L)^3+16/underset(b)(L)^3)`

4.] Determine an expression for the effective stiffness of the following systems:

Ans.

1. An expression for the effective stiffness of fig.(a) is,
   • K = `(3EI)/L^3`
2. An expression for the effective stiffness of fig.(b) is,
   • K = `(3EI)/L^3` +`(3EI)/L^3`=`((6EI)/L^3)`
3. An expression for the effective stiffness of fig.(c) is,
   • K = `(12EI)/L^3` +`(12EI)/L^3`=`((24EI)/L^3)`