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:

Question 2:

For the following structures:

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

ANS:

(a)From the given we get one degree of freedom because there is only one 'u' is given i.e

Degree of freedom=1

Then

Equation of motion   =      mu+ku=0

Stiffness of the frame (k)=   3EI/h^3

By substituting stiffness of the frame(k) in equation of motion in the place of k in equation we get,

mu+(3EI/h^3)u=0

We know that

Natural frequency (Wn)=sqrt(k/m)

By substituting k value in Wn ,we get

Wn=root(3EI/Mh^3)

ANS:

(b)Similar to previous question there is only one u mentioned here so,therefore

Degree of Freedom =      1

Eqaution of motion =      mu+ku=0

Stiffness of frame(k) = 12EI/h^3  +  12EI/h^3  + 3EI/h^3

                         (k) = 27EI/h^3

By substituting k value in stiffness of frame we get,

    mu+ku=0

    mu+(27EI/h^3)u=0

 And, Natural frequency (Wn)= sqrt(k/m)

by substituting k value in Wn equation we get,

                                 Wn=sqrt(27EI/Mh^3)

Question 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:

This figure includes one simple supported beam and one cantilever beam.

FOR SIMPLY SUPPORTED BEAM THE K VALUE :

K=48EI/Lb^3  -(i)

FOR CANTILEVER BEAM THE K VALUE:

K=3EI/Lc^3  -(ii)

Then k=(i) + (ii)

that is k= 48EI/Lb^3  +  3EI/Lc^3

          k= 3EI[(16/Lb^3)+(1/Lc^3)]

Natural frequency(Wn)= sqrt(k/m)

                          (Wn)=sqrt({3EI((16/Lb^3)+(1/Lc^3))}/M)

Then time period(Tn) = (2pi)/Wn

                           Tn=(2pi) x sqrt[M/{3EI((16/Lb^3)+(1/Lc^3))}]

Question 4

Determine an expression for the effective stiffness of the following systems:

1) Effective stiffness of (a) is  K=   3EI/L^3

2) Effective stiffness of (b) is  K=  3EI/L^3+3EI/L^3=  6EI/L^3