Calculation of Stiffness in Structural elements

1^st question: find the stiffness co efficient of the system.

    In the given figure there are three degree of freedoms as fiven 2 are rotational and one is translate motion.

Lets consider the unit displacement.

Given conditions

L=2h

EI of beam=EI of column

For the displacement u1

1)K for the column in translation=2x  for 2 columns

K for the beam =

2) For the displacement u2

1)K for the column in rotation at the right end= 

K for the beam in rotation at left enda nd column =+

Replacing h l with 2h

+=

2) For the displacement u3

1)K for the column in rotation at the left end= 

K for the beam in rotation at right end  and column =+

Fs=KxU

By simplifying the above matrix

K=(calculations are attached below.

2^nd question:

                      find degree of freedom

                       Equation of motion And natural frequency.

Given system have only one free moment in horizontal direction so the degree of freedom will be SINGLE DEGREE OF FREEDOM.

As the column is pinned at the end i.e simply supported then the stifness factor will be

2)motion of equation is Mu+Ku=0

3)Ω==

2b)                      find degree of freedom

                            Equation of motion And natural frequency

1. In the given figure the system have single degree of freedom at the right end column which is simpely supported where as other two are fixed.
2. 2)motion of equation is Mu+Ku=0
3. 3)Ω=

3^rd answer:

Effective stifness of sysetem= ke=kb+kc

Kb         Kc            Ke+

natural frequency formula =  Ω=

time period T=/ Ω