Calculation of Stiffness in Structural elements

Question-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

Solution:

Above structure has 3 DOF's

Consider
case-1 u1 and u2=u3=0

case-2 u2 and u1=u3=0

case-3 u3 and u1=u2=0

k11=(24EIc)/h^3

k12=6EIc/h^2

k21=6EI/h^3

k22=(4EIc/h)+(4EIc/h)

k31=6EIc/h^2

k32=6EIb/6

Substitue Equation b in a

Lateral stifness of frame

k=(96EIc)/(7h^3)

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

Answer

-DOF for dynamic analysis is 1
-Equation motion mu+ku+0
-Natural frequencies- 2*pi*root(24EI)/root(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.

Answer

Wn = 2*pi*root(M)/(3EI((16/Lb^3)+(1/Lc^3))

Question 4

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

Answer

a) Expression for effective stiffness K= 3EI/L^3

b) Expression for effective stiffness K=6EI/L^3

c) Expression for effective stiffness K=6EI/L^3