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

Answer:

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:

(a)

DOF= 1

Equation of motion is,

Mu+ku=0

Substitute k=3EI/H^3

• mu+(3EI/H^3)u=0

Let natural frequency be Wn(omega n),

Wn= root(k/m)

• Wn=root((3EI/H^3)/m)

Answer:

(b)

DOF=1

Equation of motion is,

Mu+ku=0

Substitute k=12EI/H^3+12EI/H^3=24EI/H^3

• mu+(24EI/H^3)u=0

Let natural frequency be Wn(omega n),

Wn=root(k/m)

Wn=root((24EI/H^3)/m)

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:

K for simply supported beam is 6EI/H^3

Let natural frequency be Wn(omega n). Then,

Wn=root(k/m)=root((6EI/H^3)/m)

Time period, Tn =2π/Wn

Tn=2π/root(6EI/H^3)

Question 4

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

Answer:

a) Effective stiffness is 3EI/H^3.

b)Effective stiffness is 3EI/H^3 + 3EI/H^3 = 6EI/H^3

c)Effective stiffness is 3EI/H^3 + 3EI/H^3 = 6EI/H^3