ETABS_Fundamentals of Structural Dynamics_ Week 2

1a) Lateral stiffness of the one-story frame:

This structure can be analyzed by any of the standard methods, including moment distribution. Here we use the definition of stiffness influence coefficients to solve the problem. The system has three Degree of freedom. To obtain the first column of the 3×3 stiffness matrix, we impose unit displacement in DOF u1, with u2 = u3 = 0.

From the second and third equations, the joint rotations can be expressed in terms of lateral
displacement as follows:

Substituting Eq. (2) into the first of three equations in Eq. (1) gives

Therefore the lateral equation of frame is

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:

2a)

Degree of freedom = 1

Equation of motion = mu + Ku =0

Stiffness of this frame, k = 3EI/h^3

Now, Substitute the K value in the equation of motion.

Therefore, Equation of Motion = mu + 3EI/h^3 = 0

For Natural frequency, wn = `root`k/m`

In this frame m = Mass (M)

Substitute the value of m and K in wn formula

Therefore, Natural frequency of wn =`root`3EI/h^3/M `

2b)

Degree of freedom = 1

Equation of motion = mu + Ku =0

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

= 27EI/h^3

Substitute the value of k in natural frequency formula

`herefore, Natural frequency of wn =`root` (K/M`

= `root` (27EI/h^3/M)

3a) Natural Period of the System:

In this question two beams connected with one another. One is Cantilever beam and other one is Simply supported beam.

so,

For lateral stiffness we need to add both beam stiffness

We know that,

For Simply supported beam lateral stiffness is = 48 EI/L^3

For Simply cantilever beam lateral stiffness is = 3 EI/L^3

Add both beam lateral stiffness

K = (48EI/L^3)+ (3EI/L^3)

K = 51 ( EI/L^3)

We know that,

Natural frequency, wn = Root (K/M)

Substitute the value of k in natural frequency formula

Therefore, wn = Root ( 51/EI/L^3)/M

4) Effective Stiffness:

a)

An effective stiffness for the cantilever beam is

K = 48EI/L^3

b)

An effective stiffness for the Simply supported beam is

K = 3EI/L^3

c)

An effective stiffness for the Simply supported beam is

K = (3EI/L^3) + (3EI/L^3)

= 6 (EI/L^3)