Calculation of Stiffness in Structural elements

1):

Aim: To compute the lateral stiffness of the building with give parameters.

Procedure:

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.

Step 1:

The system has the three DOFs. To obtain the first column of the 3×3 stiffness matrix, we impose unit displacement in DOF u1, with u2 = u3 = 0. The forces ki1 required to maintain this deflected shape

F.B.D

The equation of motion for step 1

Step 2:

The elements ki2 in the second column of the stiffness matrix are determined by imposing u2 = 1 with u1 = u3 = 0

F.B.D

The equation of motion for step 2

Step 3:

Similarly, the elements ki3 in the third column of the stiffness matrix can be determined by imposing displacements u3 = 1 with u1 = u2 = 0.

The equation of motion for step 3 is reverse that of step 2

Thus the 3 × 3 stiffness matrix of the structure is known and the equilibrium equations can be written. For a frame with Ib = Ic subjected to lateral force fS, they are.

eq (a)

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

eq (b)

Substituting Eq. (b) into the first of three equations in Eq. (a) gives

eq (c)

Thus the lateral stiffness of the frame is

2):

Aim:

• To determine the number of degrees-of-freedom for dynamic analysis
• To establish the equation of motion
• To calculate their natural frequencies

For the given structures

a):

Procedure:

• Degree of freedom is = 1
• Equation of motion = mu + k = 0

Where stiffness of the frame

Now substitute the k value in equation of motion

Therefore

• Calculation of natural frequencies

Natural frequencies =

Where m = mass and k = stiffness

Now substitute the and k value

Therefore =

b):

Procedure:

• Degree of freedom is = 1
• Equation of motion = mu + k = 0

Where

• Calculation of natural frequencies

Natural frequencies =

Where m = mass and k = stiffness

Now substitute the and k value

Therefore

3):

Aim: To determine the natural period of given system

Procedure:

In this system two beams are connected with each other. One is cantilever and other is simply supported.

W.K.T

For simply supported beam lateral stiffness =

For simply cantilever beam lateral stiffness =

now

For natural frequencies =

By substituting the value of k we get

=

4):

Aim: To determine an expression for the effective stiffness of the following systems

a): Effective stiffness for cantilever is

b): Effective stiffness for simply supported beam

c): Effective stiffness for simply supported beam