Calculation of Stiffness in Structural elements

Objectives of the challenge:

• Impact of structural configuration in lateral stiffness
• Calculation of natural frequency
• Effective stiffness concept for group of members.

AIM-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:

given data

• L = 2h
• El_b = El_c

_case-1

• unit displacement u1=1
• Stiffness co-efficent column transulation (2 similar) = 2x 12EI/h*3
• Stiffness co-efficent beam rotation (2 ends) = EI/h*2

cse-2:

• unit displacement u2=1
• Stiffness co-efficent column transulation = 6EI/h*2
• Stiffness co-efficent COLUMN + beam rotation left end side = 4EI/h+4EI/L =4EI/h+4EI/2h =6EI/h

cse-3:

• unit displacement u3=1
• Stiffness co-efficent column transulation = 6EI/h*2
• Stiffness co-efficent of column + beam rotation right end = 4EI/h+4EI/L =4EI/h+4EI/2h =6EI/h

we arrange farmet in to mattrix

24EI/h*3 6EI/h*2 6EI/h*2

=( 6EI/h*2 6EI/h*2 EI/h )

6EI/h*2 EI/h 6EI/h

fs = k.u

24 6h 6h u1 fs

=EI/h*3 ( 6h 6h*2 h*2 ) x { u2 } x { 0 }

6h h*2 6h*2 u3 0

by applying static condensation method we are deducing the 3 unknown expression into 2 unknown equation

u2 6h*2 h*2 6h 1

{ u3 } = - [ h*2 6h*2 ] x[ 6h ] u1 =6/7h [1] u1

1

fs = {(24EI/h*3) - (EI/H*3) (6/7h) ( 1 ) }

=96/7 EIc/h*3 u1.

AIM 2:

For the following structures:

• Determine the number of degrees-of-freedom for dynamic analysis
• Establish the equation of motion
• Calculate their natural frequencies

2a.Aim:TO find the below solution

Determine the number of degrees-of-freedom for dynamic analysis

Establish the equation of motion

Calculate their natural frequencies

DOF =1

Equation of the motion

=mu+ku=0

stiffness of the frame=3EI/h^3

3. For natural frequency,wn=√k/m

m=mass(M)

Sub the value of m and k in wn formula

therefore,Natural of frequency of wn=root 3EI/h^3/m2b.

DOF =1

Equation of the motion

=mu+ku=0 (mu=mass&ku=stiffness of the frame)

stiffness of the frame k=12EI/h^3+12EI/h^3+3EI/h^3

=27EI/h^3

3. For natural frequency,wn=√k/m

m=mass(M)

sub the value of k in wn formula

therefore,Natural of frequency of wn=root (27EI/h^3/m)

3.AIM: Consider the propped cantilever . 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.

The lateral stiffness we need to add the both stiffness of beam

therefore we know,

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

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

Add both beam sstiffness,

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

K=51(EI/L^3)

The natural frequency,wn=root(k/m)

wn =root(51(EI/L^3)/m

4. Expression for the effective stiffness :

The effective stiffness of cantilever beam is =K=48EI/L^3

The effective stiffness of simply supported beam is =K=3EI/L^3

The effective stiffness of 2 simply supported beam is =K=3EI/L^3+3EI/L^3

=6EI/L^3