Calculation of Stiffness in Structural elements

AIM 1 : Compute lateral stiffness of te 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 .

Lateral stifness of the one story frame:

• This structure can be analyzed by any of the standard methods, including momentdistribution. Here we use the definition of stiffness influence co efficients to solve the problem.
• Thus the 3 x 3 stiffness matrix of the structure is known and the equilibrium equations can be written. For a frame with lb=ic subjected to lateral force fs,they are,
• The degeer of freedom is 3.

Step-1:

U1=U2=0,U3=0

The stiffness matrix is calculated,

K11=24 El_c /h^3

K12=6 El_c /h^2

K13=6 El_c /h^2

Step-2

U1=0, U2=1, U3=0

The stiffness matrix is calculated

K21= 6 El_c /h^2

K22=4 El_c /h+4Elb/L

K23=2Elb/L

Step-3:

U1=0,U2=0,U3=1

The stiffness matrix is calculated,

K31=6 El_c /h^2

K32= 2Elb/L

K33= 4 El_c /h+4Elb/L

Therefore,

K= matrix of [K11 K12 K13

K21 K22 K23

K31 K32 K33]

K=Elc/h^3 matrix of [24 6h 6h {u1 ={fs

6h 6h^2 h^2 u2 0

6h h^2 6h^2] u3} 0}

The equation written as,

24u1+6hu2+6hu3= fs----------1

6hu1+6h^2u2+h^2u3=0-------2

6hu1+h^2u2+6h^2u3=0--------3

The equation can be solved by two method are Elimination methid or stastic condition method.

Now we solve

[ 6h^2 h^2 6h{U1 6/7h[1 u1 u3 ] h^2 6h^2]-1 (inverse) 6h 1]

Fs= 24Elc/h^3-Elc/h^3*6/7h(6h 6h)1by 1) u1=96/7Elc/h^3u1

AIM 2: To find the below solution

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

1.DOF = 1

2.Equation of the motion= mu+ku =0 (mu=mass& ku= stifness of the frame)

Stiffness of the frame k=12El/h^3+12El/h^3+3El/h^3=27 El/h^3

3.For natural frequency, wn=√K/M m=mass(M)

Sub the valu of k in wn formula therefore, Natural of frequency of wn=root(27El/h^3/m)

AIM 3 : Consider the propped cantilever. The beam 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 beam at point A. The weight of the beams comparision with the mass m is very small.

So, The lateral stiffness we need to add the both stifness of beam therefore we know ,

For simple supported beam lateral stifness is =48El/L^3

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

Add both beam stiffness,

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

K=51(El/L6#)

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

Wn =root (51(El/L^3)/m

4.Expression for the effective stiffness:

1. The effective stiffness of cantilever beam is =K= 48El/L^3
2. The effective stifness of simply supported beam is =K=3El/L^3

The effective stifness of 2 simply supported beam =K= 3El/L^+3El/L^3= 6El/L^3