Calculation of Stiffness in Structural elements
Que-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 Elb = Elc.
ANS.:
- Assume that Degree of freedom u1. so, u2=u3=0.
`(EIc)/h^2[[24,6h,6h],[6h,6h^2,h^2],[6h,h^2,6h^2]][[u1],[u2],[u3]]=[[fs],[0],[0]]`
- From the above equation, the joint rotations can be expressed in terms of lateral displacement into strain.
`[[u2],[u3]]=-[[6h^2,h^2],[h^2,6h^2]]^-1 [[6h],[6h]]u1=-6/(7h) [[1],[1]]u1`
- Now by submitting the second equation into the first equation
`fs=((24EIc)/h^3 - (EIc)/h^3 6/(7h) (6h,6h)[[1],[1]])u1=96/7 (EIc)/h^3u1`
- Thus the lateral stiffness of the one storey frame is
`k=96/7(EIc)/h^3`.
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Que-2
For the following structures:
- Determine the number of degrees-of-freedom for dynamic analysis
- Establish the equation of motion
- Calculate their natural frequencies
ANS.:
A)
- Degree of freedom for this structure is 1.
- Equation of motion is as below,
mu + fd + fs = p(t)
mu + ku = p(t)
`((12EI)/h^3)U+Mu=p(t)`
- Natural frequency is as below,
`fn=1/(Tn)`
`Tn=(2pi)/(Wn)`
`Fn=(Wn)/(2pi)=sqrt(k/M). 1/(2pi)`
`Wn=sqrt(k/M)`
`Wn=sqrt((12EI)/(h^3m))`
B)
- Degree of freedom for this structure is 1.
- As the beam is rigid due to as flexural rigidity moment of inertia = infinity
stiffness, `K=sum_(column)(12EI)/h^3`
`=(36EI)/h^3`
- Equation of motion is as below,
mu + fd + fs = p(t)
mu + ku = p(t)
`((36EI)/h^3)U+Mu=p(t)`
- Natural frequency is as below,
`fn=1/(Tn)`
`Tn=(2pi)/(Wn)`
`Fn=(Wn)/(2pi)=sqrt(k/M). 1/(2pi)`
`Wn=sqrt(k/M)`
`Wn=sqrt((36EI)/(h^3m))`
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Que-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.
ANS.:
- Natural frequency is as below,
`fn=1/(Tn)`
`Tn=(2pi)/(Wn)`
`Fn=(Wn)/(2pi)=sqrt(k/M). 1/(2pi)`
`Wn=sqrt(k/M)`
`Wn=sqrt(g/u)`
`Tn=2pisqrt(u/g)`
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Que-4 Determine an expression for the effective stiffness of the following systems:
ANS.:
A)
`(fs)/(Ke)=(fs)/(Ks)+(fs)/(Kb)`
`Ke=(KsKb)/(Ks+Kb)`
Now, `Kb=(3EI)/L^3`
so, `Ke=(K((3EI)/L^3))/(K+(3EI)/L^3)`
`:. Ke=(3EIK)/(3EI+KL^3)`
B)
AS M, EI = `infty`, so Beam is regid,
`K=sum_(column)(12EI)/h^3`
`K(column)=(24EI)/h^3`
`Ke=K+K(column)`
`Ke=K+(24EI)/h^3`
`:. Ke=(Kh^3+24EI)/h^3`
C)
`Ke=Ka+Kb`
`:. Ke=(EA)/sqrt(h^2+L^2)+(48EI)/L^3`