Calculation of Stiffness in Structural elements
1)
AIM:
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
Solution :
if `U_1` = 1
`U_1` = 24 `(EI_c) /h^3`
`'U_2'`= 6 `(EI_C)/h^2`
`U_3` = 6 `(EI_C)/h^2`
1 st equation :
If `U_2` = 1
`U_1 = (6 EI_C)/h^2`
`U_2 =(4EI_b )/(2h) + (4EI_C)/ h`
`U_3 = (2EI_b)/(2h)`
2nd equation :
If `U_3 = 1`
`U_1 = (6 EI_C)/h^2`
`U_3 =(4EI_b )/(2h) + (4EI_C)/ h`
`U_2= (2EI_b)/(2h)`
This is solved using a 3 X 3 matrix
*According to stiffness matrix method :
`((U_2 ), (U_3 ))` = - `[[6h^2 , h^2 ],[h^2,6h^2]]^-1``((6h),(6h))``U_1`
`-(1/(36h^4-h^4)) [[-6h^2,h^2],[h^2,-6h^2]] 6h ((1),(1))U_1`
`-(6h)/(35h^4) [[6h^2,-h^2],[-h^2,6h^2]]((1),(1))U_1`
`-(6h)/(35h^4)((5h^2),(5h^2))U_1`
= `-(6h)/(35h^4) xx 5h^2 ((1),(1))U_1``=((U_2),(U_3))`
`((U_2),(U_3))`= `-6/(7h) ((1),(1))``U_1``
substituting the above equation in the equation given below
`f_s=(24U_1 +6hU_2 + 6hU_3) (EI_C)/h^3`
= `f_s = (24 (EI_C)/h^3 -6/(7h) (EI_C)/h^3 [6h ,6h] ((1),(1)))U_1``
=`((24EI_C)/h^3 -6/(7h) (EI_C)/(h^3) . 12h)U_1`
=`((EI_c)/h^3 (24-6/(7h)).12h) U_1`
=`((EI_C)/h^3 (24)(-6/(7h)).12h)U_1`
= `(EI_C)/h^3 (24-72/7)U_1`
=`F_S = (EI_C)/h^3 96/7 U_1`
2)
AIM:
For the following structures:
- Determine the number of degrees-of-freedom for dynamic analysis
- Establish the equation of motion
- Calculate their natural frequencies
A)
*The degree of freedom is 1
*Equation of motion
= mu + fd+fs= p(t)
= mu + ku = P(t)
= `((12EI)/ h^3 ) U + Mu = P(t)`
* Natural frequency
= `F_n = 1/T_n`
= `T_n = (2 pi )/w_n`
= `F_n = w_n /(2pi)` = `sqrt ( k/M)``. 1/(2pi)`
=`w_n = sqrt (k/m`
= `w_n =sqrt (((12EI)/ (h^3 m))`
B)
*The degree of freedom is 1
*as the beam is rigid ( i.e flexural rigidity M.I = infinity
* K= `sum_(column) (12EI )/h^3`
= `(36EI)/h^3`
*Equation of motion
= mu + fd+fs= p(t)
= mu + ku = P(t)
=mu + `(36EI)/h^3` u = p(t)
* Natural frequency
= `F_n = 1/T_n`
= `T_n = (2 pi )/w_n`
= `F_n = w_n /(2pi)` = `sqrt ( k/M)``. 1/(2pi)`
= `w_n=sqrt(k/m`
= `w_n=sqrt((36EI)/(mh^3))`
3)
* Natural frequency
= `F_n = 1/T_n`
= `T_n = (2 pi )/w_n`
= `F_n = w_n /(2pi)` = `sqrt ( k/M)``. 1/(2pi)`
= `w_n = sqrt(k/m`
`therefore w_n =sqrt(g/u`
`T_n = 2pi sqrt(u/g`
`T_n = 2pi sqrt(u/g`
4)
a)
`f_s/K_e = f_s/k_s + f_s/k_b`
`K_e = (K_s K_b )/(K_s+K_b`
`K_b = (3EL)/L^3`
`K_e = (K(3EI)/L^3 )/(K + (3EI)/L^3`
`K_e = (3EIK )/((3EI) +L^3K)`
b)
*as the beam is rigid i.e EI = infinity
K= `sum_(column) (12EI )/h^3`
k ( column ) = `(24EI)/h^3`
`K_e = K + K ( column )`
`K_e = K+ (24 EI)/h^3`
`K_e = (Kh^3 + 24 EI )/h^3`
C)
`Ke = K_a + K _b`
`K_e =(EA)/(sqrt (h^2+L^2)``+ (48EI)/L^3`