Calculation of Stiffness in Structural elements

1)

ans:

unit displacement u1=1

stiffness coefficient of two columns is 2*`(12EI)/(H^2)`

stiffness coefficient of beam `(EI)/H^2`

U2=1 (unit rotation)

stiffness coefficient of column =`(6EI)/H^2`

Stiffness coefficent of u2 rotation column and beam =`(4EI)/H+(4EI)/L`

where L=2H

``(4EI)/H+(4EI)/2H``

``(6EI)/H`

U3=1 (unit rotation)

stiffness coefficient of column =`(6EI)/H^2`

Stiffness coefficent of u3 rotation column and beam =`(4EI)/H+(4EI)/L`

where L=2H

`[[24(EI)/H^3,6(EI)/H^2,6(EI)/H^2],[6(EI)/H^2,6(EI)/H,(EI)/H],[6(EI)/H^2,(EI)/H,6(EI)/H]]`

Fs=k*u

k=`96/7*(EI)/H^3`

2)

ans:a)

• it is single degree of freedom
• column one end is fixed ,the stiffness is k=`(3(EI)/H^3)`
• the equation of a motion is mu+ku=0
• the natural frequency id w=`sqrt(k/m)`
• w=`sqrt(3(EI)/(mh^3)`

ans:b)

• it is single degree of freedom
• column are fixed at one end the stiffness is k1=`(12(EI)/H^3)`
• k2=`(12(EI)/H^3)`
• k3=`(3(EI)/H^3)`
• the equation of a motion is mu+ku=0
• columns are in parallel ke=k1+k2+k3
• Ke=`(12(EI)/H^3)`+`(12(EI)/H^3)`+`(3(EI)/H^3)`
• ke=`(27(EI)/H^3)`
• the natural frequency id w=`sqrt(k/m)`

• w=`sqrt((27(EI)/(mh^3))`

w=3/h`sqrt((3(EI)/(mh))`

3)

ans:

Lc member one end is fixed and other end is free the stiffness is K=`(3(EI)/H^3)`

where h=LC

``=`(3(EI)/(Lc)^3)`

Lb member assume it as one end is hinged and other end is hinged the stiffness is K=`(48(EI)/H^3)`

where h=Lb

k``=`(48(EI)/(Lb)^3)`

ke=`(3(EI)/(Lc)^3)`+`(48(EI)/(Lb)^3)`

Ke=3EI`(1/(Lc)^3+(16/(Lc)^3))`

`he natural time period T= sqrt(2pi/w)`

w`=sqrt(k/m)`

w`=sqrt(3EI(1/(Lc)^3+(16/(Lb)^3))/m)`

4)

ans:a)

one end is fixed and other end is free k1=`(3(EI)/L^3)`

for spring stiffness is 'K'=K2

both are in series

`1/(Ke)=(1/(k1)+1/(k2))`

`1/(Ke)=(L^3/(3EI)+1/(k))`

`1/(Ke)=(((L^3K)+(3EI))/(3KEI))`

`(Ke)=((3KEI)/((L^3K)+(3EI)))`

ans:b)

for two columns stiffness

K=`(3(EI)/h^3)+(3(EI)/h^3)`

`=(6(EI)/h^3)`

the stiffness for inclined spring for load acting by u deflection

k`=(p/(ucos^2theta))`

`Ke= (6(EI)/h^3)+(p/(ucos^2theta))`

ans:c)

K=`(48(EI)/h^3)`

`=(48(EI)/(h^3cos(theta)))`

the u deformation is by applying load

K=`(48(EIP)/(uh^3))`

Ke=`(48(EI)/(h^3cos(theta)))+(48(EIP)/(uh^3))`

`ke=(48(EI)/(h^3))*(((P)/(u))+cos(theta)`