Calculation of Stiffness in Structural elements
1. ) LATERAL STIFFNESS OF THE ONE STOREY FRAME WITH AN INTERMIDIATE REALISTIC STIFFNESS OF THE BEAM :
ANSWER :
a)
Unit displacement u1= 1.
Stiffness coefficient of column translation (2 similar) = 2 x 12EI/h^3.
Stiffness coefficient of the beam rotation (2 Ends) = EI/h^2
b)
Unit rotation U2=1
Stiffness coefficient of column rotation at right end = 6 EI/h^2.
Stiffness coefficient of column + Beam rotation at left end = 4EI/h + 4EI/L
4EI/h + 4EI/2h = 6EI/h
c)
Unit rotation U3 = 1
Stiffness coefficient of column rotation at the left end = 6 EI/h^2.
Stiffness coefficient of column + Beam rotation at right end = 4EI/h + 4EI/L
4EI/h + 4EI/2h = 6EI/h
24EIc/h^3 6EIc/h^2 6EIc/h^2
6EIc/h^2 6EIc/h EIc/h
6EIc/h^2 EIc/h 6EIc/h
fs = k.u
24 6h 6h u1 fs
EIc/h^3 6h 6h^2 h^2 u2 = 0
6h h^2 6h^2 u3 0
By applying static condensation method, we are deducing the 3 unknown expression into 2 unknown expression.
u2 6h^2 h^2 -1 6h 1
= u1 = - 6/7h u1
u3 6h^2 h^2 6h 1
1
fs = (24EIc/h^3 - EIc/h^3 6/7h (6h 6h) u1 = 96/7 EIc/h^3 u1
1
Stiffness is amount of force required for unit displacement. So interchanging the translantion u1 to the left side, we arrive the stiffness coefficient as follow ,
k = 96/7 EIc/h^3.
2. ) TO DETERMINE THE FOLLOWING STRUCTURES :
- Position of the element can be discribed by 1 coordinate only u(t), because beam with infinite stiffness will induce uniform translation in all columns.
- Two column with both end fixed & one column has pinned-fixed ends. So the stiffness coefficient will be as follow.
k1 = 12EI/h^3
k2 = 12EI/h^3
k3 = 12EI/h^3
- For parallel stiffness elements Ke = (K1+K2+K3) = 27EI/h^3.
- ) The number of degrees-of-freedom for dynamic analysis :
- This system has single degree of freedom structure.
b. ) Established the equation of motion :
..
- Equation of motion shall be M u + ku = 0
c. ) Calculated their natural frequencies :
- Natural frequency formula ω = √k/m
ω = √27EI/Mh^3
3. ) THE NATURAL PERIOD OF THIS SYSTEM IF A LARGE MASS "M" IS PLACED AT THE INTERSECTION OF THE BEAMS AT POINT "A" :
Effective stiffness of the system ke = Kb+Kc
Kb = 48EI/lb^3
Kc = 3EI/lc^3
Ke = 3EI (1/lc^3 + 16/lb^3)
Natural frequency formula ω = √k/m
ω = √3EI (1/lc^3 + 16/lb^3)/M
Time Period T = 2π/ω
4. ) THE EFFECTIVE STIFFNESS :
a ) An expression for the effective stiffness of the following systems :
The stiffness coefficient of the beam element is 3EI/L^3.
The stiffness coefficient of the spring is K.
The stiffness elements are in series. So,
Ke = 3EI/L^3 + K
1/Ke = L^3/3EI + 1/K
1/Ke = KL^3+3EI/3EIK
Ke = 3EIK/KL^3+3EI.
b) An expression for the effective stiffness of the following systems :
- The stiffness coefficient of the both the column elements is 3EI/h^3.
- The stiffness coefficient of the spring is K, but is inclined. So the horizontal force (P) and the deformation (u) has to be resolved into inclined components to arrive actual coefficient of the spring.
Inclined component of the load 'P' = P/cosθ.
Inclined component of the displacement 'u' = u cosθ.
Stiffness coefficient of the spring,
K = P/u cos^2θ
P/u = Ks= K cos^2θ
cosθ = L/√L^2+h^2.
cos^2θ = L/L^2+h^2.
Ks = K L/L^2+h^2.
The effective stiffness of the system, Ke = 6EI/h^3 + kL^2/L^2+h^2.
c) An expression for the effective stiffness of the following systems :
The stiffness coefficient of the pinned-pinned beam is k1 = 48EI/L^3.
The effective stiffness coefficient for the system has to be formed by adding the relative axial stiffness provided by the inclined strut connected at the right hand.
Force acting at the right end is 'mg/2' and deformation caused by the strut at the point of 'm' is half of its right end nodal deformation.
mg/2 = Kea, Vert x 2 x △2
mg/△2 = 4 Kea,Vert.
K2 = 4Kea, Vert.
We assumed vertical axial stiffness in our calculation. But we have an inclined axial stiffness in the system. So we have to resolve the (Kea, Vert) into (Kea).
K2 = 4 Kea h^2/L^2 + h^2.
The effective stiffness shall be arrived as, 1/Ke = 1/K1 + 1/K2.
1/Ke = 1/(48EI/L^3) + 1/(4 Kea h^2/L^2 + h^2).
1/Ke = 1/(48EI/L^3) + 1/(4 EA h^2/(L^2 + h^2)^15).
Ke = (4 EA h^2/(L^2 + h^2)^3/2) x (48EI/L^3)
(4 EA h^2/(L^2 + h^2)^3/2) x (48EI/L^3)