Calculation of Stiffness in Structural elements
note : i have uploaded some images of calculation and equation by writing on paper. beacause i couldnt type on the keyboard.
Question 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-
we have three DOFs, so we are asuuming one by one
a. first case
- unit displacement u1=1
- stiffness coefficiant of column translation (2 similar) = 2 x 12EI/h^3
- stiffness coefficiant of beam rotation (2ends) = EI/h^2
b. second case
- unit rotation u2=1
- stiffness coefficiant of column rotation at right end = 6EI/h^2
- stiffness coefficiant of column + Beam rotation at left end = 4EI/h + 4EI/L where L = 2h as given
= 4EI/h + 4EI/2h
= 6EI/h
c. third case
- unit rotation u3=1
- stiffness coefficiant of column rotation at left end = 6EI/h^2
- stiffness coefficiant of column + Beam rotation at right end = 4EI/h + 4EI/L (where L = 2h as given)
= 4EI/h + 4EI/2h
= 6EI/h
- for a frame Ib=Ic subjected to lateral force fs,they are
fs= k.u
by applying static condition method, we are deducing the 3 unknown expression into 2 unknown equation.
from the second and third equation, the joint rotation can be expressed in terms of lateral displacement as follows :
substituting above equation into the first of three in equation gives
thus the lateral stiffness of frame is
Question 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.
b.
Question 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-
- both the material has same material and section. it meant they have same modulas of elasticity and second moment of inertia. but the length are different.
- when mass M is applied at the point, both element will respond simultaneously. it means they are connected in parellal ( as depicted in the sketch below).
Question 4
Determine an expression for the effective stiffness of the following systems:
a).
- the stiffness coefficiant of the beam element is 3EI/L^3
- the stiffness coefficiant of the spring is K
- the stiffness element are in series. so,
b).
- the stiffness coefficiant of both the column element is 3EI/h^3
- the stiffness coefficiant of the spring is K, but is inclined. so the horizontal force (P) and the deformation (u) has to be resolved into the inclined components to arrive actual coefficiant of the spring
inclined component of the load 'P'= p/cos ⊖
Iinclined component of deformation 'u'= u cos ⊖
stiffness coefficiant of spring
c).
- the stiffness coefficiant of the pinned-pinned beam is k1= 48EI/L^3
- the effective stiffness coefficiant for the system has to be formed adding the relative axial stiffness provided by the inclined strut connected at the right end\.
- 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.
