Calculation of Stiffness in Structural elements
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
(ANSWER):
The system has 3 DOFs
L=2h,
`El underset b`= `El underset c`
From a give diagram:
`u underset 1, u underset 2, u underset 3` are the three DOF
we know that `u underset1"=0,``u underset2"=0,``u underset3"=0,`
`k underset 11"= (2(12El c))/h^3`, `k underset 12"= (6Elc)/h^2`, `k underset 13 "= (6Elc)/h^2`
`kunderset 21" = (6Elc)/h^3`, `k underset 22"= (4Elc + 4Elb)/(h+h)`, `k underset23" = (2Elc)/h`
`k underset 31" = (6Elc)/h^2`, `k underset 32"= (2Elb)/h`, `k underset 33"= (4Elc + 4Elb)/(h+h)`
`(i.e,)`
For the first column:
`u underset 1" = 1;` `u underset 2"=``u underset3"=0,`
For second column:
`u underset2"=1;``u underset 3"=``u underset1"=0.`
For the third column:
`u underset3"=1;``u underset 1"=``u underset2"=0.`
By the method: 3X3 matrix multiplication:-
`(El c)/h^3[(24,6h,6h),(6h,6h^2,h^2),(6h,h^2,6h^2)][(u underset 1,),(u underset 2,),(u underset 3,)]=[(f underset 3,),(0),(0)]`-----equation (1).
Now, for second and third equations:
`[(u underset 2, ), (u underset 3")] = -[(6h^2,h^2),(h^2,6h^2)][(6h),(6h)] u underset1,=(-6)/7[(1),(1)]^u,underset 1`-------equation (2).
Substitute the equation (2) in (1):
i.e,
`f underset s "= [(24Elc)/h^3-(Elc)/h^3 . 6/(7h)(6h.6h)[(1),(1)] ]u underset1`
`= 96/7(Elc)/h^3 u underset 1`
| Answer: `k = 96/7.(Elc)/h^3` |
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.
(ANSWER):
a).
To determine the degree of freedom of dynamic anlysis for a cantilever support beam.
DOF=1
The degree of motion is `m u + k u =0 ---------equation (1)
and also `k = (3El)/h^3` ---------equation (2)
substitute (1) in (2):
i.e., equation of motion `m u + (3EI)/h^3(u) = 0`
`=>``m u + (3EI)/h^3(1) = 0`
Natural frequency, `omega underset n "= sqrt (k/m )`
hence m mass = M.
`omega underset n"= sqrt((3EI)/h^3/M`
b).
DOF =1
Equation of motion = `m ddot u + k u = 0`
i.e., stiffness of first column = `k underset 1"=(12EI)/h^3`
stiffness of second column = `k underset 1"=(12EI)/h^3`
stiffness of third column = `kunderset 1"=(3EI)/h^3`
`therefore``k = (27EI)/h^3`
Equation of motion = `m ddot u+ (27EI)/h^3(u)=0`
Natural frequency, `omega underset n "= sqrt (k/m )`
hence m mass = M.
`omega underset n "= sqrt ((27EI)/h^3/M`
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.
(ANSWER):
In this above diagram, there is a two beams are connected eachother
one is cantilever beam
another one is simply supported beam.
To calculate the lateral stiffness of cantilever beam :
`k underset c"= (3EI)/(l^3 underset c`
To calculate the lateral stiffness of simply supported beam:
`k underset b"= (48EI)/(l^3 underset b`
`therefore` Total stiffness is
`k= kunderset b"+ k underset c`
`k= (3EI)/(l^3 c )+(48EI)/(l^3b`
`k= 51(EI)/L^3`
Natural frequency, `omega underset n "=sqrt(k/m`
substitute the value of k in natural frequency formula
`omega underset n "= sqrt ((51EI)/L^3/M`
Question 4
Determine an expression for the effective stiffness of the following systems:
(ANSWER):
a). The expression for effective stiffness of cantilever beam is
`k = (48EI)/(l^3`
b). The expression for effective stiffness of simply supported beam is
`k= (3EI)/L^3`
c). The expression for effective stiffness of simply supported beam is
`k= (3EI)/L^3 +(3EI)/L^3`
`therefore k= (6EI)/L^3`.