I am seeking help from someone with a good grasp on beam theory (Euler–Bernoulli, whatever), such as a structural engineer, to work out the stiffness of a beam in a particular situation:
Where I am so far:
A steel-frame building has long-span girders between two columns (c44, left and c79, right).
On one floor, the connection of the girder on c79 fails, and the girder (loaded with tributary beams and a floor slab) falls - or actually rotates around a hinge which is the seat at c44. The falling end then impacts the girder below, exerting a force on that girder:
(Nomenclatura: I will consistently speak of the "falling girder" and the "girder below", and will denote their properties with subscribt-f) and subscript-b in this thread)
This bending of the girder belowabsorbs some of the kinetic energy of the falling girder (and its load, which I'll not refer to explicitly in the remainder for simplicities sake).
At the same time, the falling girder, too, bends down as it experiences an equal upward force at the impact point, and that absorbs kinetic energy, too.
The following physical quantities are involved here:
Now assume we have already worked out
This assumes that the girder below absorbs all the kinetic energy, but he forgets that the falling girder would absorb some energy, too:
KE = SEb + SEf
The two girders are springs in series, with an effective stiffness
Keff = Kb*Kf/(Kb+Kf)
I have done all the algebra to work out the resulting force F at the moment that the two girders have absorbed all the KE:
F = SQRT(2*KE*Keff)
What I want to learn:
How do I determine the stiffness of the falling girder?
It is attached at its left end to a hinge about which it rotates freely, and is loaded with F perpendicular to its long axis.
...Help...?
Where I am so far:
A steel-frame building has long-span girders between two columns (c44, left and c79, right).
On one floor, the connection of the girder on c79 fails, and the girder (loaded with tributary beams and a floor slab) falls - or actually rotates around a hinge which is the seat at c44. The falling end then impacts the girder below, exerting a force on that girder:
(Nomenclatura: I will consistently speak of the "falling girder" and the "girder below", and will denote their properties with subscribt-f) and subscript-b in this thread)
This bending of the girder belowabsorbs some of the kinetic energy of the falling girder (and its load, which I'll not refer to explicitly in the remainder for simplicities sake).
At the same time, the falling girder, too, bends down as it experiences an equal upward force at the impact point, and that absorbs kinetic energy, too.
The following physical quantities are involved here:
- PE = Potential Energy differential of the falling girder between its original position and the height of its center of mass as it impacts the girder below. Unit: kip*in
- KE = Kinetic Energy of the falling girder at the moment of impact. Unit is kip*inches
- K = stiffness of the girders (Kb and Kf); a function of elastic modulus of the steel, it's size and shape, and it's boundary conditions unit is kip/inch
- F = the force of impact on the girder below; equal and opposite to the force of impact on the falling girder: Fb = -Ff. Unit is kip
- D = Deflection of a girder (Db and Df) due to force and elastic bending = F/K. Unit is inches.
- SE = Strain Energy of a girder (SEb and SEf) = 1/2 K D2. Unit: kip*in
Now assume we have already worked out
- KE: is PE minus various losses due to inelastic deformation: Tearing and breaking of the floor slab, plastic deformation of the girder's end, etc = 3473 kip*in
- Kb: The stiffness of the girder below, given that it is a beam of length L supported on both ends with a point load only 10 inches from the support at c79 = 7627 kip/in.
This assumes that the girder below absorbs all the kinetic energy, but he forgets that the falling girder would absorb some energy, too:
KE = SEb + SEf
The two girders are springs in series, with an effective stiffness
Keff = Kb*Kf/(Kb+Kf)
I have done all the algebra to work out the resulting force F at the moment that the two girders have absorbed all the KE:
F = SQRT(2*KE*Keff)
What I want to learn:
How do I determine the stiffness of the falling girder?
It is attached at its left end to a hinge about which it rotates freely, and is loaded with F perpendicular to its long axis.
...Help...?
via International Skeptics Forum http://ift.tt/1P5yziS
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