Free Field motion vs. motion under structure

[WARose]

Just a note before I start: The issue I discuss here has to do with ground motion from artificial sources of vibration rather than than (earthquake) seismic motions.....but these situations are analogous (to me) since we are talking (in both situations) ground motions from (predominantly) Rayleigh waves.

So here we go....I routinely handle vibration problems....sometimes the effects on buildings from an adjacent source of vibration (i.e. machinery, traffic, etc.). Sometimes I estimate the motion of some structures by saying the (attenuated) ground displacements just adjacent to the structure will be how the foundation will move. (In other words, it's like a support motion problem from structural dynamics.)

I have to admit a level of discomfort with this model.....in some cases, it seems excessively conservative. Forgetting about wave reflection and other losses......saying the free field displacements are going to be what occurs under a heavy structure doesn't seem right to me. I've thought about thinking about it more as a force transfer situation (than a assumed displacement problem)......but I'm not sure about that.

I've also thought about refining the Rayleigh wave guide to somehow reflect this.....but every paper I've seen on this.....the math is just about impenetrable.

Does anyone have any (practical) ideas?

 

[swearingen]

Possibly think about treating the soil between the forcing function generator and the structure as a spring with some damping in it.

I don't think it would take too long to work out some damping numbers that will help from being excessively conservative. If I'm reading it right, your current model essentially says there is an infinitely stiff object between what's causing the vibration and your structure.


-5^2 = -25 ;-)

http://www.eng-tips.com/supportus.cfm

 

[WARose]

Possibly think about treating the soil between the forcing function generator and the structure as a spring with some damping in it.

I don't think it would take too long to work out some damping numbers that will help from being excessively conservative. If I'm reading it right, your current model essentially says there is an infinitely stiff object between what's causing the vibration and your structure.

That is kind of what I was talking about before with the "refining the Rayleigh wave guide" comment. But I'm not sure how exactly to do that. (I'm certainly no expert on soil-structure interaction.)

Thinking about it in terms of conservation of energy.....I could say it is exerting X amount of force on a structure (based on the unbalanced force at the source) and allow the base of the structure to resist it depending on the dynamic spring constant. Again, it's a logical approach.....but I don't have a comfort level with it because you are (essentially) changing what the Rayleigh wave is doing without a mathematical justification.

 

[WARose]

Doing some research.....I found some interesting approaches.

One reference[sup][blue]1[/blue] [/sup] calculated the time varying force on a structural base as a integral of the (spring constant weighted) sinusoidal amplitudes. (Saying: The right-hand side of equation (1) describes a generalized force now requiring the integration of displacements given by (3), weighted by the distributed spring-constant factor k(x). Thus it may be observed that longitudinal variation of earthquake displacement can cause varying amounts of input to the structure, depending both on distributed local soil properties and the x-phasing of the wave displacement.)

The equation(s) they come out with in this regard are:

F(t)=∫[sup]L[/sup][sub]0[/sub] k [ΣA[sub]n[/sub] cos (ω[sub]n[/sub]t + φ[sub]n[/sub] -ω[sub]n[/sub]x/C[sub]p[/sub]) dx (4)

at the nth component of the above equation that reduces to Fn= kLζ[sub]n[/sub](t) (5)

where:

Fn= Force transfer to foundation
k= spring constant
ζ=displacement at point x
L=Length of foundation

Equation 5 may seem intuitive......but I was uncomfortable with saying the wave would be effected without back up.

At another point in the article it says this:

In the case of structures such as nuclear power plants the length or width of the structure may be comparable in size to certain wavelengths of the seismic waves assumed to traverse the site. For such wave- lengths there exist horizontal ground motions of different surface particles occurring simultaneously in opposite directions in the vicinity of the structure, and these actions may produce a net ‘self-cancelling’ effect when integrated over the whole structural base area.

Another paper I found talked about a averaging effect.[sup][[blue]2[/blue]][/sup] It said:

Consider a large relatively rigid mat foundation located on this soil. This foundation will move less than the ‘free-field’ value if the size of the foundation is comparable to or larger than significant wavelengths of the incoming wave. In fact it will move in some sort of average value of the differential wave motion over its area.

A lot of these papers are dealing with foundation sizes approximately equal to the wave length. So they are considering the so-called "Tau effect"[sup][[blue]3[/blue]][/sup]. But I feel like this is also applicable to a attenuating wave under a decent sized foundation (with a wavelength obviously much shorter than the foundation size) from looking at the equations given in [sup][[blue]1[/blue]][/sup].

So in conclusion, I don't see much relief in terms of: the support won't move less than the free field value.....but I do see a case where you can average the effect for short wave lengths (i.e. attenuating waves) or perhaps use the "Tau effect" for wave lengths about equal to the foundation size. For very small foundations.....you appear to be along for the ride.

Cited References:

[[blue]1[/blue]] R. H. Scanlan, ‘Seismic wave effects of soil structure interaction’, Earthqu. Eng Struct. Dyn. 4, 379-388 (1976).

[[blue]2[/blue]] G. N. BYCROFT, 'SOIL-FOUNDATION INTERACTION AND DIFFERENTIAL GROUND MOTIONS', EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 8, 397-404, (1980)

[[blue]3[/blue]] R. W. Clough & J. Penzien, 'Dynamics of Structures', 3rd Ed. , 2003, p.670-673.