# Foundations Subjected to Vibration Loads by ProQuest

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[...] the corrected subgrade modulus for a B x B foundation K^sub BxB^, where B is the smaller of the actual foundation plan dimensions in ft (m), is determined using one of the following two equations.2 For sandy soils: ... ... Other parameters also affect the natural frequency of the system, such as the slab dimensions and the equipment weight. [...] two graphs have been developed to approximate the natural frequency of the actual system.

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```									Foundations Subjected
to Vibration Loads
A practical design tool for sizing equipment mats

By leonel I. AlmAnzAr mIchelI And KhAlId motIwAlA

T   he structural design of a foundation system supporting
dynamic equipment is complicated by the need to
model the periodic load, foundation, and soil conditions;
weight values were varied from 0 to 500 lb/ft2 (0 to 24 kPa).
In addition to equipment weight, models included induced
vibration forces uniformly distributed over the slab area.
calculate the structural response of the system; and
interpret the resulting displacements. Designers therefore     MODEL PARAMETERS
may follow historical rules of thumb—for example,              Slab and subgrade
ensuring that the total weight of the mat foundation              The slab was modeled as an isotropic material with
(slab) is three times the weight of a rotating machine or      gross section properties. The modulus of elasticity was
five times the weight of a reciprocating machine1—but          assumed to be 3605 ksi (24,900 MPa). Elements sizes were
these rules may lead to oversized foundations.                 constant at 1 x 1 ft (0.3 x 0.3 m) except when varied to
To obtain an optimum slab size while avoiding resonance     verify convergence of the results.
with the supported machine and ensuring minimal human             The modulus of subgrade reaction from a 1 x 1 ft
perception of vibrations, it’s important to track a slab’s     (0.3 x 0.3 m) reaction plate test, K1, must be corrected for
natural frequency and its displacement amplitudes. In this     the size and configuration of the actual foundation. First,
article, we present practical guidelines and recommendations   the corrected subgrade modulus for a B x B foundation
for design of slabs subject to induced vibration forces        KBxB, where B is the smaller of the actual foundation plan
from steady-state frequency equipment without vibration        dimensions in ft (m), is determined using one of the
isolators. Using graphs and tables, we also show the effects   following two equations.2
of changes in dimensions of the foundation, operational           For sandy soils:
speed of the equipment, and modulus of subgrade reaction                                 2
 B + 1 ft 
on a slab’s natural frequency and peak displacement.           K BxB = K1                                (1) (in.-lb units)
 2B 
FOUNDATION MODEL                                                                             2
 B + 0.3 m 
A finite element model incorporating 4-node plate           K BxB = K1                                   (1) (SI units)
elements with linear spring supports located at each                          2B     
node was used to find the natural frequencies and
For clays:
vibration amplitudes for slabs (Fig. 1). Slab thickness,
corrected modulus of subgrade reaction, and distributed                   1
K BxB = K1                                (2) (in.-lb units)
equipment weight values were varied to develop parametric                 B
curves. Thickness values were varied from 6 to 48 in.
(150 to 1200 mm), modulus values were varied from 25
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