Abstract
A mathematical model is presented which describes a nondestructive testing procedure for determining buckling criteria for structures. The procedure requires identification of the structure's support boundary conditions using vibration data. Column-buckling experiments are presented which validate the model. The results illustrate the feasibility of using such models to predict the buckling load for structures whose support boundary conditions are not known in advance of service.
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Abbreviations
- A :
-
cross-sectional area of beam-column
- c :
-
rotational spring constant
- C :
-
dimensionless rotational spring parameter≡cL/EI
- e :
-
Young's modulus
- E{ }:
-
expectation of a random variable
- F :
-
dimensionless frequency parameter ≡ (ω2ϱA/EI)1/4 L
- I :
-
second moment of cross-sectional area; number of ensemble members
- J :
-
number of frequency measurements
- k :
-
translational spring constant
- K :
-
dimensionless translational spring parameter≡kL 3/EI
- L :
-
length of beam-column
- m :
-
concentrated mass
- M :
-
dimensionless mass parameter≡m/ϱAL; number of natural frequencies measured
- p :
-
axial load
- p cr :
-
fundamental critical value of axial load
- P :
-
dimensionless axial-load parameter≡(p/EI)1/2 L
- q 1 (F) :
-
1+cosF coshF
- q 2 (F) :
-
sinF coshF−cosF sinhF
- q 3 (F) :
-
sinF coshF+cosF sinhF
- q 4 (F) :
-
sinF sinhF
- q 5 (F) :
-
1−cosF coshF
- s 2{ }:
-
estimator for variance of a random variable
- t :
-
time
- u :
-
lateral displacement of beam-column's neutral axis
- Var{ }:
-
variance of a random variable
- x :
-
coordinate along axis of beam-column
- X i :
-
random component of rotational restraint
- Y i :
-
random component of translational restraint
- Z i (k),Z ij (k) :
-
random component of frequency-parameter measurement
- α:
-
column parameter≡(p/EI)1/2
- δ ij :
-
Kronecker delta
- λ:
-
beam parameter ≡ (ω2ϱA/EI)1/4
- Π:
-
quasistatic measurement of dimensionless axial-load parameter
- ϱ:
-
density
- ω:
-
angular frequency
- Ω:
-
measurement of dimensionless frequency parameter
- i=1, ...I :
-
ith assembly
- j=1, ...J :
-
jth measurement
- t :
-
partial differentiation with respect to time
- x :
-
partial differentiation with respect to spatial coordinate
- k :
-
kth mode
- m=1, ...M :
-
mth mode used in identification
- ∧:
-
estimator
- −:
-
mean value
- ∼:
-
value of frequency or buckling load corresponding to the mean value of restraint in an ensemble
References
Sweet, A.L. andGenin, J., “Identification of a Model for Predicting Elastic Buckling,”J. of Sound and Vibration,14 (3),317–324 (1971).
Sweet, A.L., Genin, J. andMlakar, P., “Vibratory Identification of Beam Boundary Conditions,”ASME J. Dynamic Systems, Measurement and Control,98 (4),387–394 (Dec. 1976).
Ariaratnam, S.T., “The Southwell Method for Predicting Critical Loads of Elastic Structures,”J. of Mech. and Ap. Math.,XIV,Part 2,137–153 (1961).
Mlakar, P., “Vibratory Determination of Beam Compliance and Prediction of Column Buckling,” Ph.D. Thesis, Purdue University (Aug. 1975).
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Additional information
P.F. Mlakar was Assistant Professor, Department of Mechanics, U.S. Military Academy, West Point, NY; is currently Research Engineer, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS 39180.
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Sweet, A.L., Genin, J. & Mlakar, P.F. Determination of column-buckling criteria using vibratory data. Experimental Mechanics 17, 385–391 (1977). https://doi.org/10.1007/BF02324205
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DOI: https://doi.org/10.1007/BF02324205