Natural Periods of an Elastic Column

Note

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A column of elastic material resting on a rigid base has natural periods of vibration, depending on the mode of oscillation and the confining conditions. Three cases are examined: an unconfined column; a confined column in compression; and a column in shear.

The column is loaded by applying gravity in either the \(x\)- or \(y\)-direction and observing the oscillations with zero damping. The case of confined compression is modeled by inhibiting lateral displacement along the vertical boundaries, which prevents lateral deformation of the mesh. For unconfined compression, lateral displacement is not inhibited. For the column in shear, vertical motion is inhibited to eliminate bending modes; the loading is applied laterally.

The theoretical value for the natural period of oscillation, \(T\), is given by Equation (1),

(1)\[T = 4 L\ \sqrt{\rho\over E^*}\]

where \(E^*\) is the appropriate modulus selected from Table 1.


Table 1: Moduli Appropriate to Various Deformation Modes

Confined Compression

Unconfined Compression

Shear

\(K + (4/3)\ G\)

\({{9KG} \over {3K + G}}\)

\(G\)

(Young’s modulus)

2.5714 × 104

1.2 × 104

1.0 × 104

Material properties are given in Table 2.


Table 2: Material Properties

Properties

Symbol

Value

Comment

bulk modulus

\(K\)

2.0 × 104

for compression tests

shear modulus

\(G\)

0.428562 × 104

Poisson’s ratio

\(\nu\)

0.4

bulk modulus

\(K\)

1.0 × 104

for shear tests

shear modulus

\(G\)

1.0 × 104

Poisson’s ratio

\(\nu\)

0.125

density

\(\rho\)

1.0

applied gravity

\(g_z\)

-10.0

for compression tests

\(g_x\)

0.1

for shear tests

column height

\(L\)

800

column width

\(W\)

100

The theoretical periods and calculated (FLAC3D) natural periods of oscillation averaged over several periods by the FISH function average-period are compared in Table 3.


Table 3: Comparison of Dynamic Period of Oscillation for Three Modes

Confined Compression

Unconfined Compression

Shear

Theoretical

19.96

29.21

32.00

FLAC3D

19.96

29.26

31.98

Data Files

ConfinedCompression.dat

;-----------------------------------------------------------------------
; Script file for dynamic problem 'Natural periods of an Elastic Column'
; case 1) Confined compression
;-----------------------------------------------------------------------
model new
model large-strain off
model title "Natural periods of an Elastic Column - Confined Compression"
model configure dynamic
;; Create zones
zone create brick point 0 (0,0,0) point 1 (100,0,0) ...
                  point 2 (0,100,0) point 3 (0,0,800) ...
                  size 1,1,8
;; Assign material model and properties
zone cmodel assign elastic
zone property bulk 2e4 shear 0.428562e4 density 1
;; Boundary Conditions
zone gridpoint fix velocity-x
zone gridpoint fix velocity-y
zone gridpoint fix velocity-z range position-z 0
;; Remaining setup
model gravity 0 0 -10
history interval=1 
zone dynamic damping local 0
zone history velocity-z position (0,0,800)
model history dynamic time-total
;; Solve to 200 seconds
model solve time-total=200
model save 'ConfinedCompression'

UnconfinedCompression.dat

;-----------------------------------------------------------------------
; Script file for dynamic problem 'Natural periods of an Elastic Column'
;   case 2) unconfined compression
;-----------------------------------------------------------------------
model new
model large-strain off
model title "Natural periods of an Elastic Column - Unconfined Compression"
model configure dynamic
;; Create zones
zone create brick point 0(0,0,0) point 1(100,0,0) ...
                  point 2(0,100,0) point 3(0,0,800) ...
                  size 1,1,8
;; Assign model and properties
zone cmodel assign elastic
zone property bulk 2e4 shear 0.428562e4 density 1
;; Boundary Conditions
zone gridpoint fix velocity-z range position-z -1,1
;; Remaining setup
model gravity 0 0 -10
history interval=1
zone dynamic damping local 0
zone history velocity-z position (0,0,800)
model history dynamic time-total
;; Solve to 200 seconds
model solve time-total=200
model save 'UnconfinedCompression'

Shear.dat

;-----------------------------------------------------------------------
; Script file for dynamic problem 'Natural periods of an Elastic Column'
;  case 3) Shear
;-----------------------------------------------------------------------
model new
model large-strain off
model title "Natural periods of an Elastic Column -- Shear"
model configure dynamic
;; Create zones
zone create brick point 0(0,0,0) point 1(100,0,0) ...
                  point 2(0,100,0) point 3(0,0,800) ...
                  size 1,1,8
;; Assign model and properties
zone cmodel assign elastic
zone property bulk 1e4 shear 1e4 density 1
;; Boundary Conditions
zone gridpoint fix velocity range position-z -1,1
zone gridpoint fix velocity-z
;; Remaining setup
model gravity 0.1 0 0
history interval=1
zone dynamic damping local 0
zone history velocity-x position (0,0,800)
model history dynamic time-total
;; Solve to 200 seconds
model solve time-total=200
model save 'Shear'