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),
where \(E^*\) is the appropriate modulus selected from Table 1.
Confined Compression 
Unconfined Compression 
Shear 

\(K + (4/3)\ G\) 
\({{9KG} \over {3K + G}}\) 
\(G\) 
(Young’s modulus) 

2.5714 × 10^{4} 
1.2 × 10^{4} 
1.0 × 10^{4} 
Material properties are given in Table 2.
Properties 
Symbol 
Value 
Comment 

bulk modulus 
\(K\) 
2.0 × 10^{4} 
for compression tests 
shear modulus 
\(G\) 
0.428562 × 10^{4} 

Poisson’s ratio 
\(\nu\) 
0.4 

bulk modulus 
\(K\) 
1.0 × 10^{4} 
for shear tests 
shear modulus 
\(G\) 
1.0 × 10^{4} 

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 averageperiod are compared in Table 3.
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 largestrain 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 velocityx
zone gridpoint fix velocityy
zone gridpoint fix velocityz range positionz 0
;; Remaining setup
model gravity 0 0 10
history interval=1
zone dynamic damping local 0
zone history velocityz position (0,0,800)
model history dynamic timetotal
;; Solve to 200 seconds
model solve timetotal=200
model save 'ConfinedCompression'
UnconfinedCompression.dat
;
; Script file for dynamic problem 'Natural periods of an Elastic Column'
; case 2) unconfined compression
;
model new
model largestrain 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 velocityz range positionz 1,1
;; Remaining setup
model gravity 0 0 10
history interval=1
zone dynamic damping local 0
zone history velocityz position (0,0,800)
model history dynamic timetotal
;; Solve to 200 seconds
model solve timetotal=200
model save 'UnconfinedCompression'
Shear.dat
;
; Script file for dynamic problem 'Natural periods of an Elastic Column'
; case 3) Shear
;
model new
model largestrain 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 positionz 1,1
zone gridpoint fix velocityz
;; Remaining setup
model gravity 0.1 0 0
history interval=1
zone dynamic damping local 0
zone history velocityx position (0,0,800)
model history dynamic timetotal
;; Solve to 200 seconds
model solve timetotal=200
model save 'Shear'
⇐ Verification Problems  Comparison of FLAC3D to SHAKE for a Layered, LinearElastic Soil Deposit ⇒
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