using connectivity array.
So the ith element has node i and 1+i.
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Analysis of Pile Behavior due to Damped Vibration by Finite Element Method (FEM)
Rasel M. Chowdhury*, Tanvir M.I.S., Riyad, A.S.M., Faqrul, A.K., and Momeen-Ul-Islam
—————————— ——————————
inite element analysis is a method for numerical solution of a differential equation. Most engineering problems are expressed in terms of certain governing equation and bounda- ry conditions. The basic idea behind the finite element method is to replace a continuous function by means of piecewise pol- ynomials. Such an approximation is called piecewise polyno- mial approximation. In finite element method, the domain of integration is subdivided into a number of smaller pieces or regions- this piece is called element and over each of these elements the continuous function is approximated by a suita-
ble piecewise polynomial [1].
tion prevails. In finite element method the continuum is sub-
divided into a finite number of sub-regions often called finite
element mesh. This sub-region is called element, and over
each element the variation formulation of the given differen-
tial equations constructed using simple interpolation function
for approximation. The finer mesh the more approximation of
————————————————
• Rasel M. Chowdhury, Post-graduate Student; Dept. of Civil Enineering, Khulna University of Engineering & Twchnology, Khulna-9203, Bangla- desh, PH-+8801713173437. E-mail: civilizedrasel073@yahoo.com.com
• Tanvir M.I.S., Undergraduate Student; Dept. of Civil Enineering, Khulna
University of Engineering & Twchnology, Khulna-9203, Bangladesh, PH-
+8801733547957. E-mail: ishtiakshaon@gmail.com
• Riyad, A.S.M., Undergraduate Student; Dept. of Civil Enineering, Khulna
University of Engineering & Twchnology, Khulna-9203, Bangladesh, PH-
+8801825281736. E-mail: riyadtowhid@yahoo.com.com
• Faqrul, A.K., Undergraduate Student; Dept. of Civil Enineering, Khulna
University of Engineering & Twchnology, Khulna-9203, Bangladesh, PH-
+8801683292501. E-mail: kuet.sajib25@gmail.com
• Momeen-Ul-Islam, Undergraduate Student; Dept. of Civil Enineering, Khulna University of Engineering & Twchnology, Khulna-9203, Bangla- desh, PH-+8801678641195. E-mail: momeen.civil2k8@gmail.com
the result to exact solution.
tinuum into a finite number of elements is called descretiza-
tion or meshing.
called elements.
function is approximated by discrete model which is com-
posed of interpolation polynomials.
mial. The shape function is usually denoted by N. The shape
function is written for each node of the element and has a
magnitude 1 at the node and 0 for all other nodes in that ele-
ment.
finite elements in matrix form is called global stiffness matrix.
by local identification and then they are connected globally by
using connectivity array.
So the ith element has node i and 1+i.
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Where,
β = beta damping
ωi = undamped natural circular frequency of the ith mode
ξc = damping ratio
Nm = number of materials
{φi } = displacement vector for mode i
[K ] = stiffness matrix of part of structure of material j
= modal damping ratio of mode
= damping constant stiffness matrix multiplier for material
j
strain energy
Note that the material dependent damping contribution is
computed in the modal expansion phase, so that this damping
contribution must be included there.
crete, and/or steel those are used to transmit surface loads to
lower levels in the soil mass. This transfer may be by vertical
loads to lower levels in the soil mass. This transfer may be by
vertical distribution of the load along the pile shaft or a direct
application of load to a lower stratum through the pile point.
A vertical distribution of the load is made using a friction pile
and a direct load application is by a point, or end-bearing, pile
[2].
subjected to a certain amount of horizontal force due to wind
earth pressure, earthquake, etc., or traction force from auto-
mobiles or trains. It is unsafe to assume that frictional resists
between the bottom of the pile cap and the soil because in this
type of foundation the vertical load is transmitted through the
piles to the lower stratum, not to the soil immediately below
the pile cap. In extreme cases, the soil may even settle away
from the pile caps and leaving a small space in between. Un-
less the structure is supported laterally by other means, the
pile should be designed to resist such lateral loads [3]
To create the finite element model in ANSYS there are multi- ple tasks that have to be completed for the model to run properly. Models can be created using command prompt line input or graphical user interface (GUI). For the simulation of the model, the GUI method was utilized to create the model. The section describes the different tasks and entries into used to create the FE calibration model.
The element types for this model are shown in Table 1.
TABLE 1
ELEMENTS USED IN THIS STUDY
Two different SOLID45 elements were used for pile and soil.
SOLID45 element is an eight nodes hexahedral element with
three translational degrees of freedoms at each node- transla-
tions in the x, y, and z directions. SOLID45 is used for the 3-D
modeling of solid structures. The element is defined by eight
nodes having three degrees of freedom at each node: transla-
tions in the nodal x, y, and z directions.
The element has plasticity, creep, swelling, stress stiffening,
large deflection, and large strain capabilities. A reduced inte-
gration option with hourglass control is available. The geome-
try, node locations, and the coordinate system for this element
are shown in Fig. 1. The element is defined by eight nodes and
the orthotropic material properties. Orthotropic material di-
rections correspond to the element coordinate directions. The
element coordinate system orientation is as described in Co-
ordinate Systems.
Fig. 1. SOLID45 geometry
Fig. 2. SOLID45 stress output
Element loads are described in Node and Element Loads. Pressures may be input as surface loads on the element faces as shown by the circled numbers on Fig. 2. Positive pressures act into the element. Temperatures and fluencies may be input as element body loads at the nodes.
Zero volume elements are not allowed.
Elements may be numbered either as shown in Fig. 3.
“SOLID45 Geometry” or may have the planes IJKL and
MNOP interchanged.
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The element may not be twisted such that the element has two separate volumes. This occurs most frequently when the elements are not numbered properly.
All elements must have eight nodes.
A prism-shaped element may be formed by de-
fining duplicate K and L and duplicate O and P
node numbers (see Triangle, Prism and Tetrahe-
dral Elements).
A tetrahedron shape is also available. The extra
shapes are automatically deleted for tetrahedron
elements.
It may be called as a spring damper. COMBIN14 as shown in
Fig. 4 has longitudinal or torsional capability in 1-D, 2-D, or 3-
D applications. The longitudinal spring-damper option is a
uniaxial tension-compression element with up to three de-
grees of freedom at each node: translations in the nodal x, y,
and z directions. No bending or torsion is considered. The
torsional spring-damper option is a purely rotational element
with three degrees of freedom at each node: rotations about
the nodal x, y, and z axes. No bending or axial loads are con-
sidered.
Fig. 3. COMBIN14 geometry
tively. For a 2-D axisymmetric analysis, these values should be on a full 360° basis.
The length of the spring-damper element must not be ze- ro, i.e., nodes I and J should not be coincident, since the node locations determine the spring orientation.
The longitudinal spring element stiffness acts only along its length. The torsion spring element stiffness acts only about its length, as in a torsion bar.
The element allows only a uniform stress in the spring.
In a thermal analysis, the temperature or pressure degree
of freedom acts in a manner analogous to the displace-
ment.
Some option supports stress stiffening or large deflection.
Also, there is also some options which is used with large
deflection, the coordinates will not be updated.
The spring or the damping capability may be deleted from
the element by setting K or CV equal to zero, respectively.
If CV2 is not zero, the element is nonlinear and requires an
iterative solution
There is no real constant set for Solid45 element. The real con-
stant set for COMBI14 requires two values.
Spring constant: The spring constant is the study variable
of this study. So the values of different equivalent spring
constant for soil are given to obtain desired output. It
ranges from 1500 kN/m to 2000 kN/m.
Damping coefficient: Damping coefficient may be de-
fined as the amount of force exerted by a damping medi-
um per unit velocity of the simple harmonic damping sys-
tem. There is a large range of damping coefficient of soil.
Damping coefficient of soil depends on the soil properties.
As stated earlier, pile and soil are both defined with SOLID45
element in the model. The properties of SOLID45 elements are
listed below:
Liner isotropic properties:
EX= modulus of elasticity
PRXY= Poisson’s ratio [for soil= 0.35, for pile= 0.2]
Moduli of elasticity’s of pile and soil are the parts of design
parameters
Density:
Fig. 4. COMBIN14 stress output
The spring-damper element has no mass. Masses can be added by using the appropriate mass element. The spring or the damping capability may be removed from the element.
The element is defined by two nodes, a spring constant (k) and damping coefficients (cv )1 and (c v) 2 . The damping capability is not used for static or undamped modal analyses. The longitu- dinal spring constant should have units of Force/Length, the damping coefficient units are Force*Time/Length. The torsion- al spring constant and damping coefficient have units of Force*Length/Radian and Force*Length*Time/Radian, respec-
Mass densities of the pile= 2450 kg/m3
Mass density of soil= 1280 kg/m3
Modeling is the most important part in the simulation through
GUI method. The features of the pile-soil finite model are
shown in the Fig. 5 below:
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Fig. 5. Pile-soil model geometry
For the use of symmetric conditions, only one fourth of the model is prepared in this case. The volume is dimensioned for
6m, 4m and 10m at x, z & y directions respectively. The vol- ume is symmetric about xy and zy planes. To make the l/d ratio of the pile, the x & z dimensions of the pile is given as
0.25m. Remaining of the volume is for soil element. Soil vol- ume is little bit complex so it may be prepared by 3 different volumes as given in Table 2 below.
TABLE 1
COORDINATES FOR PILE AND SOIL VOLUME
Por- tions | Volume number | x-ordinate | y-ordinate | z-ordinate | |||
soil | 1 | 0 | 5.75 | 0 | 10 | 0 | 3.75 |
soil | 2 | 5.75 | 6 | 0 | 10 | 0 | 3.75 |
soil | 3 | 0 | 5.75 | 0 | 10 | 3.75 | 4 |
pile | 4 | 5.75 | 6 | 0 | 10 | 3.75 | 4 |
Then the next step is to provide spring elements at the two faces of the volume. Spring element, COMBIN14 provides a certain advantages because of its spring constant as a real con- stant. The unit of spring constant is force per unit length so, the length of the provided spring doesn’t create any effect on this analysis. Offset nodes are made at the two boundary faces are provided, and a line is created between the boundary node
& its offset node. This line is further defined as the COMBIN14 element.
Meshing is the important part of this model. Size control of the
mesh element is also necessary because, huge numbers of ele-
ments are prepared with many nodes, the matrix required to
solve the nodal solution will be large, and its solution may fail
for some circumstances.
Variable mesh sizes of the volume are required for this pur-
pose. The results desired for this study depends on the pile
behavior, the accuracy required may be obtained by providing
smaller mesh at the vicinity of the pile as shown in the Fig. 6.
Fig. 1. Mesh configuration for the model
The command merge items merge separate entities that have
the same location. These items will then be merged into single
entities. Caution must be takes when merging entities in a
model that has already been meshed because the order in
which merging occurs is significant. Merging key points be-
fore nodes can result in some of nodes become “orphaned”,
that is, the nodes lose their association with the solid model.
The orphaned nodes can cause certain operations (such as
boundary condition transfers, surface load transfers, and so
on) to fail. Care must be taken to always merge in the order
that the entities appeared. All precautions were taken to en-
sure that everything was merged in the proper order. Also, the
lowest number was retained during merging.
The pile is assumed to be bearing on the bed rock, all the bear-
ing nodes are taken as fixed, i.e., all nodes along the base in
fixed in all three directions. For this purpose, the plane at the
bottom of the model is given three degrees of freedom, namely
UX, UY & UZ. Values of these DOFs are given as constant val-
ue, and the constant value is 0.
Fig. 7. (a) Provided boundary conditions (fixed at bearing stratum) at the bottom of the model
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Fig. 7. (b) Symmetry boundary conditions at the two inner faces (denoted by the letters S)
Fig. 7. (c) Equivalent soil boundary condition
The symmetry boundary conditions are then set. The model being used is symmetric about two planes. The symmetric boundary conditions are applied at the planes xz and yz, where the planes cut the pile section to make a quarter mod- els. Providing symmetry boundary conditions enables one to reduce the node numbers, element numbers, help in forming of simple element matrices, & make the solution swift.
Soil boundary conditions are then provided. No initial dis- placement is provided because the spring elements transfer stiffness of the soil elements and hence the infinite soil bound- ary region is described in this way shown in Fig. 7.
The model has been verified with reported literature for static
loading (Fig. 8). The verification has been discussed in the fol-
lowing sections.
Static loading: The model is verified by loading in the x direc-
tion at top and the horizontal deflection of the pile head. Hori-
zontal deflections at the pile head are computed for different
amplitudes of applied load for the elastic case.
The results are compared with those presented in Bently and
El Naggar (2000), Poulos and Davis (1980), Maheshwari et al.
(2005) and Rajib Sarkar (2008). The comparison is shown in
Fig. 9 below.
Fig. 8. Three dimensional finite element models used by Rajib Sarkar: (a)
Eight node block element for soil and pile, (b) Finite element quarter mod-
el.
Fig. 9. Response of a single pile fixed at base for static loading
The finite element model for this analysis is a pile model un-
der horizontal loading. For the purpose of this model, the stat-
ic analysis type is utilized.The restart command is utilized to
restart an analysis after the initial run or load step has been
completed. The use of restart option will be detailed in the
analysis portion for the discussion.
The Sol’n Controls command dictates the use of a linear or non
linear solution for the finite element model. Typical com-
mands utilized in a non linear static analysis are shown in Ta-
ble 3 below.
TABLE 2
COMMANDS USED TO CONTROL NONLINEAR ANALYSIS
Analysis Options | Small Dispalcement |
Calculate prestress effects | No |
Time at end of load step | 8400 |
Automatic time steping | On |
Number of substeps | 100 |
Max no. of substeps | 200 |
Min no. of substeps | 1 |
Write Items to results file | All solution items |
Frequency | Write every substep |
In the particular case considered in this portion of this study, the analysis is small displacement and static. The time at the end of load step refers to the ending load per load step. Table
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below shows the first load step taken. The substeps are set to indicate load increments used for this analysis. The commands used to control the solver and outputs are shown in another Table 4.
TABLE 3
COMMANDS USED TO CONTROL OUTPUT
TABLE 5
ADVANCED NONLINEAR CONTROL SETTINGS USED
All these values are set to ANSYS (SAS 2005) defaults. The commands used for the nonlinear algorithm and convergence criteria are shown in table below. All values for the nonlinear algorithm are set to defaults shown in Table 5.
TABLE 4
NONLINEAR ALGORITHM AND CONVERGENCE CRITERIA PARAMETER
The FE analysis of the model was set up to examine three dif- ferent behaviors: the displacement at the pile head, multidirec- tional strain components and multidirectional stress compo- nents. The Newton-Raphson method of analysis was used to compute the nonlinear response.
The applications of the loads are done incrementally as re- quired by Newton-Raphson procedure. After each load incre- ment was applied, the restart option was used to go to the next step after convergence.
After the verification of the model, it is ready for the seismic
vibration analysis. Form the command of analysis type, we
need to accomplish spectrum analysis, but for the purpose of
modal calculation, direction of excitations etc. modal analysis
is chosen first.
Modal analysis is the prerequisite for the spectrum analysis.
Spectrum analysis requires the mode combination and the
element matrices, so modal analysis is required shown in Ta-
ble 7.
TABLE 6
MODAL ANALYSIS OPTION PROCEDURES
The values for the convergence criteria are set to defaults ex- cept for the tolerances. The tolerances for force and displace- ment are set as 5 times the default values. Table below shows the commands used for the advanced nonlinear settings. The program behavior upon non-convergence for this analysis was set such that the program will but not exit. The rest of the commands were set to defaults shown in Table 6.
Master degree of freedom: The master degree of freedom (master DOF’s) are to be set. A user defined master degree of freedom to be selected. As, all of the elements will be subjected for seismic excitations, all of the nodes are taken for 1st degree
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of freedom.
Another option, named additional degree of freedom, may be
selected when a degree of freedom is not selected as the mas-
ter degree of freedom.
The model is then solved for modal analysis (Fig. 10).
Fig. 10. Model with master degree of freedom in all derections.
After the modal analysis, the model is to be solved for certain
predominant frequency. New analysis is to be chosen as spec-
trum analysis. In the load step option, one can give the excita-
tions from a single point. The settings required are given in the
following Table 8.
TABLE 7
SETTINGS FOR SINGLE POINT RESPONSE SPECTRA
Type of response spectr | Seismic displac |
Scale factor | 1 |
Excitation direction SEDX, SEDY, SEDZ | 1, 1 ,1 respectively |
Other options should remain as default |
Different frequencies are given in the frequency table. The excitation frequency is another variable of this study. So, fre- quencies of different magnitude are to be put at each analysis. The frequency table can accept 20 frequency values with dif- ferent magnitude. The conditions for the frequency input are: the input frequency value must not be equal to zero; and the values of frequencies must be of ascending order. If a small frequency value is put after a greater value, the frequency ta- ble will end before the smaller value.
Value of damping ratio is to be put. As the model prepared is a damping system itself, the damping of the frequency table may be neglected at this stage. So, 0(zero) is to be put for damping ratio. The spectrum values for the different frequen- cies are 0.44.
The system is then solved for spectrum analysis for the first time.
Another spectrum analysis is required for mode combination.
The modes are to be combined and hence, a new spectrum
analysis is required. Mode combination methods for the single point response spectra are given in the Table 9 below.
TABLE 8
METHOD OF MODE COMBINATION
Mode combination method | SRSS |
Significant threshold | 0.15 |
Type of output | Displacement |
Note: SRSS method: Specifies the square root of sum of squares mode combination method.
The system is then solved for spectrum analysis for the 2nd
time.
ANSYS 10.0 gives many facilities to find out the desired re- sults at any point of interest in the model. Corresponding sub- step of the result from result-list is to be read by pick. Both graphical and listing result system is a unique feature for the feature. Graphical results representation system is of two types: contour plot & vector plot. Contour plot gives certain advantages to understand and utilize results with different color stripes having different value ranges for each color. Vec- tor plot gives arrow diagram with the direction of changes of values with different color.
Fig. 11. Maximum deformed shape of the model
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Fig. 3. Degree of freedom solution- vector sum of all displacement
Fig. 2. Degree of freedom solution- x-component displacement 
Fig. 16. Total strain plot- x-direction
Fig. 13. Degree of freedom solution- y-component displacement 
Fig. 17. total strain plot- y-direction
Fig. 14. Degree of freedom solution- z-component displacement
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f n = natural frequency of vibration, in this study, frequency range 5 Hz to 30 Hz is taken. Frequencies below 5 Hz don’t create much effect, so ignored. Equivalent spring constant of soil (k) ranges from 1500 kN/m to 2000 kN/m.
Fig. 18. Total strain plot- z-direction 
Fig. 20. Top displacement vs. Ep/Es plot with varying frequencies. No vertical load. Es= 7000000 kN/m2. k= 1500 kN/m
Fig. 19. Total strain intensity plot
The model prepared for this study is verified for horizontal static loading at the pile top. A similar model was prepared by Rajib Sarkar, research scholar, University of Roorki, India. For the dynamic analysis, following assumptions were taken,
ρp /ρs = 1.92 ν p /ν s = 0.60
ρp = mass density of pile ρs = mass density of soil
ν p = Poisson’s ratio of pile ν s = Poisson’s ratio of soil.
Mass density of pile is taken as 2450kg/m3. So, mass density of soil is taken as 1280 kg/m3.
Poisson’s ratio of soil varies from 0.3 to 0.4, with lower values for firm soil and higher values for soft soil. A value of Pois- son’s ratio of soil is taken as 0.35 for this study. So, the poisons ratio of pile is 0.2.
Ep/Es is one of the study variables. It ranges from 500 to 1000, a dimensionless number.
Ep = modulus of elasticity of pile material
Es = modulus of elasticity of soil
The value of soil modulus of elasticity ranges from 7000000
kN/m3 to 35000 kN/m3. The value of Ep is calculated from the
desired ratio.
Fig. 21. Top displacement vs. Ep/Es plot with varying frequencies. No vertical load. Es= 1000000 kN/m2. k= 1500 kN/m
Fig. 22. Top displacement vs. Ep/Es plot with varying frequencies. No vertical load. Es= 7000000 kN/m2. k= 1750 kN/m
Fig. 23. Top displacement vs. Ep/Es plot with varying frequencies. No vertical load. Es= 1000000 kN/m2. k= 1750 kN/m
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Fig. 24. Top displacement vs. Ep/Es plot with varying frequencies. No vertical load. Es= 7000000 kN/m2. k= 2000 kN/m
Fig. 25. Top displacement vs. Ep/Es plot with varying frequencies. No vertical load. Es= 1000000 kN/m2. k= 2000 kN/m
Fig. 26. Top displacement vs. Vertical load plot with varying frequencies.
Ep/Es= 500. Es= 7000000 kN/m2. k= 1500 kN/m
Fig. 27. Top displacement vs. Vertical load plot with varying frequencies.
Ep/Es= 1000. Es= 7000000 kN/m2. k= 1500 kN/m
Fig. 28. Top displacement vs. Vertical load plot with varying frequencies.
Ep/Es= 500. Es= 1000000 kN/m2. k= 1500 kN/m
Fig. 29. Top displacement vs. Vertical load plot with varying frequencies.
Ep/Es= 1000. Es= 1000000 kN/m2. k= 1500 kN/m
Fig. 30. Top displacement vs. Vertical load plot with varying frequencies.
Ep/Es= 500. Es= 7000000 kN/m2. k= 2000 kN/m
Fig.31. Top displacement vs. Vertical load plot with varying frequencies.
Ep/Es= 1000. Es= 7000000 kN/m2. k= 2000 kN/m
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Fig. 32. Top displacement vs. Vertical load plot with varying frequencies.
Ep/Es= 500. Es= 1000000 kN/m2. k= 2000 kN/m
Fig. 33. Top displacement vs. Vertical load plot with varying frequencies.
Ep/Es= 1000. Es= 1000000 kN/m2. k= 2000 kN/m
Linear interpolation between the suitable ranges is applicable
for the values of the variables for which the displacement are
not plotted. Suitable factor of safety must be considered to
overcome the risk of any accidental case. The soil mass was
assumed to be homogeneous and isotropic throughout the
analysis but it is a rare case. Values of Ep/Es and k are not con-
stant in any practical pile-soil system. Better solution may be
obtained if the variables are computed for the soil in the vicini-
ty of the pile.
Displacement at the top of the pile is increased up to
10% for the increment of frequency of 5 Hz, where
other parameters such as modulus of elasticity of soil,
soil spring constants remain the same, and vertical
concentric loads are absent.
While the modulus of elasticity of soil remains con-
stant, the pile may be strengthened to increase the
Ep/Es value. 50% increase of Ep/Es ratio will result in
70 to 80% decrease of deflection at pile top, lower per-
centage of deflection decrement occur at lower fre-
quencies, while larger percentage of decrement oc-
curs at higher frequencies.
Increase of Ep/Es value may not be efficient, for ex-
ample, 100% increase of Ep/Es ratio will results 75 to
85% decrease of deflection at top of pile.
From pile top displacement vs. Ep/Es plot it is ob-
served that, the slopes of the curves are steep at the
lower values of Ep/Es. As the Ep/Es value increases
the slope decreases.
No significant displacement variations are found if the Ep/Es value increase after a value of 750. So it would be worthwhile to strengthen the pile to obtain an Ep/Es value greater than 750 when the earthquake protection is the only purpose.
Concentric vertical loads reduce the pile top dis- placement due to vibration up to 20 to 60% when the pile is designed for service load equal to 80% of the pile capacity. Higher displacement reduction is ob- tained when the Ep/Es value is small.
The major purpose of this study was to prescribe some proba- ble solution to counteract the detrimental effects of earthquake vibrations. The study is expected to generate reasonable solu- tions of focused problem defined under some parametric con- ditions. Some of the variables are chosen which may be im- proved for better solutions. To be familiar with the graphical representations, one has to examine for the probable vibration frequencies that may occur during the design period of the structures. Then he has to investigate the allowable pile deflec- tion that may not cause the serious threat to the proposed structures while vibrating with investigated frequency. The graphical representations would help to chose the counter measures, such as increase of vertical load, soil improvement to increase its spring constant or Young’s modulus of elasticity, modification of pile dimension, or provision of high steel con- tribution to increase the modulus of elasticity of pile. Cost concern to adopt suitable technique should be another varia- ble. In recent, most of the tall structures are constructed for residential purposes. Only a few of the high rise structures are constructed for commercial or industrial purposes. Residential buildings are not equipped with too much valuable materials, so, earthquake proof design for residential buildings is done to save people from death, and it is the intangible benefit of this system, leaving no opportunity to evaluate the benefit-cost analysis. When the buildings are equipped with valuable ma- chineries, economic analysis must be carried out to investigate the proper solution.
50% increase of Ep/Es ratio will result in 70 to 80% decrease of deflection at pile top, lower percentage of deflection decrement occur at lower frequencies, while larger percentage of decrement occurs at high- er frequencies.
Concentric vertical loads reduce the pile top dis- placement due to vibration up to 20 to 60% when the pile is designed for service load equal to 80% of the pile capacity.
Displacement of the pile depends on the moduli of elasticity of pile, soil. Frequency, vertical loads, vis- cous damping properties of soil.
Provision of higher steel area would reduce the pile top displacement.
Vertical loads provide stiffness and so give the inertia effect of the structure.
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Equivalent spring constant of soil materials is an im- portant variable. Viscous damping is useful for miti- gation of vibrations.
[1] ANSYS 2005, Documentation for ANSYS, Version 10.0
[2] Bowels, J. E., (1997), Foundation Analysis and Design, 5th ed.,
McGraw Hill, pp. 929-963.
[3] Kuhlemeyer R.L., (1979). Vertical vibration of piles, Jour-
nal of geotechnical engineering division, ASCE, 105(GT2),
273-287.
[4] Nilson, A. H., Darwin, D., Dolan, C. W., (2003), Design of
concrete structures, 13th ed., McGraw Hill, pp. 700-723.
[5] ANSYS workbench tutorial.
[6] Sarkar, R., Maheshwari, B. K., (2008), Three dimensional
seismic analysis of pile groups, 12th international conference
of IACMAG, Goa, India.
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