14.4.1. Laterally-Loaded Pile FoundationΒΆ

  1. The original model can be found here.
  2. The Python code is converted by Pavan Chigullapally from University of Auckland, Auckland (pchi893@aucklanduni.ac.nz), and shown below, which can be downloaded here.
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# -*- coding: utf-8 -*-
"""
Created on Thu Jan 10 18:24:47 2019

@author: pchi893
"""




##########################################################
#                                                         #
# Procedure to compute ultimate lateral resistance, p_u,  #
#  and displacement at 50% of lateral capacity, y50, for  #
#  p-y springs representing cohesionless soil.            #
#   Converted to openseespy by: Pavan Chigullapally       #
#                               University of Auckland    # 
#                                                         #
#   Created by:   Hyung-suk Shin                          #
#                 University of Washington                #
#   Modified by:  Chris McGann                            #
#                 Pedro Arduino                           #
#                 Peter Mackenzie-Helnwein                #
#                 University of Washington                #
#                                                         #
###########################################################

# references
#  American Petroleum Institute (API) (1987). Recommended Practice for Planning, Designing and
#   Constructing Fixed Offshore Platforms. API Recommended Practice 2A(RP-2A), Washington D.C,
#   17th edition.
#
# Brinch Hansen, J. (1961). "The ultimate resistance of rigid piles against transversal forces."
#  Bulletin No. 12, Geoteknisk Institute, Copenhagen, 59.
#
#  Boulanger, R. W., Kutter, B. L., Brandenberg, S. J., Singh, P., and Chang, D. (2003). Pile 
#   Foundations in liquefied and laterally spreading ground during earthquakes: Centrifuge experiments
#   and analyses. Center for Geotechnical Modeling, University of California at Davis, Davis, CA.
#   Rep. UCD/CGM-03/01.
#
#  Reese, L.C. and Van Impe, W.F. (2001), Single Piles and Pile Groups Under Lateral Loading.
#    A.A. Balkema, Rotterdam, Netherlands.

import math

def get_pyParam ( pyDepth, gamma, phiDegree, b, pEleLength, puSwitch, kSwitch, gwtSwitch):
    
    #----------------------------------------------------------
    #  define ultimate lateral resistance, pult 
    #----------------------------------------------------------
    
    # pult is defined per API recommendations (Reese and Van Impe, 2001 or API, 1987) for puSwitch = 1
    #  OR per the method of Brinch Hansen (1961) for puSwitch = 2
    
    pi = 3.14159265358979
    phi = phiDegree * (pi/180)
    zbRatio = pyDepth / b
    
    #-------API recommended method-------
    
    if puSwitch == 1:
    
      # obtain loading-type coefficient A for given depth-to-diameter ratio zb
      #  ---> values are obtained from a figure and are therefore approximate
        zb = []
        dataNum = 41
        for i in range(dataNum):
            b1 = i * 0.125
            zb.append(b1)
        As = [2.8460, 2.7105, 2.6242, 2.5257, 2.4271, 2.3409, 2.2546, 2.1437, 2.0575, 1.9589, 1.8973, 1.8111, 1.7372, 1.6632, 1.5893, 1.5277, 1.4415, 1.3799, 1.3368, 1.2690, 1.2074, 1.1581, 
            1.1211, 1.0780, 1.0349, 1.0164, 0.9979, 0.9733, 0.9610, 0.9487, 0.9363, 0.9117, 0.8994, 0.8994, 0.8871, 0.8871, 0.8809, 0.8809, 0.8809, 0.8809, 0.8809] 
      
      # linear interpolation to define A for intermediate values of depth:diameter ratio
        for i in range(dataNum):
            if zbRatio >= 5.0:
                A = 0.88
            elif zb[i] <= zbRatio and zbRatio <= zb[i+1]:
                A = (As[i+1] - As[i])/(zb[i+1] - zb[i]) * (zbRatio-zb[i]) + As[i]
                
      # define common terms
        alpha = phi / 2
        beta = pi / 4 + phi / 2
        K0 = 0.4
        
        tan_1 = math.tan(pi / 4 - phi / 2)        
        Ka = math.pow(tan_1 , 2) 
    
      # terms for Equation (3.44), Reese and Van Impe (2001)
        tan_2 = math.tan(phi)
        tan_3 = math.tan(beta - phi)
        sin_1 = math.sin(beta)
        cos_1 = math.cos(alpha)
        c1 = K0 * tan_2 * sin_1 / (tan_3*cos_1)
        
        tan_4 = math.tan(beta)
        tan_5 = math.tan(alpha)
        c2 = (tan_4/tan_3)*tan_4 * tan_5
        
        c3 = K0 * tan_4 * (tan_2 * sin_1 - tan_5)
        
        c4 = tan_4 / tan_3 - Ka
    
        # terms for Equation (3.45), Reese and Van Impe (2001)
        pow_1 = math.pow(tan_4,8)
        pow_2 = math.pow(tan_4,4)
        c5 = Ka * (pow_1-1)
        c6 = K0 * tan_2 * pow_2
    
      # Equation (3.44), Reese and Van Impe (2001)
        pst = gamma * pyDepth * (pyDepth * (c1 + c2 + c3) + b * c4)
    
      # Equation (3.45), Reese and Van Impe (2001)
        psd = b * gamma * pyDepth * (c5 + c6)
    
      # pult is the lesser of pst and psd. At surface, an arbitrary value is defined
        if pst <=psd:
            if pyDepth == 0:
                pu = 0.01
              
            else:
                pu = A * pst
              
        else:
            pu = A * psd
          
      # PySimple1 material formulated with pult as a force, not force/length, multiply by trib. length
        pult = pu * pEleLength
    
    #-------Brinch Hansen method-------
    elif puSwitch == 2:
      # pressure at ground surface
        cos_2 = math.cos(phi)
        
        tan_6 = math.tan(pi/4+phi/2) 
        
        sin_2 = math.sin(phi)
        sin_3 = math.sin(pi/4 + phi/2)
        
        exp_1 = math.exp((pi/2+phi)*tan_2)
        exp_2 = math.exp(-(pi/2-phi) * tan_2)
        
        Kqo = exp_1 * cos_2 * tan_6 - exp_2 * cos_2 * tan_1
        Kco = (1/tan_2) * (exp_1 * cos_2 * tan_6 - 1)
    
      # pressure at great depth
        exp_3 = math.exp(pi * tan_2)
        pow_3 = math.pow(tan_2,4)
        pow_4 = math.pow(tan_6,2)
        dcinf = 1.58 + 4.09 * (pow_3)
        Nc = (1/tan_2)*(exp_3)*(pow_4 - 1)
        Ko = 1 - sin_2
        Kcinf = Nc * dcinf
        Kqinf = Kcinf * Ko * tan_2
    
      # pressure at an arbitrary depth
        aq = (Kqo/(Kqinf - Kqo))*(Ko*sin_2/sin_3)
        KqD = (Kqo + Kqinf * aq * zbRatio)/(1 + aq * zbRatio)
    
      # ultimate lateral resistance
        if pyDepth == 0:
            pu = 0.01
        else:
            pu = gamma * pyDepth * KqD * b
               
      # PySimple1 material formulated with pult as a force, not force/length, multiply by trib. length
        pult  = pu * pEleLength
        
    #----------------------------------------------------------
    #  define displacement at 50% lateral capacity, y50
    #----------------------------------------------------------
    
    # values of y50 depend of the coefficent of subgrade reaction, k, which can be defined in several ways.
    #  for gwtSwitch = 1, k reflects soil above the groundwater table
    #  for gwtSwitch = 2, k reflects soil below the groundwater table
    #  a linear variation of k with depth is defined for kSwitch = 1 after API (1987)
    #  a parabolic variation of k with depth is defined for kSwitch = 2 after Boulanger et al. (2003)
    
    # API (1987) recommended subgrade modulus for given friction angle, values obtained from figure (approximate)
    
    ph = [28.8, 29.5, 30.0, 31.0, 32.0, 33.0, 34.0, 35.0, 36.0, 37.0, 38.0, 39.0, 40.0]    
   
    # subgrade modulus above the water table
    if gwtSwitch == 1:
        k = [10, 23, 45, 61, 80, 100, 120, 140, 160, 182, 215, 250, 275]
        
    else:
        k = [10, 20, 33, 42, 50, 60, 70, 85, 95, 107, 122, 141, 155]
    
    dataNum = 13  
    for i in range(dataNum):
        if ph[i] <= phiDegree and phiDegree <= ph[i+1]:
            khat = (k[i+1]-k[i])/(ph[i+1]-ph[i])*(phiDegree - ph[i]) + k[i]            
            
    # change units from (lb/in^3) to (kN/m^3)
    k_SIunits = khat * 271.45
    
    # define parabolic distribution of k with depth if desired (i.e. lin_par switch == 2)
    sigV = pyDepth * gamma
    
    if sigV == 0:
         sigV = 0.01
         
    if kSwitch == 2:
       # Equation (5-16), Boulanger et al. (2003)
        cSigma = math.pow(50 / sigV , 0.5)
       # Equation (5-15), Boulanger et al. (2003)
        k_SIunits = cSigma * k_SIunits
    
    # define y50 based on pult and subgrade modulus k
    
    # based on API (1987) recommendations, p-y curves are described using tanh functions.
    #  tcl does not have the atanh function, so must define this specifically
    
    #  i.e.  atanh(x) = 1/2*ln((1+x)/(1-x)), |x| < 1
    
    # when half of full resistance has been mobilized, p(y50)/pult = 0.5
    x = 0.5
    log_1 = math.log((1+x)/(1-x))
    atanh_value = 0.5 * log_1
    
    # need to be careful at ground surface (don't want to divide by zero)
    if pyDepth == 0.0:
        pyDepth = 0.01

    y50 = 0.5 * (pu/ A)/(k_SIunits * pyDepth) * atanh_value
    # return pult and y50 parameters
    outResult = []
    outResult.append(pult)
    outResult.append(y50)
    
    return outResult

#########################################################################################################################################################################

#########################################################################################################################################################################

###########################################################
#                                                         #
# Procedure to compute ultimate tip resistance, qult, and #
#  displacement at 50% mobilization of qult, z50, for     #
#  use in q-z curves for cohesionless soil.               #
#   Converted to openseespy by: Pavan Chigullapally       #  
#                               University of Auckland    #
#   Created by:  Chris McGann                             #
#                Pedro Arduino                            #
#                University of Washington                 #
#                                                         #
###########################################################

# references
#  Meyerhof G.G. (1976). "Bearing capacity and settlement of pile foundations." 
#   J. Geotech. Eng. Div., ASCE, 102(3), 195-228.
#
#  Vijayvergiya, V.N. (1977). "Load-movement characteristics of piles."
#   Proc., Ports 77 Conf., ASCE, New York.
#
#  Kulhawy, F.H. ad Mayne, P.W. (1990). Manual on Estimating Soil Properties for 
#   Foundation Design. Electrical Power Research Institute. EPRI EL-6800, 
#   Project 1493-6 Final Report.

def get_qzParam (phiDegree, b, sigV, G):
    
    # define required constants; pi, atmospheric pressure (kPa), pa, and coeff. of lat earth pressure, Ko
    pi = 3.14159265358979
    pa = 101
    sin_4 = math.sin(phiDegree * (pi/180))
    Ko = 1 - sin_4

  # ultimate tip pressure can be computed by qult = Nq*sigV after Meyerhof (1976)
  #  where Nq is a bearing capacity factor, phi is friction angle, and sigV is eff. overburden
  #  stress at the pile tip.
    phi = phiDegree * (pi/180)

  # rigidity index
    tan_7 = math.tan(phi)
    Ir = G/(sigV * tan_7)
  # bearing capacity factor
    tan_8 = math.tan(pi/4+phi/2)
    sin_5 = math.sin(phi)
    pow_4 = math.pow(tan_8,2)
    pow_5 = math.pow(Ir,(4*sin_5)/(3*(1+sin_5)))
    exp_4 = math.exp(pi/2-phi)
    
    Nq = (1+2*Ko)*(1/(3-sin_5))*exp_4*(pow_4)*(pow_5)  
  # tip resistance
    qu = Nq * sigV
  # QzSimple1 material formulated with qult as force, not stress, multiply by area of pile tip
    pow_6 = math.pow(b, 2)  
    qult = qu * pi*pow_6/4

  # the q-z curve of Vijayvergiya (1977) has the form, q(z) = qult*(z/zc)^(1/3)
  #  where zc is critical tip deflection given as ranging from 3-9% of the
  #  pile diameter at the tip.  

  # assume zc is 5% of pile diameter
    zc = 0.05 * b

  # based on Vijayvergiya (1977) curve, z50 = 0.125*zc
    z50 = 0.125 * zc

  # return values of qult and z50 for use in q-z material
    outResult = []
    outResult.append(qult)
    outResult.append(z50)
    
    return outResult

#########################################################################################################################################################################

#########################################################################################################################################################################
##########################################################
#                                                         #
# Procedure to compute ultimate resistance, tult, and     #
#  displacement at 50% mobilization of tult, z50, for     #
#  use in t-z curves for cohesionless soil.               #
#   Converted to openseespy by: Pavan Chigullapally       #
#                               University of Auckland    #
#   Created by:  Chris McGann                             #
#                University of Washington                 #
#                                                         #
###########################################################

def get_tzParam ( phi, b, sigV, pEleLength):

# references
#  Mosher, R.L. (1984). "Load transfer criteria for numerical analysis of
#   axial loaded piles in sand." U.S. Army Engineering and Waterways
#   Experimental Station, Automatic Data Processing Center, Vicksburg, Miss.
#
#  Kulhawy, F.H. (1991). "Drilled shaft foundations." Foundation engineering
#   handbook, 2nd Ed., Chap 14, H.-Y. Fang ed., Van Nostrand Reinhold, New York

    pi = 3.14159265358979
    
  # Compute tult based on tult = Ko*sigV*pi*dia*tan(delta), where
  #   Ko    is coeff. of lateral earth pressure at rest, 
  #         taken as Ko = 0.4
  #   delta is interface friction between soil and pile,
  #         taken as delta = 0.8*phi to be representative of a 
  #         smooth precast concrete pile after Kulhawy (1991)
  
    delta = 0.8 * phi * pi/180

  # if z = 0 (ground surface) need to specify a small non-zero value of sigV
  
    if sigV == 0.0:
        sigV = 0.01
    
    tan_9 = math.tan(delta)
    tu = 0.4 * sigV * pi * b * tan_9
    
  # TzSimple1 material formulated with tult as force, not stress, multiply by tributary length of pile
    tult = tu * pEleLength

  # Mosher (1984) provides recommended initial tangents based on friction angle
	# values are in units of psf/in
    kf = [6000, 10000, 10000, 14000, 14000, 18000]
    fric = [28, 31, 32, 34, 35, 38]

    dataNum = len(fric)
    
    
	# determine kf for input value of phi, linear interpolation for intermediate values
    if phi < fric[0]:
        k = kf[0]
    elif phi > fric[5]:
        k = kf[5]
    else:
        for i in range(dataNum):
            if fric[i] <= phi and phi <= fric[i+1]:
                k = ((kf[i+1] - kf[i])/(fric[i+1] - fric[i])) * (phi - fric[i]) + kf[i]
        

  # need to convert kf to units of kN/m^3
    kSIunits =  k * 1.885

  # based on a t-z curve of the shape recommended by Mosher (1984), z50 = tult/kf
    z50 = tult / kSIunits

  # return values of tult and z50 for use in t-z material
    outResult = []
    outResult.append(tult)
    outResult.append(z50)

    return outResult


#########################################################################################################################################################################

#########################################################################################################################################################################

###########################################################
#                                                         #
# Static pushover of a single pile, modeled as a beam on  #
#  a nonlinear Winkler foundation.  Lateral soil response #
#  is described by p-y springs.  Vertical soil response   #
#  described by t-z and q-z springs.                      #
#   Converted to openseespy by: Pavan Chigullapally       #
#                               University of Auckland    #
#   Created by:  Chris McGann                             #
#                HyungSuk Shin                            #
#                Pedro Arduino                            #
#                Peter Mackenzie-Helnwein                 #
#              --University of Washington--               #
#                                                         #
# ---> Basic units are kN and meters                      #
#                                                         #
###########################################################


from openseespy.opensees import *

op.wipe()

#########################################################################################################################################################################

#########################################################################################################################################################################

# all the units are in SI units N and mm

#----------------------------------------------------------
#  pile geometry and mesh
#----------------------------------------------------------

# length of pile head (above ground surface) (m)
L1 = 1.0
# length of embedded pile (below ground surface) (m)
L2 = 20.0
# pile diameter
diameter = 1.0

# number of pile elements
nElePile = 84
# pile element length 
eleSize = (L1+L2)/nElePile

# number of total pile nodes
nNodePile =  1 + nElePile

#----------------------------------------------------------
#  create spring nodes
#----------------------------------------------------------
# spring nodes created with 3 dim, 3 dof
op.model('basic', '-ndm', 3, '-ndf', 3) 

# counter to determine number of embedded nodes
count = 0

# create spring nodes

#1 to 85 are spring nodes

pile_nodes = dict()

for i in range(nNodePile):
    zCoord = eleSize * i
    if zCoord <= L2:
        op.node(i+1, 0.0, 0.0, zCoord)
        op.node(i+101, 0.0, 0.0, zCoord)
        pile_nodes[i+1] = (0.0, 0.0, zCoord)
        pile_nodes[i+101] = (0.0, 0.0, zCoord)
        count = count + 1

print("Finished creating all spring nodes...")

# number of embedded nodes
nNodeEmbed = count

# spring node fixities
for i in range(nNodeEmbed):
    op.fix(i+1, 1, 1, 1)
    op.fix(i+101, 0, 1, 1)
    
print("Finished creating all spring node fixities...")

#----------------------------------------------------------
#  soil properties
#----------------------------------------------------------

# soil unit weight (kN/m^3)
gamma = 17.0
# soil internal friction angle (degrees)
phi = 36.0
# soil shear modulus at pile tip (kPa)
Gsoil = 150000.0

# select pult definition method for p-y curves
# API (default) --> 1
# Brinch Hansen --> 2
puSwitch = 1 

# variation in coefficent of subgrade reaction with depth for p-y curves
# API linear variation (default)   --> 1
# modified API parabolic variation --> 2
kSwitch = 1

# effect of ground water on subgrade reaction modulus for p-y curves
# above gwt --> 1
# below gwt --> 2
gwtSwitch = 1

#----------------------------------------------------------
#  create spring material objects
#----------------------------------------------------------

# p-y spring material

for i in range(1 , nNodeEmbed+1):
    # depth of current py node
    pyDepth = L2 - eleSize * (i-1)
    # procedure to define pult and y50
    pyParam = get_pyParam(pyDepth, gamma, phi, diameter, eleSize, puSwitch, kSwitch, gwtSwitch)
    pult = pyParam [0]
    y50 = pyParam [1]    
    op.uniaxialMaterial('PySimple1', i, 2, pult, y50, 0.0)
    

# t-z spring material    
for i in range(2, nNodeEmbed+1):
  # depth of current tz node
    pyDepth = eleSize * (i-1)
  # vertical effective stress at current depth    
    sigV = gamma * pyDepth
  # procedure to define tult and z50
    tzParam = get_tzParam(phi, diameter, sigV, eleSize)
    tult = tzParam [0]
    z50 = tzParam [1]
    op.uniaxialMaterial('TzSimple1', i+100, 2, tult, z50, 0.0)


# q-z spring material
    
  # vertical effective stress at pile tip, no water table (depth is embedded pile length)
sigVq = gamma * L2
  # procedure to define qult and z50
qzParam = get_qzParam (phi, diameter, sigVq, Gsoil)
qult = qzParam [0]
z50q = qzParam [1]

#op.uniaxialMaterial('QzSimple1', 101, 2, qult, z50q) #, 0.0, 0.0
op.uniaxialMaterial('TzSimple1', 101, 2, qult, z50q, 0.0)

print("Finished creating all p-y, t-z, and z-z spring material objects...")


#----------------------------------------------------------
#  create zero-length elements for springs
#----------------------------------------------------------

# element at the pile tip (has q-z spring)
op.element('zeroLength', 1001, 1, 101, '-mat', 1, 101, '-dir', 1, 3)

# remaining elements
for i in range(2, nNodeEmbed+1):
    op.element('zeroLength', 1000+i, i, 100+i, '-mat', i, 100+i, '-dir', 1, 3)
    
print("Finished creating all zero-Length elements for springs...")

#----------------------------------------------------------
#  create pile nodes
#----------------------------------------------------------

# pile nodes created with 3 dimensions, 6 degrees of freedom
op.model('basic', '-ndm', 3, '-ndf', 6) 

# create pile nodes
for i in range(1, nNodePile+1):
    zCoord = eleSize * i
    op.node(i+200, 0.0, 0.0, zCoord)

print("Finished creating all pile nodes...")

# create coordinate-transformation object
op.geomTransf('Linear', 1, 0.0, -1.0, 0.0)


# create fixity at pile head (location of loading)
op.fix(200+nNodePile, 0, 1, 0, 1, 0, 1)


# create fixities for remaining pile nodes
for i in range(201, 200+nNodePile): 
    op.fix(i, 0, 1, 0, 1, 0, 1)
    
print("Finished creating all pile node fixities...")

#----------------------------------------------------------
#  define equal dof between pile and spring nodes
#----------------------------------------------------------

for i in range(1, nNodeEmbed+1):
    op.equalDOF(200+i, 100+i, 1, 3)
        
print("Finished creating all equal degrees of freedom...")

#----------------------------------------------------------
#  pile section
#----------------------------------------------------------
##########################################################################################################################################################################

#########################################################################################################################################################################

#----------------------------------------------------------
#  create elastic pile section
#----------------------------------------------------------

secTag = 1
E = 25000000.0
A = 0.785
Iz = 0.049
Iy = 0.049
G = 9615385.0
J = 0.098

matTag = 3000
op.section('Elastic', 1, E, A, Iz, Iy, G, J)

# elastic torsional material for combined 3D section
op.uniaxialMaterial('Elastic', 3000, 1e10)

# create combined 3D section
secTag3D = 3
op.section('Aggregator', secTag3D, 3000, 'T', '-section', 1)


#########################################################################################################################################################################

##########################################################################################################################################################################

# elastic pile section
#import elasticPileSection

#----------------------------------------------------------
#  create pile elements
#----------------------------------------------------------
op.beamIntegration('Legendre', 1, secTag3D, 3)  # we are using gauss-Legendre  integration as it is the default integration scheme used in opensees tcl (check dispBeamColumn)

for i in range(201, 201+nElePile):    
    op.element('dispBeamColumn', i, i, i+1, 1, 1)  
    
print("Finished creating all pile elements...")

#----------------------------------------------------------
#  create recorders
#----------------------------------------------------------

# record information at specified increments
timeStep = 0.5

# record displacements at pile nodes
op.recorder('Node', '-file', 'pileDisp.out','-time', '-dT', timeStep, '-nodeRange', 201, 200 + nNodePile, '-dof', 1,2,3, 'disp')

# record reaction force in the p-y springs
op.recorder('Node', '-file', 'reaction.out','-time', '-dT', timeStep, '-nodeRange', 1, nNodePile, '-dof', 1, 'reaction')

# record element forces in pile elements
op.recorder('Element', '-file', 'pileForce.out','-time', '-dT', timeStep, '-eleRange', 201, 200+nElePile, 'globalForce')

print("Finished creating all recorders...")

#----------------------------------------------------------
#  create the loading
#----------------------------------------------------------

op.setTime(10.0) 

# apply point load at the uppermost pile node in the x-direction
values = [0.0, 0.0, 1.0, 1.0]
time = [0.0, 10.0, 20.0, 10000.0]

nodeTag = 200+nNodePile
loadValues = [3500.0, 0.0, 0.0, 0.0, 0.0, 0.0]
op.timeSeries('Path', 1, '-values', *values, '-time', *time, '-factor', 1.0)

op.pattern('Plain', 10, 1)
op.load(nodeTag, *loadValues)

print("Finished creating loading object...")

#----------------------------------------------------------
#  create the analysis
#----------------------------------------------------------
op.integrator('LoadControl', 0.05)
op.numberer('RCM')
op.system('SparseGeneral')
op.constraints('Transformation')
op.test('NormDispIncr', 1e-5, 20, 1)
op.algorithm('Newton')
op.analysis('Static')

print("Starting Load Application...")
op.analyze(201)

print("Load Application finished...")
#print("Loading Analysis execution time: [expr $endT-$startT] seconds.")

#op.wipe

op.reactions()
Nodereactions = dict()
Nodedisplacements = dict()
for i in range(201,nodeTag+1):
    Nodereactions[i] = op.nodeReaction(i)
    Nodedisplacements[i] = op.nodeDisp(i)
print('Node Reactions are: ', Nodereactions)    
print('Node Displacements are: ', Nodedisplacements)