1. The document describes the Beam on Nonlinear Winkler Foundation (BNWF) model, which models soil-structure interaction using nonlinear springs. It captures both material and geometric nonlinearity in the soil. 2. The BNWF model consists of vertical and horizontal springs distributed below the foundation. Vertical springs capture settlement, rocking, and uplift. Horizontal springs model frictional sliding resistance and passive resistance. 3. The document provides equations to calculate the properties of the vertical Q-z and horizontal P-y and T-z springs based on soil parameters, including ultimate capacity, stiffness, and displacement at 50% capacity. It also shows hysteretic load-displacement curves for the different spring models.

1. Beam on Nonlinear Winkler Foundation
(BNWF)
July 30, 2018
1 Introduction
In 2009, Raychowdhury and Hutchinson developed a new approach to model the interaction
between the soil and the foundation that can capture both material and geometric nonlin-
earity. Material nonlinearity is associated with the inelastic behavior of the material after
reaching the yielding point while the geometric nonlinearity is associated withe the sliding
and uplifting of the foundation. BNWF model was validated by comparing the theoretical
results of a square and strip footing by their experimental results.
2 Description of BNWF Model
Fig.1 shows a sample of BNWF model applcatios which represents the soil-structure inter-
action of a shallow foundation. The model consists of a set of vertical and horizontal springs
with different stiffnesses. The vertical springs are distributed on the base with different
stiffnesses to capture the settlement, rocking and the uplift of the foundation. Horizontal
springs are intended to capture the frictional sliding resistance and passive resistance.
Using OpenSees software, BNWF can be modelled as follow:
• The foundation can be modeled by a series of horizontal elasticBeamColumn elemets
with high rigidity.
• The soil can be modeled by a set of closely spaced nonlinear springs with different
stiffnesses in vertical direction and another two nonlinear springs in the horizontal
direction to account for the passive and frictional sliding resistances.
1. Vertical Q-z springs are modeled using zeroLength elements with uniaxialmate-
rial QzSimple2.
2. Horizontal P-y spring is modeled using zeroLength element with uniaxialmate-
rial PySimple2.
3. Horizontal T-z spring is modeled using zeroLength element with uniaxialmate-
rial TzSimple2.
1

2. .
Figure 1: Representation of the BNWF model
”Prishati Raychowdhury1 and Tara C. Hutchinson2, 2008”
Figures 5, 7, and 8 show the hysteretic responses of the vertical and horizontal springs
considering the following soil properties:
• Soil Type: 1 ”Clay”
• Cohesion factor C: 100 KPa
• Friction Angle φ: 0.0
• Unit weight γ: 16 KN/m3
• Shear Modulus G: 30 MPa
• Poisson’s ratio ν: 0.4
• Radiation Damping Crad: 0.05
• Tension Capacity Tp: 0.1
Using the above defined soil parameters and the assumed spacings between the vertical
springs, the material properties of the vertical and horizontal springs (Qult, Z50) can be
calculated as follow.
2

3. 1. QzSimple 2
The ultimate bearing capacity can be calculated using any of the methods listed in
Fig.2. For accurate values, at least two methods should be used to calculate qult and
if there is a big discrepancy between the two values, a third method should be used
to calculate an average value of qult. In this example we used only Terzaghi (1943)
[1] method for its simplicity and it is suitable for horizontal footings subjected to
concentric load.
Figure 2: Bearing-Capacity Equations [2]
3

4. Figure 3: Bearing-Capacity Factors for Terzaghi Equations
Figure 4: Shape, Depth, and Inclination Factors
4

5. The ultimate load capacity of the footing can be calculated from Eq.1:
Qult = qultBfootingLfooting (1)
The vertical stiffness of the footing can be estimated by Eq.2 which was introduced by
Gazetas [3] in 1991. The vertical stiffness should be divided into closely spaced springs
distributed along the base with different stiffnesses.
Kv =
GL
1 − ν
0.73 + 1.54
B
L
0.75
(2)
The displacement at which 50% of the ultimate load is mobilized can be calculated
from eq.3, where Ks is the spring stiffness.
Z50 = j
qult
Ks
(3)
Table 1: Values of j factor
Soil Type 1 Soil Type 2
jQz 0.525 1.39
jPy 8.00 0.524
jTz 0.708 2.05
i
All previous values are from OpenSees documenta-
tion.
Using OpenSees software, QzSimple2 material can be defined as:
uniaxialMaterial QzSimple2 matTag soilType Qult Z50 Tp Crad
Fig.7 shows the hysteretic response of the QzSimple2 material considering tension
capacity of 10% of the compression capacity, the vertical displacement is normalized
to Z50 and the vertical load is normalized to qult. Fig.6 shows the hysteretic response
where tension capacity is neglected.
5

6. -20 -15 -10 -5 0 5 10 15 20
Normalized Vertical Displacement, s/z50
-0.2
0
0.2
0.4
0.6
0.8
1
NormalizedVerticalLoad,q/qult
Figure 5: Hysteretic Response of Q-z Springs (Axial and Rotational)
0 2 4 6 8 10 12 14 16 18 20
Normalized Vertical Displacement, s/z50
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
NormalizedVerticalLoad,q/qult
Figure 6: Hysteretic Response of Q-z Springs (Axial and Rotational)-Zero Tension
6

7. 2. PySimple 2
Using OpenSees software, PySimple2 material can be defined as:
uniaxialMaterial PySimple2 matTag soilType Pult Xp50 Tp Crad
The ultimate lateral passive capacity of the footing can be calculated using Eq.4 fol-
lowing Rankin’s passive earth pressure theory, where Kp can be determined from Fig.4.
Pult = 0.5KpGsDf K2
p Lf (4)
The stiffness of the Py spring can be estimated using Gazetas [3] equations:
Kp =
GL
2 − ν
2 + 2.5
B
L
0.85
(5)
The displacement at which 50% of the ultimate load is mobilized (Xp50) can be calcu-
lated using eq.2 and Table 1.
-20 -15 -10 -5 0 5 10 15 20
Normalized Lateral Displacement, u/x50
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
NormalizedLateralLoad,V/Pult
Figure 7: Hysteretic Response of P-y Spring (Passive Resistance)
7

8. 3. TzSimple 2
Using OpenSees software, TzSimple2 material can be defined as:
uniaxialMaterial TzSimple2 matTag soilType Tult Xt50 Tp Crad
The sliding resistance of the foundation can be calculated using Eq.6, where: Wg is
total weight above the foundation, δ is the friction angel between the soil and the
foundation, Ab is the area of the footing base, and C is the cohesion factor. For
cohesionless soil, the second term can be neglected.
tult = Wgtanδ + AbC (6)
The stiffness of the Tz spring can be calculated using Eq.5. The displacement at which
50% of the ultimate load is mobilized (Xt50) can be calculated using eq.2 and Table 1.
-20 -15 -10 -5 0 5 10 15 20
Normalized Lateral Displacement, u/x50
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
NormalizedLateralLoad,V/Tult
Figure 8: Hysteretic Response of T-z Springs (Sliding Resistance)
8