Chapter 2: Three–Dimensional Lateral Spreading Model
2.5 Surface Load Elements
In order to achieve the goals of this research, it is necessary to have models in which the liquefied layer is located at varying depths within the soil system. Due to the computational limitations induced by working with three–dimensional finite element models, the mesh is selectively refined in the area of the liquefied layer as opposed to a uniform refinement. This strategy, while sensible for computational optimization, necessitates the creation of customized meshes and input files for each case. In lieu of laboriously creating a series of models for each pile for liquefied layers at various depths, a uniform overburden pressure is applied to the ground (free) surface of the soil elements in the model. This overburden pressure, applied as a surface load, creates stress conditions in the soil which are equivalent to moving the liquefied layer to a deeper location.
In OpenSees, there is no built–in surface loading capability, forces can only be applied at a nodal level. Due to the irregular shape of elements in the lateral spreading model, as shown in Figures 2.3 and 2.4, determination of the equivalent nodal forces for a uniform surface load is a tedious task. One that would need to be performed for every increment of overburden pressure in every model. Instead of pursuing that somewhat tedious strategy for the application of overburden pressure in the finite element model, a new element is developed in OpenSees which is able to determine the appropriate nodal forces for a given magnitude of uniform pressure in a quadrilateral element. No additional stiffness is provided by these surface load elements, and the increase in computational cost is minimal, as only the assembly phase is affected. This new element eases the implementation of surface loads for the purpose of this research, while also increasing the capabilities of the OpenSees platform.
2.5.1 Formulation of Surface Load Elements
The developed surface load element is based upon a relatively simple strategy. The internal force vector for each element is replaced by the external force vector that would result from the application of a uniform surface loading. By creating a force imbalance in the elements, for equilibrium to be satisfied, there must exist an equal–but–opposite set of external forces which is applied to each of the elements on the surface. This set of external
forces is manifested in the application of energetically conjugate nodal forces representing the uniform surface loading.
The formulation of the surface load elements begins with the weak form of the principle of virtual displacements − Z V σ:∇sηdV + Z V b·ηdV + Z ∂Vσ t·ηdS =0 (2.1)
where σ is the stress tensor, ∇s is the symmetric vector operator, η is an arbitrary dis-
placement function, V is the volume of the body,b is the body force acting on the body,t is the surface traction vector, ∂Vσ is the portion of the surface of the body with prescribed
stresses, and S is the surface of the body.
Equation (2.1) expresses equilibrium for the system in terms of an arbitrary displacement function η. The vector of external forces, which follows from the third term of (2.1), can be determined by, first, expressing the arbitrary displacement as
η=X
I
NI(ξ, η)·ηI (2.2)
where NI are the (linear) shape functions, the subscript I refers to each of the four nodes
for the element, and ηI are arbitrary nodal displacements. The linear shape functions,NI, can be expressed for these quadrilateral elements as
NI =
1
4(1 +ξIξ)(1 +ηIη) (2.3)
by mapping a bi–unit square onto the quadrilateral surface patch. The normalized coordi- nates ξ andη, whereξI and ηI represent the nodal coordinates on the bi–unit square.
Applying (2.2) to the external force term of (2.1) results in
X I Z t·NI(ξ, η)Jdξdη ·ηI =:X I fIext·ηI (2.4)
which must hold for any arbitrary nodal displacement ηI, thus uniquely defining the nodal force
fIext=
Z
t·NI(ξ, η)Jdξdη (2.5)
where J is the Jacobian determinant necessary for the coordinate transformation toξ and
η, and the integration is performed over the bi–unit square to which the element has been mapped.
For a uniform surface pressure applied perpendicular to a given surface, the traction vector,t, is
t=−pn(ξ, η) (2.6)
in which p is the scalar magnitude of the pressure and n is the unit vector defining the outward normal of the surface. To establish the outward normal for the surface elements, general base vectors are defined for each element. There are two general base vectors, one in theξ direction,gξ, and the other in theη direction on the element,gη. This general base
can be found from the nodal position vector as x(ξ, η) =X
I
NI(ξ, η)·xI (2.7)
wherexI are the nodal position vectors. The base vectors follow as
gξ = ∂x ∂ξ = X I ∂NI ∂ξ ·xI = X I ξI 4(1 +ηIη)·xI (2.8) and gη = ∂x ∂η = X I ∂NI ∂η ·xI = X I ηI 4 (1 +ξIξ)·xI (2.9)
A local normal vector, ˆn, for each element is defined by the cross product of the two base vectors as
ˆ
n=gξ×gη (2.10)
This vector contains information about both the area and the direction of the outward normal for its respective element. It can be shown that the norm of ˆnis equal to the surface Jacobian determinant,J. Thus, the relationship between ˆn and n can be expressed as
n= 1
Jnˆ (2.11)
This relation is used to express the surface traction (2.6) in terms of the local normal vector as
t=−p
Jnˆ (2.12)
Applying (2.12) to (2.5), the external force acting at node I is obtained as
fIext=−p Z
ˆ
n(ξ, η)NI(ξ, η)dξdη (2.13)
The integral of (2.13) is evaluated using four–point Gaussian integration of the form
fIext= 2 X α=1 2 X β=1 −pnˆ(ξα, ηβ)NI(ξα, ηβ)wαwβ (2.14)
in which the Gaussian quadrature and weights are (ξα, wα) = (ηβ, wβ) = (±
1 √
3,1).
In each surface element, the internal force vector is set equal to the vector of forces resulting from a uniform surface traction as determined by (2.14). To satisfy equilibrium, the elements must be subject to a set of nodal forces in opposition to the prescribed external force.
2.5.2 Validation of Surface Elements
A simple model is created in order to validate the successful implementation of the sur- face load elements in OpenSees. The model is meshed using irregularly–shaped elements as depicted in Figure 2.5. The newly created surface load elements are applied to the up- per surface of this model over four layers of linear–elastic isotropic brick elements. The irregular shape of the mesh allows the elements to be tested for generality, as they should create energetically consistent nodal loads across the surface regardless of the shape of the quadrilateral elements, and thus create a constant stress in the solid.
Figure 2.6: Distribution of vertical stress in the surface load element validation model.
The validation test used for the surface load elements, consists of the application a uniform loading of 10 kPa to the upper surface of the model. The base of the model is held fixed against translation in the direction of the loading. As shown in Figure 2.6, which shows a three–dimensional view of the validation model, the surface load elements are able to create nodal forces which create an equivalent loading to the assumed uniform load, resulting in a constant vertical stress distribution of −10 kPa. This result validates that the surface load elements created for use in OpenSees perform as intended.