3. Macro command description#
The macro command POST_NEWMARK makes it possible to obtain an estimate of the irreversible lateral displacement of a potentially slippery area of an embankment structure (dam/dike) via the Newmark [bia1] _ method.
The Newmark method is based on the idea that the potentially slippery area of the structure can be approached by a sliding block on an inclined plane. During the earthquake, this block slides along the inclined plane when the average acceleration (\({a}_{m}\)) of the block exceeds a fixed value, called critical acceleration (\({a}_{y}\)). The method considers that the residual displacement of the sliding block can be obtained by integrating twice the times of the average acceleration exceeding the critical acceleration. This operation is carried out in the following manner:
obtaining the speed signal by integrating the average acceleration exceeding the critical acceleration,
clipping of the signal in speed to v>0 and then integration to obtain the signal while moving.
From a dynamic finite element calculation, the average acceleration of a potentially slippery zone is defined as the quotient of the resultant of lateral forces \({F}_{L}\) along the interface between the potentially slippery zone and the rest of the structure and the mass \(m\) of this zone:
Critical acceleration is defined as the acceleration that leads to a safety factor of 1.0 for the potentially slippery area. From the critical acceleration, we define the seismic coefficient \({k}_{y}\) by relating the critical acceleration to the value of the acceleration of gravity \(g\):
The POST_NEWMARK macro command only accepts 2D meshes and two types of sliding zones:
a simple circular shape, whose position is provided by the user. In this case, the resultant of the lateral forces is calculated from the cells of the mesh on which the dynamic calculation was carried out and part of which belongs to the circle defined by the user.
a shape defined from an auxiliary mesh, which must be positioned on the geometric location of the sliding mass. In this case, the resultant of lateral forces on the cells corresponding to the geometric location of the auxiliary mesh is calculated.
The seismic coefficient \({k}_{y}\) can be provided by the user as input to the macro-command, if obtained for example from a pseudo-static stability calculation with a non-linear behavior law integrating a failure criterion, or by a limit analysis methodology.
The user may also not provide seismic coefficient \({k}_{y}\). In this case, it is estimated from the evolution of the safety factor with the average acceleration (cf. test case sdlp200). The regression is done for a range between 0.8 and 1.2 of the value of the safety factor, and the critical acceleration value is retained by default conservatively because it is at -3 standard deviations from the mean of the regression.
3.1. Calculation of the safety factor in statics and dynamics#
The macro command POST_NEWMARK makes it possible to estimate the static and dynamic safety factor of a potentially slippery zone defined from:
of an auxiliary mesh defining the potentially sliding zone or directly of the data provided by the user of the position of the sliding circle. In the latter case, the macro-command builds the auxiliary mesh directly,
of a static field of constraints resulting from gravity.
For this we use the definition of global safety factor \({F}_{s}^{g}\) from Kulhawy [bib2] _:
where \(c\text{'}\) and \(\mathrm{\varphi }\text{'}\) are the effective cohesion and friction angle of the material, \(\mathrm{\sigma }{\text{'}}_{n,s}\) is the normal stress and \({\mathrm{\sigma }}_{t}\) is the stress tangential to the sliding line, both obtained from a static gravity calculation, \(N\) the spatial discretization of the sliding line, and \(\mathrm{\Delta }L\) the length of each discretized element.
For a dynamic calculation, the estimation of the safety factor takes into account the variations in normal and tangential stresses along the sliding line, using the following formula:
: label: eq-4
{F} _ {s} ^ {g} =frac {sum _ {i} ^ {N} (ctext {”} +mathrm {sigma} {text {“}}} _ {n, s}mathrm {tan}}mathrm {tan}} (mathrm {tan}} (mathrm {varphi}} ^ {n}})mathrm {delta}}} _ {n, s}mathrm {tan} (mathrm {tan}} (mathrm {varphi}text {“}))mathrm {Delta} L+sum _ {tan} (mathrm {tan}} (mathrm {varphi}}text {“}) ^ {N} (mathrm {sigma} {sigma} {text {“}} {text {“}}} _ {n, d}mathrm {varphi}text {“}))mathrm {Delta} L} {text {Delta} L} L} {text {” Delta} L} {text {”} L} {text {”} L} {text {“})mathrm {Delta} L} {text} L} {text {”} L} {text {”} L} {text {”} L} {text {”} L} {text} L} {text {”} L} {text {”} L} {text} L} {text {”} L mathrm {Delta} L+sum _ {i} ^ {i} ^ {N} {mathrm {sigma}} _ {t, d}mathrm {.} mathrm {Delta} L}
where \(\mathrm{\sigma }{\text{'}}_{n,d}\) is the normal stress and \({\mathrm{\sigma }}_{t,d}\) is the stress tangential to the sliding line for dynamic calculation. The user must therefore provide the field of static constraints resulting from gravity. The macro-command estimates the safety factor for each time step.
3.2. Test cases#
sslp119 [V3.02.119] Calculation of the static safety factor for a homogeneous embankment.
zzzz402 [V1.01.402] Dynamic response of an embankment structure during an earthquake.
sdlp200 [V2.03.200] Comparison of the earthquake resistance of a homogeneous embankment between POST_NEWMARK and GeoSlope
3.3. Bibliography#
Newmark, N.M. 1965. Effects of earthquakes on dams and embankments. Géotechnique, 15(2): 139-160.
Kulhawy. F.H. 19,69. Finite element analysis of the behavior of embankments .PhD. Thesis, the Universiry of Califomia, at. Berkley, California. U,S.A.