Brian Zelenko, P.E., Erik Loehr, Ph.D., P.E., Andy Boeckmann, P.E., Chris Dorney, AICP, and Elizabeth Godfrey, P.E. WSP USA, Inc. FHWA-HIF-23-008 February 2023
Geohazards—such as landslides, liquefaction, rockfalls, subsidence, expansive/collapsible soils, and erosion—can pose major threats to transportation assets. Extreme weather events can also trigger and/or exacerbate geohazards, and the increasing incidence of such events is a concern in certain regions of the United States. A geohazards management program that evaluates the associated risk can help transportation agencies manage these threats by making comprehensive system performance decisions. Such a program can help agencies manage cost-effective methods for characterizing and reducing risk, develop adaptation methods, and establish a resilient transportation system.
Betsy Godfrey, Brian Zelenko, P.E., Erik Loehr, Ph.D., P.E. Parsons Brinckerhoff FHWA-HIF-23-009 May 2016, updated February 2023
A Peer Exchange was held March 22–23, 2016, in Atlanta, Georgia, to gather experts from different fields relating to geologic hazards, climate change, extreme weather events, and geotechnical asset management (GAM). The purpose was to share perspectives and brainstorm ideas on how FHWA can best continue its geologic hazards program and research. Feedback from the Peer Exchange contributed to updating a draft work plan from Phase I of this study on geohazards, extreme weather events and climate change resilience. The Peer Exchange consisted of presentations from attendees, discussion based on these presentations, breakout sessions for more focused discussion, and brainstorming about suggested changes to the provided draft work plan for a future Phase III study on geologic hazards, extreme events, and climate change. This report provides a summary of the Peer Exchange meeting.
On this site we feature the U.S. Army Corps of Engineers publication Retaining and Flood Walls, which details the design of several types of retaining walls. As it was published a good while back, it details the design of these walls using hand calculations. Sometimes these can get tedious, especially when the aptly named “trial wedge” solutions are employed.
These days it’s more likely that a computer solution–be it a true finite element analysis or simply the automation of those tedious hand calculations–is used to finalise the design of a retaining wall. An example of such an analysis is shown above, and addresses in particular an issue that gets the short shrift in classical retaining wall analysis of any kind: global stability failure. An idea of a hand solution to this problem is shown at the right. Global stability failures still happen and are generally disastrous; beyond a conventional slope their analysis is fairly complex.
But in the meanwhile, what do we do to verify that we’ve got it right with a computer solution? Or what can we do to start with a reasonable design that we can refine with numerical analysis? Back in the “slide rule era” we used quick, “back of the envelope” methods to design things, and we can use them today both to get started and to gain a basic understanding of the elements of the design.
In this case we’re going to discuss the design of a cantilever retaining wall, an example of which is shown below.
The example we’ll look at is below. We need to design this wall to prevent failure against three events: sliding, overturning and bearing capacity failure.
Defining the Geometry, Weights and Centres of Gravity
Our datum/coordinate origin is at the toe, which is convenient since we assume that the overturning moment is computed around the toe. Since this is a cantilever/gravity wall, the weight of the wall is part of the resistance force and moment. Included with that is the backfill that is trapped behind the wall (shown above.)
We start by dividing up the wall into sections, each of which has a weight and a centre of gravity. The first section we take up is the backfill, which is a simple rectangle shown above.
At this point we need to make one correction to the Corps’ work: the weights and moments are in pounds per unit length of wall and ft-pounds per unit length of wall, respectively. Leaving those per unit length designations is a common shortcut among practitioners but is confusing for students, who frequently find the unit length concept difficult to grasp at first. The weight of the backfill is properly 18,000 lbs/ft of wall and the moment around the toe ((clockwise) is 162,000 ft-lbs/ft of wall.
Computing the weights and moments of the various sections of the wall itself yields the following results.
Taking all of this information and processing it yields the total weight, total moment around the toe, and moment arm around the toe:
Now comes the tricky part: in retaining wall design, we traditionally define the factor of safety as
(1)
Where the “F” values can be forces or moments and FS is the factor of safety. Sometimes this is not the optimal way of applying factors to account for uncertainties, especially when we get to LRFD. Another approach is either to increase the driving force or reduce the resisting force. We do the latter with sheet piling (where there there are earth forces to resist) but here we’ll to the former. To make this happen we first define a shear mobilisation factor (SMF) thus
(2)
The value of can be computed thus:
(3)
For this problem, assuming SMF = 2/3 and φ’ = 35°, by substitution φ’mob = 25°. We will discuss the effect of cohesion later.
Now we turn to computing the force of the soil behind the wall on the wall.
We note the following:
Rankine theory is used. Methods shown in both Retaining and Flood Walls and the Soils and Foundations Reference Manual use Coulomb and/or log-spiral for these computations. The problem with this is that the value of wall friction δ is sometimes difficult to determine without data or experience, and in reality most of this pressure bears on soil, not the wall. So Rankine is easier to start with, and is more conservative.
The backfill is level, the formula for Ka only applies in that case. We will discuss sloping backfill below.
Note that φ’mob (not φ’ ) is being used to compute the soil force.
Analysing Sliding and the Location of the Resultant/Overturning
At this point we can compute the forces on the base (both the forces T’ and N’ and the location of the resultant xN’,) which are shown below.
The calculations are shown below.
Turning to the sliding problem, the driving force is the earth pressure force and the resisting force is the maximum Coulomb friction of the base/soil interface. In this case the value of δ is equated to the unmodified value of φ’, although that isn’t always the case. (The basis for this is that there is a thin layer of backfill sand under the wall, under which is a different foundation soil.) We then apply Equation (1) and determine the factor of safety against sliding, which checks out against the Corps criteria.
Knowing the location and magnitude of the resultant, we compute the maximum and minimum pressures on the base. Since both pressures are compressive, the resultant is in the middle third, and thus we can proceed with the base design.
One thing that is missing from this analysis is a specific analysis for overturning. In this case we make a common assumption that, as long as the resultant force of the wall is within the kern and there are no negative pressures on the base, overturning will not be experienced. It is certainly possible to do an explicit overturning analysis to check this result.
Bearing Capacity Analysis
With the wall’s sliding and overturning established, we turn to the bearing capacity analysis of the base. The complete bearing capacity equation, from the Soils and Foundations Reference Manual (with modification,) is
(4)
where
qult = ultimate unit upper bound bearing capacity
sc, sγ and sq = shape correction factors. These are unity in this case since it is a continuous foundation (generally the case with retaining walls)
bc , bγ and bq = base inclination correction factors (unity in this case since the foundation is level)
Cwγ and Cwq are groundwater correction factors (unity in this case since groundwater isn’t an issue, a rare event with retaining and especially flood walls)
Ic, Iγ and Iq = load inclination factors, discussed below
Nc , Nq and Nγ are bearing capacity factors that are a function of the friction angle of the soil. Nc , Nq and Nγ are shown in the table below. These are handled differently when a slope is present. They are given in the Soils and Foundations Reference Manual. There is a general consensus for Nc and Nq but not Nγ. In this case we will use Vesić’s values for it, following AASHTO/FHWA practice. For this case the base soil φ’ = 40° we have Nc = 75.3, Nq = 64.2 and Nγ = 109.4
Bf = Base width of the foundation. In this case, with an eccentrically loaded foundation, this must be reduced to the equivalent foundation width by the formula B’f = Bf – 2e = 13 – (2)(1.27) = 10.46’.
q0 = overburden pressure on the base from the dredge (low) side of the wall. For walls such as this we neglect all effects of this, both any potential passive lateral pressure and overburden pressure.
c = cohesion of the soil = 0
Because of this and the previous point, we can neglect the first two terms of Equation (4) and only concern ourselves with the last one.
Load inclination is the result of two perpendicular loads acting on the base of the foundation. It is illustrated in the sketch at the left.
The load inclination factors are given as follows:
(5a)
(5b)
where the load inclination angle is given as follows
(6)
Substituting yields 𝞭’ = tan-1 (10,137.5 lbs/26,625 lbs) = 20.8°. The friction angle of the base soil proper is 𝟇 = 40°. Substituting into Equation (5b) yields l𝞬 = (1-20.8/40)2 = 0.23. We can neglect the factors for Equation (5a) as those terms do not apply to this situation, but for completeness lc = lq = (1-20.8/90)2 = 0.591.
Settlement of retaining wall is an important topic, as settlement of walls and levees has led to overtopping (as we found out the hard way during Hurricane Katrina.) Instead of picking a method and doing it “by hand,” we will use the USACOE software package CSANDSET, developed by Virginia Knowles. To accomplish this we need to do the following:
The foundation with we use is the reduced foundation width, or B’f = Bf – 2e = 13 – (2)(1.27) = 10.46’.
The normal load is N = 26,625 lbs.
The unit load on the foundation is thus 26625/(2000*10.46)=1.27 tsf (the units used by the program.)
We assume a length of 50′; this puts the L/B > 10, which is an assumption for continuous foundations.
The input data is shown in the screenshot below.
The SPT and CPT are taken from “typical” values as they were not given in the problem statement. The option data is generated by the program. The horizontal (at-rest) earth pressure is Jaky’s Equation for normally consolidated soils.
The supplemental data generated by the program is at the end of the post.
Dealing with Sloping Backfill
The example problem above has a level backfill. Sloping backfills–usually positive (up from the wall,) occasionally negative, are common with retaining walls. The problem of the sloping backfill is illustrated at the right.
Without going into the actual solution of the problem using a sloping backfill, the following changes must be made in order to accommodate the effects of this condition:
The Rankine equation for sloping backfill needs to be applied. This can be found in the post Rankine and Coulomb Earth Pressure Coefficients. The coefficient computed is actually Coulomb theory applied without wall friction; the differences between this and “extended Rankine” theory for sloping backfill are discussed in the post on the coefficients.
An additional region needs to be defined with the soil below the sloping backfill (assuming it’s positive) as shown in the diagram below.
The length a of this region is already defined. The length b is given by the equation b = a tan (β). That length is important in a) computing the weight of the region and b) computing the additional length of the height the soil bears on the wall, or H + b.
The backfill force must be separated into horizontal and vertical components, as shown in the diagram above. The vertical component actually increases both the weight on the foundation and the resisting moment of the weight.
Assuming a β = 10° and applying φ’mob = 25°, with the geometry shown we note the following:
With the two angles shown, Ka= 0.462, which is higher than the level backfill value.
The value of b = (8)(tan(10°))=1.41′. This means that the total height is 20 + 1.41 = 21.41′.
The lateral pressure at the base is (0.462)(21.41)(125) = 1236.5 psf.
The total lateral force on the wall Fh is (1236.4)(21.41)/2 = 13,236 lb/ft of wall. The resultant of that force is 21.41/3 = 7.14′ above the base of the foundation.
That lateral force has a horizontal component of (13236)(cos((10°)) = 13,035 lb/ft and a vertical component of (13236)(sin(10°)) = 2298 lb/ft. The latter has a moment arm of 13′ from the toe.
The area added by the sloping backfill has a weight Wba = (125)(8)(1.41)/2 = 705 lb/ft. Its centroid is located 5 + (2)(8)/3 = 10.3′ from the toe.
We will leave working out the effects of this backfill slope to the reader.
The Shear Mobilisation Factor (SMF) and Cohesive Soils
If we have soils with cohesion in the backfill, the cohesion should be modified in a similar way to the friction angle thus:
One thing any academic does (except those at grand institutions where they get to have someone else do it for them) is evaluate students through grading. Most institutions afford students the opportunity to retaliate through the faculty evaluation system in place. That usually takes places at the end of the term, although my institution now had mid-term evaluations in place.
Student evaluations of faculty always take me back to this incident in my own undergraduate saga:
One of the things that the Mechanical Engineering department required* its majors to take was Logic, which was offered by the Philosophy Department. Most of the engineers did pretty well in this course, which was doubtless a source of secret frustration to liberal arts’ professors.
One day I went up to pick up a test from the professor. The professor looked at the grade, noted that I had nearly aced it, looked at me, and exclaimed, “You’re not as dumb as you look!”
The purpose of student evaluations of me is to determine whether they agree with my Logic teacher’s opinion or not.
*Originally posted here. Since then I discovered that there was another option available, but my advisor at the time did not avail me of that choice.
The impulse–response (IR) test is the most commonly used field procedure for assessing the structural integrity of piles embedded in soil. The IR test uses the response of the pile to waves induced by an impulse load applied at the pile head in order to assess the condition of the pile. However, due to the contact between the pile and the soil, the recorded response at the pile head carries information not only about the pile, but about the soil as well, thus creating the as-yet-unexplored opportunity to characterize the properties of the surrounding soil. In effect, such dual use of the IR test data renders piles into probes for characterizing the near-surface soil deposits and/or soil erosion along the pile–soil interface. In this article, we discuss a systematic full-waveform-based inversion methodology that allows imaging of the soil surrounding a pile using conventional IR test data. We adopt a heterogeneous Winkler model to account for the effect of the soil on the pile’s response, and the pile’s end is assumed to be elastically supported, thus also accounting for the underlying soil. We appeal to a partial differential equation (PDE)-constrained-optimization approach, where we seek to minimize the misfit between the recorded time-domain response at the pile head (the IR data), and the response due to trial distributions of the spatially varying soil stiffness, subject to the coupled pile–soil wave propagation physics. We report numerical experiments involving layered soil profiles for piles founded on either soft or stiff soil, where the inversion methodology successfully characterizes the soil.
Over the years I have been looking at many different aspects of the problem of pile dynamics, which includes both prediction of drivability of piles and the inverse problem of estimating the static resistance of piles based on their performance during driving. In the course of working with all of this many ideas have come to mind; two of those are as follows:
Is it possible to use the pile hammer as a geotechnical sounding tool to determine the properties of soil layers into which the pile is being driven? In Improved Methods for Forward and Inverse Solution of the Wave Equation for Piles the soil at any given point was defined by the “Mohr-Coulomb triple” (unit weight, internal friction angle and cohesion) along with other properties, which is the point for most soil testing. The inverse method returned those properties.
The present paper does some interesting things to get to those solutions but ultimately doesn’t quite get to reaching a solution to these problems.
The Strong Point
Probably the strongest point of the paper is the entire mathematical presentation, from the development of the method to its execution. It shows computational proficiency of a high order. For example, it is the first time in pile dynamics that I have seen the use of the conjugate gradient technique. As the product of a PhD program with a heavy emphasis on computational fluid mechanics, this technique was well familiar to me (along with GMRES, which was the method of choice for my colleagues.) One of the challenges the geotechnical industry faces moving forward is the proper application of numerical techniques to geotechnical problems which are non-linear in ways which are unknown in other fields. We have people who are specialists with the geomechanics and people who are specialists in numerical methods, but few are those who are proficient in both.
One thing I would like to mention is that my use of a polytope method–which had many drawbacks–was driven by the difficulties in optimising geotechnical problems. Those difficulties are caused largely by the existence of false minima and maxima in the solution. It is why we still see, for example, grid optimisation used for slope stability problems: the use of, say a Newton’s Method type of optimisation may easily result in finding a false minimum. Although I think the author’s techniques have great promise of solving these problems, it is something they will have to watch for moving forward.
The Soil Property Issue
Probably the greatest weakness of the paper is the way soil properties are characterised.
Simplified Hammer-Pile-Soil System (from Warrington (1997))
In this diagram the ram mass M impacts the cushion with a velocity Vo. There are several stiffnesses and damping coefficients k and c respectively. The accessory has a mass of m. The pile has an acoustic speed c, an impedance Z and a geometry ratio rg. For inverse analysis the hammer, cushion and driving accessory can be deleted and a force F(t) and velocity V(t) (both a function of time) substituted at the pile head.
The equation of motion u(x,t), which is a function of both the distance from the pile head x and time t, is given by the equation
(1)
The variables a and b are stiffness and damping related coefficients which are related to k and c by geometric and material property considerations, as discussed in Closed Form Solution of the Wave Equation for Piles.
Although the present paper allows for k(x), the one major difference between the governing equation presented above and the one in the paper is the omission of damping by the latter. (This omission is also repeated at the pile toe.) The soil damping is for the most part a representation of the propagation of wave energy from the pile as it is dynamically loaded. It is impossible to avoid in one form or another. First derivatives like that are always a danger in problems such as this. One way to get around that is to redistribute the damping into the spring and mass terms using Rayleigh damping. This is very frequency dependent and can be tricky to accurately apply; however, if it can be done successfully (and the authors’ note of wavespeed changes with soil interaction may be part of the solution) it would bypass the problems created by the first derivative. (The same comments regarding the shaft also apply at the toe, where an additional mass would have to be applied to achieve Rayleigh damping.)
But that doesn’t address what is, in some ways, the more serious issue: applying a linear model to a very non-linear problem. Concentrating on the shaft resistance, let us begin by noting the results in Estimating Load-Deflection Characteristics for the Shaft Resistance of Piles Using Hyperbolic Strain Softening, and stipulating that, even with hyperbolic stress-strain considerations, up to the time of separation between the shaft and the soil the load-deflection relationship is essentially linear. This study showed that, for the specific case in question, the geometric nonlinearity of the deflecting soil around the shaft and the material nonlinearlity of the soil offset each other to a large degree. Obviously more study needs to be done but this is a start.
Having said that, let us look at a diagram I have used and modified over the years:
At zero strain we have the small strain elastic or shear modulus. As strain increases, if we use a linear model for load-deflection the only way the model can simulate that kind of response is to do some kind of “secant modulus” estimate. What this means is that the elastic modulus/shear modulus/spring constant is strain dependent, which will vary with loading condition (to put it in classical geotechnical terms, to what extent the shaft resistance is mobilised.) It is also worth noting that this mobilisation is not identical in static and dynamic testing, even on the same pile.
Based on all of this, it is difficult to see how the results of the inverse method can be used to accurately characterise the load-deflection characteristics of the pile, let alone the properties of the soil.
This paper is an interesting study of the problem at hand. While some significant advances have been done in the numerical treatment of the problem, the physics of the pile-soil system need to be re-examined and improved.
(1)
(2)
can be computed thus:
(3)
(4)
(5a)
(5b)
(6)
(7)
(1)
vulcanhammer.net 