The Challenge of Foundations on Layered Soils
One of the major challenges for all geotechnical engineers when considering foundations on layered soil, is that “design codes” don’t provide closed form solutions for calculating the bearing capacity. The design methods specified in such codes are typically for drained or undrained conditions including linearly increasing strength gradient if one is lucky. It is normally left to the designer to satisfy him/herself that the chosen design method and assumed failure mechanism are suitable for the foundation geometry, soil conditions and loading under consideration.
There are some approaches available for considering load distribution on layered soil i.e. onto an underlying layer, and these are sometimes used for checking for critical failure mechanisms and evaluating the bearing capacity of foundations of mobile drilling units (Figure 1).
Figure 1: Alternative Failure Mechanisms for A Foundation on Layered Soil
When installing a jack-up rig footing (spud can) the loading is quite clearly dominated by vertical loading and the SNAME guidelines on site specific assessment for jack-up mobile drilling units (MODUs) provide a useful commentary when considering installation on layered soils. Some would argue though that the method outlined in this document it is still some way from being a robust approach.Given that the guidelines are currently being revised, this might be a correct view to take.
Particular Concerns for Subsea Structures
For geotechnical engineers practicing in the the offshore environment, there is an added complication, in that for typical foundations, the design case is normally defined by horizontal and / or moment loading. The starting point for our analysis is very similar to the case for vertical loading, however we have a tricky question to contend with: how do we model the response of the foundation system to increasing horizontal load?
We can start by making some assumptions:
- The limiting horizontal load will be controlled by the shear stress mobilised on the interface between the foundation and the upper soil unit i.e. Hmax = W’ tan(δ)/γm on sand and Hmax = A’. suDSS/γm on clay;
- The limiting vertical load (Vmax) will be controlled by the the clay layer, if the clay resistance is less than the resistance of the sand (but we would need to check for punch-through i.e. brittle failure of the sand layer;
- The limiting vertical load will be controlled by the sand layer if the clay is “strong” i.e. heavily overconsolidated relative to the sand.
If we wish to try to apply existing closed form limit equilibrium solutions to the problem, then we might start by considering either of case i or ii in Figure 1 by performing a modified calculation for the appropriate soil unit. In doing so, we would need to take into account the effective bearing area and any effective embedment. In case 3. we might also be concerned with the thickness of the sand layer and whether a fully developed failure surface could be mobilised in this unit.
For simplicity and to aid my commentary, we shall consider that assumptions 1 and 2 hold for our foundation and state that the clay layer is relatively “weak” compared to the overlying sand. We shall also make some assumptions on the soil properties:
- Sand: φcrit = 30 deg, γ’ = 7 kN/m3
- Clay: Suave= 10 kPa, γ’ = 7 kN/m3
The sand layer is taken as been silty and the clay layer slightly overconsolidated,a rather common occurrence for North sea applications in fact
Constructing A Stability Envelope
If we turn to ISO, DNV or API codes for assistance in performing our calculations, then we can use the well established interaction formula to apply modifications to vertical bearing capacity based on the ratio of vertical to horizontal loading. The end product of such a calculation is a stability envelope in vertical-horizontal (VH) load space as presented in Figure 2:
Figure 2: A typical VH stability envelope for clay soil
The curves presented in Figure 2 were generated for a 3m wide strip footing using the formulation presented in DNV Classification notes 30.4.
The stability envelope for clay generated following assumption 2. and the sliding limit for sand based on assumption 1, are used in this case to generate a composite failure envelope that resepects both constraints. The area inside the bounded zone can be taken as stable for all combinations of vertical and horizontal loading. Outside of the bounded zone, the foundation will be unstable. It is also possible to estimate the effect of “load spread” through the surficial sand layer by comparing the stability curves for clay – in this case a load spread of 1h:2v is taken and it is clearly a beneficial effect. The typical range of accepted load spread factors for vertical loading is between 1h:2v and 1h:4v.
It is worth noting that this curve is valid for a fixed moment, and so it represents a slice through a three dimensional yield envelope that describes the permissible combinations of VHM loading.
Without any further guidance from design codes we might be quite content with the approach outlined. It is not unusual to see such a method being used by practitioners in the offshore industry, perhaps we can conclude that the design codes provide sufficient reassurance to not challenge our assumptions further. It is worth noting however, that we have taken a closed form solution and applied it to a problem that has a different set of boundary conditions. For me this creates a certain level of anxiety.
A Few Points to Ponder
If we were to take failure mode i) from Figure 1 and the stability envelope for the “load spread” case from Figure 2 and apply it to our design we might be interested in understanding whether the assumed “load spread” holds for an (or any?) incremental increase in horizontal load up to the sliding limit of the structure.
If the only tools at our disposal were the design code and a pen and paper, we might choose to check whether alternative failure mechanisms exist that yield a lower capacity. We could do this using limit analysis, attempt to guess at some feasible failure mechanisms and compare these results with the predictions for the limit equilibrium check.
If we had at our disposal one of the new limit analysis software tools, such as LimitState:Geo, we could actually do this quite rapidly and with some confidence (Figure 3).
Figure 3: LimitState:Geo User Interface
We could also achieve the same using finite element analysis with a programme such as Plaxis 2D, however, as we are not concerned with displacements at this stage we could be quite content with a faster and simpler analysis tool.
Generating a Problem Specific Failure Envelope
Using either Limit Analysis or Finite Element methods, we can avoid the uncertainties in applying a closed form Limit Equilibrium solution to layered soil. In Figure 4 below, I have presented a stability curve generated using a combination of Limit Analysis and Finite Element calculations using the same problem geometry and inputs as used for the Limit Equilibrium check. Apparently my anxiety was well justified.
Figure 4: Check of Stability Envelope using Limit Analysis
The result indicates that not only is the assumption on “load spread” grossly unconservative, the assumed failure mode for the V/Vmax = 1 condition (vertical loading only) is also incorrect.
Why might this be? Well the answer can be found on Figure 5 below. I have presented the failure mechanisms for a range of load combinations and referenced them to the stability envelope. We can see as we move around the envelope stepwise from V/Vmax = 1 to V/Vmax = 0.4 a couple of interesting features:
Figure 5: Comparison of failure mechanisms for various load combinations
1) Under pure vertical loading (V/Vmax = 1) the surficial sand layer yields locally at the edges of the footing such that a full “load spread” can not be achieved. The feature is apparent in both Limit Analysis and Finite Element output. The cause of this is a low soil unit weight and friction angle in the sand. Our assumption on the sand layer being “strong” compared to the clay was inaccurate and as the failure mechanism in the sand layer is constrained by the clay layer a “deep” failure mechanism can not be generated in the sand layer. Equally a full punching mechanism isn’t observed to occur;
2) For a small applied high horizontal load (V/Vmax = 0.9 to 0.8) the failure mechanism seems to develop into a combined sliding-bearing failure at the sand/ clay interface – again not a mechanism that was postulated to occur at the start of our exercise, and not something that is readily checked using Limit Equilibrium calculations.
It may seem obvious in hindsight, but applying traditional Limit Equilibrium solutions to layered soil is a risky business. Never-the-less engineers do so on a surprisingly regular basis. Until design codes catch up with the types of problems encountered in day to day projects the best solution is to equip oneself with the tools needed to quickly and efficiently develop a robust geotechnical design.
In my view Limit Analysis is an invaluable tool for most offshore designs, particularly as we are not so concerned with serviceability conditions.
Edited 17/5/2011: Corrected gramatical errors and the logic in bullet items 1) & 2)