Posted in Academic Issues, Geotechnical Engineering

Lower and Upper Bound Solutions for Bearing Capacity

Although today we have finite element methods which can combine elastic and plastic components of soil response to loading, the use of lower and upper bound plasticity is important in enhancing our understanding of plasticity in soils and many of the methods we use in geotechnical design. This is an overview of both lower and upper bound solutions to the classic bearing capacity problem. Much of this presentation is drawn from Tsytovich (1976) but the equations have been re-derived and checked.

Definitions (from Verruijt)

  1. Lower bound theorem.The true failure load is larger than the load corresponding to an equilibrium system.
  1. Upper bound theorem.The true failure load is smaller than the load corresponding to a mechanism, if that load is determined using the virtual work principle.

For our purposes, since we’re assuming an elastic/perfectly plastic type of soil model, the lower bound solution is where the stress at some point reaches the elastic limit, while the upper bound solution has the stress fully plastic to the boundaries of the system, at which point the capacity of the system to resist further stress has been exhausted (reached its upper limit.)


  • Foundation is very rigid relative to the soil
  • No sliding occurs between foundation and soil (rough foundation)
  • Applied load is compressive and applied vertically to the centroid of the foundation
  • No applied moments present
  • Foundation is a strip footing (infinite length)
  • Soil beneath foundation is homogeneous semi-infinite mass. For the derivations here, we additionally assume that the properties of the soil above the base of the foundation are the same as those below it
  • Mohr-Coulomb model for soil
  • General shear failure mode is the governing mode
  • No soil consolidation occurs
  • Soil above bottom of foundation has no shear strength; is only a surcharge load against the overturning load
  • The effective stress of the soil weight acts in a hydrostatic fashion, i.e., the horizontal stresses are the same as the vertical ones.

These are fairly standard assumptions for basic bearing capacity theory; the “additions” from these are workarounds that have been developed. That includes the analysis of finite foundations (squares, rectangles, circles, etc.)

Theory of Elasticity of Infinite Strip Footings

Let us begin by considering the system below of a strip footing with a uniform load. The variables are defined in the figure.

Figure 1 Elastic Model of Stresses of Strip Loads (adapted from Tsytovich (1976))

It can be shown that the stresses at a point of interest can be defined as follows:

\sigma_{{z}}={\frac {p\left (\alpha+\sin(\alpha)\cos(2\,\beta)\right )}{\pi }} (1)
\sigma_{{y}}={\frac {p\left (\alpha-\sin(\alpha)\cos(2\,\beta)\right )}{\pi }} (2)
\tau={\frac {p\sin(\alpha)\sin(2\,\beta)}{\pi }} (3)

It can also be shown that the principal axis of the stresses at the point are along a line in the middle of the angle \alpha . This is the dashed line in the diagram above. Along this line the angle \beta = 0 (and thus \frac{\alpha}{2}=-\beta' ) and the principal stresses due to the load become

\sigma_{{1}}={\frac {p\left (\alpha+\sin(\alpha)\right )}{\pi }} (4)
\sigma_{{3}}=-{\frac {p\left (-\alpha+\sin(\alpha)\right )}{\pi }} (5)

Lower Bound Solution

Shallow foundations are seldom built with the base of the foundation at the same elevation as the groundline. They are customarily built to a depth from the surface, as shown below.

Figure 2 Geometry and Nomenclature of a Typical Shallow Foundation (from Soils and Foundations Manual)

At this point, for analysis purposes, we transform the effect of the depth into an overburden stress, which is the product of the the unit weight of the soil \gamma and the depth of the foundation base from the surface D (or h,) as shown below:

Figure 3 Strip Foundation with Surrounding Overburden (from Tsytovich (1976))

The effective stress at any point below the surface is given by the equation

\sigma_{{0}}={\it \gamma}\,\left (h+z\right ) (6)

At the point the hydrostatic stress assumption becomes important. The transformation from Equations (1-3) to (4-5) involved an axis rotation. Assuming the soil acts hydrostatically means that, no matter how we rotate the axis, the addition of the effective stress to the principal stress is independent of direction.

Doing just that yields the following:

\sigma_{{1}}={\frac {\left (p-{\it \gamma}\,h\right )\left (\alpha+\sin(\alpha)\right )}{\pi }}+{\it \gamma}\,\left (h+z\right ) (7)
\sigma_{{3}}={\frac {\left (p-{\it \gamma}\,h\right )\left (\alpha-\sin(\alpha)\right )}{\pi }}+{\it \gamma}\,\left (h+z\right ) (8)

At this point we state the failure function for Mohr-Coulomb theory:

\sigma_{{1}}-\sigma_{{3}}-2\,c\cos(\phi)-\left (\sigma_{{1}}+\sigma_{{3}}\right )\sin(\phi)=0 (9)

Substituting Equations (7) and (8) into Equation (9) yields

-2\,{\frac {-p\sin(\alpha)+{\it \gamma}\,h\sin(\alpha)+c\cos(\phi)\pi+\sin(\phi)p\alpha-\sin(\phi){\it \gamma}\,h\alpha+\sin(\phi){\it \gamma}\,\pi \,h+\sin(\phi){\it \gamma}\,\pi \,z}{\pi }}=0 (10)

Solving for z, we have

z={\frac {p\sin(\alpha)}{\sin(\phi){\it \gamma}\,\pi }}-{\frac {h\sin(\alpha)}{\sin(\phi)\pi }}-{\frac {c\cos(\phi)}{\sin(\phi){\it \gamma}}}-{\frac {p\alpha}{{\it \gamma}\,\pi }}+{\frac {h\alpha}{\pi }}-h (11)

At this point we want to find the maximum value of z at which point plasticity first sets in. We do this by taking the derivative of z relative to \alpha and setting it to zero, or

{\frac {p\cos(\alpha)}{\sin(\phi){\it \gamma}\,\pi }}-{\frac {h\cos(\alpha)}{\sin(\phi)\pi }}-{\frac {p}{{\it \gamma}\,\pi }}+{\frac {h}{\pi }}=0 (12)

It can be shown that this condition is fulfilled when \alpha = \frac{\pi}{2} . Substituting that value back into Equation (11) gives us the value of z at which point plasticity is first induced, or

z_{{\max}}={\frac {p\cos(\phi)}{\sin(\phi){\it \gamma}\,\pi }}-{\frac{h\cos(\phi)}{\sin(\phi)\pi }}-{\frac {c\cos(\phi)}{\sin(\phi){\it \gamma}}}-1/2\,{\frac {p}{{\it \gamma}}}+{\frac {p\phi}{{\it \gamma}\,\pi }}-1/2\,h-{\frac {h\phi}{\pi }} (13)

If we solve for the pressure p , that pressure will be in reality the critical pressure at which plasticity is first induced. Solving for that pressure,

p_{{{\it cr}}}={\frac {2\,z_{{\max}}\sin(\phi){\it \gamma}\,\pi +2\,h\cos(\phi){\it \gamma}+2\,c\cos(\phi)\pi +\sin(\phi){\it \gamma}\,\pi\,h+2\,h\phi\,\sin(\phi){\it \gamma}}{2\,\cos(\phi)-\sin(\phi)\pi +2\,\phi\,\sin(\phi)}} (14)

At this point we need to face reality and note that, if the point we’re looking for is the point at which plastic deformation begins, then it cannot be at any depth other than the base of the foundation, or z_{max} = 0 . Making that final substitution yields at last

p_{{{\it cr}}}={\frac {2\,h\cos(\phi){\it \gamma}+2\,c\cos(\phi)\pi +\sin(\phi){\it \gamma}\,\pi \,h+2\,h\phi\,\sin(\phi){\it \gamma}}{2\,\cos(\phi)-\sin(\phi)\pi +2\,\phi\,\sin(\phi)}} (15)

Upper Bound

The upper bound solution is a well-worn path in geotechnical engineering and only the highlights will be shown here.

In 1920-1 Prandtl and Reissener solved the problem for a soil by neglecting its own weight, i.e., Equation (6) They determined that the failure pattern and surface can be represented by the following configuration.

Figure 3 Slip Lines and Failure Surface for Upper Bound Bearing Capacity Failure (from Tsytovich (1976))

They determined that the upper bound critical pressure was given by the equation

p_{{{\it cr}}}={\frac {\left (q+c\cot(\phi)\right )\left (1+\sin(\phi)\right ){e^{\pi \,\tan(\phi)}}}{1-\sin(\phi)}}-c\cot(\phi) (16)

If we define

N_{{q}}={\frac {\left (1+\sin(\phi)\right ){e^{\pi \,\tan(\phi)}}}{1-\sin(\phi)}} (17)


p_{{{\it cr}}}=qN_{{q}}+c\cot(\phi)\left (N_{{q}}-1\right ) (18)

If we further define

N_{{c}}=\left (N_{{q}}-1\right )\cot(\phi) (19)

we have

p_{{{\it cr}}}=qN_{{q}}+cN_{{c}} (20)

The only thing missing from this equation is the effect of the weight of the soil bearing on the failure surface at the bottom of the failure region shown in Figure 3, and thus the bearing capacity equation can be written thus:

p_{{{\it cr}}}=qN_{{q}}+cN_{{c}}+1/2\,{\it \gamma}\,bN_{{{\it \gamma}}} (21)


N_{{{\it \gamma}}}=2\,\left (N_{{q}}+1\right )\tan(\phi) (22)

This last bearing capacity factor has been the subject of variable solutions over the years; the one shown here is that of Vesić, which is enshrined in FHWA/AASHTO recommended practice. Verruijt discusses this issue in detail.

Worked Example

We can take an example from the Soils and Foundations Manual, shown below

It would probably be useful to state the bearing capacity equations in nomenclature that’s more consistent with American practice (and the diagram above.) In both cases this is, for the lower bound solution,

p_{{{\it cr}}}={\frac {2\,D\cos(\phi){\it \gamma}+\sin(\phi){\it \gamma}\,\pi \,D+2\,D\phi\,\sin(\phi){\it \gamma}+2\,c\cos(\phi)\pi }{2\,\cos(\phi)-\sin(\phi)\pi +2\,\phi\,\sin(\phi)}} (15a)

and for the upper bound solution,

p_{{{\it cr}}}=qN_{{q}}+cN_{{c}}+1/2\,{\it \gamma}\,BN_{{{\it \gamma}}} (21a)

One important practical difference between the two is the way the overburden is handled. With the lower bound solution, it is equal to \gamma D , while with the upper bound solution it is simply the pressure q . For a uniform soil above the foundation base with no water table to complicate things, q = \gamma D = (125)(5) = 625\,psf .

Direct substitution into Equation (15a) of all of the variables with show that the lower bound critical pressure is 4740.5 psf.

The upper bound is a little more complicated. The three bearing capacity factors are N_q = 6.4,\,N_c = 14.8,\,and\,N_{\gamma} = 5.39 . Substituting these, q and the other variables yield an upper bound critical pressure of 13,436.8 psf.

If the lower bound is a reduction from the upper bound using a factor of safety, then the FS = 2.83. The lower bound solution is conservative.


Although the lower bound solution may be too conservative for general practice, it is at least an interesting exercise to show the variations in critical pressure from the onset of plastic yielding to its final failed state.

Posted in Academic Issues, Geotechnical Engineering, STADYN

Presentation of “Estimating Load-Deflection Characteristics for the Shaft Resistance of Piles Using Hyperbolic Strain Softening”

Last year we posted the paper Estimating Load-Deflection Characteristics for the Shaft Resistance of Piles Using Hyperbolic Strain Softening. Today it’s presented at the University of Tennessee at Chattanooga’s Research Dialogues, and a slide show of the presentation is below.

Posted in Academic Issues, Geotechnical Engineering

The Invertibility of the p-q Diagram System

We know that we can transform the traditional Mohr-Coulomb \sigma-\tau system to the p-q system by using the equations




Stated formally, this means that, for every set of principal stresses, there is a unique pair of p and q values.

But did you know you can go the other way, if you need to? Let’s start by putting these equations into matrix format, which yields

\left[\begin{array}{cc} 1/2 & 1/2\\ {\medskip}1/2 & -1/2 \end{array}\right]\left[\begin{array}{c} \sigma_{{1}}\\ {\medskip}\sigma_{{3}} \end{array}\right]=\left[\begin{array}{c} p\\ {\medskip}q \end{array}\right]

Inverting the matrix and premultiplying the right hand side yields

\left[\begin{array}{c} \sigma_{{1}}\\ {\medskip}\sigma_{{3}} \end{array}\right]=\left[\begin{array}{cc} 1 & 1\\ {\medskip}1 & -1 \end{array}\right]\left[\begin{array}{c} p\\ {\medskip}q \end{array}\right]

The inversion is the key step. The fact that the matrix is invertible, square and of the same rank as the vectors means that the transformation is linear, one-to-one and onto. We can also say that, for every set of p and q values, there is a unique set of principal stresses.

Those principal stresses are

\left[\begin{array}{c} \sigma_{{1}}\\ {\medskip}\sigma_{{3}} \end{array}\right]=\left[\begin{array}{c} p+q\\ {\medskip}p-q \end{array}\right]

As an example, consider the first set of p and q values computed in my original post on the subject. Substituting those into the last equation yields

\left[\begin{array}{c} \sigma_{{1}}\\ {\medskip}\sigma_{{3}} \end{array}\right]=\left[\begin{array}{c} 200\\ {\medskip}70 \end{array}\right]

which of course are the original values given.

Posted in Geotechnical Engineering

A Geotechnical View of the Effect of Explosions in the Earth

Certainly relevant with events in the Ukraine, from Tsytovich’s Soil Mechanics text:

Explosions may cause a whole series of rapid mechanical processes in soils: appearance of an explosion gas chamber within a rather short interval of time (sometimes a few thousandths of a second), which exerts an enormous pressure (of the order of a few hundred thousand atmospheres), causes the formation and propagation of explosion waves which change the stressed state of a soil mass and cause its particles to move with velocities varying from a few thousand metres per second to zero.

Explosion impulses are characterized by the maximum pressure p_{max} the rise time t_1 during which this pressure is formed, the fall time t_2 during which the pressure drops from the maximum to zero, and the total time of explosion action t_{\sigma} .

As seen from experiments of Prof. G. M. Lyakhov*, the gas chambers formed in soil through explosion of deep concentrated charges of explosives are almost spherical in shape. With time, a gas chamber (the cavity in soil) is destroyed, but the time period of its destruction may be very different , from a few minutes (in sands) to several months (in dense clays).

As has been shown by the experiments, the radius of an explosion gas chamber R_{ch} , after it has been formed completely, is determined by the following relationship:

R_{ch} = \aleph \sqrt{C}


  • C = weight of explosive charge, kg
  • \aleph  = proportionality factor depending on the properties of the soil

According to G. M. Lyakhov, numerical values of this factor are:

for saturated sands\aleph = 0.4-0.7
for loams (according to G. I. Pokrovsky)\aleph = 0.45
for loess soils\aleph = 0.35
for clayey soils\aleph = 0.6-0.7

Explosion of a concentrated charge in a soil results in the formation of normal (radial) pressures p , lateral (tangential) pressures p_{\tau} , and the motion of particles with a velocity u .

For non-saturated soils and rocks, all these three parameters are determined in calculations as functions of time, i.e.,

p = p(t);\,p_{\tau} = p_{\tau}(t);\,\dot u=\dot u(t)

For saturated soils and liquid media, it is sufficient to investigate only two of these parameters, for instance,

p = p(t);\,\dot u=\dot u(t)

The parameters of stress waves in soils caused by explosions and the parameters of velocities of their propagation are determined by special field tests. Using the results of such tests, empirical formulae are established for determination of the design parameters of explosion, waves in soils depending on the weight of charge, the distance from explosion centre, etc.

* Lyakhov G. M. Osnovy dinamiki vzryva v gruntakh i zhidkikh sredakh (Fundamentals of Dynamics of Explosion in Soils and Liquid Media), Moscow, Nedra Publishers, 1964.

Although geotechnical engineering has always had a military application (witness the prominence of documents such as NAVFAC DM 7 and the many others offered on this site,) this is the only elementary level soil mechanics text where I can recall seeing such a presentation.

Posted in Geotechnical Engineering

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Designing gravity or MSE retaining walls with the attractive Redi-Rock precast modular block (PMB) facing just got a lot easier! The latest version of RRWall+ design software was created by Fine Software, sponsors, and […]

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