Visualize Mohr circles and Coulomb failure envelopes in real time. Calculate Terzaghi bearing capacity and consolidation settlement for shallow foundations interactively.
The Mohr-Coulomb failure criterion defines the shear strength of the soil. Failure occurs when the shear stress on a plane reaches a value that depends linearly on the normal stress on that same plane.
$$\tau_f = c + \sigma \tan(\phi)$$Where $\tau_f$ is the shear stress at failure, $c$ is the cohesion (soil's "stickiness"), $\sigma$ is the normal stress, and $\phi$ is the internal friction angle. In the simulator, $c$ and $\phi$ are set by their respective sliders.
Terzaghi's bearing capacity equation calculates the ultimate pressure a shallow foundation can support before a shear failure mechanism develops in the underlying soil.
$$q_u = c N_c + \gamma D_f N_q + 0.5 \gamma B N_\gamma$$Here, $q_u$ is the ultimate bearing capacity, $\gamma$ is the soil unit weight, $D_f$ is the foundation depth, $B$ is the foundation width, and $N_c$, $N_q$, $N_\gamma$ are bearing capacity factors that depend only on the friction angle $\phi$. The simulator solves this when you adjust the geometry and soil strength parameters.
Shallow Foundation Design: This is the direct application. Every house, building, or bridge pier resting on footings or rafts requires a bearing capacity check to prevent catastrophic sinking and a settlement analysis to ensure doors won't jam and pipes won't crack from differential movement.
Slope Stability Analysis: The same Mohr-Coulomb failure criterion is used to assess the risk of landslides. Engineers analyze potential failure planes within a slope to calculate a factor of safety, helping to design safe road cuts, embankments, and open-pit mines.
Retaining Wall Design: The lateral earth pressure exerted on a retaining wall, which determines how thick and deep the wall must be, is derived directly from the soil's shear strength parameters (cohesion and friction angle) visualized in the simulator.
Construction Planning: Predicting consolidation settlement over time is critical for large infrastructure projects like embankments for highways or pre-loading of port terminals. It informs how much surcharge to apply and how long to wait before building the final structure.
First, note that cohesion and internal friction angle are not determined independently. While the simulator lets you adjust them with separate sliders, in real soil—especially cohesive soil (clay)—the internal friction angle φ can be an apparent value. There's the concept of the "effective internal friction angle φ'" used in effective stress analysis and the "φ=0" approach used in total stress analysis. For example, a simplified method evaluating stability of soft clay using only cohesion c with "φ=0" is often used. After learning about the two extremes of sandy soil (high φ, low c) and clay layers (low φ, high c) with the tool, consider how both parameters interact in intermediate soils like "silt" or "sandy clay".
Next, Terzaghi's formula is not a "universal solution". It assumes a continuous footing (strip foundation). For square or circular isolated footings, you need to apply shape factors. For instance, the bearing capacity of a square footing with width B=2m is about 1.3 times that of a continuous footing with the same width. Also, the formula assumes ideal conditions of homogeneous, horizontal soil. In reality, soil is layered, and inclined loads or seismic forces may act. Use the "ultimate bearing capacity" obtained from the tool for understanding the basic concept and for parameter sensitivity analysis (to see which factors are influential).
Finally, understand the limitation that the "consolidation time" calculation is based on a one-dimensional model. The tool uses Terzaghi's one-dimensional consolidation theory; once the drainage distance H (e.g., half the clay layer thickness for double drainage) is determined, the settlement time history can be calculated. However, in the field, vertical drains like sand drains are installed to shorten the drainage path and accelerate consolidation. You can confirm in the tool that reducing the "drainage distance" dramatically shortens consolidation time—this is precisely one of the theoretical bases for ground improvement.
The core concepts of this tool—the Mohr-Coulomb failure criterion and ultimate bearing capacity—are directly applied in slope stability analysis. A potential slip surface in a slope is assumed, and the factor of safety is calculated by comparing the shear strength of the soil on that surface (τ_f = c + σ tanφ) with the actual acting shear stress. The state in NovaSolver where the circle touches the failure line is exactly the limit equilibrium state with a factor of safety = 1.0.
They are also deeply related to underground structure design. The stress state of soil around tunnels or buried pipes changes significantly due to excavation, sometimes causing the Mohr's circle to expand and approach the failure line. To prevent this, support is provided with linings or shotcrete. Conversely, in design of excavation retaining walls, the internal friction angle φ becomes a critical parameter when calculating the earth pressure (active earth pressure) acting on the back. It's surprising how a difference of just 5 degrees in φ can change the acting earth pressure by tens of percent.
Looking further ahead, it connects to dynamic analysis of ground (earthquake engineering). During earthquakes, cyclic shear forces act on the ground, particularly in saturated sand layers, causing pore water pressure to rise and effective stress σ' to decrease. This means a decrease in σ in the Mohr-Coulomb equation, which can be understood as a strength reduction phenomenon (liquefaction). Understanding static strength is the first step toward learning about dynamic behavior.
As a next step, I recommend thoroughly learning to distinguish between the two analytical approaches: "effective stress" and "total stress". The tool simplifies this by referring to "cohesion c," but in reality, there are various types like "apparent cohesion," "cohesion under drained conditions," and "cohesion under undrained conditions." Understanding this will help you grasp why the "φ=0" approximation works in clay stability calculations and the concept of "undrained shear strength S_u" behind it.
If you want to delve a bit deeper into the mathematical background, try deriving the Mohr's circle itself and calculating the coordinates of its tangent point with the failure line. When vertical stress σ and horizontal stress Kσ act on soil at a certain depth (K is the earth pressure coefficient), the major and minor principal stresses become the endpoints of the Mohr's circle. The stress state at failure, derived from the condition that this circle is tangent to the failure line, gives the relation: $$ \sigma_1 = \sigma_3 \tan^2(45+\phi/2) + 2c \tan(45+\phi/2) $$. This equation is also fundamental to Rankine's earth pressure theory.
For learning closer to practical work, be mindful of the bridge to geotechnical analysis using the Finite Element Method (FEM). Limit equilibrium methods like NovaSolver directly find the "failure load," whereas FEM tracks the continuous process of soil stress-strain-deformation. Even in such FEM analysis, the "Mohr-Coulomb yield condition" is set as the most basic material model (constitutive law). In other words, the intuition you gain from playing with this simulator becomes the foundation for understanding the input parameters of advanced numerical simulations.