Mohr's Circle, a graphical method used in engineering to determine the state of stress at a point, is derived using the normal and shear stress components on a plane. To construct it, plot the normal stress (σ) on the horizontal axis and the shear stress (τ) on the vertical axis. The circle's center is at the average normal stress, (σx + σy)/2, and its radius is the square root of the sum of the square of the half difference of the normal stresses and the square of the shear stress. This graphical representation helps in understanding the maximum and minimum stresses and their orientations.«Stability analysis of a non-circular tunnel face in soils characterized by modified mohr-coulomb yield criterion »
The center of Mohr's circle is located at (σx + σy)/2 on the σ axis, with no shear stress component, meaning it lies on the horizontal axis. The radius of the circle is calculated using the formula √[(σx - σy)/2]^2 + τxy^2, where σx and σy are the normal stresses on the x and y axes, respectively, and τxy is the shear stress. The circle's faces represent different stress states on various planes through the point of interest.«Failure of rammed earth walls: from observations to quantifications »
| Parameter | Description | Typical Range | Typical Applications/Scenarios | Factors Affecting Values |
|---|---|---|---|---|
| Normal Stress | Stress perpendicular to a plane | 20 - 200 kPa | Foundation design, slope stability | Soil type, depth, water content |
| Shear Stress | Stress parallel to a plane | 10 - 100 kPa | Assessing soil shear strength, retaining wall design | Material cohesion, internal friction |
| Principal Stress | Maximum principal stress | 100 - 300 kPa | Earth pressure analysis, tunneling | Geological conditions, overburden pressure |
| Principal Stress | Minimum principal stress | 50 - 150 kPa | Subsurface structure analysis, excavation | Geostatic stress, anisotropy of soil |
| Angle of Rotation | Angle at which principal stresses occur | 5 - 75 ° | Stress transformation, failure criteria analysis | Stress state, loading conditions |
In conclusion, Mohr's Circle is a transformative tool that ingeniously maps the state of stress at a point within a material, allowing engineers and geologists to visualize and predict failure modes efficiently. By applying this method, one can deduce the principal stresses and the maximum shear stress, which are critical for designing safe and robust structures. This graphical representation simplifies complex stress analysis, making it an indispensable technique in the field of materials science and structural engineering.«Introducing stress transformation and mohr’s circle»
The Mohr circle is a circle because it graphically represents the state of stress at a point by plotting normal stress (\(\sigma\)) on the x-axis and shear stress (\(\tau\)) on the y-axis for all possible plane orientations through that point. This method, based on the principle of equilibrium and the relationships between normal and shear stresses on inclined planes, results in a circle when these stresses are plotted. The circle's geometric properties allow for the visualization of stress states, demonstrating how normal and shear stresses change with plane orientation, thus inherently forming a circle due to the mathematical relationship between these stresses.«New concept for the stress analysis on rock burst failure based on modified mohr's stress circle representation u.s. rock mechanics/geomechanics symposium onepetro»
Each point on Mohr's circle represents the state of stress (both normal and shear stresses) on a plane at a certain orientation within a material. The x-coordinate of any point on the circle indicates the magnitude of the normal stress (\(\sigma\)), and the y-coordinate represents the shear stress (\(\tau\)) on that plane. By rotating around the circle, one can find the stresses acting on planes at different orientations, thus providing a comprehensive view of how stresses vary within the material under consideration. This allows engineers to analyze and design materials and structures more effectively by understanding stress distributions.«Construction by contour crafting using sulfur concrete with planetary applications»
In Mohr's circle, Sigma N (\(\sigma_n\)) represents the normal stress acting on a specific plane within a material. It is the stress component perpendicular to the plane, responsible for deformation without shear. The value of \(\sigma_n\) varies with the orientation of the plane, and it can be found at any point on Mohr's circle by projecting that point onto the horizontal axis. This parameter is crucial for understanding and predicting how materials will react under various loading conditions, as different orientations of planes will experience different magnitudes of normal stress, affecting the material's overall behavior.«Behaviors of stresses on materials at different angle of inclination using stress and mohr’s circle computation and transforma»
One of the disadvantages of Mohr's method is its complexity in application to three-dimensional stress states, requiring extensive calculations and graphical interpretations that can be time-consuming and prone to errors. Additionally, it assumes homogeneity and isotropy in materials, which may not be the case in real-world scenarios, leading to inaccuracies in stress analysis. The method also relies heavily on graphical solutions, which can limit precision, especially for complex stress conditions or when precise numerical answers are needed for engineering design and analysis.«Streamlining teaching of mohr circle in geotechnical classrooms»
