Combined Stresses and Mohr's Circle - Concept Map
Stress Transformation and Principal Stresses at a Point
Related concepts
Superposition of Stresses
Sub concepts
- Axial stress (P/A)
- Bending stress (Mc/I)
- Torsional shear (Tc/J)
- Biaxial bending
- Load combination rules
Relationship to central
Foundation step—combines multiple load effects to establish the initial stress state (σx, σy, τxy) at a point before transformation
Plane Stress Transformation Equations
Sub concepts
- Normal stress on inclined plane (σx′)
- Shear stress on inclined plane (τx′y′)
- Angle convention (θ from x-axis)
- Double-angle relationships (2θ)
Relationship to central
Mathematical method to find stress components on any inclined plane; the analytical equivalent of Mohr's circle
Principal Stresses
Sub concepts
- Principal stress formula (σ1, σ2)
- Principal plane orientation (θp)
- Physical meaning (extreme normal stresses)
- Ordering convention (σ1 ≥ σ2)
Relationship to central
The maximum and minimum normal stresses; occur where shear stress = 0; represent the critical failure planes
Maximum In-Plane Shear Stress
Sub concepts
- τmax formula and calculation
- Relationship to principal stresses (σ1 − σ2)/2
- Plane of maximum shear (45° offset)
- Normal stress on max-shear plane (σavg)
Relationship to central
The largest shear stress in the plane; equals the radius of Mohr's circle; acts 45° from principal planes
Mohr's Circle Construction and Interpretation
Sub concepts
- Center location on σ-axis
- Circle radius (equals τmax)
- Plotting point X and point Y
- Diameter as principal axis
- Reading principal stresses and angles
- Physical rotation ↔ 2θ on circle
Relationship to central
Graphical representation of all possible stress states at a point; transforms algebra into intuitive geometry
Combined Bending and Torsion of Shafts
Sub concepts
- Surface stress state of shaft (σx, σy=0, τxy)
- Equivalent bending moment (Me)
- Equivalent torsional torque (Te)
- Diameter formulas using equivalents
- Normal-stress vs. shear-stress theories
Relationship to central
Practical application where bending moment and torque both act; yields equivalent moment and equivalent torque for design
Failure Theories Under Combined Stress
Sub concepts
- Maximum normal stress (Rankine)—brittle materials
- Maximum shear stress (Tresca)—ductile metals
- Distortion energy (von Mises)—most accurate for ductile
- Relationship to yield strength and safety factor
Relationship to central
Criteria to determine if a material will yield or fail under the combined principal stress state
Special Stress States
Sub concepts
- Uniaxial tension (σ1 = σ; σ2 = 0)
- Pure shear (σ1 = −σ2 = τxy)
- Biaxial equal tension (σ1 = σ2; τmax = 0)
- Hydrostatic (σ1 = σ2 = σ3; no shear)
Relationship to central
Common cases with simplified analysis; often examined on licensure boards
Concept connections
- Superposition of StressesInitial Stress State (σx, σy, τxy)Superposition combines multiple load effects to establish the three components needed for transformation
- Initial Stress State (σx, σy, τxy)Plane Stress Transformation EquationsThe initial state is the input to transformation equations; different angles yield different stress components
- Plane Stress Transformation EquationsPrincipal StressesPrincipal stresses are found where shear = 0; they are the extreme values found via transformation
- Initial Stress State (σx, σy, τxy)Mohr's Circle Construction and InterpretationPoints X and Y are plotted directly from (σx, τxy) and (σy, −τxy); the circle graphically solves all transformations
- Mohr's Circle Construction and InterpretationPrincipal StressesPrincipal stresses are read where the circle crosses the σ-axis; geometric equivalent to solving transformation equations
- Mohr's Circle Construction and InterpretationMaximum In-Plane Shear Stressτmax equals the radius of Mohr's circle; acts at the top or bottom of the circle
- Principal StressesMaximum In-Plane Shear Stressτmax = (σ1 − σ2)/2; derived directly from principal stresses
- Combined Bending and Torsion of ShaftsInitial Stress State (σx, σy, τxy)Shaft loading produces a specific stress state: σx from bending, τxy from torque, σy = 0 on surface
- Combined Bending and Torsion of ShaftsPrincipal StressesEquivalent moment and torque are derived from principal stress formulas; they simplify shaft design
- Principal StressesFailure Theories Under Combined StressFailure theories use principal stresses (σ1, σ2) as input to compare against allowable or yield strength
- Maximum In-Plane Shear StressFailure Theories Under Combined StressMaximum shear stress theory (Tresca) uses τmax directly; compared to half the yield strength
- Special Stress StatesPrincipal StressesSpecial cases (pure shear, uniaxial, biaxial) are limiting forms that simplify principal stress formulas
- Special Stress StatesMohr's Circle Construction and InterpretationSpecial cases have distinctive circle geometries (e.g., pure shear is a circle centered at origin)
- Plane Stress Transformation EquationsMohr's Circle Construction and InterpretationMohr's circle is the graphical representation of transformation equations; both yield identical results
- Combined Bending and Torsion of ShaftsFailure Theories Under Combined StressShaft design applies failure theories to check if the combined stress state exceeds the material's strength
Diagrams
Complete Concept Hierarchy: Combined Stresses and Mohr's Circle
Comprehensive mind map showing the hierarchical relationships among all major topics in combined stresses and Mohr's circle. Flows from basic superposition through transformation methods to practical applications and failure theories. This is the master concept map used to organize the entire chapter.
Mohr's Circle Construction and Interpretation Process
Step-by-step flowchart for constructing and using Mohr's circle. Shows the decision points and calculations needed to extract principal stresses, maximum shear, angles, or stress on an inclined plane. This is the procedural guide used in solved problems on the PRC boards.
Combined Bending and Torsion Shaft Design Procedure
Iterative design process for circular shafts under combined bending and torsion. Demonstrates how to compute equivalent moment or torque, apply the appropriate failure theory, and check against allowable stresses. Commonly tested in shaft design problems on the PRC Civil Engineer Licensure Examination.
Stress Transformation: From Initial State to Principal Stresses
Comparison of analytical and graphical methods for stress transformation. Shows both the equation-based approach and the Mohr's circle geometric approach converging to the same principal stresses. Helps reviewees choose the most efficient method for exam problems.
Material Failure Criteria Selection Based on Stress State
State diagram showing the decision path for selecting and applying failure theories based on material type and required accuracy. Guides the reviewer through material classification, theory selection, and failure criterion check. Essential for safe design decisions under combined stress.
Typical Board Exam Question Distribution: Combined Stresses Topics
Distribution of combined stresses topics across PRC Civil Engineer Licensure Examination. Shows that Mohr's circle and principal stress problems are the most frequently tested (50%), followed by shaft design (18%). This informs study priority and practice problem selection.