CELE Strength of Materials — Combined Stresses and Mohr's CircleConcept Map

Concept

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

Concept

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

Concept

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

Concept

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

Concept

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

Concept

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

Concept

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

Concept

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

To

Initial Stress State (σx, σy, τxy)

From

Superposition of Stresses

Strength

strong

Relationship

Superposition combines multiple load effects to establish the three components needed for transformation

To

Plane Stress Transformation Equations

From

Initial Stress State (σx, σy, τxy)

Strength

strong

Relationship

The initial state is the input to transformation equations; different angles yield different stress components

To

Principal Stresses

From

Plane Stress Transformation Equations

Strength

strong

Relationship

Principal stresses are found where shear = 0; they are the extreme values found via transformation

To

Mohr's Circle Construction and Interpretation

From

Initial Stress State (σx, σy, τxy)

Strength

strong

Relationship

Points X and Y are plotted directly from (σx, τxy) and (σy, −τxy); the circle graphically solves all transformations

To

Principal Stresses

From

Mohr's Circle Construction and Interpretation

Strength

strong

Relationship

Principal stresses are read where the circle crosses the σ-axis; geometric equivalent to solving transformation equations

To

Maximum In-Plane Shear Stress

From

Mohr's Circle Construction and Interpretation

Strength

strong

Relationship

τmax equals the radius of Mohr's circle; acts at the top or bottom of the circle

To

Maximum In-Plane Shear Stress

From

Principal Stresses

Strength

strong

Relationship

τmax = (σ1 − σ2)/2; derived directly from principal stresses

To

Initial Stress State (σx, σy, τxy)

From

Combined Bending and Torsion of Shafts

Strength

strong

Relationship

Shaft loading produces a specific stress state: σx from bending, τxy from torque, σy = 0 on surface

To

Principal Stresses

From

Combined Bending and Torsion of Shafts

Strength

moderate

Relationship

Equivalent moment and torque are derived from principal stress formulas; they simplify shaft design

To

Failure Theories Under Combined Stress

From

Principal Stresses

Strength

strong

Relationship

Failure theories use principal stresses (σ1, σ2) as input to compare against allowable or yield strength

To

Failure Theories Under Combined Stress

From

Maximum In-Plane Shear Stress

Strength

moderate

Relationship

Maximum shear stress theory (Tresca) uses τmax directly; compared to half the yield strength

To

Principal Stresses

From

Special Stress States

Strength

moderate

Relationship

Special cases (pure shear, uniaxial, biaxial) are limiting forms that simplify principal stress formulas

To

Mohr's Circle Construction and Interpretation

From

Special Stress States

Strength

moderate

Relationship

Special cases have distinctive circle geometries (e.g., pure shear is a circle centered at origin)

To

Mohr's Circle Construction and Interpretation

From

Plane Stress Transformation Equations

Strength

strong

Relationship

Mohr's circle is the graphical representation of transformation equations; both yield identical results

To

Failure Theories Under Combined Stress

From

Combined Bending and Torsion of Shafts

Strength

strong

Relationship

Shaft design applies failure theories to check if the combined stress state exceeds the material's strength