CELE Strength of Materials — Combined Stresses and Mohr's CircleCheat Sheet
Combined Stresses and Mohr's Circle cheat sheet for CELE aspirants. If you could only take one sheet of paper into your review session, this is what it would look like. Professional Regulation Commission (PRC) — Board of Civil Engineering's most-tested concepts, all in one place.
Exam context
On the CELE 2026, the Strength of Materials subtest carries a "Core" weight in Professional Regulation Commission (PRC) — Board of Civil Engineering's pattern. Combined Stresses and Mohr's Circle lands at position 6th out of 8 in the standard review order. Target score is 70% weighted average, no sub-test below 50%, and roughly a meaningful share of items come from Strength of Materials on a typical CELE paper.
Combined Stresses and Mohr's Circle - Cheat Sheet
Your final 30-minute companion for stress transformation, principal stresses, and Mohr's circle. Every formula, definition, and pitfall you need to ace the PRC exam on this topic.
Sections
Formulas
Formula
σ = P/A ± Mc/I
Meaning
P = axial load; A = cross-sectional area; M = bending moment; c = distance from neutral axis; I = second moment of inertia
Watch Out
Sign of bending stress depends on which fibre (top or bottom); compression and tension add as signed quantities
When To Use
Combined axial and bending loads on a member
Formula
σ = P/A ± (M_x c_y)/I_x ± (M_y c_x)/I_y
Meaning
Biaxial bending: moments M_x and M_y about perpendicular axes; c_x, c_y = distances; I_x, I_y = second moments
Watch Out
Track your signs carefully; use right-hand rule for moment directions; all terms must be in same unit system
When To Use
Eccentric loading or simultaneous bending about two axes
Section Title
Superposition of Stresses
Important Facts
- Stresses must be calculated at the same point and in the same coordinate system before superposition
- All stresses must be converted to the same units before adding (typically MPa or N/mm²)
- Bending stresses vary linearly across the section; maximum occurs at the outer fibre farthest from neutral axis
- For combined axial and bending, the neutral axis shifts if the axial force is eccentric
Key Definitions
Term
Superposition
Example
A shaft bent by a lateral load and twisted by a torque: total stress = bending stress + torsional shear stress
Definition
The algebraic sum of stresses at a point caused by individual loads acting separately.
Term
Plane Stress
Example
A thin-walled pressure vessel or a flat plate loaded in its plane
Definition
A stress state where stresses exist only in a 2D plane (σ_z = τ_xz = τ_yz = 0); the third dimension is stress-free.
Term
Biaxial Stress
Example
Pressurized thin pipe with hoop and longitudinal stresses
Definition
A state where normal stresses act in two perpendicular directions (σ_x and σ_y both ≠ 0, σ_z = 0).
Diagrams To Know
- Cross-section showing stress distribution for combined axial + bending
- Element showing σ_x, σ_y, and τ_xy on opposite faces
Formulas
Formula
σ_x' = [(σ_x + σ_y)/2] + [(σ_x - σ_y)/2]cos(2θ) + τ_xy sin(2θ)
Meaning
σ_x' = normal stress on plane at angle θ; θ = counterclockwise from x-axis (in degrees or radians)
Watch Out
Angle is DOUBLED in the formula (2θ); use consistent angle units (radians preferred in formulas)
When To Use
Find normal stress on an inclined plane at a specific angle
Formula
τ_x'y' = -[(σ_x - σ_y)/2]sin(2θ) + τ_xy cos(2θ)
Meaning
τ_x'y' = shear stress on plane at angle θ to x-axis
Watch Out
Shear stress is zero at principal planes; maximum shear occurs at 45° from principal axes
When To Use
Find shear stress on an inclined plane; used to verify Mohr's circle
Formula
σ_n = (σ_x + σ_y)/2
Meaning
Average normal stress (center of Mohr's circle on σ-axis); σ_n = constant for all planes
Watch Out
This is NOT the principal stress unless one principal equals the other
When To Use
Identify the normal stress that always exists on maximum-shear planes
Section Title
Plane Stress Transformation Equations
Important Facts
- Transformation equations rotate the stress coordinate system; the stresses themselves are properties of the material point
- θ is measured counterclockwise from the x-axis in the reference frame
- The sum σ_x' + σ_y' = σ_x + σ_y = invariant (always constant regardless of plane orientation)
- On any plane, σ_n + σ_t = σ_x' + σ_y' (both normal stresses sum to invariant)
- Plane stress assumes σ_z = 0; if out-of-plane stress exists, three principal stresses must be considered
Key Definitions
Term
Principal Plane
Example
On a principal plane, only normal stress acts; used to identify critical failure directions
Definition
A plane perpendicular to a principal stress direction where shear stress is zero.
Term
Principal Stress
Example
σ_1 = 92.4 MPa and σ_2 = 7.6 MPa are the two principal stresses in a 2D state
Definition
The extreme (maximum and minimum) normal stresses at a point; shear stress = 0 on these planes.
Term
Maximum In-Plane Shear Stress
Example
τ_max = 42.4 MPa acts on planes inclined 45° to the principal axes
Definition
The largest shear stress magnitude in the plane; occurs 45° away from principal planes and carries average normal stress.
Diagrams To Know
- Infinitesimal square element showing σ_x, σ_y, τ_xy on reference axes
- Same element rotated by angle θ showing σ_x', σ_y', τ_x'y' on inclined planes
Formulas
Formula
σ_1, σ_2 = [(σ_x + σ_y)/2] ± √{[(σ_x - σ_y)/2]² + τ_xy²}
Meaning
σ_1 (max) and σ_2 (min) principal stresses; the ± gives both extremes
Watch Out
The term under the square root uses (σ_x − σ_y)/2, NOT the full difference; HALVING is critical
When To Use
Always use this to find the principal stresses from σ_x, σ_y, τ_xy
Formula
τ_max = √{[(σ_x - σ_y)/2]² + τ_xy²} = (σ_1 - σ_2)/2
Meaning
Maximum in-plane shear stress; equals the radius of Mohr's circle
Watch Out
This is in-plane maximum; if σ_1 and σ_2 have same sign, absolute max shear may involve σ_z = 0
When To Use
Find the largest shear stress (equals radius R); verify with principal stresses
Formula
tan(2θ_p) = 2τ_xy / (σ_x - σ_y)
Meaning
Angle θ_p to the principal plane (counterclockwise from x-axis); gives two angles 90° apart
Watch Out
Result gives 2θ_p, not θ_p; there are always TWO solutions 90° apart; use arctan(…) with correct quadrant
When To Use
Find the orientation of principal stresses; remember the circle solution is 2θ_p, divide by 2
Formula
θ_s = θ_p ± 45°
Meaning
Orientation of maximum-shear planes; always 45° from principal planes
Watch Out
Two planes 90° apart carry equal and opposite maximum shear; both carry the average normal stress
When To Use
Identify the plane where maximum shear acts
Section Title
Principal Stresses and Maximum Shear
Important Facts
- Principal stresses are always perpendicular to each other (in 2D, 90° apart)
- At a principal plane, shear stress MUST be zero by definition
- The average normal stress σ_avg = (σ_x + σ_y)/2 acts on ALL maximum-shear planes
- In pure shear (σ_x = σ_y = 0), principal stresses are equal in magnitude but opposite in sign (σ_1 = −σ_2)
- Maximum in-plane shear = radius of Mohr's circle; equals (σ_1 − σ_2)/2 always
Key Definitions
Term
Principal Stress σ_1
Example
σ_1 = 92.4 MPa in a brittle material failure
Definition
The algebraically largest (most positive) normal stress; governs failure in tension-governed materials.
Term
Principal Stress σ_2
Example
σ_2 = 7.6 MPa or σ_2 = −50 MPa in pure shear
Definition
The algebraically smallest (most negative, or least positive) normal stress.
Term
Maximum Shear Plane
Example
On this plane: σ_n = (σ_1 + σ_2)/2 and τ = (σ_1 − σ_2)/2
Definition
A plane oriented 45° to the principal planes where shear stress equals τ_max and normal stress equals (σ_x + σ_y)/2.
Diagrams To Know
- Infinitesimal element showing principal stresses σ_1 and σ_2 aligned with principal axes
- Same element rotated 45° showing maximum shear τ_max and average normal stress σ_avg
Formulas
Formula
Center = [(σ_x + σ_y)/2, 0] on σ-axis; Radius R = τ_max
Meaning
Center horizontal coordinate is average stress; vertical coordinate is zero (on σ-axis); radius is max shear
Watch Out
Circle is drawn in the (σ, τ) plane, NOT the physical (x, y) plane; σ-axis horizontal, τ-axis vertical
When To Use
Start here to construct any Mohr's circle from σ_x, σ_y, τ_xy
Formula
Point X: (σ_x, τ_xy); Point Y: (σ_y, −τ_xy)
Meaning
Plot the reference-axis stresses as two points; line XY is a diameter
Watch Out
Y has the NEGATIVE of τ_xy (not positive); XY passes through center; these are diametrically opposite
When To Use
Quick way to find center and radius without calculation
Formula
σ_1 = Center + R; σ_2 = Center − R
Meaning
Principal stresses are where circle intersects σ-axis (τ = 0)
Watch Out
These are the points where the circle crosses the horizontal axis
When To Use
Read principal stresses directly off the circle or calculate from center and radius
Formula
Physical rotation θ ⟷ Circle rotation 2θ (same sense)
Meaning
Rotating the element by θ counterclockwise rotates the point on Mohr's circle by 2θ counterclockwise
Watch Out
This is the most common source of angle errors; if you rotate element by 30°, rotate on circle by 60°
When To Use
Convert between physical element orientation and circle representation; always use 2θ on circle
Section Title
Mohr's Circle — Construction and Use
Important Facts
- The circle is always plotted with σ on the horizontal axis and τ on the vertical axis
- Tension (positive σ) is to the right; compression (negative σ) is to the left
- Positive τ is typically upward; negative τ is downward (follow your convention)
- Any point on the circle represents the stresses on some plane through the element
- The two points where the circle touches the σ-axis (τ = 0) are the principal stresses
- The top and bottom of the circle give the maximum and minimum (most negative) shear stresses
- A rotation of the physical element by 2θ moves around the circle by 2θ (same angular sense)
Key Definitions
Term
Mohr's Circle
Example
A circle in (σ, τ) space centered at (50 MPa, 0) with radius 42.4 MPa represents all stress states at a point
Definition
A graphical representation of all possible (σ, τ) states at a point; every point on the circle represents a plane through the material point.
Term
Diameter XY
Example
X at (80, 30) and Y at (20, −30) define a diameter; midpoint (50, 0) is the center
Definition
The line connecting the two reference-state points X(σ_x, τ_xy) and Y(σ_y, −τ_xy); passes through center.
Term
Pole (or Origin) of the Circle
Example
A line drawn from pole parallel to a physical plane intersects circle at that plane's stress state
Definition
A special point on the circle used to construct planes geometrically; less common in exams but useful for graphical solutions.
Diagrams To Know
- Mohr's circle showing center, radius, points X and Y, principal stresses, and maximum shear points
- Stress element and corresponding Mohr's circle side by side showing correspondence between physical planes and circle points
Formulas
Formula
σ_x = 32M / (πd³)
Meaning
Bending stress on the outer surface of a solid circular shaft; M = bending moment; d = diameter
Watch Out
This assumes the stress point is on the outer fibre; for hollow shafts use different formula; σ_y = 0
When To Use
Calculate bending stress in a shaft (acts on top and bottom of cross-section)
Formula
τ_xy = 16T / (πd³)
Meaning
Torsional shear stress on outer fibre of solid circular shaft; T = torque
Watch Out
Maximum torsional stress is at the outer surface; τ = 0 at centre; always acts tangent to radius
When To Use
Shear stress from torsion only; combine with bending via Mohr's circle
Formula
T_e = √(M² + T²)
Meaning
Equivalent torque (for maximum-shear-stress theory); combines bending M and actual torque T
Watch Out
This is NOT the principal stress; it is a design tool for shear-based failure theories
When To Use
Use in shaft design codes; τ_max formula applies to T_e instead of T
Formula
M_e = (1/2)[M + √(M² + T²)]
Meaning
Equivalent moment (for maximum-normal-stress theory); yields σ_1 directly
Watch Out
M_e > M always; gives more conservative result than maximum-shear method for ductile materials
When To Use
Convert combined bending and torsion to an equivalent pure-bending moment for normal-stress theory
Formula
σ_1 = 32M_e / (πd³)
Meaning
Maximum principal stress on shaft using equivalent moment
Watch Out
Assume σ_y = 0 and σ_z = 0 for surface element in plane stress
When To Use
Failure check against yield/rupture strength using normal-stress theory
Formula
τ_max = 16T_e / (πd³)
Meaning
Maximum shear stress on shaft using equivalent torque
Watch Out
Common exam question: which theory (normal vs shear) gives larger allowable diameter? (shear is more conservative)
When To Use
Failure check against max-shear criterion (especially for ductile materials)
Section Title
Combined Bending and Torsion of Circular Shafts
Important Facts
- On the outer fibre of a circular shaft, bending creates normal stress σ_x and torsion creates shear τ_xy
- Bending moment M varies along the shaft; torque T may vary or be constant (check FBD)
- Maximum combined stress occurs where M and T are simultaneously large (often at a concentrated load point)
- Hollow shafts use T = 16T_o / (πd_o³) and M formula derived from larger I; more efficient than solid
- Two failure theories give different answers: normal-stress (Rankine) is more conservative for brittle materials; max-shear (Tresca) suits ductile metals
- Von Mises equivalent stress σ_v = √(σ_1² − σ_1 σ_2 + σ_2²) is most accurate for ductile materials but rarely asked directly on old PRC exams
Key Definitions
Term
Equivalent Torque T_e
Example
M = 1.5 kN·m, T = 2.0 kN·m ⟹ T_e = √(1.5² + 2.0²) = 2.5 kN·m
Definition
A single torque that produces the same maximum shear stress as the combined bending and torsion.
Term
Equivalent Moment M_e
Example
M = 1.0, T = 1.0 ⟹ M_e = 0.5(1.0 + √2) ≈ 1.207
Definition
A single bending moment that produces the same maximum normal stress as combined M and T, per normal-stress theory.
Term
Combined Bending and Torsion State
Example
Upper fibre of shaft in bending has tension; this fibre also experiences torsional shear
Definition
A plane-stress condition on a shaft surface: σ_x = 32M/(πd³), σ_y = 0, τ_xy = 16T/(πd³); both σ_x and τ_xy are nonzero.
Diagrams To Know
- Circular shaft cross-section showing bending stress distribution (linear, max at top and bottom) and torsional shear distribution (linear, max at outer fibre)
- Combined stress element on shaft outer fibre showing σ_x (tension or compression from bending) and τ_xy (shear from torsion)
- Free-body diagram of shaft segment showing bending moment M and torque T vectors
Formulas
Formula
Maximum Normal Stress: σ_1 ≤ σ_y (or σ_ut for brittle)
Meaning
Failure when largest principal stress reaches material yield (ductile) or ultimate (brittle) strength
Watch Out
Ignores shear effects; does NOT account for the interaction of σ_1 and σ_2; too conservative for ductile metals
When To Use
Brittle materials (cast iron, concrete, stone) and simple normal-stress problems
Formula
Maximum Shear Stress (Tresca): τ_max ≤ σ_y/2
Meaning
Failure when maximum shear stress reaches half the yield strength; τ_max = (σ_1 − σ_2)/2
Watch Out
Conservative (underpredicts actual failure) compared to von Mises; still widely used in practice
When To Use
Ductile metals (steel); common in shaft design and pressure vessel codes
Formula
Von Mises (Distortion Energy): σ_v = √[σ_1² − σ_1 σ_2 + σ_2²] ≤ σ_y
Meaning
Equivalent uniaxial stress; accounts for interaction between σ_1 and σ_2; most accurate for ductile metals
Watch Out
More complex than Tresca but closer to experimental data; less common in older PRC exams, more in modern codes
When To Use
Design of ductile steel structures under multiaxial stress; slightly predicts failure better than Tresca
Common Values
Value
250 MPa
Symbol
σ_y
Quantity
Yield strength of mild steel (Grade 250)
Value
350 MPa
Symbol
σ_y
Quantity
Yield strength of high-strength steel (Grade 350)
Value
20–40 MPa
Symbol
f_c'
Quantity
Typical concrete compressive strength
Value
2–4 MPa
Symbol
f_t
Quantity
Typical concrete tensile strength
Section Title
Failure Theories (Material Strength Criteria)
Important Facts
- Maximum-normal-stress theory fails for ductile materials in pure shear (predicts failure at τ = σ_y, but actual is τ = σ_y/2)
- Von Mises gives σ_v = σ_1 for uniaxial stress (check: √(σ_1² − 0 + 0) = σ_1) ✓
- In pure shear, τ_max = τ: von Mises gives σ_v = √(τ² − (−τ)τ + τ²) = τ√3 (yield at τ = σ_y/√3 ≈ 0.577 σ_y) ✓
- NSCP 2015 and ACI 318 typically use allowable-stress design (ASD); AISC 360 permits both ASD and LRFD
- PRC exam often tests whether a design (shaft, beam, etc.) satisfies the chosen failure criterion; know both limits
Key Definitions
Term
Yield Strength σ_y
Example
Mild steel: σ_y ≈ 250 MPa; high-strength steel: σ_y ≈ 350–400 MPa
Definition
The stress at which a ductile material begins permanent deformation; typically 0.2% plastic strain.
Term
Ultimate Strength σ_ut
Example
Concrete: σ_ut (compression) ≈ 20–40 MPa; σ_ut (tension) ≈ 2–4 MPa
Definition
Maximum stress a material can withstand before fracture; used for brittle materials.
Term
Factor of Safety (FOS)
Example
FOS = 2: σ_allow = σ_y / 2 = 250/2 = 125 MPa for mild steel
Definition
Ratio of material strength to working (allowable) stress; e.g., FOS = 2 means stress ≤ σ_y/2.
Diagrams To Know
- Mohr's circle overlaid with the failure envelope (straight lines for max-stress or hexagon for von Mises)
- σ_1 vs σ_2 plot showing yield curves for different failure theories (Rankine = straight lines; Tresca = hexagon; von Mises = ellipse)
Formulas
Formula
Pure Shear: σ_x = σ_y = 0, τ_xy ≠ 0 ⟹ σ_1 = τ, σ_2 = −τ, τ_max = τ
Meaning
Equal tension and compression at 45° to shear planes; use for torsion-only problems
Watch Out
Principal stresses are equal in magnitude, opposite in sign; they are ±τ (NOT ±τ/2)
When To Use
Shaft in torsion alone (no bending) or a thin element in pure shear
Formula
Uniaxial Tension: σ_x = σ (σ_y = τ_xy = 0) ⟹ σ_1 = σ, σ_2 = 0, τ_max = σ/2
Meaning
All stress is normal in one direction; principal axis aligned with load
Watch Out
σ_2 = 0, NOT the negative of σ; maximum shear is σ/2 on 45° planes
When To Use
Simple tension or compression member (no bending or torsion)
Formula
Hydrostatic (Isotropic): σ_x = σ_y = σ_z = σ, τ = 0 ⟹ σ_1 = σ_2 = σ, τ_max = 0
Meaning
Equal stress in all directions; no shear anywhere; no distortion, only volume change
Watch Out
This state can never cause failure by shear; only by bulk compression or tension
When To Use
Pressure vessel in all directions or deep-ocean point (uniform hydrostatic pressure)
Formula
Biaxial Tension: σ_x = σ_y = σ (τ = 0) ⟹ σ_1 = σ, σ_2 = σ, τ_max = 0
Meaning
Equal stress in two directions; shear is zero everywhere
Watch Out
No shear stress in the plane; if third stress exists, it may drive shear
When To Use
Thin-walled spherical pressure vessel (hoop and meridional stresses nearly equal)
Section Title
Special Cases and Quick Recognition
Important Facts
- In pure shear, the material wants to fail on a 45° helix (in 3D) because that is where the maximum normal stresses develop
- Shaft failures in torsion typically follow a 45° spiral pattern along the shaft length — this is NOT random
- Uniaxial stress is the simplest stress state; it defines the baseline for all other conditions
- Isotropic (hydrostatic) stress does NOT cause failure by shear or distortion; only by compression or tension of the bulk
- The principal stress formula and Mohr's circle handle all these special cases automatically (no separate logic needed)
Key Definitions
Term
Pure Shear State
Example
A shaft in torsion exhibits pure shear on the outer fibre (ignoring any bending)
Definition
A stress condition where σ_x = σ_y = 0 and only τ_xy is present; equivalent to equal tension and compression at 45°.
Term
Hydrostatic State
Example
Submersed object at uniform depth experiences hydrostatic stress from all sides
Definition
A pressure state where σ_x = σ_y = σ_z and all shear stresses are zero; changes volume but not shape.
Diagrams To Know
- Stress element and corresponding Mohr's circle for pure shear (circle centered at origin with radius τ)
- Stress element for uniaxial tension and its Mohr's circle (degenerate circle with center at σ/2 and radius σ/2)
Must Remember
- PRINCIPAL STRESSES = Center ± Radius on Mohr's circle: σ_1,2 = [(σ_x + σ_y)/2] ± √{[(σ_x − σ_y)/2]² + τ_xy²}; the term under the root uses HALF the difference (σ_x − σ_y)/2.
- MAXIMUM SHEAR = Radius of Mohr's circle = τ_max = √{[(σ_x − σ_y)/2]² + τ_xy²} = (σ_1 − σ_2)/2; it ALWAYS acts on planes inclined 45° to the principal axes.
- PHYSICAL ROTATION θ on element = CIRCLE ROTATION 2θ on Mohr's plot (same angular sense); this is the #1 angle mistake on exams.
- MOHR'S CIRCLE CENTER = [(σ_x + σ_y)/2, 0] on the σ-axis; RADIUS = τ_max; PRINCIPAL STRESSES = where circle crosses the σ-axis (where τ = 0).
- ON MAXIMUM-SHEAR PLANES, the normal stress is ALWAYS (σ_x + σ_y)/2 (the average); this average stress is constant for all planes through the element.
- COMBINED BENDING AND TORSION of a shaft: use equivalent torque T_e = √(M² + T²) for max-shear design, or equivalent moment M_e = 0.5[M + √(M² + T²)] for normal-stress design.
- PURE SHEAR (σ_x = σ_y = 0, τ ≠ 0) gives principal stresses σ_1 = +τ and σ_2 = −τ (equal and opposite); this is why shafts fail on a 45° helix under torsion.
- IN-PLANE vs OUT-OF-PLANE shear: the in-plane maximum (from Mohr) may not be the absolute maximum if the third principal stress σ_z ≠ 0; for plane stress σ_z = 0 so absolute max shear = τ_max (in-plane).
- FAILURE CRITERIA: Rankine (max normal stress) for brittle; Tresca (max shear ≤ σ_y/2) for ductile; von Mises (σ_v = √(σ_1² − σ_1 σ_2 + σ_2²) ≤ σ_y) most accurate for ductile but rarely asked directly.
- PLOT POINTS X(σ_x, τ_xy) and Y(σ_y, −τ_xy); their midpoint is the circle centre; their distance is the diameter = 2R; this is the fastest way to construct Mohr's circle by hand.
Last Minute Tips
- ANGLE TRAP: If the exam asks for stress at θ = 30°, use 2θ = 60° in the transformation equation. Most errors come from forgetting to double the angle.
- HALVING TRAP: The term (σ_x − σ_y)/2 uses HALF the difference, not the full difference. This halving appears in both the principal-stress formula AND the max-shear formula. Check your denominator.
- MOHR'S CIRCLE vs ELEMENT: The circle is plotted in (σ, τ) space, NOT in physical x–y space. Points on the circle represent different planes through the material point; a rotation of the element by θ appears as 2θ on the circle.
- SHAFT DESIGN: If a problem gives M and T, first calculate T_e = √(M² + T²), then use τ_max = 16T_e/(πd³) for max-shear design. Using T directly (forgetting the equivalent) is a common error.
- SIGN CONVENTIONS: When plotting Mohr's circle, use (σ_x, τ_xy) and (σ_y, −τ_xy) for X and Y. The NEGATIVE on Y is not an error — it ensures XY is a diameter. Mixing signs here ruins the circle.
Comparison Tables
Rows
Values
- τ = 0 (by definition)
- τ = ±τ_max (maximum magnitude)
Property
Shear on that plane
Values
- σ = σ_1 or σ_2 (extreme value)
- σ = (σ_x + σ_y)/2 (always the average)
Property
Normal stress on plane
Values
- At angle θ_p where tan(2θ_p) = 2τ_xy/(σ_x − σ_y)
- At θ_p ± 45° (always 45° from principal planes)
Property
Plane orientation
Values
- Rankine (normal-stress) failure theory
- Tresca (max-shear) and von Mises theories
Property
Used in
Values
- Points where circle intersects σ-axis (τ = 0)
- Top and bottom of circle (maximum |τ|)
Property
Graphical location on Mohr circle
Columns
- Property
- Principal Stress σ_1, σ_2
- Maximum Shear Stress τ_max
Table Title
Principal Stresses vs Maximum Shear Stress
Rows
Values
- σ_1 ≥ σ_ut
- Brittle materials (cast iron, concrete, stone)
- σ_1 ≤ σ_y
Property
Maximum Normal Stress (Rankine)
Values
- τ_max ≥ σ_y/2 OR (σ_1 − σ_2)/2 ≥ σ_y/2
- Ductile metals; simple and conservative
- τ_max ≤ σ_y/2
Property
Maximum Shear (Tresca)
Values
- σ_v ≥ σ_y where σ_v = √(σ_1² − σ_1 σ_2 + σ_2²)
- Ductile metals; most accurate
- σ_v ≤ σ_y
Property
Von Mises (Distortion Energy)
Columns
- Theory
- Failure Criterion
- Best For
- Formula
Table Title
Failure Theories Compared
Rows
Values
- σ_x = P/A ± Mc/I, σ_y = 0, τ = 0
- σ_max = P/A + Mc/I; σ_min = P/A − Mc/I
- Eccentric load on column or post
Property
Axial + Bending
Values
- σ_x = 32M/(πd³), σ_y = 0, τ_xy = 16T/(πd³)
- τ_max (from Mohr) or use T_e, M_e
- Pulley shaft, coupling, propeller shaft
Property
Bending + Torsion (Shaft)
Values
- σ_hoop = pD/(2t), σ_long = pD/(4t), σ_axial = P/A
- Add axial term to hoop and long. stresses
- Pressurized pipe under tension/compression
Property
Internal Pressure + Axial Load
Values
- σ_x = 0, σ_y = 0, τ_xy = 16T/(πd³)
- σ_1 = τ, σ_2 = −τ, τ_max = τ
- Shaft rotating under torque alone
Property
Pure Torsion (no bending)
Columns
- Scenario
- Stress Components
- Key Formula
- Typical Application
Table Title
Combined Loading: Common Scenarios
Rows
Values
- σ = 32M/(πd³)
- —
- Maximum at outer fibre
Property
Solid Circular Shaft (bending)
Values
- —
- τ = 16T/(πd³)
- Maximum at outer surface; zero at centre
Property
Solid Circular Shaft (torsion)
Values
- Hoop: σ_h = pD/(2t); Long: σ_l = pD/(4t)
- —
- p = internal pressure; D = diameter; t = thickness
Property
Thin-Wall Circular Pressure Vessel
Values
- σ = My/I = M c/I at extreme fibre
- τ = (3V)/(2A) for rectangular cross-section
- y = distance from neutral axis; V = shear force
Property
Rectangular Beam (bending)
Columns
- Member Type
- Normal Stress Formula
- Shear Stress Formula
- Notes
Table Title
Reference Stress Formulas for Common Members
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