CELE Strength of Materials — Combined Stresses and Mohr's CircleRevision Notes

Formulas

Exam Tips

• Identify σ_x, σ_y, and τ_xy FIRST before touching any principal-stress formula. Write them down explicitly.
• For rectangular sections: I = bh³/12; for circular sections: I = πd⁴/64, J = πd⁴/32.
• At the neutral axis of a beam, σ_bending = 0, so only axial and shear stresses need combining.
• At the extreme fiber of a beam, τ_transverse = 0 (Q = 0 at the extreme fiber), so only normal stresses and torsional shear combine.

Key Points

• Superposition is valid only when the material is linearly elastic and deformations are small (Hooke's Law region).
• Only stresses of the SAME TYPE (both normal, or both shear) on the SAME plane can be added directly.
• Axial stress and bending stress are both normal stresses acting on the cross-sectional plane — they add algebraically.
• Bending shear and torsional shear are both shear stresses on the cross-section — they add algebraically at the same point.
• After superposition, you obtain σ_x, σ_y, and τ_xy at the critical point — the inputs to Mohr's Circle.
• Always identify the critical point (maximum combined stress) before applying transformation equations.

Definitions

Section Title

Superposition of Stresses

Common Mistakes

• Adding a normal stress (σ) and a shear stress (τ) directly — they act on different faces and CANNOT be summed; they must be combined via the transformation equations.
• Forgetting to check all critical points (top fiber, bottom fiber, neutral axis) when bending and axial stresses combine.
• Using the wrong sign for axial stress: compressive P gives negative σ = –P/A.
• Assuming the neutral axis location does not shift when axial load is combined with bending.

Formulas

Exam Tips

• Memorize: normal stress equation has cosine of 2θ; shear stress equation has sine of 2θ — opposite from what many expect.
• When asked for stress on a plane at θ = 45°, use 2θ = 90°: cos90° = 0, sin90° = 1. This simplifies the equations greatly.
• The transformation equations are rarely needed in full for board problems — Mohr's Circle is faster and less error-prone.
• Check: at θ = 0, you should recover σ_x and τ_xy from the transformation equation — verify your formula.

Key Points

• Stress is NOT a scalar — its value depends on the orientation of the plane on which it acts.
• The transformation equations give the normal stress σ_x' and shear stress τ_x'y' on a plane whose normal is rotated θ counterclockwise from the x-axis.
• The sum (σ_x + σ_y) is INVARIANT under rotation — it does not change with plane orientation.
• As θ varies from 0° to 180°, the stresses trace one full circle on the Mohr's Circle diagram.
• A rotation of θ in the physical element corresponds to a rotation of 2θ on Mohr's Circle.
• The transformation equations are derived from equilibrium of a wedge element — no material property is involved.

Definitions

Section Title

Stress Transformation Equations

Common Mistakes

• Using θ instead of 2θ in the trigonometric terms — the equations use cos2θ and sin2θ, not cosθ and sinθ.
• Mixing up which sign of τ_xy to use when setting up the equations — always establish and follow one convention.
• Forgetting that σ_x' + σ_y' = σ_x + σ_y (invariant) — use this to find σ_y' after computing σ_x' without re-solving.
• Not checking the answer by confirming the computed (σ_x', τ_x'y') lies on Mohr's Circle.

Formulas

Exam Tips

• QUICK CHECK: σ_1 + σ_2 must equal σ_x + σ_y. If your computed sum differs, you made an arithmetic error.
• For PURE SHEAR (σ_x = σ_y = 0): σ_1 = +τ_xy, σ_2 = –τ_xy, θ_p = 45°. Memorize this — it is a classic board case.
• For UNIAXIAL STRESS (σ_y = 0, τ_xy = 0): σ_1 = σ_x, σ_2 = 0, τ_max = σ_x/2 at 45°. Also memorize.
• When the board gives you principal stresses and asks for σ_x, σ_y: use the invariant σ_x + σ_y = σ_1 + σ_2 and the R formula.

Key Points

• Principal stresses σ_1 and σ_2 are the MAXIMUM and MINIMUM normal stresses at the point, acting on planes where shear stress is ZERO.
• The algebraic maximum is σ_1 (could be tension or compression); the algebraic minimum is σ_2.
• Maximum in-plane shear stress τ_max equals the RADIUS of Mohr's Circle, R.
• The principal planes are oriented at θ_p from the reference x-face; the max-shear planes are at θ_p ± 45°.
• The normal stress on the planes of maximum shear equals the average stress σ_avg = (σ_x + σ_y)/2 — it is NOT zero unless σ_x + σ_y = 0.
• For a complete 3D analysis, the absolute maximum shear may be larger than the in-plane maximum if both principal stresses have the same sign.

Definitions

Section Title

Principal Stresses and Maximum In-Plane Shear Stress

Common Mistakes

• Halving error: writing τ_max = √[(σ_x–σ_y)² + τ_xy²] — missing the division by 2 inside the radical. The correct term is (σ_x–σ_y)/2, not (σ_x–σ_y).
• Assuming zero normal stress on the maximum shear plane — it equals σ_avg, not zero.
• Forgetting the second principal plane is at θ_p + 90° — and that the sign of τ reverses.
• Using τ_max = (σ_1 + σ_2)/2 instead of (σ_1 – σ_2)/2.
• Not checking whether the absolute maximum shear exceeds the in-plane maximum when both principal stresses are of the same sign.

Formulas

Exam Tips

• FASTEST METHOD for board exams: Write C and R, then σ_1 = C + R, σ_2 = C – R, τ_max = R. Four lines of work.
• To find stress on a specific plane at angle θ from the x-face: move 2θ along the circle from point X.
• The top and bottom of the circle (maximum and minimum τ) are always at 90° from the x-intercepts on the circle = 45° in physical space.
• Draw a rough sketch of Mohr's Circle even for calculation problems — it confirms your answers and prevents sign errors.

Key Points

• Mohr's Circle is a graphical representation of the stress transformation equations — every point on the circle represents the stress state on a specific plane through the point.
• The horizontal axis is the normal stress σ (positive = tension, rightward); the vertical axis is shear stress τ (sign convention: see below).
• CONVENTION: Plot τ downward as positive (some texts use upward) — the key is CONSISTENCY within the problem.
• The center C is always on the σ-axis at coordinates (σ_avg, 0).
• The radius R of the circle equals τ_max (in-plane).
• Principal stresses are the rightmost and leftmost intercepts of the circle on the σ-axis.
• A physical rotation of angle θ corresponds to a circle rotation of 2θ in the SAME direction.

Definitions

Section Title

Mohr's Circle — Construction and Application

Common Mistakes

• Plotting both X and Y with the SAME sign of τ — they must be plotted with OPPOSITE signs of τ (one above, one below the σ-axis).
• Measuring physical angles θ directly on the circle diagram — remember to divide by 2 to get the actual plane orientation.
• Confusing the direction of rotation: CCW rotation in the physical element corresponds to CCW rotation on the circle (if using the standard convention).
• Misidentifying which intercept is σ_1 (maximum) and which is σ_2 — always σ_1 is to the right (larger algebraic value).

Formulas

Exam Tips

• Memorize T_e = √(M² + T²) — it appears on nearly every shaft-related board problem involving combined loads.
• The formula τ_max = 16T_e/(πd³) is the single most-tested shaft formula in combined stress problems.
• If the problem says 'factor of safety = 2 against yielding': S_sy(Tresca) = S_y/2, so τ_allow = S_y/4 after applying F.S.
• For hollow shafts: replace πd³/16 with 16T_e·D/(π(D⁴–d⁴)) where D = outer diameter, d = inner diameter.

Key Points

• A circular shaft under simultaneous bending moment M and torque T has a plane-stress state at the surface: σ_x = 32M/(πd³), σ_y = 0, τ_xy = 16T/(πd³).
• The critical point is at the top or bottom of the cross-section (maximum bending stress) OR at the side (where bending stress is zero but torsional shear is maximum) — always check both.
• Equivalent moment M_e is used with the MAXIMUM NORMAL STRESS theory (Rankine/brittle).
• Equivalent torque T_e is used with the MAXIMUM SHEAR STRESS theory (Tresca/ductile).
• These equivalent quantities allow shaft sizing using standard bending and torsion formulas without explicitly running through Mohr's Circle each time.
• The NSCP 2015 and machine design references use these equivalent moments extensively for power transmission shaft checks.

Definitions

Section Title

Combined Bending and Torsion of Shafts

Common Mistakes

• Using M_e instead of T_e for ductile shaft design — the Tresca criterion requires T_e (maximum shear), not M_e (maximum normal).
• Forgetting σ_y = 0 on a circular shaft surface — bending stress acts only in one direction at the extreme fiber.
• Not converting units: M in kN·m must become N·mm (×10⁶) before substituting into the formula with d in mm.
• Using d³ when the formula requires d³ — double-check: the section modulus for bending is πd³/32 and for torsion is πd³/16.

Formulas

Exam Tips

• For board exam: if material is STEEL → use Tresca or von Mises. If CAST IRON → use Rankine.
• Tresca yield surface is a hexagon inscribed in the von Mises ellipse — Tresca is always ≤ von Mises limit.
• F.S. = S_y / σ_v (von Mises) or S_y / (2τ_max) (Tresca). Know both forms for factor-of-safety problems.
• Quick comparison: For pure shear (σ_1 = –σ_2 = τ), Tresca gives yield at τ = S_y/2; von Mises gives τ = S_y/√3 ≈ 0.577 S_y — von Mises is less conservative in pure shear.

Key Points

• Failure theories predict when a material under combined stress will yield or fracture by comparing the multi-axial stress state to a uniaxial strength value.
• Maximum Normal Stress Theory (Rankine): Failure when σ_1 = S_u (ultimate strength). Best for BRITTLE materials like cast iron.
• Maximum Shear Stress Theory (Tresca): Failure when τ_max = S_y/2. Conservative (safe) estimate for DUCTILE materials.
• Distortion Energy Theory (von Mises): Failure when von Mises stress σ_v = S_y. Most ACCURATE for ductile materials; used in AISC 360 and modern codes.
• For exam purposes: Tresca is conservative (gives larger required cross-section), von Mises is less conservative (smaller cross-section) — both are tested.
• For plane stress with σ_3 = 0: Tresca gives 2τ_max = σ_1 – σ_2 (if opposite signs) or max(σ_1, σ_2) (if same sign) ≤ S_y.

Definitions

Section Title

Failure Theories

Common Mistakes

• Applying Rankine (maximum normal stress) to ductile steel — it is only appropriate for brittle materials.
• Using 2τ_max = S_y when both principal stresses are the same sign — the Tresca criterion in this case is max(σ_1, σ_2) = S_y, not (σ_1 – σ_2) = S_y.
• Forgetting the factor of 3 in the von Mises component form: the shear term is 3τ_xy², not τ_xy².
• Confusing allowable stress (S_y / F.S.) with the failure stress S_y when setting up failure equations.