Mechanics of Materials Equation Sheet - Chapters 1-14 - formulas and key relations

Nov 2, 2026

Mechanics of Materials Equation Sheet compiles core governing relations for stress, strain, elasticity, and structural response. It covers simple stress and strain, elastic properties (E, G, ν), axial deformation, torsion, bending moment–shear–deflection links, transverse shear, thin-walled pressure vessels, failure criteria (distortion energy and brittle limits), and plane stress transformations using Mohr’s circle. Additional sections include beam deflections, strain energy, generalized Hooke’s law, and column buckling with critical load and stress expressions.

LanguageEnglish

CategoryTechnology

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Reading Time5 min

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General

Chapter 1-2: Simple Stress and Strain

Chapter 3: Elastic Properties of Materials

Chapter 4: Axial Load

Chapter 5: Torsion

Chapter 6: Bending

Chapter 7: Transverse Shear

Chapter 8: Thin-Walled Pressure Vessels

Chapter 9: Plane Stress Transformations

Chapter 10: Failure Criteria

Chapter 12: Beam Deflections

Chapter 13: Column Buckling

Chapter 14: Strain Energy

Mechanics of Materials Equation Sheet General sin θa sin θb sin θc = = a b c c2 = a2 + b2 − 2abcos θc W = mg g = 32.2 ft/sec2 = 9.81 m/sec2 N = kg·m slug = lb sec2 , (ft/ sec2 ) N lb Pa = psi = m2 , in2 Chapter 1 , 2: Simple Stress and Strain σ = NA , τavg = VA Acircle = πr2 = π4 d2 δ L′ − L π 􀀥avg = L = L , γ = 2 − θ Chapter 3: Elastic Properties of Materials 􀀥lat σ = E 􀀥 , ν = − ~~ ~~ 􀀥long τ = G γ , G = 2~~ ~~(1~~ ~~ν) Chapter 4: Axial Load EAdudx = N (x) δ = 􀀡0 L 􀀥 dx = 􀀡0 L NA(xE) dx Constant N, A, E : δ = 􀀥 L = NALE Displacement kinematics: Series : δ1 + δ2 + · · · Parallel : δ1 = δ2 = · · ·δB/A = uB − uA 􀀥T = α∆T, δT = α∆TL Chapter 5: Torsion τ = TJρ , τmax = TJc Jshaft = π2 c4 , Jtube = π2 (c4o − c4i) P = T ω , W = N·m/s, hp = ft·lb/s GJddφx = T (x), φ = 􀀡0 L TJ(xG) dx Constant T A G : φ = TL, , JG Twist angle kinematics: Series : φ 1 + φ2 + · · · Parallel : φ 1 = φ2 = · · · Thin-walled tube: τavg = 2tTAm , q = τavg t = 2TAm φ = 4TAL2mG 􀀢 dts Chapter 6: Bending Shear and Moment Relations: ddVx = w , ∆V = 􀀡 w (x)dx dMdx = M, ∆M = 􀀡 V (x)dx σx = − MzIzy , σmax = MIc Irect = 112 bh3 , Icircle = 14πr4 I = ¯I+ Ad σx = − MzIzy + MyIyz My Iz Iz tan α = = tan θ Mz Iy Iy Chapter 7: Transverse Shear τ = V~~ ~~QIt , Q = 􀀡A′ y dA′ = y¯′ A′ Asemicircle = 12πr2 , y¯ = 43rπ V Q q = τ t = I , Vfastener = q s Chapter 8: Thin-Walled Pressure Vessels pr pr pr σhoop = t , σlong = 2t , σsphere = 2tr is inside radius. Combined Stresses P M c V Q Tc σ = ± A ± I , τ = ± It ± J Chapter 10: Failure Criteria τabs−max occurs at max radius of Max distortion energy failure criteria:(σ2p1 − σp1σp2 + σ2p2 ) ≤ (σY )2 Brittle materials max (|σp1 | , |σp2 |) ≤ σY Chapter 9: Plane Stress Transformations 1 σavg = 2 (σx + σy ) (σy , −⌧xy ) σp2 = σavg − R σp1 = σavg + R 2✓p1 σ 2✓s (σx , ⌧xy ) ⌧ ⌧inmpalxane = R = s ✓σx~~ ~~−2~~ ~~σy~~ ~~ ◆2 +(⌧xy )2 σx′ = σx~~ ~~+2~~ ~~σy + σx~~ ~~−2~~ ~~σy cos 2θ+τxy sin 2θ σy′ = σx~~ ~~+2~~ ~~σy − σx~~ ~~−2~~ ~~σy cos 2θ−τxy sin 2θ τx′ y′ = − σx~~ ~~−2~~ ~~σy sin 2θ + τxy cos 2θ Center of Mohr’s Circle: σx + σy (σ, τ ) = (σavg , 0), where σavg = 2 Radius of Mohr’s Circle: R = 􀀥 􀀣 σx~~ ~~−2~~ ~~σy~~ ~~ 􀀤 2 + (τxy)2 Plot (σx , τxy) and (σy , −τxy), + τ ↓ . Principal plane with no shear 3 Mohr circles for three planes. (3D principal stresses, σ 1 > σ2 > σ3 ) : τ abs = max σmax − σmin 2 , σ 1 = σmax , σ3 = σmin When σp1 , σp2 same sign, τabsmax = max 􀀣 ~~ ~~|σp21~~ ~~| , |σp22~~ ~~| 􀀤 , occurs out of plane When σp1 , σp2 diﬀerent signs, σp1 − σp2 τ abs = ~~ ~~ = τ max max 2 in plane , occurs in plane Max shear stress failure criteria: τ abs ≤ σY max 2 When σp1 , σp2 same signs, max (|σp1 | , |σp2 |) ≤ σY , When σp1 , σp2 opposite signs,|σp1 − σp2| ≤ σY σp1 , σp2 = σavg ± R τxy tan 2θp = (σx − σy)/2 Check θ p with σ x′ transformation [eqn. to](eqn. to) determine if θp equals θp2 , or θp1; or use Mohr’s circle. Other angle is ±90◦ rotated. τ = 0 at principal orientations. On Mohr’s circle (σ, τ ) = (σp1 , 0), and (σ, τ ) = (σp2 , 0) . Max in-plane shear τinmaplxane = R = 􀀥 􀀣 σx~~ ~~−2~~ ~~σy~~ ~~ 􀀤 2 + (τxy)2 tan 2θs = − tan12θp = − ~~ ~~(σx~~ ~~xσyy)/2 ,θs = θp ± 45◦ Direction of τmax-in-plane determined from τx′ y′ transformation eqn. Normal stress at this orientation: σx + σy σx′ = σy′ = σavg = 2 Author: Dr. Lonny L. Thompson, Clemson University Chapter 12: Beam Deﬂections Relationships between w , V, M, θ , and v Deﬂection = v dv Slope angle: = θ dx EI d2 v Moment: M = ~~ ~~ = EI ~~ ~~ρ dx2 Shear: V = dMdx = ddx 􀀣EId2~~ ~~vdx2 􀀤 Load: w = ddVx = ddx22 􀀣EId2~~ ~~vdx2 􀀤 Method of Integration, determine internal moment as a function of position M (x) . d2 v EI ~~ ~~ = M (x) dx2 θ (x) = dvdx = 1EI (􀀡 M (x) dx + C1 ) v (x) = 1EI (􀀡 (􀀡 M (x) dx) dx+C1 x+Cs) C1 and C2 are determined from two end conditions (d

FAQs

What equations are provided for simple stress and strain in Chapters 1–2?

The sheet includes stress measures based on force and area and average shear relations based on shear force and area, along with related strain expressions such as average elongation or deformation forms.

How does the sheet handle plane stress transformations and principal stresses?

It provides stress transformation equations for σx′, σy′, and τx′y′, then uses Mohr’s circle parameters (σavg and radius R) to locate principal planes and compute maximum shear and in-plane shear relationships.

What failure criteria are summarized for combined stresses?

It lists distortion energy failure conditions for ductile behavior and maximum stress/brittle limits based on principal stresses, including max shear stress statements tied to allowable yield strength.

Nov 2, 2026

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