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Mechanics of Materials: formula sheet

Strength of materials · vessels · joints & the shop floor · 78 formulas · metric edition 1

The full study guide: modules, problems, answer key
Module 1 · Stress and Strain
Mechanics of Materials-1 Normal (Axial) Stress
σ=PA\sigma = \frac{P}{A}
Axial stress in a bar or hanger rod: the internal force divided by the cross-sectional area that…
Mechanics of Materials-2 Normal Strain (ε = δ/L)
ε=δL\varepsilon = \frac{\delta}{L}
Normal strain as the change in length divided by the original length, a dimensionless ratio usually…
Mechanics of Materials-3 Young's Modulus (E = σ/ε)
E=σεE = \frac{\sigma}{\varepsilon}
Young's modulus as the ratio of normal stress to normal strain, the stiffness constant of a…
Mechanics of Materials-4 Axial Deformation (δ = PL/AE)
δ=PLAE\delta = \frac{P L}{A E}
Elongation of an axially loaded bar from load, length, area and Young's modulus: the workhorse δ =…
Mechanics of Materials-5 Average Shear Stress (τ = V/A)
τ=VA\tau = \frac{V}{A}
Average shear stress on a bolt, pin or weld throat: the transverse force divided by the area…
Mechanics of Materials-6 Shear Modulus (G = τ/γ)
G=τγG = \frac{\tau}{\gamma}
Shear (rigidity) modulus as shear stress divided by shear strain, roughly 0.38 of Young's modulus…
Mechanics of Materials-7 Poisson's Ratio
ν=εlatεax\nu = \frac{\varepsilon_{lat}}{\varepsilon_{ax}}
Poisson's ratio, the lateral contraction per unit of axial extension: close to 0.30 for steel and…
Mechanics of Materials-8 Relation Between E, G and ν
E=2G(1+ν)E = 2G(1 + \nu)
The isotropic elastic identity E = 2G(1 + ν), linking Young's modulus, the shear modulus and…
Mechanics of Materials-9 Factor of Safety
FS=σuσallowFS = \frac{\sigma_{u}}{\sigma_{allow}}
Factor of safety as ultimate or yield strength divided by the allowable working stress, the…
Mechanics of Materials-10 Thermal Stress in a Restrained Member
σ=EαΔT\sigma = E \alpha \Delta T
Stress raised in a fully restrained member that is heated or cooled: the cause of rail sun kinks…
Module 2 · Torsion and Shafts
Mechanics of Materials-11 Polar Moment of Inertia: Solid Shaft
J=πd432J = \frac{\pi d^{4}}{32}
Polar moment of inertia of a solid round shaft, J = πd⁴/32, in m⁴: exactly twice the diametral…
Mechanics of Materials-12 Torsional Shear Stress (τ = Tr/J)
τ=TrJ\tau = \frac{T r}{J}
Torsional shear stress at radius r in a round shaft, τ = Tr/J, peaking at the surface, with J…
Mechanics of Materials-13 Angle of Twist (φ = TL/JG)
φ=TLJG\varphi = \frac{T L}{J G}
Angle of twist of a round shaft under torque, φ = TL/JG, the stiffness check that governs long…
Mechanics of Materials-14 Shaft Torque from Power and Angular Speed
T=PωT = \frac{P}{\omega}
The torque in a rotating shaft: transmitted power divided by angular speed, with ω in radians per…
Mechanics of Materials-15 Rotational Power (P = τω)
P=τωP = \tau \omega
Mechanical power delivered by a torque turning at angular velocity ω.
Mechanics of Materials-16 Shaft Diameter from Allowable Torsional Shear
d=16Tπτ3d = \sqrt[3]{\frac{16 T}{\pi \tau}}
The diameter a solid round shaft needs so that the torsional shear stress at its surface stays…
Mechanics of Materials-17 Shear Stress in a Parallel Key
τ=2Td w L\tau = \frac{2T}{d \, w \, L}
The average shear stress across a parallel key in a shaft keyway: the tangential force at the shaft…
Module 3 · Bending and Beams
Mechanics of Materials-18 Area Moment of Inertia: Rectangle
I=bh312I = \frac{b h^{3}}{12}
Area moment of inertia of a rectangle about its centroidal axis, I = bh³/12, returned in m⁴ and…
Mechanics of Materials-19 Area Moment of Inertia: Solid Round Bar
I=πd464I = \frac{\pi d^{4}}{64}
Area moment of inertia of a solid round bar about a diameter, I = πd⁴/64, returned in m⁴ and…
Mechanics of Materials-20 Centroid of a Composite Area (ȳ = ΣAȳ / ΣA)
yˉ=A1y1+A2y2A1+A2\bar{y} = \frac{A_1 y_1 + A_2 y_2}{A_1 + A_2}
Centroid of an area built from two parts: the area-weighted average of the parts' own centroids,…
Mechanics of Materials-21 Parallel Axis Theorem (I = I_c + Ad²)
I=Ic+Ad2I = I_c + A d^{2}
Second moment of area of a shape about an axis parallel to its own centroidal axis: add A times the…
Mechanics of Materials-22 Moment of Inertia: I-Beam or Built-Up Section
I=BH3−(B−tw)(H−2tf)312I = \frac{B H^{3} - (B - t_w)(H - 2t_f)^{3}}{12}
Strong-axis second moment of area of a doubly symmetric I-shape, taken as the full bounding…
Mechanics of Materials-23 Elastic Section Modulus (S = I/c)
S=IcS = \frac{I}{c}
Elastic section modulus S = I/c, in m³, the single number that turns a bending moment straight into…
Mechanics of Materials-24 Max Bending Moment: Centre Point Load
M=PL4M = \frac{P L}{4}
Maximum bending moment in a simply supported beam carrying one point load at midspan, M = PL/4,…
Mechanics of Materials-25 Max Bending Moment: Uniform Load
M=wL28M = \frac{w L^{2}}{8}
Maximum bending moment at midspan of a simply supported beam under a uniformly distributed load,…
Mechanics of Materials-26 Bending Stress (σ = Mc/I)
σ=McI\sigma = \frac{M c}{I}
Bending stress at a distance c from the neutral axis of a beam, with the area moment of inertia I…
Mechanics of Materials-27 Bending Stress from Section Modulus (σ = M/S)
σ=MS\sigma = \frac{M}{S}
Bending stress straight from the moment and a tabulated section modulus S in m³, the everyday form…
Mechanics of Materials-28 Transverse Shear Stress (τ = VQ/Ib)
τ=VQIb\tau = \frac{V Q}{I b}
Shear stress at any height in a beam cross-section, peaking at the neutral axis.
Mechanics of Materials-29 Shear Flow (q = VQ/I)
q=VQIq = \frac{V Q}{I}
Longitudinal shear force per unit length that must cross a joint in a built-up beam: the number…
Mechanics of Materials-30 Beam Deflection: Simply Supported, Centre Load
δ=PL348EI\delta = \frac{P L^{3}}{48 E I}
Maximum midspan deflection of a simply supported beam with a central point load, δ = PL³/48EI, with…
Mechanics of Materials-31 Beam Deflection: Simply Supported, Uniform Load
δ=5wL4384EI\delta = \frac{5 w L^{4}}{384 E I}
Maximum midspan deflection of a simply supported beam under a uniform load, δ = 5wL⁴/384EI, with I…
Mechanics of Materials-32 Cantilever Deflection: End Load
δ=PL33EI\delta = \frac{P L^{3}}{3 E I}
Tip deflection of a cantilever carrying a point load at its free end, δ = PL³/3EI, with I entered…
Mechanics of Materials-33 Cantilever Deflection: Uniform Load
δ=wL48EI\delta = \frac{w L^{4}}{8 E I}
Tip deflection of a cantilever carrying a uniformly distributed load along its whole length, δ =…
Module 4 · Columns, Vessels, Combined Stress
Mechanics of Materials-34 Radius of Gyration (r = √(I/A))
r=IAr = \sqrt{\frac{I}{A}}
Radius of gyration of a cross-section, r = √(I/A), the shape property that decides how slender a…
Mechanics of Materials-35 Slenderness Ratio (KL/r)
λ=KLr\lambda = \frac{K L}{r}
Slenderness ratio KL/r of a compression member, the single number that decides whether a column…
Mechanics of Materials-36 Euler Critical Buckling Load
Pcr=π2EI(KL)2P_{cr} = \frac{\pi^{2} E I}{(K L)^{2}}
Euler's critical buckling load for a slender column, using the end-condition factor K and the area…
Mechanics of Materials-37 Hoop Stress in a Thin-Walled Cylinder
σh=pd2t\sigma_{h} = \frac{p d}{2 t}
Hoop (circumferential) stress in a thin-walled pipe or pressure vessel, σ = pd/2t: exactly twice…
Mechanics of Materials-38 Longitudinal Stress in a Thin-Walled Cylinder
σl=pd4t\sigma_{l} = \frac{p d}{4 t}
Longitudinal (axial) stress in a thin-walled cylinder under internal pressure, σ = pd/4t: exactly…
Mechanics of Materials-39 Membrane Stress in a Thin-Walled Sphere (σ = pr/2t)
σ=pr2t\sigma = \frac{p r}{2 t}
Membrane stress in the wall of a thin-walled sphere under internal pressure, σ = pr/2t.
Mechanics of Materials-40 Lamé Hoop Stress in a Thick-Walled Cylinder
σθ=piri2ro2−ri2(1+ro2r2)\sigma_{\theta} = \frac{p_i r_i^{2}}{r_o^{2} - r_i^{2}} \left(1 + \frac{r_o^{2}}{r^{2}}\right)
Circumferential (hoop) stress at any radius r in a thick-walled cylinder with internal pressure…
Mechanics of Materials-41 Lamé Radial Stress in a Thick-Walled Cylinder
σr=piri2ro2−ri2(1−ro2r2)\sigma_{r} = \frac{p_i r_i^{2}}{r_o^{2} - r_i^{2}} \left(1 - \frac{r_o^{2}}{r^{2}}\right)
Radial stress at any radius r in a thick-walled cylinder with internal pressure only.
Mechanics of Materials-42 ASME Required Wall Thickness (t = PR/(SE − 0.6P))
t=PRSE−0.6Pt = \frac{P R}{S E - 0.6 P}
Minimum required wall thickness of a cylindrical shell under internal pressure, from ASME Section…
Mechanics of Materials-43 Combined Axial and Bending Stress
σ=PA+McI\sigma = \frac{P}{A} + \frac{M c}{I}
Extreme-fibre stress where an axial force and a bending moment act together, as in an eccentrically…
Mechanics of Materials-44 Maximum Principal Stress (Mohr's Circle)
σ1=σx+σy2+(σx−σy2)2+τxy2\sigma_1 = \frac{\sigma_x + \sigma_y}{2} + \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}
Larger of the two principal stresses for a plane-stress element, from the normal stresses σx and σy…
Mechanics of Materials-45 Minimum Principal Stress (Mohr's Circle)
σ2=σx+σy2−(σx−σy2)2+τxy2\sigma_2 = \frac{\sigma_x + \sigma_y}{2} - \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}
Smaller of the two principal stresses for a plane-stress element: the circle's centre minus its…
Mechanics of Materials-46 Maximum In-Plane Shear Stress
τmax=(σx−σy2)2+τxy2\tau_{max} = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}
Radius of Mohr's circle: the largest shear stress on any plane through a plane-stress element,…
Mechanics of Materials-47 Von Mises Equivalent Stress (Plane Stress)
σv=σx2−σxσy+σy2+3τxy2\sigma_v = \sqrt{\sigma_x^{2} - \sigma_x \sigma_y + \sigma_y^{2} + 3\tau_{xy}^{2}}
Von Mises equivalent stress for a plane stress state: the single tensile stress that would do the…
Mechanics of Materials-48 Stress Concentration (σmax = Kt σnom)
σmax=Kt σnom\sigma_{max} = K_t \, \sigma_{nom}
Peak elastic stress at a hole, notch, groove or fillet: the nominal stress on the net section…
Module 5 · Bolts, Welds and the Shop Floor
Mechanics of Materials-49 Thread Tensile Stress Area
At=π4(d−ktp)2A_{t} = \frac{\pi}{4} \left( d - k_{t} p \right)^{2}
The effective cross-section a threaded fastener pulls on: a circle of diameter partway between the…
Mechanics of Materials-50 Bolt Preload from Torque (T = KDF)
T=KDFT = K D F
Bolt preload from tightening torque using the nut factor K, T = KDF, the field method behind every…
Mechanics of Materials-51 Joint Stiffness Ratio of a Bolted Joint
C=kbkb+kmC = \frac{k_{b}}{k_{b} + k_{m}}
The share of an external load that lands on the bolt rather than on the clamped members, set by the…
Mechanics of Materials-52 External Load That Separates a Preloaded Joint
P0=Fi1−CP_{0} = \frac{F_{i}}{1 - C}
The external tensile load at which the clamped members lose all their compression and the joint…
Mechanics of Materials-53 Bearing Stress on a Pin or Bolt (σ = P/dt)
σb=Pd t\sigma_{b} = \frac{P}{d \, t}
Bearing stress where a pin or bolt presses on the side of its hole: the load divided by the…
Mechanics of Materials-54 Power Screw Raising Torque (Square Thread)
TR=Fdm2(l+πμdmπdm−μl)T_{R} = \frac{F d_{m}}{2} \left( \frac{l + \pi \mu d_{m}}{\pi d_{m} - \mu l} \right)
Torque needed at the nut or the handle to drive a square-thread power screw AGAINST its load: a…
Mechanics of Materials-55 Power Screw Lowering Torque and Self-Locking
TL=Fdm2(πμdm−lπdm+μl)T_{L} = \frac{F d_{m}}{2} \left( \frac{\pi \mu d_{m} - l}{\pi d_{m} + \mu l} \right)
Torque to let a square-thread power screw down under its load, and the sign of the answer, which is…
Mechanics of Materials-56 Fillet Weld Effective Throat
a=0.707 za = 0.707 \, z
The shortest distance from the root of an equal-leg fillet weld to its face: the plane the weld…
Mechanics of Materials-57 Fillet Weld Capacity from Throat Area
F=τ AtF = \tau \, A_t
The load a fillet weld can carry: the allowable stress on the weld metal multiplied by the area of…
Mechanics of Materials-58 Fillet Weld Size for a Load per Unit Length
w=f0.707 τaw = \frac{f}{0.707 \, \tau_{a}}
The leg size an equal-leg fillet weld needs to carry a stated load per unit length at a stated…
Mechanics of Materials-59 Shear Stress on a Fillet Weld Throat
τ=F0.707 wL\tau = \frac{F}{0.707 \, w L}
Shear stress on the effective throat of an equal-leg fillet weld: the load divided by 0.707 times…
Mechanics of Materials-60 Welding Heat Input
H=η V ISH = \frac{\eta \, V \, I}{S}
The energy the arc delivers into every millimetre of joint: volts times amperes gives the power at…
Mechanics of Materials-61 Carbon Equivalent (IIW)
CE=C+Mn6+Cr+Mo+V5+Ni+Cu15CE = C + \frac{Mn}{6} + \frac{Cr + Mo + V}{5} + \frac{Ni + Cu}{15}
The International Institute of Welding's screening number for weldability: every alloying element…
Mechanics of Materials-62 Cutting Speed and Spindle Speed
V=π D NV = \pi \, D \, N
The bridge between the speed the cutting edge wants and the rpm the machine is set to:…
Mechanics of Materials-63 Milling Table Feed Rate
vf=N z fzv_f = N \, z \, f_z
The speed the table actually travels under a milling cutter: spindle speed times the number of…
Mechanics of Materials-64 Material Removal Rate: Turning
Q=V f apQ = V \, f \, a_p
How fast metal is leaving the part on a lathe: cutting speed times feed per revolution times depth…
Mechanics of Materials-65 Machining Time: Turning Pass
tm=Lf Nt_m = \frac{L}{f \, N}
How long one turning pass takes: the length to be travelled divided by how fast the tool advances,…
Mechanics of Materials-66 Taylor Tool Life Equation
V T n=CV \, T^{\,n} = C
The relation Frederick Taylor drew out of twenty-six years of cutting trials: cutting speed times…
Module 6 · Materials, Fatigue, Fracture
Mechanics of Materials-67 Vickers Hardness
HV=2Fsin⁡(136∘/2)d2=1.8544 Fd2HV = \dfrac{2F \sin(136^\circ/2)}{d^{2}} = \dfrac{1.8544\,F}{d^{2}}
Press a square-based diamond pyramid into the metal, measure the two diagonals of the mark it…
Mechanics of Materials-68 Hall–Petch Relation
σy=σ0+ky d−1/2\sigma_y = \sigma_0 + k_y \, d^{-1/2}
Finer grains make a stronger metal, and the strength climbs as the inverse square root of the grain…
Mechanics of Materials-69 True Stress from Engineering Stress
σt=σe (1+e),εt=ln⁡(1+e)\sigma_t = \sigma_e \, (1 + e), \qquad \varepsilon_t = \ln(1 + e)
A tensile machine divides load by the ORIGINAL area, because that is the only area it knows.
Mechanics of Materials-70 Goodman Fatigue Criterion
σaSe+σmSu=1n\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_u} = \frac{1}{n}
The modified Goodman line for infinite life under a fluctuating stress: alternating stress over…
Mechanics of Materials-71 Basquin S-N Relation
σa=σf′ (2Nf)b\sigma_a = \sigma_f' \, (2N_f)^{b}
The straight line an S-N curve becomes on log-log paper: alternating stress against reversals to…
Mechanics of Materials-72 Miner's Cumulative Damage Rule (Three Blocks)
D=n1N1+n2N2+n3N3D = \frac{n_1}{N_1} + \frac{n_2}{N_2} + \frac{n_3}{N_3}
Linear damage summation over three blocks of a variable-amplitude load history.
Mechanics of Materials-73 Stress Intensity Factor
K=Y σ πaK = Y \, \sigma \, \sqrt{\pi a}
Irwin's 1957 result, and the single most useful equation in fracture mechanics: the entire elastic…
Mechanics of Materials-74 Paris Law Crack Growth Rate
dadN=C(ΔK)m\dfrac{da}{dN} = C \left( \Delta K \right)^{m}
Paris and Erdogan's 1963 observation that fatigue crack growth plots as a straight line on log-log…
Mechanics of Materials-75 Plastic Section Modulus: Rectangle
Z=bh24Z = \frac{b h^{2}}{4}
Plastic section modulus of a solid rectangle, the first moment of the two half-areas about the…
Mechanics of Materials-76 Plastic Moment Capacity (Mp = Z fy)
Mp=ZfyM_p = Z f_y
Moment at which a compact steel section is fully plastic, with every fibre at yield.
Mechanics of Materials-77 LRFD Load Combination (1.2D + 1.6L)
U=1.2D+1.6LU = 1.2 D + 1.6 L
The everyday strength-design load combination from ASCE 7 (combination 2) and the National Building…
Mechanics of Materials-78 Elastic Modulus from Compressive Strength
Ec=kfc′E_c = k \sqrt{f'_c}
The square-root correlation every concrete code carries: stiffness estimated from compressive…