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Engineering Formula Sheet Engineering Formula Sheet Statistics Mode Place data in ascending order. Mean Mode = most frequently occurring value ∑ x If two values occur at the maximum frequency the data set is bimodal. If three or more values occur at the maximum µ = mean value frequency the data set is multi-modal. Σxi = sum of all data values (x1, x2, x3, … n = number of data values Median Place data in ascending order. Standard Deviation If n is odd, median = central value If n is even, median = mean of two central values ∑(x ) √ n = number of data values σ = standard deviation Range x = individual data value ( x , x , x , … i 1 2 3 Range = xmax - xmin n = number of data values xmax = maximum data value x = minimum data value min Probability Independent Events Frequency P (A and B and C) = PAPBPC x P (A and B and C) = probability of independent x events A and B and C occurring in sequence PA = probability of event A x x Mutually Exclusive Events fx = relative frequency of outcome x nx = number of events with outcome x P (A or B) = PA + PB n = total number of events P = probability of outcome x P (A or B) = probability of either mutually exclusive x event A or B occurring in a trial fa = frequency of all events PA = probability of event A Σxi = sum of all data values (x1, x2, x3, … Binomial Probability (order doesn’t matter) n = number of data values Conditional Probability ( ) ( | ) Pk = binomial probability of k successes in n trials ( | ) ( ) ( | ) ( ) ( | ) p = probability of a success q = 1 – p = probability of failure k = number of successes P (A|D) = probability of event A given event D P(A) = probability of event A occurring n = number of trials P(~A) = probability of event A not occurring P(D| ~A) = probability of event D given event A did not occur PLTW, Inc. Engineering Formulas IED POE DE CEA AE BE CIM EDD 1 Plane Geometry Ellipse Rectangle 2b Perimeter = 2a + 2b Circle 2a Area = ab B Triangle Parallelogram c Area = ½ bh a h h 2 2 2 a = b + c – 2bc·cos∠A Area = bh 2 2 2 A b = a + c – 2ac·cos∠B C b b c2 = a2 + b2 – 2ab·cos∠C s Right Triangle Regular Polygons f 2 2 2 c = a + b c a n = number of sides θ b Trapezoid a h Area = ½(a + b)h h h b Solid Geometry h Cube Sphere s Volume = s3 r Surface Area = 6s2 Volume r3 s s Surface Area = 4 r2 Rectangular Prism Cylinder h r Volume = wdh 2 h Surface Area = 2(wd + wh + dh) Volume = r h w d 2 Surface Area = 2 r h+2 r Right Circular Cone h Irregular Prism r h √ Volume = Ah A = area of base Pyramid h Constants A = area of base g = 9.8 m/s2 = 32.27 ft/s2 G = 6.67 x 10-11 m3/kg·s2 π = 3.14159 PLTW, Inc. Engineering Formulas IED POE DE CEA AE BE CIM EDD 2 Conversions Mass Area Force Energy 2 1 kg = 2.205 lbm 1 acre = 4047 m 1 N = 0.225 lbf 1 J = 0.239 cal 2 -4 1 slug = 32.2 lbm = 43,560 ft 1 kip = 1,000 lbf = 9.48 x 10 Btu 2 1 ton = 2000 lbm = 0.00156 mi = 0.7376 ft·lbf Pressure 1kW h = 3,6000,000 J Length Volume 1 atm = 1.01325 bar = 33.9 ft H2O 1 m = 3.28 ft 1L = 0.264 gal = 29.92 in. Hg Defined Units 1 km = 0.621 mi = 0.0353 ft3 = 760 mm Hg 1 in. = 2.54 cm = 33.8 fl oz = 101,325 Pa 1 mi = 5280 ft 1mL = 1 cm3 = 1 cc 1 J = 1 N·m = 14.7 psi 2 1 yd = 3 ft 1 N = 1 kg·m / s 1psi = 2.31 ft of H2O 1 Pa = 1 N / m2 Time 1 V = 1 W / A Temperature Change 1 d = 24 h Power 1 W = 1 J / s 1 K = 1 ºC 1 h = 60 min 1 W = 1 V / A 1 W = 3.412 Btu/h -1 1 min = 60 s 1 Hz = 1 s = 1.8 ºF = 0.00134 hp 1 yr = 365 d 1 F = 1 A·s / V = 1.8 ºR = 14.34 cal/min 1 H = 1 V·s / V = 0.7376 ft lb /s · f SI Prefixes Numbers Less Than One Numbers Greater Than One Power of 10 Prefix Abbreviation Power of 10 Prefix Abbreviation 10-1 deci- d 101 deca- da 10-2 centi- c 102 hecto- h 10-3 milli- m 103 kilo- k 10-6 micro- µ 106 Mega- M 10-9 nano- n 109 Giga- G 10-12 pico- p 1012 Tera- T 10-15 femto- f 1015 Peta- P 10-18 atto- a 1018 Exa- E 10-21 zepto- z 1021 Zetta- Z 10-24 yocto- y 1024 Yotta- Y Equations Temperature Force F = ma TK = TC + 273 Mass and Weight F = force TR = TF + 460 m = mass M = VDm a = acceleration W = mg W = VDw Equations of Static Equilibrium V = volume ΣFx = 0 ΣFy = 0 ΣMP = 0 TK = temperature in Kelvin Dm = mass density TC = temperature in Celsius m = mass Fx = force in the x-direction TR = temperature in Rankin Fy = force in the y-direction Dw = weight density TF = temperature in Fahrenheit g = acceleration due to gravity MP = moment about point P PLTW, Inc. Engineering Formulas IED POE DE CEA AE BE CIM EDD 3 Equations (Continued) Electricity Ohm’s Law Energy: Work Fluid Mechanics V = IR P = IV W = work RT (series) = R1 + R2+ ··· + Rn F = force ’ L d = distance (Guy-L ’ L Power Kirchhoff’s Current Law P V = P V B y ’ L 1 1 2 2 IT = I1 + I2 + ··· + In Q = Av or ∑ A v = A v Kirchhoff’s Voltage Law 1 1 2 2 VT = V1 + V2 + ··· + Vn or ∑ P = power E = energy W = work absolute pressure = gauge pressure V = voltage t = time + atmospheric pressure VT = total voltage τ = torque I = current P = absolute pressure rpm = revolutions per minute I = total current F = Force T R = resistance A = Area R = total resistance V = volume T Efficiency P = power T = absolute temperature Q = flow rate y v = flow velocity Thermodynamics ′ ∆T Pout = useful power output Mechanics Pin = total power input (where acceleration = 0) ∆ Energy: Potential (where acceleration = 0) L U = potential energy m =mass L g = acceleration due to gravity A v = A v h = height 1 1 2 2 v = v0 + at Energy: Kinetic d = d + v t + ½at2 P = rate of heat transfer 0 0 2 2 Q = thermal energy v = v0 + 2a(d – d0) A = Area of thermal conductivity τ = dFsinθ U = coefficient of heat conductivity K = kinetic energy (U-factor) m = mass s = speed ∆T = change in temperature v = velocity v = velocity a = acceleration R = resistance to heat flow ( R-value) Energy: Thermal X = range k = thermal conductivity t = time v = velocity d = distance Pnet = net power radiated g = acceleration due to gravity -8 = 5.6696 x 10 Q = thermal energy d = distance m = mass θ = angle e = emissivity constant c = specific heat τ = torque T1, T2 = temperature at time 1, time 2 ∆T = change in temperature F = force v = flow velocity PLTW, Inc. Engineering Formulas POE 4 DE 4 Section Properties Moment of Inertia Rectangle Centroid h x x x̅ and y̅ xx b Right Triangle Centroid x̅ and y̅ I = moment of inertia of a rectangular section xx about x-x axis Semi-circle Centroid Complex Shapes Centroid ∑ x ∑ y x̅ y̅ x̅ and y̅ ∑ ∑ x̅ x x̅ x y̅ y y̅ y x = x distance to centroid of shape i i y = y distance to centroid of shape i i A = Area of shape i i Structural Analysis Material Properties Beam Formulas Stress (axial) Reaction B L Moment (at point of load) x L Deflection x (at point of load) = stress L Reaction F = axial force B A = cross-sectional area L Moment x (at center) L Strain (axial) Deflection x (at center) Reaction B L Moment x (between loads) = strain Deflection ( L - ) (at center) x L0 = original length δ = change in length Reaction and B L L Moment (at Point of Load) x L ( )√ ( ) Modulus of Elasticity Deflection ( ) (at √ ) Deformation: Axial Truss Analysis ( L 2J = M + R E = modulus of elasticity δ = stress J = number of joints = strain δ = deformation M =number of members A = cross-sectional area F = axial force R = number of reaction forces F = axial force L0 = original length δ = deformation A = cross-sectional area E = modulus of elasticity PLTW, Inc. Engineering Formulas POE 5 AE 4 CEA 4 Simple Machines Inclined Plane Mechanical Advantage (MA) L y ( ) Wedge IMA = Ideal Mechanical Advantage AMA = Actual Mechanical Advantage L DE = Effort Distance DR = Resistance Distance FE = Effort Force FR = Resistance Force Lever Screw 1st IMA = Class Pitch = 2nd C = Circumference Class r = radius Pitch = distance between threads TPI = Threads Per Inch 3rd Class Compound Machines MA = (MA ) (MA ) (MA ) . TOTAL 1 2 3 Wheel and Axle Gears; Sprockets with Chains; and Pulleys with Belts Ratios Effort at Axle ( ) Compound Gears B Effort at Wheel GRTOTAL = ( ) ( ) GR = Gear Ratio in = Angular Velocity - driver Pulley Systems out = Angular Velocity - driven Nin = Number of Teeth - driver IMA = Total number of strands of a single string Nout = Number of Teeth - driven supporting the resistance din = Diameter - driver dout = Diameter - driven = Torque - driver IMA = in = Torque - driven out PLTW, Inc. Engineering Formulas POE 6 Structural Design Steel Beam Design: Shear Steel Beam Design: Moment Spread Footing Design qnet = qallowable - pfooting Vn = 0.6FyAw Mn = FyZx M = allowable bending moment Va = allowable shear strength a M = nominal moment strength Vn = nominal shear strength n q = net allowable soil Ω = 1.67 = factor of safety for net Ωv = 1.5 = factor of safety for shear b bearing pressure bending moment Fy = yield stress qallowable = total allowable soil Fy = yield stress Aw = area of web bearing pressure Zx = plastic section modulus about pfooting = soil bearing pressure neutral axis due to footing weight Storm Water Runoff tfooting = thickness of footing q = soil bearing pressure Rational Method Runoff Coefficients P = column load applied Storm Water Drainage Categorized by Surface A = area of footing Q = CfCiA Forested 0.059—0.2 Asphalt 0.7—0.95 Brick 0.7—0.85 Concrete 0.8—0.95 Q = peak storm water runoff rate (ft3/s) Shingle roof 0.75—0.95 Cf = runoff coefficient adjustment Lawns, well drained (sandy soil) factor Up to 2% slope 0.05—0.1 C = runoff coefficient 2% to 7% slope 0.10—0.15 i = rainfall intensity (in./h) Over 7% slope 0.15—0.2 A = drainage area (acres) Lawns, poor drainage (clay soil) Up to 2% slope 0.13—0.17 Runoff Coefficient 2% to 7% slope 0.18—0.22 Adjustment Factor Over 7% slope 0.25—0.35 Driveways,
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