IED-Review Engineering Formula Sheet

Engineering Formula Sheet

Statistics Mode

Mean Place data in ascending order.
Mode = most frequently occurring value
∑x
If two values occur at the maximum frequency the
µ = mean value data set is bimodal.
Σxi = sum of all data values (x1, x2, x3, … If three or more values occur at the maximum
n = number of data values frequency the data set is multi-modal.

Standard Deviation Median
Place data in ascending order.
√∑(x ) If n is odd, median = central value
If n is even, median = mean of two central values
σ = standard deviation
xi = individual data value ( x1, x2, x3, … n = number of data values
n = number of data values
Range
Probability Range = xmax - xmin
xmax = maximum data value
Frequency xmin = minimum data value

x Independent Events
x
P (A and B and C) = PAPBPC
x
x P (A and B and C) = probability of independent
events A and B and C occurring in sequence
fx = relative frequency of outcome x
nx = number of events with outcome x PA = probability of event A
n = total number of events
Px = probability of outcome x Mutually Exclusive Events
fa = frequency of all events
P (A or B) = PA + PB
Binomial Probability (order doesn’t matter)
P (A or B) = probability of either mutually exclusive
Pk = binomial probability of k successes in n trials event A or B occurring in a trial
p = probability of a success
q = 1 – p = probability of failure PA = probability of event A
k = number of successes Σxi = sum of all data values (x1, x2, x3, …
n = number of trials n = number of data values

Conditional Probability

(| ) () ( |) )
() (|) ( ) (|

P (A|D) = probability of event A given event D
P(A) = probability of event A occurring
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 2b Rectangle

Circle 2a Perimeter = 2a + 2b
Area = ab

Parallelogram h Triangle B A
Area = bh c
Area = ½ bh
b a2 = b2 + c2 – 2bc·cos∠A ah
b2 = a2 + c2 – 2ac·cos∠B
c2 = a2 + b2 – 2ab·cos∠C Cb

Right Triangle Regular Polygons s
c2 = a2 + b2
n = number of sides f
ac Trapezoid
Area = ½(a + b)h a
θ h
b h
h
Solid Geometry b
h

Cube s Sphere r
Volume = s3 ss
Surface Area = 6s2 Volume r3 r
Surface Area = 4 r2 h
Rectangular Prism h d
w Cylinder
Volume = wdh Volume = r2 h
Surface Area = 2(wd + wh + dh) Surface Area = 2 r h+2 r2

Right Circular Cone

√ h Irregular Prism h
r
Volume = Ah
A = area of base

Pyramid

h Constants

A = area of base g = 9.8 m/s2 = 32.27 ft/s2

PLTW, Inc. G = 6.67 x 10-11 m3/kg·s2

π = 3.14159

Engineering Formulas IED POE DE CEA AE BE CIM EDD 2

Conversions

Mass Area Force Energy

1 kg = 2.205 lbm 1 acre = 4047 m2 1 N = 0.225 lbf 1 J = 0.239 cal
1 slug = 32.2 lbm = 43,560 ft2 1 kip = 1,000 lbf = 9.48 x 10-4 Btu
1 ton = 2000 lbm = 0.00156 mi2 = 0.7376 ft·lbf

Pressure 1kW h = 3,6000,000 J

Length Volume 1 atm = 1.01325 bar
1psi = 33.9 ft H2O
1m = 3.28 ft 1L = 0.264 gal = 29.92 in. Hg Defined Units
1 km = 0.621 mi 1mL = 0.0353 ft3 = 760 mm Hg
1 in. = 2.54 cm = 101,325 Pa 1J = 1 N·m
1 mi = 5280 ft = 33.8 fl oz = 14.7 psi 1N = 1 kg·m / s2
1 yd = 3 ft = 1 cm3 = 1 cc = 2.31 ft of H2O 1 Pa = 1 N / m2
1V
Time 1W =1W/A
1W
Temperature Change 1d = 24 h Power 1 Hz =1J/s
1h = 60 min 1F
1 K = 1 ºC 1 min = 60 s 1 W = 3.412 Btu/h 1H =1V/A
= 1.8 ºF 1 yr = 365 d = 0.00134 hp = 1 s-1
= 1.8 ºR = 14.34 cal/min = 1 A·s / V
= 0.7376 ft·lbf/s
= 1 V·s / V

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
TK = TC + 273
Mass and Weight TR = TF + 460 F = ma

M = VDm TK = temperature in Kelvin F = force
W = mg TC = temperature in Celsius m = mass
TR = temperature in Rankin a = acceleration
W = VDw TF = temperature in Fahrenheit
V = volume Equations of Static Equilibrium
Dm = mass density
m = mass ΣFx = 0 ΣFy = 0 ΣMP = 0
Dw = weight density
g = acceleration due to gravity Fx = force in the x-direction
Fy = force in the y-direction
MP = moment about point P

PLTW, Inc. Engineering Formulas IED POE DE CEA AE BE CIM EDD 3

Equations (Continued) Fluid Mechanics Electricity

Energy: Work ’L Ohm’s Law

W = work (Guy-L ’L V = IR
F = force
d = distance P1V1 = P2V2 B y ’ L P = IV
Q = Av RT (series) = R1 + R2+ ··· + Rn
Power A1v1 = A2v2
Kirchhoff’s Current Law
P = power absolute pressure = gauge pressure
E = energy + atmospheric pressure IT = I1 + I2 + ··· + In
W = work or ∑
t = time P = absolute pressure
τ = torque F = Force Kirchhoff’s Voltage Law
rpm = revolutions per minute A = Area
V = volume VT = V1 + V2 + ··· + Vn
Efficiency T = absolute temperature or ∑
y Q = flow rate
v = flow velocity V = voltage
Pout = useful power output VT = total voltage
Pin = total power input Mechanics I = current
IT = total current
Energy: Potential (where acceleration = 0) R = resistance
RT = total resistance
U = potential energy (where acceleration = 0) P = power
m =mass
g = acceleration due to gravity v = v0 + at Thermodynamics
h = height ′ ∆T
d = d0 + v0t + ½at2
Energy: Kinetic ∆
v2 = v02 + 2a(d – d0)
K = kinetic energy L
m = mass τ = dFsinθ
v = velocity L
s = speed A1v1 = A2v2
Energy: Thermal v = velocity
a = acceleration P = rate of heat transfer
Q = thermal energy X = range Q = thermal energy
m = mass t = time A = Area of thermal conductivity
c = specific heat d = distance U = coefficient of heat conductivity
∆T = change in temperature g = acceleration due to gravity
d = distance (U-factor)
θ = angle ∆T = change in temperature
τ = torque
F = force R = resistance to heat flow ( R-value)
k = thermal conductivity
v = velocity
Pnet = net power radiated

= 5.6696 x 10-8
e = emissivity constant
T1, T2 = temperature at time 1, time 2

PLTW, Inc. Engineering Formulas v = flow velocity POE 4 DE 4

Section Properties

Moment of Inertia h Rectangle Centroid
x̅ and y̅
xx xx Right Triangle Centroid
x̅ and y̅
b
Semi-circle Centroid
Ixx = moment of inertia of a rectangular section
about x-x axis x̅ y̅

Complex Shapes Centroid x̅ x
y̅ y
x̅ ∑x and y̅ ∑y
∑ ∑

x̅ x
y̅ y

xi = x distance to centroid of shape i
yi = y distance to centroid of shape i
Ai = Area of shape i

Structural Analysis

Material Properties Beam Formulas

Stress (axial) Reaction B
Moment
= stress Deflection x L (at point of load)
F = axial force x L (at point of load)
A = cross-sectional area Reaction
Moment L
Strain (axial) Deflection B
Reaction
L Moment x L (at center)
= strain Deflection x L (at center)
L0 = original length
δ = change in length Reaction B
Moment
Modulus of Elasticity Deflection x (between loads)

x ( L - ) (at center)

L and B L

x L (at Point of Load)

( )√ ( )

(at √ ( ) )

( Deformation: Axial Truss Analysis

E = modulus of elasticity L 2J = M + R
= stress δ
= strain J = number of joints
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
L
Mechanical Advantage (MA)

y( )

IMA = Ideal Mechanical Advantage Wedge
L
AMA = Actual Mechanical Advantage

DE = Effort Distance DR = Resistance Distance
FE = Effort Force FR = Resistance Force

Lever

1st Screw
Class IMA =

Pitch =

2nd C = Circumference
Class r = radius
Pitch = distance between

threads
TPI = Threads Per Inch

3rd Compound Machines
Class MATOTAL = (MA1) (MA2) (MA3) . . .

Wheel and Axle Gears; Sprockets with Chains; and Pulleys with
Effort at Axle Belts Ratios

( )

Effort at Wheel Compound Gears

Pulley Systems GRTOTAL = (B) ( )
IMA = Total number of strands of a single string
supporting the resistance GR = Gear Ratio
IMA = in = Angular Velocity - driver
out = Angular Velocity - driven

Nin = Number of Teeth - driver
Nout = Number of Teeth - driven
din = Diameter - driver
dout = Diameter - driven
in = Torque - driver
out = Torque - driven

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 qnet = net allowable soil
bearing pressure
Va = allowable shear strength Ma = allowable bending moment
Vn = nominal shear strength Mn = nominal moment strength qallowable = total allowable soil
Ωv = 1.5 = factor of safety for shear Ωb = 1.67 = factor of safety for bearing pressure
Fy = yield stress
Aw = area of web bending moment pfooting = soil bearing pressure
Fy = yield stress due to footing weight
Storm Water Runoff Zx = plastic section modulus about
tfooting = thickness of footing
Storm Water Drainage neutral axis q = soil bearing pressure
Q = CfCiA P = column load applied
Rational Method Runoff Coefficients A = area of footing

Categorized by Surface

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
C = runoff coefficient Up to 2% slope 0.05—0.1
i = rainfall intensity (in./h) 0.10—0.15
A = drainage area (acres) 2% to 7% slope 0.15—0.2
Over 7% slope

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

Return Driveways, 0.75—0.85

Period Cf walkwayCs ategorized by Use

1, 2, 5, 10 1.0 Farmland 0.05—0.3

25 1.1 Pasture 0.05—0.3

50 1.2 Unimproved 0.1—0.3

100 1.25 Parks 0.1—0.25

Cemeteries 0.1—0.25

Water Supply Railroad yard 0.2—0.40

Playgrounds 0.2—0.35

Hazen-Williams Formula (except asBpuhsainlteosrs Districts
L
Ncoenigchrebtoer)hood 0.5—0.7

City (downtown) 0.7—0.95

Residential

hf = head loss due to friction (ft of H2O) Single-family 0.3—0.5
L = length of pipe (ft)
Q = water flow rate (gpm) Multi-plexes, 0.4—0.6
C = Hazen-Williams constant
d = diameter of pipe (in.) Mdeutaltic-hpeledxes, 0.6—0.75

Dynamic Head Satutabcuhrbeadn 0.25—0.4

dynamic head = static head – head loss Apartments, 0.5—0.7

condominiumsIndustrial

Light 0.5—0.8

Heavy 0.6—0.9

PLTW, Inc. Engineering Formulas CEA 5

PLTW, Inc. Hazen-Williams Constants
Equivalent Length of (Generic) Fittings
Engineering Formulas

CEA 6

555 Timer Design Equations

T = 0.693 (RA + 2RB)C

yy B
B
T = period
f = frequency
RA = resistance A
RB = resistance B
C = capacitance

Boolean Algebra

Boolean Theorems Commutative Law Consensus Theorems
X• 0 = 0 X•Y = Y•X ̅
X•1 = X X+Y = Y+X ̅̅ ̅
X• X =X
Associative Law ̅ ̅̅̅̅
̅ X(YZ) = (XY)Z ̅ ̅ ̅̅
X + (Y + Z) = (X + Y) + Z
X+0=X DeMorgan’s Theorems
X+1=1 Distributive Law ̅̅̅̅̅ ̅ ̅
X+X=X X(Y+Z) = XY + XZ ̅̅̅̅̅̅̅ ̅ • ̅
(X+Y)(W+Z) = XW+XZ+YW+YZ
̅
̿

Speeds and Feeds

()

fm = ft·nt·N
Plunge Rate = ½·fm
N = spindle speed (rpm)
CS = cutting speed (in./min)
d = diameter (in.)
fm = feed rate (in./min)
ft = feed (in./tooth)
nt = number of teeth

PLTW, Inc. Engineering Formulas DE 5 CIM 4

Aerospace Equations Propulsion Orbital Mechanics
() √
Forces of Flight

√√

L FN = net thrust = eccentricity
W = air mass flow b = semi-minor axis
L vo = flight velocity a =semi-major axis
vj = jet velocity T = orbital period
CL = coefficient of lift I = total impulse a = semi-major axis
CD = coefficient of drag Fave = average thrust force
L = lift gravitational parameter
D = drag t = change in time (thrust F = force of gravity between two
A = wing area duration)
bodies
density Fnet = net force G = universal gravitation constant
Re = Reynolds number Favg = average force M =mass of central body
v = velocity Fg = force of gravity m = mass of orbiting object
l = length of fluid travel vf = final velocity r = distance between center of two
a = acceleration
= fluid viscosity objects
F = force t = change in time (thrust
m = mass duration) Ber oulli’s L w )
g = acceleration due to gravity ( )(
M = moment NOTE: Fave and Favg are
d = moment arm (distance from easily confused.

datum perpendicular to F) Energy

PS = static pressure

v = velocity
y

Atmosphere Parameters

K = kinetic energy [( )]
m =mass
v = velocity ( )
U = gravitational potential energy
G = universal gravitation constant T = temperature
M =mass of central body h = height
m = mass of orbiting object p = pressure
R = Distance center main body to
density
center of orbiting object
E = Total Energy of an orbit

PLTW, Inc. Engineering Formulas AE 5