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Engineering Formula Sheet Cheat Sheet, Cheat Sheet of Engineering

Statistics, Probability, Geometry, Conversions, Section Properties, Structural Design, Boolean Algebra and more chapters formulas

Typology: Cheat Sheet

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Engineering Formulas
Engineering Formula Sheet
Probability
Binomial Probability (order doesn’t matter)
Pk = binomial probability of k successes in n trials
p = probability of a success
q = 1 p = probability of failure
k = number of successes
n = number of trials
Independent Events
P (A and B and C) = PAPBPC
P (A and B and C) = probability of independent
events A and B and C occurring in sequence
PA = probability of event A
Mutually Exclusive Events
P (A or B) = PA + PB
P (A or B) = probability of either mutually exclusive
event A or B occurring in a trial
PA = probability of event A
Σxi = sum of all data values (x1, x2, x3,
n = number of data values
x x
x x
Frequency
fx = relative frequency of outcome x
nx = number of events with outcome x
n = total number of events
Px = probability of outcome x
fa = frequency of all events
( | ) ( ) ( | )
( ) ( | ) ( ) ( | )
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
Mode
Place data in ascending order.
Mode = most frequently occurring value
If two values occur at the maximum frequency the
data set is bimodal.
If three or more values occur at the maximum
frequency the data set is multi-modal.
x
Mean
µ = mean value
Σxi = sum of all data values (x1, x2, x3, …
n = number of data values
∑(x )
Standard Deviation
σ = standard deviation
xi = individual data value ( x1, x2, x3, …
n = number of data values
Range
Range = xmax - xmin
xmax = maximum data value
xmin = minimum data value
Median
Place data in ascending order.
If n is odd, median = central value
If n is even, median = mean of two central values
n = number of data values
Statistics
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Engineering Formula Sheet

Probability

Binomial Probability (order doesn’t matter)

Pk = binomial probability of k successes in n trials p = probability of a success q = 1 – p = probability of failure k = number of successes n = number of trials

Independent Events P (A and B and C) = PAPBPC P (A and B and C) = probability of independent events A and B and C occurring in sequence PA = probability of event A

Mutually Exclusive Events P (A or B) = PA + PB P (A or B) = probability of either mutually exclusive event A or B occurring in a trial PA = probability of event A Σxi = sum of all data values (x 1 , x 2 , x 3 , … n = number of data values

x^ x

x^ x

Frequency

fx = relative frequency of outcome x nx = number of events with outcome x n = total number of events Px = probability of outcome x fa = frequency of all events

( | ) ( ) ( |( ) )^ ((^ |^ )) ( | )

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

Mode Place data in ascending order. Mode = most frequently occurring value If two values occur at the maximum frequency the data set is bimodal. If three or more values occur at the maximum frequency the data set is multi-modal.

∑ x

Mean

μ = mean value Σxi = sum of all data values (x 1 , x 2 , x 3 , … n = number of data values

√∑(x^ )

Standard Deviation

σ = standard deviation xi = individual data value ( x 1 , x 2 , x 3 , … n = number of data values

Range Range = xmax - xmin xmax = maximum data value xmin = minimum data value

Median Place data in ascending order. If n is odd, median = central value If n is even, median = mean of two central values n = number of data values

Statistics

Plane Geometry

Triangle Area = ½ bh a^2 = b^2 + c^2 – 2bc· cos ∠A b^2 = a^2 + c^2 – 2ac· cos ∠B c^2 = a^2 + b^2 – 2ab· cos ∠C

h

b

a c A

B

C

Ellipse (^) 2b

2a

Regular Polygons

n = number of sides

f

s

Rectangle Perimeter = 2a + 2b Area = ab

Circle

Parallelogram Area = bh h

b

Pyramid

A = area of base

h

Solid Geometry

Sphere

Volume r^3 Surface Area = 4 r^2

r

h

w^ d

Rectangular Prism

Volume = wdh

Surface Area = 2(wd + wh + dh) h

r

Cylinder

Volume = r^2 h Surface Area = 2 r h+2 r^2

Cube

Volume = s^3 Surface Area = 6s^2

s

s s

Right Circular Cone

r

h

Trapezoid Area = ½(a + b)h b h

h h

a h

Constants g = 9.8 m/s^2 = 32.27 ft/s^2 G = 6.67 x 10-^11 m^3 /kg·s^2

h

Irregular Prism

Volume = Ah A = area of base

a

Right Triangle

c^2 = a^2 + b^2

b

c θ

Energy: Work

W = work F = force d = distance

Fluid Mechanics

’ L

(Guy-L ’ L P 1 V 1 = P 2 V 2 B y ’ L Q = Av A 1 v 1 = A 2 v 2

absolute pressure = gauge pressure

  • atmospheric pressure P = absolute pressure F = Force A = Area V = volume T = absolute temperature Q = flow rate v = flow velocity

Equations (Continued)

Power

P = power E = energy W = work t = time τ = torque rpm = revolutions per minute

y

Efficiency

Pout = useful power output Pin = total power input

Electricity Ohm’s Law V = IR P = IV RT (series) = R 1 + R 2 + ··· + Rn

Kirchhoff’s Current Law IT = I 1 + I 2 + ··· + In orKirchhoff’s Voltage Law VT = V 1 + V 2 + ··· + Vn or ∑ V = voltage VT = total voltage I = current IT = total current R = resistance RT = total resistance P = power

Energy: Potential

U = potential energy m =mass g = acceleration due to gravity h = height

Energy: Kinetic

K = kinetic energy m = mass v = velocity

Energy: Thermal

Q = thermal energy m = mass c = specific heat ∆T = change in temperature

Mechanics (where acceleration = 0) (where acceleration = 0)

v = v 0 + at d = d 0 + v 0 t + ½at^2 v^2 = v 02 + 2a(d – d 0 ) τ = dFsinθ s = speed v = velocity a = acceleration X = range t = time d = distance g = acceleration due to gravity d = distance θ = angle τ = torque F = force

L

L

Thermodynamics ′ ∆T

A 1 v 1 = A 2 v 2

P = rate of heat transfer Q = thermal energy A = Area of thermal conductivity U = coefficient of heat conductivity (U-factor) ∆T = change in temperature R = resistance to heat flow ( R-value) k = thermal conductivity v = velocity Pnet = net power radiated = 5.6696 x 10-^8 e = emissivity constant T 1 , T 2 = temperature at time 1, time 2

Engineering Formulas^ v = flow velocity

Section Properties

x ̅ x y̅ y

Complex Shapes Centroid x ̅ ∑ ∑^ x and y ̅ ∑ ∑^ y

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

Material Properties

Stress (axial)

= stress F = axial force A = cross-sectional area

L

Strain (axial)

= strain L 0 = original length δ = change in length

Modulus of Elasticity

E = modulus of elasticity = stress = strain A = cross-sectional area F = axial force δ = deformation

Structural Analysis

Truss Analysis 2J = M + R J = number of joints M =number of members R = number of reaction forces

Beam Formulas Reaction (^) B Moment (^) x^ L^ (at point of load) Deflection (^) x L^ (at point of load) Reaction (^) B^ L Moment (^) x^ L^ (at center) Deflection (^) x L^ (at center) Reaction (^) B Moment (^) x (between loads) Deflection (^) x ( L - ) (at center) Reaction (^) L and (^) B (^) L Moment (^) x (^) L (at Point of Load) Deflection (^ )√^ (^ ) (at (^) √ (^ )^ )

x ̅ y ̅

x ̅ x y̅ y

Rectangle Centroid x ̅ and y ̅ Right Triangle Centroid x ̅ and y ̅

Semi-circle Centroid

δ

L

Deformation: Axial

δ = deformation F = axial force L 0 = original length A = cross-sectional area E = modulus of elasticity

xx

Moment of Inertia

Ixx = moment of inertia of a rectangular section about x-x axis

x

h

b

x

Structural Design

Steel Beam Design: Moment

Mn = FyZx Ma = allowable bending moment Mn = nominal moment strength Ωb = 1.67 = factor of safety for bending moment Fy = yield stress Zx = plastic section modulus about neutral axis

Spread Footing Design qnet = qallowable - pfooting

qnet = net allowable soil bearing pressure qallowable = total allowable soil bearing pressure pfooting = soil bearing pressure due to footing weight tfooting = thickness of footing q = soil bearing pressure P = column load applied A = area of footing

Steel Beam Design: Shear

Vn = 0.6FyAw

Va = allowable shear strength Vn = nominal shear strength Ωv = 1.5 = factor of safety for shear Fy = yield stress Aw = area of web

Storm Water Runoff Rational Method Runoff Coefficients Categorized by Surface Forested 0.059—0. Asphalt 0.7—0. Brick 0.7—0.8 5 Concrete 0.8—0. Shingle roof 0.75—0. Lawns, well drained (sandy soil) Up to 2% slope 0.05—0. 2% to 7% slope 0.10—0. Over 7% slope 0.15—0. Lawns, poor drainage (clay soil) Up to 2% slope 0.13—0. 2% to 7% slope 0.18—0. Over 7% slope 0.25—0. Driveways, walkways

Categorized by Use Farmland 0.05—0. Pasture 0.05—0. Unimproved 0.1—0. Parks 0.1—0. Cemeteries 0.1—0. Railroad yard 0.2—0. Playgrounds (except asphalt or concrete)

Business Districts Neighborhood 0.5—0. City (downtown) 0.7—0. Residential Single-family 0.3—0. Multi-plexes, detached

Multi-plexes, attached

Suburban 0.25—0. Apartments, condominiums

Industrial Light 0.5—0. Heavy 0.6—0.

Runoff Coefficient Adjustment Factor Return Period Cf 1, 2, 5, 10 1. 25 1. 50 1. 100 1.

Water Supply

Storm Water Drainage Q = CfCiA

Q = peak storm water runoff rate (ft^3 /s) Cf = runoff coefficient adjustment factor C = runoff coefficient i = rainfall intensity (in./h) A = drainage area (acres)

L

Hazen-Williams Formula

hf = head loss due to friction (ft of H 2 O) L = length of pipe (ft) Q = water flow rate (gpm) C = Hazen-Williams constant d = diameter of pipe (in.)

Dynamic Head

dynamic head = static head – head loss

Hazen

  • Williams Constants

Equivalent Length of (Generic) Fittings

Aerospace Equations

Ber oulli’s L w

PS = static pressure v = velocity y

Energy

K = kinetic energy m =mass v = velocity U = gravitational potential energy G = universal gravitation constant M =mass of central body m = mass of orbiting object R = Distance center main body to center of orbiting object E = Total Energy of an orbit

L

L

Forces of Flight

CL = coefficient of lift CD = coefficient of drag L = lift D = drag A = wing area density Re = Reynolds number v = velocity l = length of fluid travel = fluid viscosity F = force m = mass g = acceleration due to gravity M = moment d = moment arm (distance from datum perpendicular to F)

Orbital Mechanics

= eccentricity b = semi-minor axis a =semi-major axis T = orbital period a = semi-major axis gravitational parameter F = force of gravity between two bodies G = universal gravitation constant M =mass of central body m = mass of orbiting object r = distance between center of two objects

Propulsion

FN = net thrust W = air mass flow vo = flight velocity vj = jet velocity I = total impulse Fave = average thrust force t = change in time (thrust duration) Fnet = net force Favg = average force Fg = force of gravity vf = final velocity a = acceleration t = change in time (thrust duration)

NOTE: Fave and Favg are

easily confused.

[(^ )]

Atmosphere Parameters

T = temperature h = height p = pressure density