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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 or ∑ Kirchhoff’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
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
