Engineering Formula Sheet PDF

Preview

Summary

Download Engineering Formula Sheet PDF

Description

V20.0

PLTW Engineering Formula Sheet 2020 (v20.0) 1.0 Statistics Mean σ xi

μ=

Mode

തx =

Place data in ascending order. Mode = most frequently occurring value

σ xi

(1.1a) n N µ = population mean ത = sample mean x Σxi = sum of all data values (x1, x2, x3, …) N = size of population n = size of sample

(1.1b)

σ=ට

Place data in ascending order. (1.2)

N = size of population Range (1.5) Range = xmax - xmin

(1.3)

xmax = maximum data value xmin = minimum data value

2.0 Probability

nx

fx = relative frequency of outcome x nx = number of events with outcome x n = total number of events

σሺxi - ഥ x ሻ2

s=ට

n ‒1

(Population)

(Sample)

(1.5a)

(1.5b)

σ = population standard deviation s = sample standard deviation xi = individual data value ( x1, x2, x3, …) 𝜇 = population mean ത = sample mean x N = size of population n = size of sample

(2.3)

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

(2.4)

P (A or B) = probability of either mutually exclusive event A or B occurring in a trial PA = probability of event A

Binomial Probability (order doesn’t matter)

Pk =

N

P (A and B and C) = PAPBPC

(2.1)

n

σሺxi - μሻ2

Independent Events

Frequency

fx =

If two values occur with maximum frequency the data set is bimodal. If three or more values occur with maximum frequency the data set is multi-modal. Standard Deviation

Median If N is odd, median = central value If N is even, median = mean of two central values

(1.4)

n!(pk )(qn-k ) k!(n-k)!

(2.2)

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

© 2020 Project Lead The Way, Inc. PLTW Engineering Formula Sheet v20 0

Conditional Probability

𝑃ሺ𝐴ȁ𝐷ሻ =

𝑃ሺ𝐴ሻ∙𝑃ሺ𝐷ȁ𝐴ሻ

𝑃ሺ𝐴ሻ∙𝑃൫𝐷 ห𝐴൯+𝑃ሺ~𝐴ሻ∙𝑃ሺ𝐷ȁ~𝐴ሻ

(2.5)

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

EES IED

POE

DE

CEA

AE

ES

CIM EDD

1

V20.0

3.0 Plane Geometry

Ellipse

Circle

Rectangle

2b

Area = π a b (3.8)

Perimeter = 2a + 2b (3.9)

2a

Area = ab

(3.10)

Circumference =2 π r (3.1)

Area = π r2

(3.2)

B

Triangle (3.6) Parallelogram h

Area = bh

(3.3)

b

Area = ½ bh

(3.11)

a2 = b2 + c2 – 2bc·cos ∠A b2 = a2 + c2 – 2ac·cos ∠B c2 = a2 + b2 – 2ab ·cos∠C

a

(3.12) (3.13) (3.14)

C

1

Area = n

c2 = a2 + b2

(3.4)

sin θ =

(3.5)

cos θ = tan θ =

c b

c

2

=

ns2

(3.15)

4tan ቀ n ቁ 180

Trapezoid

b

(3.7)

s(2 f)

n = number of sides θ

(3.6)

c a b

A b

Regular Polygons

Right Triangle

a

c

h

a

Area = ½(a + b)h

(3.16)

h b

4.0 Solid Geometry Cube Volume =

Sphere s3

Surface Area = 6s2

s

(4.1) (4.2)

s

s

Rectangular Prism Volume = wdh

Volume = 3 π r3

4

(4.8)

Surface Area = 4 π r2

(4.9)

Cylinder

h

(4.3)

Surface Area = 2(wd + wh + dh) (4.4)

Volume = π r2 h

d

w

(4.10)

Surface Area = 2 π r h+2 π r2

(4.11)

Right Circular Cone Volume =

Irregular Prism

h

πr2 h

(4.5)

3

Total Surface Area = π r2 + π r ඥr2 +h2

r

Volume = Ah

(4.12)

(4.6)

A = area of base Pyramid Volume =

Ah 3

(4.7)

A = area of base

© 2020 Project Lead The Way, Inc. PLTW Engineering Formula Sheet v20 0

h

5.0 Constants g = - 9.8 m/s2 = - 32.17 ft/s2 G = 6.67 x 10-11 m3/kg·s2 π = 3.14159 EES IED

POE

DE

CEA

AE

ES

CIM EDD

2

V20.0

6.0 Conversions Mass/Weight (6.1) 1 kg 1 slug 1 ton 1 lb

= 2.205 lbm = 32.2 lbm = 2000 lb = 16 oz

= 3.28 ft = 0.621 mi = 2.54 cm = 5280 ft = 3 ft

= 24 h = 60 min = 60 s = 365 d

1 atm

Volume (6.5) = 0.264 gal = 0.0353 ft3 = 33.8 fl oz = 1 cm3 = 1 cc

1mL

1psi

1W

*Use equation in section 9.0 to convert

1 hp

= Δ 1 ºC = Δ 1.8 ºF = Δ 1.8 ºR

1 Hz = 60 rpm = 2π rad/sec

7.0 Defined Units

= 3.412 Btu/h = 0.00134 hp = 14.34 cal/min = 0.7376 ft·lbf/s = 550 ft∙lb/sec

1J 1N 1 Pa 1V 1W 1W 1 Hz 1F 1H

Energy (6.10) 1J

= 0.239 cal = 9.48 x 10-4 Btu = 0.7376 ft·lbf 1kW h = 3,600,000 J

Force (6.7) 1N 1 kip

Rotational Speed (6.11)

= 1.01325 bar = 33.9 ft H2O = 29.92 in. Hg = 760 mm Hg = 101,325 Pa = 14.7 psi = 2.31 ft of H2O

Power (6.9)

Temperature Unit Equivalents (6.6)

Δ1 K

Time (6.3) 1d 1h 1 min 1 yr

1 acre = 4047 m2 = 43,560 ft2 = 0.00156 mi2

1L

Length (6.2) 1m 1 km 1 in. 1 mi 1 yd

Pressure (6.8)

Area (6.4)

= 0.225 lb = 1,000 lb

= 1 N·m = 1 kg·m / s2 = 1 N / m2 =1W/A =1J/s =1V/A = 1 s-1 = 1 A·s / V = 1 V·s / A

8.0 SI Prefixes Numbers Less Than One Power of 10

Decimal Equivalent

Prefix

Numbers Greater Than One

Abbreviation

Power of 10

Whole Number

Prefix

Abbreviation

10

deca-

da

100

hecto-

h

Equivalent

10-1

0.1

deci-

d

101

10-2

0.01

centi-

c

102

10-3

0.001

milli-

m

103

1000

kilo-

k

10-6

0.000001

micro-

µ

106

1,000,000

Mega-

M

10-9

0.000000001

1,000,000,000

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

9.0 Equations

Temperature

Mass and Weight

TK = TC + 273

(9.4)

TR = TF + 460

(9.5)

m = VDm

(9.1)

W = mg

(9.2)

W = VDw

(9.3)

V = volume Dm = mass density m = mass Dw = weight density W = weight g = acceleration due to gravity © 2020 Project Lead The Way, Inc. PLTW Engineering Formula Sheet v20 0

Force and Moment F = ma

TF = TC =

9 5

Tc + 32

TF - 32 1.8

(9.6a) (9.6b)

M = Fd⊥ ( 9.7b)

Equations of Static Equilibrium

TK = temperature in Kelvin TC = temperature in Celsius TR = temperature in Rankin TF = temperature in Fahrenheit EES

(9.7a)

F = force m = mass a = acceleration M = moment d⊥= perpendicular distance

ΣFx = 0

ΣFy = 0

ΣMP = 0 (9.8)

Fx = force in the x-direction Fy = force in the y-direction MP = moment about point P IED

POE

DE

CEA

AE

ES

CIM EDD

3

V20.0

9.0 (Continued) Equations Energy: Work (9.9)

W = work F∥ = force parallel to direction of displacement d = displacement Power E t

=

W

(9.10)

t

P=τω

(9.11)

P = power E = energy W = work t = time τ = torque ω = angular velocity

Pout ∙100% (9.12) Pin

Pout = useful power output Pin = total power input

(9.13)

U = potential energy m =mass g = acceleration due to gravity h = height Energy: Kinetic K = 21 mv2

(9.14)

K = kinetic energy m = mass v = velocity Energy: Thermal ∆Q = mc∆T

V1 T1 p1 T1

V = IR

F

=

V2 T2 p2 T2

(9.15)

∆Q = change in thermal energy m = mass c = specific heat ∆T = change in temperature © 2020 Project Lead The Way, Inc. PLTW Engineering Formula Sheet v20 0

(9.32)

(9.16)

A

=

(Charles’ Law)

(9.17)

(Gay-Lussanc’s Law) (9.18)

P = IV

(9.33)

RT (series) = R1 + R2+ ··· + Rn

(9.34)

RT (parallel) =

1 1 1 1 + + ∙∙∙ + R R1 R2 n

p1V1 = p2V2 (Boyle’s Law)

(9.19)

Q = Av

(9.20)

A1v1 = A2v2

(9.21)

IT = I1 + I2 + ··· + In n or IT = σ k=1 Ik

P = Qp

(9.22)

Kirchhoff’s Voltage Law

VT = V1 + V2 + ··· + Vn n or VT = σ k=1Vk V = voltage VT = total voltage I = current IT = total current R = resistance RT = total resistance P = power

Mechanics

Thermodynamics

sҧ =

a= X=

d t

(9.24) (9.25)

vf − vi t vi 2 sin(2θ) -g

U= (9.26)

P= (9.27)

v = vi + at

(9.28)

d = di + vit + ½at2

(9.29)

v2 = vi2 + 2a(d – di)

(9.30)

τ = dFsinθ

(9.31)

sഥ = average speed vത = average velocity v = velocity vi = initial velocity (t =0) a = acceleration X = range t = time ∆d = change in displacement d = distance di = initial distance (t=0) g = acceleration due to gravity θ = angle τ = torque F = force

1 R

∆Q ∆t

=

(9.37)

(9.39)

k

(9.40)

L

kA∆T (9.41)

L

(9.42)

A1v1 = A2v2 Pnet =

k=

(9.36)

(9.38)

P = Q′ = AU∆T P = Q' =

∆d ∆t

(9.35)

Kirchhoff’s Current Law

p = absolute pressure F = force A = area V = volume T = absolute temperature Q = flow rate v = flow velocity P = power

vത =

Energy: Potential U = mgh

p=

absolute pressure = gauge pressure + atmospheric pressure (9.23)

Efficiency Efficiency (%) =

Ohm’s Law

Fluid Mechanics

W = F∥ ∙ d

P=

Electricity

4

σAe(T2 -T14 )

PL A∆T

(9.43) (9.44)

P = rate of heat transfer Q = thermal energy A = area of thermal conductivity U = coefficient of heat conductivity (U-factor) ∆T = change in temperature ∆t = change in time R = resistance to heat flow ( R-value) k = thermal conductivity v = velocity Pnet = net power radiated σ = 5.6696 x 10-8

W 4 m2 ∙K

e = emissivity constant L = thickness T1, T2 = temperature at time 1, time 2 EES 4 POE 4 DE 4

AE 4

CIM 4

V20.0

10.0 Section Properties y

Moment of Inertia

Ixx =

y

Rectangle Centroid h

bh3

ത= x

x

(10.1)

12

b

ത= x

σ Ai

and yഥ =

h

b 3

and yത =

h

σ Ai

4r 3π

Moment

σ = stress F = axial force A = cross-sectional area Strain (axial)

ε=

(11.2)

E=

ε (F2 -F1 )L0

ሺ𝛿2 −𝛿1 )A

4

(12.1) (12.2)

(at point of load)

3

Reaction

RA = RB = ωL

ωL

(12.4)

2 2

Moment

Mmax =

Deflection

5ωL Δmax = 384EI (at center)

Deflection

8

(12.5)

(at center)

4

(12.6)

RA = RB = P

(12.7)

Mmax = Pa

Δmax =

(12.8)

Pa 2 ቀ3L -4a2 ቁ 24EI

(12.9)

(at center) Pb

Pa

Reaction

RA =

Moment

Mmax =

Deflection

ඥ3a(a+2b) Δmax = Pab(a+2b) 27EIL

L

and RB = Pab L

(12.10)

L

(at Point of Load) (12.11)

aሺa+2bሻ

(at x = ට

(11.3)

3,

(12.12)

when a>b )

E = modulus of elasticity I = moment of inertia

(11.4)

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

Mmax =

2

PL

PL Δmax = 48EI (at point of load) (12.3)

Moment

Modulus of Elasticity σ

P

RA = RB =

Deflection

Reaction

ε = strain L0 = original length δ = change in length

E=

x

Beam Formulas

(11.1)

δ L0

(10.5)

12.0 Structural Analysis

Stress (axial) A

y

ഥ= x-distance to the centroid x തy = y-distance to the centroid

Reaction

σ=

x y

(10.4)

3

തx = r and തy =

(10.2)

xഥ= x-distance to the centroid yത = y-distance to the centroid xi = x distance to centroid of shape i yi = y distance to centroid of shape i Ai = Area of shape i

F

(10.3)

2

Semi-circle Centroid

σ yi A i

11.0 Material

and yത =

x

Complex Shapes Centroid σ xi A i

2

Right Triangle Centroid

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

xഥ =

b

Deformation: Axial

δ=

FL0 AE

Truss Analysis (12.13)

δ = deformation F = axial force L0 = original length A = cross-sectional area E = modulus of elasticity

© 2020 Project Lead The Way, Inc. PLTW Engineering Formula Sheet v20 0

2J = M + R

(12.14)

J = number of joints M =number of members R = number of reaction forces

POE 5 AE 5 CEA 4

V20.0

13.0 Simple Machines Inclined Plane Mechanical Advantage (MA)

IMA=

DE DR

(13.1)

% Efficiency= ቀ IMA ቁ 100 AMA

AMA=

FR FE

(13.2)

L

IMA=

(13.6)

H

(13.3)

IMA = ideal mechanical advantage AMA = actual mechanical advantage DE = effort distance DR = resistance distance FE = effort force FR = resistance force

Wedge

L H

IMA=

(13.7)

Lever 1st Class

Screw IMA = Pitch =

2nd Class

C Pitch

(13.8)

1 (13.9)

TPI

C = circumference r = radius Pitch = distance between threads TPI = threads per inch

3rd Class Compound Machines

MATOTAL = (MA1) (MA2) (MA 3) . . .

Wheel and Axle

(13.10)

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

GR = dout din

=

Nout Nin ωin ωout

dout din

= =

τout τin

=

ωin ωout

=

(pulleys)

τout τin

(13.11)

(13.12)

Compound Gears

Effort at Wheel

GRTOTAL = ቀAቁ ቀ Cቁ (13.13) B

Pulley Systems IMA = total number of strands of a single string supporting the resistance (13.4) IMA =

DE (string pulled) DR (resistance lifted)

© 2020 Project Lead The Way, Inc. PLTW Engineering Formula Sheet v20 0

(13.5)

D

GR = gear ratio ωin = angular velocity - driver ωout = angular velocity - driven Nin = number of teeth - driver Nout = number of teeth - driven din = diameter - driver dout = diameter - driven 𝜏in = torque - driver 𝜏out = torque - driven EES 5

IED 4 POE 6

V20.0

14.0 Structural Design Steel Beam Design: Shear

Va ≤

Vn

(14.1)

Ωv

Vn = 0.6FyAw

(14.2)

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

Steel Beam Design: Moment

Spread Footing Design

Ma ≤

qnet = qallowable - pfooting

(14.5)

pfooting = tfooting ∙150 lb3

(14.6)

P q= A

(14.7)

Ωb

(14.3)

Mn = FyZx

(14.4)

Storm Water Drainage Q = CfCiA

Cc =

(15.1)

C1 A1 + C2 A2 + ∙∙∙ A1 + A2 + ∙∙∙

(15.2)

(ft3/s)

Q = peak storm water runoff rate Cf = runoff coefficient adjustment factor C = runoff coefficient i = rainfall intensity (in./h) A = drainage area (acres) Runoff Coefficient Adjustment Factor Return Period 1, 2, 5, 10 25 50 100

Cf 1.0 1.1 1.2 1.25

𝑀𝑛

= allowable bending strength

Rational Method Runoff Coefficients Categorized by Surface Forested 0.059—0.2 Asphalt 0.7—0.95 Brick 0.7—0.85 Concrete 0.8—0.95 Shingle roof 0.75—0.95 Lawns, well drai...