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Applied Field Engineering — formula sheet

Ground, air, water & what it costs · 118 formulas · metric edition 1

The full booklet — learning zone, problems, answer key
Percent Grade from Rise and Run
G=100 ΔhLG = \frac{100\,\Delta h}{L}
Elevation from Grade and Distance
E2=E1+G L100E_2 = E_1 + \frac{G\,L}{100}
Grade to Slope Angle
θ=arctan⁡ ⁣(G100)\theta = \arctan\!\left(\frac{G}{100}\right)
Slope Ratio (H:V) to Percent Grade
G=100nG = \frac{100}{n}
Earthwork Volume by Average End Area
V=L (A1+A2)2V = \frac{L\,(A_1 + A_2)}{2}
Earthwork Volume by the Prismoidal Formula
V=L (A1+4Am+A2)6V = \frac{L\,(A_1 + 4A_m + A_2)}{6}
Swell: Loose Volume from Bank Volume
VL=VB(1+S100)V_L = V_B\left(1 + \frac{S}{100}\right)
Shrinkage: Compacted Volume from Bank Volume
VC=VB(1−Sh100)V_C = V_B\left(1 - \frac{S_h}{100}\right)
Concrete Volume with Waste Allowance
V=L W T(1+w100)V = L\,W\,T\left(1 + \frac{w}{100}\right)
Asphalt Tonnage from Area and Thickness
M=A t ρM = A\,t\,\rho
Stockpile Volume (Truncated Pyramid)
V=h3(A1+A2+A1A2)V = \frac{h}{3}\left(A_1 + A_2 + \sqrt{A_1 A_2}\right)
Radius from Degree of Curve (Arc Definition)
R=5729.578DR = \frac{5729.578}{D}
Horizontal Curve Tangent Length
T=Rtan⁡Δ2T = R\tan\frac{\Delta}{2}
Horizontal Curve Length from Degree of Curve
L=100 ΔDL = \frac{100\,\Delta}{D}
Horizontal Curve Long Chord
C=2Rsin⁡Δ2C = 2R\sin\frac{\Delta}{2}
Horizontal Curve External Distance
E=R(sec⁡Δ2−1)E = R\left(\sec\frac{\Delta}{2} - 1\right)
Horizontal Curve Middle Ordinate
M=R(1−cos⁡Δ2)M = R\left(1 - \cos\frac{\Delta}{2}\right)
Vertical Curve Length from K Value
L=K AL = K\,A
Elevation on a Parabolic Vertical Curve
E=EBVC+g1x100+A x2200 LE = E_{BVC} + \frac{g_1 x}{100} + \frac{A\,x^{2}}{200\,L}
High or Low Point on a Vertical Curve
xt=−g1LAx_t = -\frac{g_1 L}{A}
Stopping Sight Distance
d=v tr+v22ad = v\,t_r + \frac{v^{2}}{2a}
Crest Vertical Curve Length for Sight Distance
L=A S2200(h1+h2)2L = \frac{A\,S^{2}}{200\left(\sqrt{h_1} + \sqrt{h_2}\right)^{2}}
Superelevation Rate for a Horizontal Curve
e100+f=v2g R\frac{e}{100} + f = \frac{v^{2}}{g\,R}
Back Azimuth
αb=α±180∘\alpha_b = \alpha \pm 180^{\circ}
Latitude of a Traverse Leg
Lat=Lcos⁡α\text{Lat} = L\cos\alpha
Departure of a Traverse Leg
Dep=Lsin⁡α\text{Dep} = L\sin\alpha
Traverse Closure Error
Ec=(ΣLat)2+(ΣDep)2E_c = \sqrt{\left(\Sigma\text{Lat}\right)^{2} + \left(\Sigma\text{Dep}\right)^{2}}
Traverse Precision Ratio
N=PEcN = \frac{P}{E_c}
Differential Levelling Elevation
E2=E1+BS−FSE_2 = E_1 + \text{BS} - \text{FS}
Stadia Distance from Rod Intercept
D=K s+CD = K\,s + C
Area of a Three-Sided Parcel by Coordinates
A=12[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)]A = \tfrac{1}{2}\left[x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\right]
Environmental Lapse Rate
Γ=T1−T2z2−z1\Gamma = \frac{T_1 - T_2}{z_2 - z_1}
ppm to mg/m³ Conversion
C=ppm⋅M24.45C = \frac{ppm \cdot M}{24.45}
Barometric Pressure with Altitude
P=P0 e−Mgz/RTP = P_0 \, e^{-Mgz/RT}
Wind Speed at Height (Power Law)
u2=u1(z2z1)pu_2 = u_1 \left(\frac{z_2}{z_1}\right)^{p}
Stack Exit Velocity
vs=4Qvπd2v_s = \frac{4 Q_v}{\pi d^2}
Stack Draft Pressure (Chimney Effect)
Δp=h g (ρa−ρs)\Delta p = h \, g \, (\rho_a - \rho_s)
Good Engineering Practice Stack Height
HGEP=hb+1.5LH_{GEP} = h_b + 1.5 L
Briggs Buoyancy Flux
F=g vs d2(Ts−Ta)4 TsF = \frac{g \, v_s \, d^2 (T_s - T_a)}{4 \, T_s}
Briggs Plume Rise (Neutral and Unstable)
Δh=1.6 F1/3x2/3u\Delta h = \frac{1.6 \, F^{1/3} x^{2/3}}{u}
Holland Plume Rise
Δh=vsdu(1.5+2.68×10−3Pd Ts−TaTs)\Delta h = \frac{v_s d}{u}\left(1.5 + 2.68\times10^{-3} P d \, \frac{T_s - T_a}{T_s}\right)
Effective Stack Height
H=hs+ΔhH = h_s + \Delta h
Emission Rate from Stack Concentration
E=C QvE = C \, Q_v
Emission Correction to Reference Oxygen
Ccorr=Cmeas 20.9−O2,ref20.9−O2,measC_{corr} = C_{meas} \, \frac{20.9 - O_{2,ref}}{20.9 - O_{2,meas}}
Excess Air from Flue Gas Oxygen
EA=O220.9−O2EA = \frac{O_2}{20.9 - O_2}
Pasquill–Gifford Dispersion Coefficient
σ=a xb\sigma = a \, x^{b}
Gaussian Plume Ground-Level Concentration
C=Qπσyσzu e−H2/(2σz2)C = \frac{Q}{\pi \sigma_y \sigma_z u} \, e^{-H^{2}/(2\sigma_z^{2})}
Maximum Ground-Level Concentration
Cmax=2QeπuH2⋅σzσyC_{max} = \frac{2Q}{e \pi u H^{2}} \cdot \frac{\sigma_z}{\sigma_y}
Particulate Collection Efficiency
η=Cin−CoutCin\eta = \frac{C_{in} - C_{out}}{C_{in}}
Isokinetic Sampling Rate
Qn=vsAnQ_n = v_s A_n
Sound Power Level to Sound Pressure Level
Lp=LW+10log⁡10 ⁣(Q4πr2)L_p = L_W + 10\log_{10}\!\left(\frac{Q}{4\pi r^{2}}\right)
Combining Sound Levels
Lt=10log⁡10 ⁣(10L1/10+10L2/10)L_t = 10\log_{10}\!\left(10^{L_1/10} + 10^{L_2/10}\right)
Distance Attenuation from a Point Source
L2=L1−20log⁡10 ⁣(r2r1)L_2 = L_1 - 20\log_{10}\!\left(\frac{r_2}{r_1}\right)
Sabine Reverberation Time (RT60)
T60=0.161 VAT_{60} = \frac{0.161\,V}{A}
Total Absorption (Sabins)
A=S1α1+S2α2+S3α3A = S_1\alpha_1 + S_2\alpha_2 + S_3\alpha_3
Eyring Reverberation Time
T60=0.161 V−S ln⁡(1−αˉ)T_{60} = \frac{0.161\,V}{-S\,\ln(1-\bar{\alpha})}
Noise Reduction Coefficient (NRC)
NRC=α250+α500+α1000+α20004\mathrm{NRC} = \frac{\alpha_{250} + \alpha_{500} + \alpha_{1000} + \alpha_{2000}}{4}
Sound Transmission Loss
TL=10log⁡10 ⁣(IiIt)TL = 10\log_{10}\!\left(\frac{I_i}{I_t}\right)
Mass Law Transmission Loss
TL=20log⁡10 ⁣(πmfρ0c)−5TL = 20\log_{10}\!\left(\frac{\pi m f}{\rho_0 c}\right) - 5
Composite Transmission Loss
TLc=10log⁡10 ⁣(Sw+SdSw 10−TLw/10+Sd 10−TLd/10)TL_c = 10\log_{10}\!\left(\frac{S_w + S_d}{S_w\,10^{-TL_w/10} + S_d\,10^{-TL_d/10}}\right)
Allowable Noise Exposure Time
T=82(L−Lc)/qT = \frac{8}{2^{(L - L_c)/q}}
Available Water Capacity
AWC=(θfc−θpwp) DAWC = (\theta_{fc} - \theta_{pwp}) \, D
Readily Available Water from MAD
RAW=AWC×MADRAW = AWC \times MAD
Crop Evapotranspiration
ETc=Kc×ET0ET_c = K_c \times ET_0
Irrigation Interval
I=RAWETcI = \frac{RAW}{ET_c}
Net Irrigation Requirement
IRn=ETc−PeIR_n = ET_c - P_e
Irrigation Application Efficiency
Ea=WsWdE_a = \frac{W_s}{W_d}
Distribution Uniformity
DU=dˉlqdˉDU = \frac{\bar{d}_{lq}}{\bar{d}}
Irrigation Set Run Time
t=d AQt = \frac{d \, A}{Q}
Irrigation System Capacity
Q=A ETpEa fQ = \frac{A \, ET_p}{E_a \, f}
Sprayer Application Rate
V=Qw vV = \frac{Q}{w \, v}
Nozzle Output at a New Pressure
Q2=Q1p2p1Q_2 = Q_1 \sqrt{\frac{p_2}{p_1}}
Effect of Speed on Application Rate
V2=V1v1v2V_2 = V_1 \frac{v_1}{v_2}
Nozzle Output Deviation
D=100 Qm−QrQrD = 100 \, \frac{Q_m - Q_r}{Q_r}
Area Covered per Tank
A=TVA = \frac{T}{V}
Tank Loads to Cover a Field
N=A VTN = \frac{A \, V}{T}
Active Ingredient Rate
Rai=c RvR_{ai} = c \, R_v
Percent Solution in the Tank
P=100 VpTP = 100 \, \frac{V_p}{T}
Seeding Rate from Target Plant Population
S=P⋅TKW1000 G ES = \frac{P \cdot TKW}{1000 \, G \, E}
Plant Population from Row and Seed Spacing
P=1w sP = \frac{1}{w \, s}
Pure Live Seed
PLS=p×gPLS = p \times g
Field Emergence
E=PSE = \frac{P}{S}
Effective Field Capacity
C=w S eC = w \, S \, e
Field Efficiency
e=CeCte = \frac{C_e}{C_t}
Time to Cover a Field
t=ACt = \frac{A}{C}
Map Scale to Real Distance
d=m Sd = m\,S
Distance from Pace Count
d=n Ld = n\,L
Ground Distance on a Slope
g=m2+r2g = \sqrt{m^{2} + r^{2}}
Height by Clinometer
H=dtan⁡θ+eH = d\tan\theta + e
Naismith's Rule (Hiking Time)
t=d5 km/h+h600 m/ht = \frac{d}{5\,\text{km/h}} + \frac{h}{600\,\text{m/h}}
Estimated Time En Route
t=dVgt = \frac{d}{V_g}
Compass to True Heading (Variation and Deviation)
T=C+D+VT = C + D + V
Great Circle Distance (Haversine)
d=2Rarcsin⁡sin⁡2φ2−φ12+cos⁡φ1cos⁡φ2sin⁡2λ2−λ12d = 2R\arcsin\sqrt{\sin^{2}\frac{\varphi_2-\varphi_1}{2} + \cos\varphi_1\cos\varphi_2\sin^{2}\frac{\lambda_2-\lambda_1}{2}}
Initial Great Circle Bearing
θ=atan2⁡ ⁣(sin⁡Δλ cos⁡φ2,  cos⁡φ1sin⁡φ2−sin⁡φ1cos⁡φ2cos⁡Δλ)\theta = \operatorname{atan2}\!\left(\sin\Delta\lambda\,\cos\varphi_2,\; \cos\varphi_1\sin\varphi_2 - \sin\varphi_1\cos\varphi_2\cos\Delta\lambda\right)
Rhumb Line Distance
d=R(φ2−φ1)2+q2(λ2−λ1)2d = R\sqrt{\left(\varphi_2-\varphi_1\right)^{2} + q^{2}\left(\lambda_2-\lambda_1\right)^{2}}
Dead Reckoning Position
Δφ=S tcos⁡CR\Delta\varphi = \frac{S\,t\cos C}{R}
Set and Drift of the Current
Dr=Δn2+Δe2tD_r = \frac{\sqrt{\Delta n^{2} + \Delta e^{2}}}{t}
Cross Track Error
ext=Rarcsin⁡(sin⁡d13R sin⁡(θ13−θ12))e_{xt} = R\arcsin\left(\sin\frac{d_{13}}{R}\,\sin\left(\theta_{13}-\theta_{12}\right)\right)
Wind Triangle Ground Speed
Vg=V2−W2sin⁡2θ−Wcos⁡θV_g = \sqrt{V^{2} - W^{2}\sin^{2}\theta} - W\cos\theta
Wind Correction Angle
WCA=arcsin⁡(Wsin⁡θV)\mathrm{WCA} = \arcsin\left(\frac{W\sin\theta}{V}\right)
Distance to the Visible Horizon
D=h(2R+h)D = \sqrt{h\left(2R + h\right)}
Longitude to Solar Time Difference
Δt=4 Δλ\Delta t = 4\,\Delta\lambda
Present Value
PV=FV(1+r)t\mathit{PV} = \frac{\mathit{FV}}{(1 + r)^{t}}
Compound Interest (Periodic)
A=P(1+rn)ntA = P \left( 1 + \frac{r}{n} \right)^{n t}
Effective Annual Rate from a Nominal Rate
EAR=(1+rm)m−1\mathit{EAR} = \left(1 + \frac{r}{m}\right)^{m} - 1
Real Interest Rate (Fisher Equation)
rreal=1+i1+f−1r_{\text{real}} = \frac{1 + i}{1 + f} - 1
Rule of 72 (Doubling Time)
n≈0.72in \approx \frac{0.72}{i}
Future Value of an Annuity (Regular Deposits)
FV=D (1+i)n−1i\mathit{FV} = D\,\frac{(1+i)^n - 1}{i}
Net Present Value of a Uniform Annual Cash Flow
NPV=A 1−(1+i)−ni−C0\mathit{NPV} = A\,\frac{1 - (1+i)^{-n}}{i} - C_0
Capital Recovery Factor
CRF=i (1+i)n(1+i)n−1\mathit{CRF} = \frac{i\,(1+i)^{n}}{(1+i)^{n} - 1}
Loan Payment (Amortized Loan or Mortgage)
M=P i1−(1+i)−nM = \frac{P\,i}{1 - (1+i)^{-n}}
Total Interest Paid Over a Loan
I=M n−PI = M\,n - P
Equivalent Annual Cost
EAC=P i (1+i)n(1+i)n−1+M\mathit{EAC} = P\,\frac{i\,(1+i)^{n}}{(1+i)^{n} - 1} + M
Simple Payback Period
t=CSt = \frac{C}{S}
Return on Investment (ROI)
ROI=G−CC\mathit{ROI} = \frac{G - C}{C}
Break-Even Quantity
Q=Fp−vQ = \frac{F}{p - v}
Straight-Line Depreciation
D=C−SnD = \frac{C - S}{n}
Declining-Balance Depreciation (Book Value)
B=C (1−d)kB = C\,(1 - d)^k