Formulas

Structural Engineering

Stress & Strain

Axial Stress
σ = P / A
σ Axial Stress
P Axial Load
A Cross Sectional Area

Bending Stress
σ = M * y / I
σ Bending Stress
M Bending Moment
y Distance from Neutral Axis
I Moment of Inertia

Shear Stress
τ = V * Q / (I * b)
τ Shear Stress
V Shear Force
Q First Moment of Area
I Moment of Inertia
b Width

Torsion Stress
τ = T * r / J
τ Shear Stress due to Torsion
T Torque
r Radius
J Polar Moment of Inertia

Principal Stress
σ_1, σ_2 = (σ_x + σ_y)/2 ± √(((σ_x – σ_y)/2)² + τ_xy²)
σ_1 Maximum Principal Stress
σ_2 Minimum Principal Stress
σ_x Normal Stress in X Direction
σ_y Normal Stress in Y Direction
τ_xy Shear Stress

Von Mises Stress
σ_vm = √(σ_x² – σ_x * σ_y + σ_y² + 3τ_xy²)
σ_vm Von Mises Stress
σ_x Normal Stress in X Direction
σ_y Normal Stress in Y Direction
τ_xy Shear Stress

Strain
ε = δ / L
ε Strain
δ Deformation
L Original Length

Shear Strain
γ = Δx / y
γ Shear Strain
Δx Lateral Displacement
y Original Height

Thermal Strain
ε_thermal = α * ΔT
ε_thermal Thermal Strain
α Coefficient of Thermal Expansion
ΔT Change in Temperature

Modulus of Elasticity
E = σ / ε
E Modulus of Elasticity
σ Stress
ε Strain

Poissons Ratio
ν = – ε_lateral / ε_axial
ν Poissons Ratio
ε_lateral Lateral Strain
ε_axial Axial Strain

Section Properties

Moment of Inertia for Rectangle
I = b * h³ / 12
I Moment of Inertia
b Base Width
h Height

Moment of Inertia for Circle
I = π * d⁴ / 64
I Moment of Inertia
π Pi Constant
d Diameter

Radius of Gyration
r = √(I / A)
r Radius of Gyration
I Moment of Inertia
A Cross Sectional Area

Section Modulus
S = I / c
S Section Modulus
I Moment of Inertia
c Distance to Extreme Fiber

Plastic Section Modulus
Z = A * y
Z Plastic Section Modulus
A Area of Section
y Distance to Centroid

Torsional Constant
J = I_x + I_y
J Torsional Constant
I_x Moment of Inertia about X Axis
I_y Moment of Inertia about Y Axis

Warping Constant
C_w = (h² * I_y) / 4
C_w Warping Constant
h Depth of Section
I_y Moment of Inertia about Y Axis

Beam Analysis

Moment in Simply Supported Beam with UDL
M_max = w * L² / 8
M_max Maximum Bending Moment
w Uniformly Distributed Load per Unit Length
L Span Length

Shear in Simply Supported Beam with UDL
V_max = w * L / 2
V_max Maximum Shear Force
w Uniformly Distributed Load per Unit Length
L Span Length

Moment in Simply Supported Beam with Point Load at Center
M_max = P * L / 4
M_max Maximum Bending Moment
P Point Load
L Span Length

Moment in Cantilever Beam with UDL
M_max = w * L² / 2
M_max Maximum Bending Moment
w Uniformly Distributed Load per Unit Length
L Span Length

Shear in Cantilever Beam with UDL
V_max = w * L
V_max Maximum Shear Force
w Uniformly Distributed Load per Unit Length
L Span Length

Deflection for Simply Supported Beam with Point Load
δ = P * L³ / (48 * E * I)
δ Deflection
P Point Load
L Length of Beam
E Modulus of Elasticity
I Moment of Inertia

Deflection for Cantilever with End Load
δ = P * L³ / (3 * E * I)
δ Deflection
P Point Load at End
L Length of Beam
E Modulus of Elasticity
I Moment of Inertia

Shear Flow
q = V * Q / I
q Shear Flow
V Shear Force
Q First Moment of Area
I Moment of Inertia

Strain Energy
U = (1/2) * P * δ
U Strain Energy
P Applied Load
δ Deflection

Column Analysis

Slenderness Ratio
λ = K * L / r
λ Slenderness Ratio
K Effective Length Factor
L Unsupported Length
r Radius of Gyration

Effective Length
L_e = K * L
L_e Effective Length
K Effective Length Factor
L Actual Length

Euler Buckling Load
P_cr = π² * E * I / (K * L)²
P_cr Critical Buckling Load
E Modulus of Elasticity
I Moment of Inertia
K Effective Length Factor
L Actual Length of Column

Critical Buckling Stress
σ_cr = π² * E / (λ)²
σ_cr Critical Buckling Stress
E Modulus of Elasticity
λ Slenderness Ratio

Moment Magnification Factor
δ_b = C_m / (1 – (P_u / (φ * P_c))) ≥ 1.0
δ_b Moment Magnification Factor
C_m Coefficient for Moment Diagram
P_u Factored Axial Load
φ Strength Reduction Factor
P_c Critical Buckling Load

Load Calculations

Tributary Area
TA = Tributary Width * Span Length
TA Tributary Area
Tributary Width Width of Area Supported
Span Length Length of Supporting Member

Total Load on Member
w_total = w_dead + w_live
w_total Total Load per Unit Length
w_dead Dead Load per Unit Length
w_live Live Load per Unit Length

Factored Load LRFD
U = 1.2D + 1.6L
U Factored Load
D Dead Load
L Live Load

Design

Allowable Stress Design
σ_allow = σ_yield / F.S.
σ_allow Allowable Stress
σ_yield Yield Stress
F.S. Factor of Safety

Connection Strength
R_n = F_n * A
R_n Nominal Strength
F_n Nominal Stress
A Effective Area

Truss Analysis

Method of Joints Equilibrium
ΣF_x = 0 and ΣF_y = 0
ΣF_x Sum of Forces in X Direction
ΣF_y Sum of Forces in Y Direction

Statics & Equilibrium

Sum of Forces in X Direction
ΣF_x = 0
ΣF_x Sum of All Forces in X Direction

Sum of Forces in Y Direction
ΣF_y = 0
ΣF_y Sum of All Forces in Y Direction

Sum of Moments
ΣM = 0
ΣM Sum of All Moments about a Point

Construction & Materials

Concrete

Volume of Concrete
V = L * W * H
V Volume
L Length
W Width
H Height

Concrete Yardage
Yards³ = (L ft * W ft * H ft) / 27
Yards³ Cubic Yards of Concrete
L ft Length in Feet
W ft Width in Feet
H ft Height in Feet

Modular Ratio
n = E_steel / E_concrete
n Modular Ratio
E_steel Modulus of Elasticity of Steel
E_concrete Modulus of Elasticity of Concrete

Development Length
L_d = (φ * f_y * ψ_t * ψ_e) / (λ * √(f’_c)) * d_b
L_d Development Length
φ Strength Reduction Factor
f_y Yield Strength of Steel
ψ_t Coating Factor
ψ_e Reinforcement Size Factor
λ Lightweight Concrete Factor
f’_c Concrete Compressive Strength
d_b Bar Diameter

Earthwork

Volume of Cut and Fill
V = (A₁ + A₂) / 2 * L
V Volume
A₁ Cross Sectional Area at One End
A₂ Cross Sectional Area at Other End
L Distance Between Sections

Modulus of Subgrade Reaction
k = p / δ
k Modulus of Subgrade Reaction
p Applied Pressure
δ Settlement

Material Quantities

Number of Bricks
N = Wall Area / (Brick Length * Brick Height)
N Number of Bricks
Wall Area Surface Area of Wall
Brick Length Length of One Brick
Brick Height Height of One Brick

Number of Tiles
N = Floor Area / Tile Area
N Number of Tiles
Floor Area Area to be Covered
Tile Area Area of One Tile

Weight of Steel
W = Volume * Density
W Weight
Volume Volume of Steel
Density Density of Steel

Geometric & Spatial

Area Calculations

Area of a Circle
A = π * r²
A Area
π Pi Constant
r Radius

Area of a Triangle
A = b * h / 2
A Area
b Base
h Height

Area of a Trapezoid
A = (a + b) / 2 * h
A Area
a Length of First Parallel Side
b Length of Second Parallel Side
h Height

Area of an Ellipse
A = π * a * b
A Area
π Pi Constant
a Semi Major Axis
b Semi Minor Axis

Perimeter of a Circle
C = 2 * π * r
C Circumference
π Pi Constant
r Radius

Volume Calculations

Volume of a Cylinder
V = π * r² * h
V Volume
π Pi Constant
r Radius
h Height

Volume of a Pyramid
V = l * w * h / 3
V Volume
l Length of Base
w Width of Base
h Height

Volume of a Sphere
V = 4/3 * π * r³
V Volume
π Pi Constant
r Radius

Volume of a Cone
V = π * r² * h / 3
V Volume
π Pi Constant
r Radius
h Height

Volume of a Prism
V = B * h
V Volume
B Area of the Base
h Height

Trigonometry

Pythagorean Theorem
a² + b² = c²
a Length of First Side
b Length of Second Side
c Length of Hypotenuse

Sine
sin θ = Opposite / Hypotenuse
θ Angle
Opposite Length of Opposite Side
Hypotenuse Length of Hypotenuse

Cosine
cos θ = Adjacent / Hypotenuse
θ Angle
Adjacent Length of Adjacent Side
Hypotenuse Length of Hypotenuse

Tangent
tan θ = Opposite / Adjacent
θ Angle
Opposite Length of Opposite Side
Adjacent Length of Adjacent Side

Law of Cosines
c² = a² + b² – 2ab cos γ
a b c Lengths of Sides of a Triangle
γ Angle Opposite Side c

Law of Sines
a / sin α = b / sin β = c / sin γ
a b c Lengths of Sides of a Triangle
α β γ Angles Opposite Those Sides

Environmental Design

Thermal Performance

U Value
U = 1 / R_T
U U Value
R_T Total Thermal Resistance

R Value
R_T = R_si + R_1 + R_2 + … + R_so
R_T Total Thermal Resistance
R_si Internal Surface Resistance
R_1 R_2 Resistance of Each Layer
R_so External Surface Resistance

Heat Loss
Q = U * A * ΔT
Q Rate of Heat Loss
U U Value
A Area
ΔT Temperature Difference

Solar Heat Gain Coefficient
SHGC = Solar Heat Gain / Incident Solar Radiation
SHGC Solar Heat Gain Coefficient
Solar Heat Gain Total Solar Radiation Entering
Incident Solar Radiation Total Solar Radiation Striking the Glass

Daylighting

Daylight Factor
DF = (E_internal / E_external) * 100%
DF Daylight Factor
E_internal Illuminance at a Point Inside
E_external Simultaneous External Horizontal Illuminance

Lumen Method for Zonal Cavity
N = (E * A) / (Φ * UF * LLF)
N Number of Lamps
E Required Illuminance
A Area of Space
Φ Luminous Flux per Lamp
UF Utilization Factor
LLF Light Loss Factor

Ventilation

Air Changes per Hour
ACH = 60 * Q / V
ACH Air Changes per Hour
Q Air Flow Rate in Cubic Feet per Minute
V Volume of the Space

HVAC

Sensible Heat
Q_s = 1.08 * CFM * ΔT
Q_s Sensible Heat Flow Rate BTU/hr
CFM Cubic Feet per Minute
ΔT Temperature Difference

Latent Heat
Q_l = 4840 * CFM * ΔW
Q_l Latent Heat Flow Rate BTU/hr
CFM Cubic Feet per Minute
ΔW Humidity Ratio Difference

Total Heat
Q_t = 4.5 * CFM * Δh
Q_t Total Heat Flow Rate BTU/hr
CFM Cubic Feet per Minute
Δh Enthalpy Difference

Acoustics

Reverberation Time Sabines Formula
RT₆₀ = 0.049 * V / A
RT₆₀ Reverberation Time
V Volume of Room
A Total Absorption

Noise Reduction
NR = TL – 10 * log10 (A_room / S)
NR Noise Reduction
TL Transmission Loss of Barrier
A_room Total Absorption in Receiving Room
S Area of the Barrier

Sound Intensity Level
L_I = 10 * log10 (I / I₀)
L_I Sound Intensity Level
I Sound Intensity
I₀ Reference Intensity

Sound Pressure Level
L_p = 20 * log10 (p / p₀)
L_p Sound Pressure Level
p Measured RMS Sound Pressure
p₀ Reference Sound Pressure

Surveying

Pace Factor
PF = Known Distance / Number of Paces
PF Pace Factor
Known Distance Measured Length
Number of Paces Paces Taken to Cover Distance

Slope Correction
C_s = (h²) / (2 * L)
C_s Slope Correction
h Difference in Elevation
L Slope Length

Accessibility

Ramp Slope Ratio
Slope Ratio = Rise : Run = 1 : 12
Rise Vertical Change
Run Horizontal Distance

Seismic Design

Seismic Force
F = m * a
F Seismic Force
m Mass
a Seismic Acceleration

Base Shear
V = C_s * W
V Base Shear
C_s Seismic Response Coefficient
W Effective Seismic Weight

Fundamental Period
T = C_t * h_n^x
T Fundamental Period
C_t Building Period Coefficient
h_n Height of Building
x Height Exponent

Ductility Ratio
μ = Δ_max / Δ_yield
μ Ductility Ratio
Δ_max Maximum Displacement
Δ_yield Displacement at Yield

Wind Loads

Wind Pressure
p = q * G * C_p
p Wind Pressure
q Velocity Pressure
G Gust Effect Factor
C_p Pressure Coefficient

Velocity Pressure
q = 0.00256 * K_z * K_zt * K_d * V² * I
q Velocity Pressure
K_z Velocity Pressure Exposure Coefficient
K_zt Topographic Factor
K_d Wind Directionality Factor
V Basic Wind Speed
I Importance Factor

Soil Mechanics

Bearing Capacity
q_u = c N_c + q N_q + 0.5 γ B N_γ
q_u Ultimate Bearing Capacity
c Cohesion
q Surcharge
γ Unit Weight of Soil
B Width of Footing
N_c N_q N_γ Bearing Capacity Factors

Vertical Stress in Soil
σ_v = γ * z
σ_v Vertical Stress
γ Unit Weight of Soil
z Depth

Fluid Mechanics

Flow Rate
Q = A * v
Q Flow Rate
A Cross Sectional Area
v Velocity

Bernoullis Equation
P₁ + 1/2 ρ v₁² + ρ g h₁ = P₂ + 1/2 ρ v₂² + ρ g h₂
P Pressure
ρ Density
v Velocity
g Acceleration due to Gravity
h Height

Electrical

Ohms Law
V = I * R
V Voltage
I Current
R Resistance

Power
P = V * I
P Power
V Voltage
I Current

Voltage Drop
V_d = 2 * I * L * R / 1000
V_d Voltage Drop
I Current
L Length of Run
R Resistance per 1000 feet

Project Management

Cost per Square Foot
Cost = Area SF * Cost per SF
Cost Total Estimated Cost
Area SF Total Area in Square Feet
Cost per SF Unit Cost

Float or Slack Time
Float = LS – ES
Float Total Float
LS Late Start Time
ES Early Start Time