50 Civil Engineering Formulas You Must Know
As a civil engineering student, preparing for exams requires you to apply concepts to a problem, and these formulas will guide you in solving those theoretical and practical problems.
Structural Engineering
Understanding structural behavior is fundamental to designing safe and efficient structures. These formulas help you analyze stresses, deflections, and internal forces in structural elements.
1. Stress
![\[\sigma = \frac{F}{A}\]](https://civilengghub.com/wp-content/ql-cache/quicklatex.com-78f1d827dbeb82f0660da1fd94ab6780_l3.png "Rendered by QuickLaTeX.com")
Where:
- σ = normal stress
- F = applied force
- A = cross-sectional area
This fundamental equation calculates the internal force per unit area within a material, essential for ensuring structural members don’t exceed their strength capacity.
2. Shear Stress in Beams
![\[\tau=\frac{VQ}{Ib}\]](https://civilengghub.com/wp-content/ql-cache/quicklatex.com-d5b28b405f5791b7f6598d44776bc76e_l3.png "Rendered by QuickLaTeX.com")
Where:
- τ = shear stress
- V = shear force
- Q = first moment of area
- I = moment of inertia
- b = width of section
Used to determine shear stress distribution across a beam’s cross-section, critical for preventing shear failure.
3. Bending Stress
![\[\sigma=\frac{M_y}{I}\]](https://civilengghub.com/wp-content/ql-cache/quicklatex.com-682b355b001df59475147a74f37a85f1_l3.png "Rendered by QuickLaTeX.com")
Where:
- σ = bending stress
- M = bending moment
- y = distance from neutral axis
- I = moment of inertia
Also known as the flexure formula, this calculates stress in beams subjected to bending moments.
4. Moment of Inertia (Rectangular Section)
![\[I=\frac{bh^3}{12}\]](https://civilengghub.com/wp-content/ql-cache/quicklatex.com-56202e3af5980d3a5d938ece89434e27_l3.png "Rendered by QuickLaTeX.com")
Where:
- I = moment of inertia
- b = width
- h = height
Represents resistance to bending for rectangular cross-sections about the centroidal axis.
5. Hooke’s Law
![\[P_{abs}=P_{gauge}+P_{atm}\]](https://civilengghub.com/wp-content/ql-cache/quicklatex.com-8d3d3480035ce2cb52127fd51f72e1c0_l3.png "Rendered by QuickLaTeX.com")
Where:
- σ = stress
- E = modulus of elasticity
- ε = strain
Describes the linear elastic relationship between stress and strain in materials.
6. Deflection (Simply Supported Beam, Point Load at Center)
δ = PL³/(48EI)
Where:
- δ = maximum deflection
- P = point load
- L = span length
- E = modulus of elasticity
- I = moment of inertia
Essential for serviceability checks in beam design.
7. Euler’s Critical Buckling Load
![\[P_{cr}=\pi EI/(KL)^\]](https://civilengghub.com/wp-content/ql-cache/quicklatex.com-e5537b0dc6882a3675a5d978ae937bcb_l3.png "Rendered by QuickLaTeX.com")
Where:
- Pcr = critical buckling load
- E = modulus of elasticity
- I = moment of inertia
- K = effective length factor
- L = actual length
Determines the axial load at which a column will buckle.
8. Shear Force and Bending Moment Relationship
![\[\frac{dM}{dx}=V\]](https://civilengghub.com/wp-content/ql-cache/quicklatex.com-3d64c456c262cf532b69b7d723e7145a_l3.png "Rendered by QuickLaTeX.com")
This differential relationship shows that the slope of the bending moment diagram equals the shear force.
9. Second Moment-Area Theorem
δB/A = ∫(M/EI)x dx
Used to calculate deflections and rotations in beams using the moment diagram.
10. Torsional Shear Stress
![\[\tau = Tr/J\]](https://civilengghub.com/wp-content/ql-cache/quicklatex.com-cd58b49740fc3a55d44855a00ea631f7_l3.png "Rendered by QuickLaTeX.com")
Where:
- τ = shear stress
- T = applied torque
- r = radial distance from center
- J = polar moment of inertia
Calculates stress in circular shafts subjected to torsion.
Fluid Mechanics
Fluid mechanics principles are essential for water supply systems, drainage design, and hydraulic structures.
11. Reynolds Number
![\[Re= \frac{\rho vD}{\mu} = \frac{vD}{\nu} <!-- /wp:paragraph --> <!-- wp:paragraph --> Where: <!-- /wp:paragraph --> <!-- wp:list --> <ul class="wp-block-list"><!-- wp:list-item --> <li>Re = Reynolds number</li> <!-- /wp:list-item --> <!-- wp:list-item --> <li>ρ = fluid density</li> <!-- /wp:list-item --> <!-- wp:list-item --> <li>v = flow velocity</li> <!-- /wp:list-item --> <!-- wp:list-item --> <li>D = characteristic length (diameter)</li> <!-- /wp:list-item --> <!-- wp:list-item --> <li>μ = dynamic viscosity</li> <!-- /wp:list-item --> <!-- wp:list-item --> <li>ν = kinematic viscosity</li> <!-- /wp:list-item --></ul> <!-- /wp:list --> <!-- wp:paragraph --> Determines whether flow is laminar or turbulent (Re < 2300 is laminar, Re > 4000 is turbulent). <!-- /wp:paragraph --> <!-- wp:heading {"level":3,"fontSize":"medium"} --> <h3 class="wp-block-heading has-medium-font-size" id="12-froude-number">12. Froude Number</h3> <!-- /wp:heading --> <!-- wp:paragraph --> \[F_r= v/\sqrt{gL}\]](https://civilengghub.com/wp-content/ql-cache/quicklatex.com-404893fadabb0fed7a455e7620c41ad1_l3.png "Rendered by QuickLaTeX.com")
Where:
- Fr = Froude number
- v = flow velocity
- g = gravitational acceleration
- L = characteristic length
Characterizes open channel flow as subcritical (Fr < 1), critical (Fr = 1), or supercritical (Fr > 1).
13. Absolute Pressure
Where:
- Pabs = absolute pressure
- Pgauge = gauge pressure
- Patm = atmospheric pressure
Converts between gauge and absolute pressure measurements.
14. Viscosity
![\[\tau=\mu\frac{du}{dy}\]](https://civilengghub.com/wp-content/ql-cache/quicklatex.com-585617ef64cb1e3bdeb8b21a4a1bf6ba_l3.png "Rendered by QuickLaTeX.com")
Where:
- τ = shear stress
- μ = dynamic viscosity
- du/dy = velocity gradient
Defines the relationship between shear stress and velocity gradient in fluids.
15. Hydrostatic Pressure
![\[P=\rho gh\]](https://civilengghub.com/wp-content/ql-cache/quicklatex.com-41707bf60c19131f3bf7592edfe90a9d_l3.png "Rendered by QuickLaTeX.com")
Where:
- P = pressure
- ρ = fluid density
- g = gravitational acceleration
- h = depth
Calculates pressure at any depth in a static fluid.
16. Specific Weight of Fluid
![\[\gamma= \rho g=\frac{W}{V}\]](https://civilengghub.com/wp-content/ql-cache/quicklatex.com-e1aee8fbb9ac726d9a24a151699c94ed_l3.png "Rendered by QuickLaTeX.com")
Where:
- γ = specific weight
- ρ = density
- g = gravitational acceleration
- W = weight
- V = volume
Represents weight per unit volume of a fluid.
17. Discharge
![\[Q=Av\]](https://civilengghub.com/wp-content/ql-cache/quicklatex.com-9d69e8ce2200e5320b6c7baa18952c46_l3.png "Rendered by QuickLaTeX.com")
Where:
- Q = volumetric flow rate
- A = cross-sectional area
- v = average velocity
Fundamental equation for calculating flow rate through pipes and channels.
18. Bernoulli’s Equation
![\[\frac{P}{\gamma}+\frac{v^2}{2g}+z=H\]](https://civilengghub.com/wp-content/ql-cache/quicklatex.com-b827e8f41da356d05aecc436ca240aef_l3.png "Rendered by QuickLaTeX.com")
Where:
- P/γ = pressure head
- v²/(2g) = velocity head
- z = elevation head
Conservation of energy equation for fluid flow along a streamline.
19. Darcy-Weisbach Equation
![\[h_f=f\cdot \frac{L}{D} \ cdot \frac{v^2}{2g}\]](https://civilengghub.com/wp-content/ql-cache/quicklatex.com-e7fe870b7f7b0a97ef65727603e5c9db_l3.png "Rendered by QuickLaTeX.com")
Where:
- hf = head loss due to friction
- f = friction factor
- L = pipe length
- D = pipe diameter
- v = flow velocity
- g = gravitational acceleration
Calculates head loss in pipes due to friction.
20. Manning’s Equation
![\[V=\frac{1}{n} R^{2/3} S^{1/2}\]](https://civilengghub.com/wp-content/ql-cache/quicklatex.com-51190150d96005579577d59ba59f84b2_l3.png "Rendered by QuickLaTeX.com")
Where:
- V = average velocity
- n = Manning’s roughness coefficient
- R = hydraulic radius
- S = slope of energy grade line
Used for open channel flow calculations.
Concrete and Steel Design
These formulas are crucial for designing reinforced concrete and steel structural members according to modern building codes.
21. Ultimate Moment Capacity of a Singly Reinforced Beam
Mu = φAsfy(d – a/2)
Where:
- Mu = ultimate moment capacity
- φ = strength reduction factor
- As = area of steel reinforcement
- fy = yield strength of steel
- d = effective depth
- a = depth of equivalent stress block
Determines the maximum moment a reinforced concrete beam can resist.
22. Depth of Neutral Axis
![\[c=\frac{\alpha}{\beta_1}\]](https://civilengghub.com/wp-content/ql-cache/quicklatex.com-998ad4ff5da5392f2c14e8b5733eb824_l3.png "Rendered by QuickLaTeX.com")
Where:
- c = depth to neutral axis
- a = depth of equivalent stress block
- β₁ = stress block factor (typically 0.85 for normal concrete)
Locates the neutral axis in a reinforced concrete section.
23. Balanced Steel Ratio
Where:
- ρb = balanced steel ratio
- β₁ = stress block factor
- fc’ = concrete compressive strength
- fy = steel yield strength
- εcu = ultimate concrete strain
- εy = steel yield strain
Defines the reinforcement ratio that produces simultaneous crushing of concrete and yielding of steel.
24. Moment of Inertia (Cracked Section)
Where:
- Icr = cracked moment of inertia
- b = width
- c = depth to neutral axis
- n = modular ratio (Es/Ec)
- As = steel area
- d = effective depth
Used for deflection calculations in cracked reinforced concrete sections.
25. LRFD
26. Net Area in Tension
Where:
- An = net area
- Ag = gross area
- Aholes = area of bolt holes
Calculates effective area for tension members accounting for holes.
27. Block Shear Rupture
28. Euler Buckling Stress
29. Plastic Moment
30. Slenderness Ratio
Geotechnical Engineering
31. Void Ratio
32. Porosity
33. Mohr-Coulomb Failure
34. Saturated Unit Weight
35. Bearing Capacity
36. Settlement
37. Permeability (Falling Head Test)
38. Time Factor for Consolidation
39. Factor of Safety
40. Earth Pressure (Rankine-Active)
Surveying and Transportation Engineering
41. Slope Distance Correction
42. Stopping Sight Distance
43. Taping Correction Due to Sag
44. Taping Correction due to Temp
