Intermediate civil engineering: 50 formulas and methods

All 50 cards

1. 1
   Static determinacy of a beam
   Determinate if $r = 3 + c$; indeterminate if $r > 3 + c$

   $r$ is the number of reaction components and $c$ the number of extra condition equations, such as one per internal hinge. With $r < 3 + c$ the beam is unstable, and it is also unstable if all its reactions are parallel or concurrent.
2. 2
   Static determinacy of a plane truss
   Determinate if $b + r = 2j$; indeterminate if $b + r > 2j$; unstable if $b + r < 2j$

   $b$ members, $r$ reaction components, $j$ joints. The degree of indeterminacy is $b + r - 2j$.
3. 3
   Method of joints
   Isolate each joint and solve $\sum F_x = 0$ and $\sum F_y = 0$, starting at a joint with no more than two unknown member forces
4. 4
   Method of sections
   Cut the truss through the members of interest, usually no more than three unknowns, and apply the three equilibrium equations to one side

   Taking moments about the point where two of the cut members meet gives the third force directly.
5. 5
   Zero-force members in a truss
   At an unloaded, unsupported joint with two non-collinear members, both carry zero force; at one with three members, two of them collinear, the third carries zero force
6. 6
   Relationship between shear force and bending moment
   $\dfrac{dM}{dx} = V$

   So the change in moment between two points equals the area under the shear diagram, and the moment peaks where the shear passes through zero.
7. 7
   Maximum moment in a simply supported beam under a uniform load
   $M_{\max} = \dfrac{wL^2}{8}$, at midspan
8. 8
   Maximum moment in a simply supported beam with a central point load
   $M_{\max} = \dfrac{PL}{4}$, at midspan
9. 9
   Midspan deflection of a simply supported beam under a uniform load
   $\delta = \dfrac{5wL^4}{384EI}$

   $EI$ is the flexural rigidity: Young’s modulus times the second moment of area.
10. 10
    Midspan deflection of a simply supported beam with a central point load
    $\delta = \dfrac{PL^3}{48EI}$
11. 11
    Elastic curve equation
    $EI\dfrac{d^2v}{dx^2} = M(x)$

    Integrate twice and apply the support conditions to get slopes and deflections. Valid for small deflections of linear elastic beams.
12. 12
    Second moment of area of a rectangle about its centroid
    $I = \dfrac{bh^3}{12}$

    $b$ is the width and $h$ the depth in the direction of bending, so doubling the depth makes a beam eight times stiffer.
13. 13
    Parallel axis theorem
    $I = I_c + Ad^2$

    $I_c$ is the second moment about the centroidal axis, $A$ the area and $d$ the distance between the two parallel axes.
14. 14
    Flexure formula
    $\sigma = \dfrac{My}{I}$

    $y$ is the distance from the neutral axis, where bending stress is zero. In a homogeneous, linear elastic section the neutral axis passes through the centroid.
15. 15
    Elastic section modulus
    $S = \dfrac{I}{c}$, so $\sigma_{\max} = \dfrac{M}{S}$

    $c$ is the distance from the neutral axis to the extreme fiber. For a rectangle, $S = \dfrac{bh^2}{6}$.
16. 16
    Shear stress in a beam
    $\tau = \dfrac{VQ}{Ib}$

    $Q$ is the first moment, about the neutral axis, of the area beyond the point, and $b$ the width there. In a rectangle the peak is $1.5\,V/A$, at the neutral axis.
17. 17
    Euler buckling load
    $P_{cr} = \dfrac{\pi^2 EI}{(KL)^2}$

    $KL$ is the effective length. With equal effective lengths, use the smaller $I$: the column buckles about its weaker axis.
18. 18
    Effective length factor $K$
    Pinned–pinned 1.0, fixed–free 2.0, fixed–pinned 0.7, fixed–fixed 0.5

    These are theoretical values. Design guides recommend higher ones for the fixed cases, since real connections aren’t perfectly rigid.
19. 19
    Slenderness ratio of a column
    $\dfrac{KL}{r}$, where $r = \sqrt{I/A}$ is the radius of gyration

    Euler’s critical stress is $\sigma_{cr} = \dfrac{\pi^2 E}{(KL/r)^2}$. It applies only while that stays below the proportional limit; stocky columns yield or buckle inelastically first.
20. 20
    Balanced section in reinforced concrete
    The tension steel starts to yield just as the extreme compression fiber of the concrete reaches its crushing strain

    The crushing strain is taken as 0.003 in ACI 318 and 0.0035 in Eurocode 2 for normal-strength concrete.
21. 21
    Under-reinforced section
    Less steel than balanced, so the steel yields before the concrete crushes: a ductile failure, with wide cracks and large deflections as warning

    An over-reinforced section fails by sudden crushing of the concrete, so codes limit the amount of tension steel.
22. 22
    Equivalent rectangular stress block (ACI)
    A uniform stress of $0.85f'_c$ over a depth $a = \beta_1 c$ from the compression face

    $c$ is the neutral axis depth. $\beta_1 = 0.85$ for concrete up to $28\ \mathrm{MPa}$ ($4000\ \mathrm{psi}$), and decreases for stronger concrete.
23. 23
    Nominal moment strength of a singly reinforced rectangular beam
    $M_n = A_s f_y\left(d - \dfrac{a}{2}\right)$, with $a = \dfrac{A_s f_y}{0.85 f'_c b}$

    $d$ is the effective depth, from the compression face to the centroid of the tension steel. It assumes the steel has yielded.
24. 24
    Limit states of a steel tension member
    Yielding of the gross section and fracture of the net section through the bolt holes

    Block shear rupture at the connection is checked too.
25. 25
    Lateral-torsional buckling
    A beam bent about its strong axis deflects sideways and twists, because its compression flange isn’t braced enough

    Moment capacity falls as the unbraced length $L_b$ grows; bracing the compression flange prevents it.
26. 26
    Plastic moment of a steel section
    $M_p = F_y Z$

    $Z$ is the plastic section modulus: $bh^2/4$ for a rectangle, 1.5 times its elastic $S$. For wide-flange beams $Z/S$ is typically about 1.1 to 1.2.
27. 27
    Compact steel section
    A section whose flanges and web are stocky enough to reach the plastic moment before local buckling

    Compactness is judged from the width-to-thickness ratios of the flanges and web.
28. 28
    Effective stress
    $\sigma' = \sigma - u$

    Total stress minus pore water pressure (Terzaghi); it controls soil strength and compressibility. Below the water table, $u = \gamma_w z_w$, where $z_w$ is the depth below it and $\gamma_w \approx 9.81\ \mathrm{kN/m^3}$.
29. 29
    Atterberg limits
    The water contents at which a fine-grained soil changes state: shrinkage limit (solid to semisolid), plastic limit (semisolid to plastic) and liquid limit (plastic to liquid)
30. 30
    Plasticity index
    $PI = LL - PL$

    The range of water content over which the soil is plastic. With $LL$, it places a fine soil on the plasticity chart.
31. 31
    Darcy’s law
    $Q = kiA$

    $k$ is the hydraulic conductivity and $i = \Delta h / L$ the hydraulic gradient. Water in the pores moves faster, at the seepage velocity $ki/n$, where $n$ is the porosity.
32. 32
    Critical hydraulic gradient
    $i_c = \dfrac{\gamma'}{\gamma_w} \approx 1$

    $\gamma'$ is the submerged unit weight. At this upward gradient the effective stress in a cohesionless soil falls to zero and it boils (the quick condition).
33. 33
    Proctor compaction test
    Compacting soil at several water contents with a set energy to find the maximum dry density and the optimum water content

    Compaction mainly expels air. The modified test uses more energy, giving a higher maximum dry density at a lower optimum water content.
34. 34
    Terzaghi’s bearing capacity equation for a strip footing
    $q_u = cN_c + qN_q + \tfrac{1}{2}\gamma BN_\gamma$

    $c$ is the cohesion, $q$ the overburden pressure at footing level and $B$ the width; the $N$ factors depend on the friction angle. The allowable pressure is $q_u$ divided by a factor of safety, typically 3.
35. 35
    Manning’s equation
    $V = \dfrac{1}{n} R^{2/3} S^{1/2}$

    SI units; in US units the $1$ becomes $1.49$. $n$ is the roughness coefficient, $R$ the hydraulic radius and $S$ the energy slope, equal to the bed slope in uniform flow.
36. 36
    Hydraulic radius
    $R = \dfrac{A}{P}$

    Flow area over wetted perimeter. In a wide rectangular channel, $R \approx y$, the depth.
37. 37
    Froude number in open-channel flow
    $Fr = \dfrac{V}{\sqrt{gD}}$

    $D = A/T$ is the hydraulic depth, with $T$ the top width ($D = y$ in a rectangular channel). $Fr < 1$ is subcritical, $Fr = 1$ critical and $Fr > 1$ supercritical.
38. 38
    Critical depth in a rectangular channel
    $y_c = \left(\dfrac{q^2}{g}\right)^{1/3}$

    $q$ is the discharge per unit width. Specific energy $E = y + \dfrac{V^2}{2g}$ is at its minimum here: $E_{\min} = 1.5\,y_c$.
39. 39
    Hydraulic jump
    An abrupt rise in the water surface where supercritical flow changes to subcritical, dissipating energy in turbulence

    Stilling basins below spillways are designed to hold a jump.
40. 40
    Sequent depths of a hydraulic jump in a rectangular channel
    $\dfrac{y_2}{y_1} = \dfrac{1}{2}\left(\sqrt{1 + 8Fr_1^2} - 1\right)$

    $y_1$ and $Fr_1$ are the depth and Froude number just upstream of the jump.
41. 41
    Rational method for peak runoff
    $Q = CiA$

    $C$ is the runoff coefficient, $i$ the rainfall intensity for a duration equal to the time of concentration and $A$ the area. In SI, $Q = 0.278\,CiA$ gives $\mathrm{m^3/s}$ from $\mathrm{mm/h}$ and $\mathrm{km^2}$. Meant for small catchments.
42. 42
    Runoff coefficient
    The fraction of rainfall that becomes direct runoff

    Typically about 0.7 to 0.95 for pavement and roofs, and much lower for lawns and woodland.
43. 43
    Time of concentration
    The time runoff takes to travel from the hydraulically most distant point of a catchment to the outlet

    Once rain has lasted this long, the whole catchment contributes to the flow at the outlet.
44. 44
    Stopping sight distance
    $d = vt + \dfrac{v^2}{2a}$

    Reaction distance plus braking distance. AASHTO typically assumes $t = 2.5\ \mathrm{s}$ and a deceleration $a = 3.4\ \mathrm{m/s^2}$.
45. 45
    Superelevation
    Tilting a curve’s cross section toward the inside, so part of the centripetal force comes from the bank rather than tire friction

    For design, $e + f = \dfrac{v^2}{gR}$, where $e$ is the superelevation rate, $f$ the side friction factor and $R$ the curve radius.
46. 46
    Fundamental equation of traffic flow
    $q = kv$

    Flow $q$ (vehicles per hour) equals density $k$ (vehicles per kilometer) times space-mean speed $v$.
47. 47
    Maximum flow in the Greenshields model
    $q_{\max} = \dfrac{v_f k_j}{4}$

    Speed falls linearly with density, $v = v_f\left(1 - k/k_j\right)$, so flow peaks at half the jam density $k_j$ and half the free-flow speed $v_f$.
48. 48
    Steps of conventional drinking water treatment
    Coagulation, flocculation, sedimentation, filtration, disinfection

    Coagulants such as alum neutralize the charge on fine particles; gentle mixing during flocculation then grows them into flocs heavy enough to settle.
49. 49
    Surface overflow rate of a settling tank
    $v_o = \dfrac{Q}{A_s}$

    $A_s$ is the tank’s surface area. In an ideal tank, every particle that settles faster than $v_o$ is removed, whatever the depth.
50. 50
    Activated sludge process
    Aerating wastewater with a suspended culture of microorganisms that consume organic matter, then settling the biomass in a clarifier and returning part of it to the aeration tank

    A common form of secondary treatment, after screening, grit removal and primary settling.