CIVIL ENGINEERING GEOTECHNICAL Definitions c = cohesion qu = unconfined compressive strength = 2c Dr = relative density (%) = [(emax – e)/(emax – emin)] ×100 = [(1/γmin – 1/γ) /(1/γmin – 1/γmax)] × 100 emax = maximum void ratio emin = minimum void ratio γmax = maximum dry unit weight γmin = minimum dry unit weight τ = general shear strength = c + σtan φ φ = angle of internal friction σ = normal stress = P/A P = force A = area σ′ = effective stress = σ – u σ = total normal stress u = pore water pressure

= ultimate bearing capacity = cNc + γDf Nq + 0.5γBNγ Nc, Nq, and Nγ = bearing capacity factors B = width of strip footing Df = depth of footing below surface of ground qult

k

= = Q = i = A = Q = H = Nf = Nd =

coefficient of permeability = hydraulic conductivity Q/(iA) (from Darcy's equation) discharge flow rate hydraulic gradient = dH/dx cross-sectional area kH(Nf/Nd) (for flow nets, Q per unit width) total hydraulic head (potential) number of flow channels number of potential drops

Cc = compression index = ∆e/∆log p = (e1 – e2)/(log p2 – log p1) = 0.009 (LL – 10) for normally consolidated clay e1 and e2 = void ratios p1 and p2 = pressures ∆H = settlement = H [Cc /(1 + e0)] log [(σ0 + ∆p)/σ0] = H∆e/(1 + e0) H = thickness of soil layer ∆e, ∆p = change in void ratio, change in pressure e0, σ0 = initial void ratio, initial pressure cv = coefficient of consolidation = THdr2/t T = time factor t = consolidation time Hdr = length of drainage path

Cc = coefficient of curvature of gradation = (D30)2/[(D60)(D10)] D10, D30, D60 = particle diameters corresponding to 10% 30%, and 60% finer on grain-size curve Cu = uniformity coefficient = D60 /D10 e = void ratio = Vv/Vs Vv = volume of voids Vs = volume of solids w = water content (%) = (Ww/Ws) ×100 Ww = weight of water Ws = weight of solids Wt = total weight Gs = specific gravity of solids = Ws /(Vsγw) γw = unit weight of water (62.4 lb/ft3 or 1,000 kg/m3) PI = plasticity index = LL – PL LL = liquid limit PL = plastic limit S = degree of saturation (%) = (Vw/Vv) × 100 Vw = volume of water Vv = volume of voids Vt = total volume γt = total unit weight of soil = Wt/Vt γd = dry unit weight of soil = Ws/Vt = Gsγw/(1 + e) = γ /(1 + w) Gsw = Se γs = unit weight of solids = Ws / Vs n = porosity = Vv/Vt = e/(1 + e)

Ka = = Kp = = Pa = H =

Rankine active lateral pressure coefficient tan2(45 – φ/2) Rankine passive lateral pressure coefficient tan2(45 + φ/2) active resultant force = 0.5γH 2Ka height of wall

FS = factor of safety against sliding (slope stability) cL + Wcosα tanφ = W sinα L = length of slip plane α = slope of slip plane with horizontal φ = angle of internal friction W = total weight of soil above slip plane

111

CIVIL ENGINEERING (continued)

UNIFIED SOIL CLASSIFICATION SYSTEM (ASTM DESIGNATION D-2487) SOIL CLASSIFICATION CHART

Soil Classification Criteria for Assigning Group Symbols and Group Names Using Laboratory TestsA COARSEGRAINED SOILS More than 50% retained on No. 200 sieve

FINE-GRAINED SOILS 50% or more pass the No. 200 sieve

Group Symbol

Cu ≥ 4 and 1 ≤ Cc ≤ 3E Cu < 4 and/or Cc > 3E

GW GP GM GC SW SP SM SC CL

Well-graded gravelF Poorly graded gravelF Silty gravelF,G,H Clayey gravelF,G,H Well-graded sand I Poorly graded sand I Silty sand G,H,I Clayey sand G,H,I Lean clay K,L,M

PI < 4 or plots below "A" line J Liquid Limit - oven dried < 0.75 Liquid Limit - not dried

ML OL

Silt K,L,M

PI plots on or above "A" line PI plots below "A" line Liquid Limit - oven dried < 0.75 Liquid Limit - not dried

CH

Gravels More than 50% of coarse fraction retained on No. 4 sieve Sands 50% or more of coarse fraction passes No. 4 sieve

Cleans Gravels Less than 5% finesc Gravels with Fines More than 12% finesc

Silts and Clays Liquid limit less than 50

inorganic

Fines classify as CL or CH PI > 7 and plots on or above "A" line J

organic Silts and Clays Liquid limit 50 or more

HIGHLY ORGANIC SOILS

Fines classify as ML or MH Fines classify as CL or CH Cu ≥ 6 and 1 ≤ Cc ≤ 3E Cu < 6 and/or 1 > Cc > 3E

Cleans Sands Less than 5% finesD Sands with Fines More than 12% finesD

Fines classify as ML or MH

inorganic organic

Elastic silt K,L,M Organic clay K,L,M,P Organic silt K,L,M,Q Peat

PT

SP-SM poorly graded sand with silt SP-SC poorly graded sand with clay ( D 30 ) 2 E CC = C U = D 60 / D 10 D 10 X D 60 F If soil contains ≥ 15% sand, add "with sand" to group name. G If fines classify as CL-ML, use dual symbol GC-GM, or SC-SM. H If fines are organic, add "with organic fines" to group name. I If soil contains ≥ 15% gravel, add "with gravel" to group name. J If Atterberg limits plot in hatched area, soil is a CL-ML, silty clay.

Based on the material passing the 3-in. (75mm) sieve. B If field sample contained cobbles or boulders, or both, add "with cobbles or boulders, or both" to group name. C Gravels with 5 to 12% fines require dual symbols: GW-GM well-graded gravel with silt GW-GC well-graded gravel with clay GP-GM poorly graded gravel with silt GP-GC poorly graded gravel with clay D Sands with 5 to 12% fines require dual symbols: SW-SM well-graded sand with silt SW-SC well-graded sand with clay

Organic clay K,L,M,N Organic silt K,L,M,O Fat clay K,L,M

MH OH

Primarily organic matter, dark in color, and organic odor

A

Group NameB

K

If soil contains 15 to 29% plus No. 200, add "with sand" or "with gravel, "whichever is predominant. L If soil contains ≥ 30% plus No. 200, predominantly sand, add "sandy" to group name. M If soil contains ≥ 30% plus No. 200, predominantly gravel, add "gravelly" to group name. N PI ≥ 4 and plots on or above "A" line. O PI < 4 or plots below "A" line. P PI plots on or above "A" line. Q PI plots below "A" line.

60

E

Notes: (1) The A-Line separates clay classifications and silt classifications. (2) The U-Line represents an approximate upper limit of LL and PL combinations for natural soils (empirically determined).

Plasticity index, PI

50

" "U

LIN

CH

40

E

IN

H rO

"L

"A

o

30 20 10 7 4 0

CL CL - ML 0

10 16 20

or

OL

MH or OH

ML or OL 30

40

50

60

Liquid limit, LL

112

70

80

90

100

110

CIVIL ENGINEERING (continued)

GROUP SYMBOL Cc > 3

GP

Cu ≥ 4 and 1 ≤ Cc ≤ 3 GRAVEL % GRAVEL > % SAND

5-12% FINES

FINES = ML or MH

GW-GM

FINES = CL, CH, (or CL-ML)

GW-GC

FINES = ML or MH

GP-GM

FINES = CL, CH, (or CL-ML)

GP-GC

FINES =ML or MH

GM

FINES = CL or CH

GC

FINES = CL-ML

GC-GM

Cu < 4 and/or 1 > Cc > 3

>12% FINES

Cc > 3

SP

Cu ≥ 6 and 1 ≤ Cc ≤ 3 SAND % SAND ≥ % GRAVEL

FINES = ML or MH

SW-SM

FINES = CL, CH, (or CL-ML)

SW-SC

FINES = ML or MH

SP-SM

FINES = CL, CH, (or CL-ML)

SP-SC

FINES = ML or MH

SM

FINES = CL or CH

SC

FINES = CL-ML

SC-SM

5-12% FINES Cu < 6 and/or 1 > Cc > 3

>12% FINES

hf : As,max =

Av f y d

[may not exceed 8 bw d f c ' ] s Required and maximum-permitted stirrup spacing, s φVc Vu ≤ : No stirrups required 2 φVc Vu > : Use the following table ( Av given ): 2

0.85 f c ' b

⎤ ⎡ ⎛ As f y ⎞⎛ a⎞ − As ' ⎟⎟ ⎜ d − ⎟ + As ' ( d − d ' ) ⎥ Mn = fs' ⎢ ⎜⎜ 2⎠ ⎠⎝ ⎦⎥ ⎣⎢ ⎝ f s '

be

b (rectangular beams )

bw =

0.85 f c ' β1 be ⎛ 3 dt ⎞ 0.85 f c ' (be − bw ) h f ⎜ ⎟+ fy fy ⎝ 7 ⎠

Mn = 0.85 fc' [hf (be − bw) (d −

hf 2

Maximum permitted spacing

)

Smaller of: d s= 2

Smaller of: d s= OR 2 s =24"

OR s =24"

a + a bw (d − )] 2

Vs > 4 bw d Smaller of: d s= 4 s =12"

117

fc '

fc '

CIVIL ENGINEERING (continued)

SHORT COLUMNS Limits for main reinforcements: A ρ g = st Ag

Concentrically-loaded short columns: φPn ≥ Pu M1 = M2 = 0 KL ≤ 22 r Design column strength, spiral columns: φ = 0.70 φPn = 0.85φ [ 0.85 fc' ( Ag − Ast ) + Ast fy ]

0.01 ≤ ρg ≤ 0.08 Definition of a short column: 12 M 1 KL ≤ 34 − r M2 where: KL = Lcol clear height of column [assume K = 1.0]

Design column strength, tied columns: φ = 0.65 φPn = 0.80φ [ 0.85 fc' ( Ag − Ast ) + Ast fy ] Short columns with end moments: Mu = M2 or Mu = Pu e Use Load-moment strength interaction diagram to: 1. Obtain φPn at applied moment Mu 2. Obtain φPn at eccentricity e 3. Select As for Pu , Mu

r = 0.288h rectangular column, h is side length perpendicular to buckling axis ( i.e., side length in the plane of buckling ) r = 0.25h circular column, h = diameter M1 = smaller end moment M2 = larger end moment M1 M2

positive if M1, M2 cause single curvature negative if M1, M2 cause reverse curvature

LONG COLUMNS − Braced (non-sway) frames Definition of a long column: 12 M 1 KL > 34 − r M2

Long columns with end moments: M1 = smaller end moment M2 = larger end moment M1 positive if M1 , M2 produce single curvature M2

Critical load:

Pc =

π2 E I π2 EI = ( Lcol ) 2 ( KL ) 2

C m = 0.6 +

where: EI = 0.25 Ec Ig Mc =

Concentrically-loaded long columns: emin = (0.6 + 0.03h) minimum eccentricity M1 = M2 = Pu emin (positive curvature) KL > 22 r M2 Mc = Pu 1− 0.75 Pc

0.4 M 1 ≥ 0.4 M2

Cm M 2 ≥ M2 Pu 1− 0.75 Pc

Use Load-moment strength interaction diagram to design/analyze column for Pu , Mu

Use Load-moment strength interaction diagram to design/analyze column for Pu , Mu

118

CIVIL ENGINEERING (continued)

GRAPH A.11

Column strength interaction diagram for rectangular section with bars on end faces and γ = 0.80 (for instructional use only). Design of Concrete Structures, 13th ed., Nilson, Darwin, Dolan,

McGraw-Hill ISBN 0-07-248305-9 GRAPH A.11, Page 762

119

CIVIL ENGINEERING (continued)

GRAPH A.15

Column strength interaction diagram for circular section γ = 0.80 (for instructional use only). Design of Concrete Structures, 13th Edition (2004), Nilson, Darwin, Dolan

McGraw-Hill ISBN 0-07-248305-9 GRAPH A.15, Page 766

120

CIVIL ENGINEERING (continued)

STEEL STRUCTURES

References:

AISC LRFD Manual, 3rd Edition AISC ASD Manual, 9th Edition

LOAD COMBINATIONS (LRFD) Floor systems: 1.4D 1.2D + 1.6L

Roof systems:

1.2D + 1.6(Lr or S or R) + 0.8W 1.2D + 0.5(Lr or S or R) + 1.3W

0.9D ± 1.3W D = dead load due to the weight of the structure and permanent features

where:

L

= live load due to occupancy and moveable equipment

L r = roof live load S

= snow load

R

= load due to initial rainwater (excluding ponding) or ice

W = wind load

TENSION MEMBERS: flat plates, angles (bolted or welded) Gross area: Ag = bg t (use tabulated value for angles) An = (bg − ΣDh +

Net area: where:

s2 ) t across critical chain of holes 4g

bg = gross width t

= thickness

s = longitudinal center-to-center spacing (pitch) of two consecutive holes g = transverse center-to-center spacing (gage) between fastener gage lines Dh = bolt-hole diameter Effective area (bolted members):

Effective area (welded members):

U = 1.0 (flat bars)

U = 1.0 (flat bars, L ≥ 2w)

U = 0.85 (angles with ≥ 3 bolts in line)

Ae = UAn

U = 0.87 (flat bars, 2w > L ≥ 1.5w)

Ae = UAg

U = 0.75 (angles with 2 bolts in line)

U = 0.75 (flat bars, 1.5w > L ≥ w) U = 0.85 (angles)

LRFD

Yielding:

φTn = φy Ag Fy = 0.9 Ag Fy

Fracture:

φTn = φf Ae Fu = 0.75 Ae Fu

ASD

Block shear rupture (bolted tension members):

Yielding:

Ta = Ag Ft = Ag (0.6 Fy)

Fracture:

Ta = Ae Ft = Ae (0.5 Fu)

Agt =gross tension area Block shear rupture (bolted tension members):

Agv =gross shear area Ant =net tension area

Ta = (0.30 Fu) Anv + (0.5 Fu) Ant

Anv=net shear area

Ant = net tension area

When FuAnt ≥ 0.6 FuAnv:

Anv = net shear area

0.75 [0.6 Fy Agv + Fu Ant]

φRn = smaller

0.75 [0.6 Fu Anv + Fu Ant]

When FuAnt < 0.6 FuAnv: 0.75 [0.6 Fu Anv + Fy Agt]

φRn = smaller

0

0.75 [0.6 Fu Anv + Fu Ant]

121

CIVIL ENGINEERING (continued)

BEAMS: homogeneous beams, flexure about x-axis Flexure – local buckling: bf

No local buckling if section is compact:

2t f

≤

65 Fy

For rolled sections, use tabulated values of

where:

h 640 ≤ tw Fy

and

bf

and

2t f

h tw

For built-up sections, h is clear distance between flanges For Fy ≤ 50 ksi, all rolled shapes except W6 × 19 are compact.

Flexure – lateral-torsional buckling: Lb = unbraced length ASD–compact rolled shapes

LRFD–compact rolled shapes

300 ry

Lp =

Lc =

Fy Zx Table

ry X 1

Lr =

1 +

FL

1 +

76 b f Fy

M1 is smaller end moment M1 /M2 is positive for reverse curvature

EGJA 2

Ma = S Fb

W-Shapes Dimensions and Properties Table

C ⎛ S ⎞2 = 4 w ⎜ x⎟ I y ⎝ GJ ⎠

X2 φ

π Sx

Lb ≤ Lc: Fb = 0.66 Fy Lb > Lc:

= 0.90

φMp = φ Fy Zx φMr = φ FL Sx Cb =

2.5 M max

L b ≤ L p:

Fb

⎡2 Fy ( Lb / rT )2 ⎤ ⎥ ≤ 0.6 Fy ⎢ = − 1,530,000 Cb ⎥ ⎢⎣ 3 ⎦

Fb

=

Fb

=

Zx Table 12.5 M max + 3M A + 4M B + 3MC

φMn = φMp

Lp < Lb ≤ Lr:

⎡ ⎛ Lb − L p ⎞⎤ ⎟⎥ φMn = Cb ⎢φM p − ( φM p − φM r ) ⎜ ⎜ Lr − L p ⎟⎥ ⎢⎣ ⎝ ⎠⎦

For:

170 ,000 Cb

For:

See Zx Table for BF

(F1-7)

12 ,000 Cb ≤ 0.6 Fy Lb d / A f

(F1-8)

102,000 Cb L < b ≤ Fy rT

Lb > rT

510,000 Cb : Fy

510,000 Cb : Fy

Use larger of (F1-7) and (F1-8)

Lb > Lr : φC b S x X 1 2 X 12 X 2 ≤ φMp 1+ Lb /r y 2 Lb /r y 2

(

See Allowable Moments in Beams curve

)

See Beam Design Moments curve

122

(F1-6)

≤ 0.6 Fy

( Lb / rT )2

Use larger of (F1-6) and (F1-8)

= Cb [φMp − BF (Lb − Lp)] ≤ φMp

φM n =

20,000 use smaller (d / A f ) Fy

Cb = 1.75 + 1.05(M1 /M2) + 0.3(M1 /M2)2 ≤ 2.3

X 2 FL2

where: FL = Fy – 10 ksi

X1 =

or

CIVIL ENGINEERING (continued)

Shear – unstiffened beams LRFD – E = 29,000 ksi φ = 0.90

Aw = d tw

h 417 ≤ tw Fy

φVn = φ (0.6 Fy) Aw

417 Fy


1.5:

⎡ 0.877 ⎤ φFcr = φ ⎢ 2 ⎥ Fy ⎢⎣ λc ⎥⎦

2 π2 E Fy

Fa =

See Table 3-50: Design Stress for Compression Members (Fy = 50 ksi, φ = 0.85)

⎡ ( KL/r ) 2 ⎤ ⎢1 − ⎥ Fy 2 Cc 2 ⎦⎥ ⎣⎢ 5 3 ( KL/r ) (KL / r ) 3 + − 3 8 Cc 8 Cc 3

⎛ KL ⎞ When ⎜ > Cc: ⎟ ⎝ r ⎠ max

Fa =

12 π 2 E 23 ( KL / r ) 2

See Table C-50: Allowable Stress for Compression Members (Fy = 50 ksi)

123

CIVIL ENGINEERING (continued)

BEAM-COLUMNS:

Sidesway prevented, x-axis bending, transverse loading between supports (no moments at ends), ends unrestrained against rotation in the plane of bending LRFD

ASD

Pu ≥ 0.2 : φ Pn

Pu 8 Mu + ≤ 1.0 φ Pn 9 φ M n

Pu < 0.2 : φ Pn

Pu Mu + ≤ 1 .0 2 φ Pn φMn

where: Mu = B1 Mnt B1 =

fa Cm f b ≤ 1.0 + Fa ⎛ fa ⎞ ⎟ Fb ⎜⎜ 1 − Fe′ ⎟⎠ ⎝

fa ≤ 0.15 : Fa

fa f + b ≤ 1 .0 Fa Fb

where:

Cm ≥ 1.0 Pu 1− Pex

Cm = 1.0

fa > 0.15 : Fa

Cm = 1.0 Fe′ =

for conditions stated above

for conditions stated above

12 π 2 E 23 ( KLx /rx ) 2

x-axis bending

⎛ π2 E I x ⎞ ⎟ x-axis bending Pex = ⎜ ⎜ ( KL ) 2 ⎟ x ⎠ ⎝

BOLTED CONNECTIONS: A325 bolts db = nominal bolt diameter Ab = nominal bolt area s = spacing between centers of bolt holes in direction of force Le = distance between center of bolt hole and edge of member in direction of force t = member thickness Dh = bolt hole diameter = db +

1

/16" [standard holes]

Bolt tension and shear strengths: ASD

LRFD Design strength (kips / bolt): Tension:

Design strength ( kips / bolt ): Tension: Rt = Ft Ab Shear: Rv = Fv Ab Design resistance to slip at service loads (kips / bolt): Rv

φRt = φ Ft Ab

Shear: φRv = φ Fv Ab Design resistance to slip at factored loads ( kips / bolt ): φRn Bolt size Bolt strength

Bolt size

3/4"

7/8"

1"

φRt

29.8

40.6

53.0

φRv ( A325-N )

15.9

21.6

φRn (A325-SC )

10.4

14.5

Bolt strength

3/4"

7/8"

1"

Rt

19.4

26.5

34.6

28.3

Rv ( A325-N )

9.3

12.6

16.5

19.0

Rv ( A325-SC )

6.63

9.02

11.8

φRv and φRn values are single shear

Rv values are single shear

124

CIVIL ENGINEERING (continued)

Bearing strength LRFD Design strength (kips/bolt/inch thickness):

ASD Design strength (kips/bolt/inch thickness):

φrn = φ 1.2 Lc Fu ≤ φ 2.4 db Fu φ

When s ≥ 3 db and Le ≥ 1.5 db

= 0.75

rb = 1.2 Fu db

Lc = clear distance between edge of hole and edge of adjacent hole, or edge of

When Le < 1.5 db : rb =

member, in direction of force Lc = s – Dh

When s < 3 db :

Dh Lc = Le – 2

d ⎞ ⎛ ⎜⎜ s − b ⎟⎟ Fu 2 ⎠ rb = ⎝ 2

Design bearing strength (kips/bolt/inch thickness) for various bolt spacings, s, and end distances, Le: Bearing strength φrn (k/bolt/in)

Le Fu 2

Design bearing strength (kips/bolt/inch thickness) for various bolt spacings, s, and end distances, Le:

Bolt size 3/4"

7/8"

1"

Fu = 58 ksi

62.0

72.9

83.7

69.5

81.7

93.8

rb(k/bolt/in) s ≥ 3 db

s = 3" Fu = 58 ksi

78.3

91.3

101

Fu = 65 ksi

87.7

102

113

Fu = 58 ksi Fu = 65 ksi

44.0

40.8

37.5

Fu = 65 ksi

49.4

45.7

42.0

Fu = 58 ksi Fu = 65 ksi Fu = 58 ksi Fu = 65 ksi

Le = 2" Fu = 58 ksi

78.3

79.9

76.7

Fu = 65 ksi

87.7

89.6

85.9

3/4"

7/8"

1"

and Le ≥ 1.5 db 52.2 58.5

60.9 68.3

69.6 78.0

s = 2 2/3 db (minimum permitted)

Le = 1 1/4" Fu = 58 ksi

Bolt size

Bearing strength

s = 2 2/3 db ( minimum permitted ) Fu = 65 ksi

≤ 1.2 Fu db

The bearing resistance of the connection shall be taken as the sum of the bearing resistances of the individual bolts.

125

47.1 52.8

55.0 61.6

62.8 70.4

Le = 1 1/4" 36.3 [all bolt sizes] 40.6 [all bolt sizes]

CIVIL ENGINEERING (continued)

Area Depth Web Shape

A 2

d

tw

Flange bf

tf

Compact

X1

section

X2 x 10

6

rT

d/Af

**

**

Axis X-X I

S 4

r 3

Axis Y-Y Z 3

I

r 4

in.

in.

in.

in.

in.

bf/2tf

h/tw

ksi

1/ksi

in.

1/in.

in.

in.

in.

in.

in.

in.

W24 × 103

30.3

24.5

0.55

9.00

0.98

4.59

39.2

2390

5310

2.33

2.78

3000

245

9.96

280

119

1.99

W24 × 94

27.7

24.3

0.52

9.07

0.88

5.18

41.9

2180

7800

2.33

3.06

2700

222

9.87

254

109

1.98

W24 × 84

24.7

24.1

0.47

9.02

0.77

5.86

45.9

1950

12200

2.31

3.47

2370

196

9.79

224

94.4

1.95

W24 × 76

22.4

23.9

0.44

8.99

0.68

6.61

49.0

1760

18600

2.29

3.91

2100

176

9.69

200

82.5

1.92

W24 × 68

20.1

23.7

0.42

8.97

0.59

7.66

52.0

1590

29000

2.26

4.52

1830

154

9.55

177

70.4

1.87

W24 × 62

18.3

23.7

0.43

7.04

0.59

5.97

49.7

1730

23800

1.71

5.72

1560

132

9.24

154

34.5

1.37

W24 × 55

16.3

23.6

0.40

7.01

0.51

6.94

54.1

1570

36500

1.68

6.66

1360

115

9.13

135

29.1

1.34

W21 × 93

27.3

21.6

0.58

8.42

0.93

4.53

32.3

2680

3460

2.17

2.76

2070

192

8.70

221

92.9

1.84

W21 × 83

24.3

21.4

0.52

8.36

0.84

5.00

36.4

2400

5250

2.15

3.07

1830

171

8.67

196

81.4

1.83

W21 × 73

21.5

21.2

0.46

8.30

0.74

5.60

41.2

2140

8380

2.13

3.46

1600

151

8.64

172

70.6

1.81

W21 × 68

20.0

21.1

0.43

8.27

0.69

6.04

43.6

2000

10900

2.12

3.73

1480

140

8.60

160

64.7

1.80

W21 × 62

18.3

21.0

0.40

8.24

0.62

6.70

46.9

1820

15900

2.10

4.14

1330

127

8.54

144

57.5

1.77

*

W21 × 55

16.2

20.8

0.38

8.22

0.52

7.87

50.0

1630

25800

---

---

1140

110

8.40

126

48.4

1.73

*

W21 × 48

14.1

20.6

0.35

8.14

0.43

9.47

53.6

1450

43600

---

---

959

93.0

8.24

107

38.7

1.66

W21 × 57

16.7

21.1

0.41

6.56

0.65

5.04

46.3

1960

13100

1.64

4.94

1170

111

8.36

129

30.6

1.35

W21 × 50

14.7

20.8

0.38

6.53

0.54

6.10

49.4

1730

22600

1.60

5.96

984

94.5

8.18

110

24.9

1.30

W21 × 44

13.0

20.7

0.35

6.50

0.45

7.22

53.6

1550

36600

1.57

7.06

843

81.6

8.06

95.4

20.7

1.26

* LRFD Manual only

** AISC ASD Manual, 9th Edition

126

CIVIL ENGINEERING (continued)

Table 1-1: W-Shapes Dimensions and Properties (continued)

Area Depth Web Shape

A

Flange

Compact

X2

rT

d/Af

Axis X-X

**

**

I

ksi

1/ksi

in.

1/in.

in.

33.4 37.8 32.4 35.7 38.7 41.1 45.2 44.6 50.9 53.5

2460 2180 2690 2470 2290 2110 1920 2060 1810 1590

4060 6520 3290 4540 6080 8540 12400 10100 17200 30800

2.97 2.95 1.98 1.97 1.96 1.95 1.94 1.54 1.52 1.49

2.15 2.43 2.99 3.22 3.47 3.82 4.21 4.93 5.67 6.94

5.92 6.77 7.70 4.98 5.61 6.23 6.93 8.12 6.28 7.97

25.9 29.9 34.4 33.0 37.4 41.1 46.5 48.1 51.6 56.8

3160 2770 2440 2650 2340 2120 1890 1700 1740 1480

1460 2460 4040 3400 5530 8280 12700 20400 19900 40300

2.79 2.77 2.75 1.86 1.84 1.83 1.82 1.79 1.39 1.36

0.94 0.86 0.78 0.71 0.86 0.79 0.72 0.65 0.66 0.60

7.80 8.49 9.34 10.2 5.92 6.41 6.97 7.75 6.11 6.75

19.3 21.7 23.5 25.9 22.4 25.4 27.5 30.4 30.9 33.6

3830 3490 3190 2900 3590 3280 3020 2720 2830 2580

601 853 1220 1750 849 1200 1660 2470 2250 3250

0.99 0.90 0.81 0.74 0.67 0.61 0.64 0.58 0.64 0.58 0.52

6.17 6.76 7.48 8.22 8.99 9.92 7.82 8.69 6.31 7.00 7.77

15.9 17.7 18.9 20.7 22.6 24.9 27.0 28.1 26.8 29.6 33.6

4660 4250 3880 3530 3230 2940 3070 2820 3120 2820 2530

285 407 586 839 1180 1720 1470 2100 1500 2210 3360

tw

bf

tf

in.

in.

in.

in.

in.

bf/2tf

h/tw

W18 × 86 W18 × 76 W18 × 71 W18 × 65 W18 × 60 W18 × 55 W18 × 50 W18 × 46 W18 × 40 W18 × 35

25.3 22.3 20.8 19.1 17.6 16.2 14.7 13.5 11.8 10.3

18.4 18.2 18.5 18.4 18.2 18.1 18.0 18.1 17.9 17.7

0.48 0.43 0.50 0.45 0.42 0.39 0.36 0.36 0.32 0.30

11.1 11.0 7.64 7.59 7.56 7.53 7.50 6.06 6.02 6.00

0.77 0.68 0.81 0.75 0.70 0.63 0.57 0.61 0.53 0.43

7.20 8.11 4.71 5.06 5.44 5.98 6.57 5.01 5.73 7.06

W16 × 89 W16 × 77 W16 × 67 W16 × 57 W16 × 50 W16 × 45 W16 × 40 W16 × 36 W16 × 31 W16 × 26

26.4 22.9 20.0 16.8 14.7 13.3 11.8 10.6 9.1 7.7

16.8 16.5 16.3 16.4 16.3 16.1 16.0 15.9 15.9 15.7

0.53 0.46 0.40 0.43 0.38 0.35 0.31 0.30 0.28 0.25

10.4 10.3 10.2 7.12 7.07 7.04 7.00 6.99 5.53 5.50

0.88 0.76 0.67 0.72 0.63 0.57 0.51 0.43 0.44 0.35

W14 × 120 W14 × 109 W14 × 99 W14 × 90 W14 × 82 W14 × 74 W14 × 68 W14 × 61 W14 × 53 W14 × 48

35.3 32.0 29.1 26.5 24.0 21.8 20.0 17.9 15.6 14.1

14.5 14.3 14.2 14.0 14.3 14.2 14.0 13.9 13.9 13.8

0.59 0.53 0.49 0.44 0.51 0.45 0.42 0.38 0.37 0.34

14.7 14.6 14.6 14.5 10.1 10.1 10.0 9.99 8.06 8.03

W12 × 106 W12 × 96 W12 × 87 W12 × 79 W12 × 72 W12 × 65 W12 × 58 W12 × 53 W12 × 50 W12 × 45 W12 × 40

31.2 28.2 25.6 23.2 21.1 19.1 17.0 15.6 14.6 13.1 11.7

12.9 12.7 12.5 12.4 12.3 12.1 12.2 12.1 12.2 12.1 11.9

0.61 0.55 0.52 0.47 0.43 0.39 0.36 0.35 0.37 0.34 0.30

12.2 12.2 12.1 12.1 12.0 12.0 10.0 9.99 8.08 8.05 8.01

2

X1

6

d

section

x 10

S

Axis Y-Y

r

Z

in.

in.

in.

in.

in.

1530 1330 1170 1070 984 890 800 712 612 510

166 146 127 117 108 98.3 88.9 78.8 68.4 57.6

7.77 7.73 7.50 7.49 7.47 7.41 7.38 7.25 7.21 7.04

186 163 146 133 123 112 101 90.7 78.4 66.5

175 152 60.3 54.8 50.1 44.9 40.1 22.5 19.1 15.3

2.63 2.61 1.70 1.69 1.68 1.67 1.65 1.29 1.27 1.22

1.85 2.11 2.40 3.23 3.65 4.06 4.53 5.28 6.53 8.27

1310 1120 970 758 659 586 518 448 375 301

157 136 119 92.2 81.0 72.7 64.7 56.5 47.2 38.4

7.05 7.00 6.97 6.72 6.68 6.65 6.63 6.51 6.41 6.26

177 152 132 105 92.0 82.3 73.0 64.0 54.0 44.2

163 138 119 43.1 37.2 32.8 28.9 24.5 12.4 9.59

2.48 2.46 2.44 1.60 1.59 1.57 1.57 1.52 1.17 1.12

4.04 4.02 4.00 3.99 2.74 2.72 2.71 2.70 2.15 2.13

1.05 1.14 1.25 1.36 1.65 1.79 1.94 2.15 2.62 2.89

1380 1240 1110 999 881 795 722 640 541 484

190 173 157 143 123 112 103 92.1 77.8 70.2

6.24 6.22 6.17 6.14 6.05 6.04 6.01 5.98 5.89 5.85

212 192 173 157 139 126 115 102 87.1 78.4

495 447 402 362 148 134 121 107 57.7 51.4

3.74 3.73 3.71 3.70 2.48 2.48 2.46 2.45 1.92 1.91

3.36 3.34 3.32 3.31 3.29 3.28 2.72 2.71 2.17 2.15 2.14

1.07 1.16 1.28 1.39 1.52 1.67 1.90 2.10 2.36 2.61 2.90

933 833 740 662 597 533 475 425 391 348 307

145 131 118 107 97.4 87.9 78.0 70.6 64.2 57.7 51.5

5.47 5.44 5.38 5.34 5.31 5.28 5.28 5.23 5.18 5.15 5.13

164 147 132 119 108 96.8 86.4 77.9 71.9 64.2 57.0

301 270 241 216 195 174 107 95.8 56.3 50.0 44.1

3.11 3.09 3.07 3.05 3.04 3.02 2.51 2.48 1.96 1.95 1.94

4

3

3

** AISC ASD Manual, 9th Edition

127

I

r 4

CIVIL ENGINEERING (continued)

Table 5-3 W-Shapes Selection by Zx

Fy = 50 ksi φb = 0.9 φv = 0.9

Zx

X-X AXIS

Shape

Zx in.3

Ix in.4

φbMp kip-ft

φbMr kip-ft

Lp ft

Lr ft

BF kips

φvVn kips

W 24 × 55 W 18 × 65 W 12 × 87 W 16 × 67 W 10 × 100 W 21 × 57

135 133 132 131 130 129

1360 1070 740 963 623 1170

506 499 495 491 488 484

345 351 354 354 336 333

4.73 5.97 10.8 8.65 9.36 4.77

12.9 17.1 38.4 23.8 50.8 13.2

19.8 13.3 5.13 9.04 3.66 17.8

252 224 174 174 204 231

W 21 × 55 W 14 × 74 W 18 × 60 W 12 × 79 W 14 × 68 W 10 × 88

126 126 123 119 115 113

1140 796 984 662 722 534

473 473 461 446 431 424

330 336 324 321 309 296

6.11 8.76 5.93 10.8 8.69 9.29

16.1 27.9 16.6 35.7 26.4 45.1

14.3 7.12 12.9 5.03 6.91 3.58

211 173 204 157 157 176

W 18 × 55

112

890

420

295

5.90

16.1

12.2

191

W 21 × 50 W 12 × 72

111 108

989 597

416 405

285 292

4.59 10.7

12.5 33.6

16.5 4.93

213 143

W 21 × 48 W 16 × 57 W 14 × 61 W 18 × 50 W 10 × 77 W 12 × 65

107 105 102 101 97.6 96.8

959 758 640 800 455 533

401 394 383 379 366 363

279 277 277 267 258 264

6.09 5.65 8.65 5.83 9.18 11.9

15.4 16.6 25.0 15.6 39.9 31.7

13.2 10.7 6.50 11.5 3.53 5.01

195 190 141 173 152 127

W 21 × 44 W 16 × 50 W 18 × 46 W 14 × 53 W 12 × 58 W 10 × 68 W 16 × 45

95.8 92.0 90.7 87.1 86.4 85.3 82.3

847 659 712 541 475 394 586

359 345 340 327 324 320 309

246 243 236 233 234 227 218

4.45 5.62 4.56 6.78 8.87 9.15 5.55

12.0 15.7 12.6 20.1 27.0 36.0 15.1

15.0 10.1 12.9 7.01 4.97 3.45 9.45

196 167 176 139 119 132 150

W 18 × 40 W 14 × 48 W 12 × 53 W 10 × 60

78.4 78.4 77.9 74.6

612 485 425 341

294 294 292 280

205 211 212 200

4.49 6.75 8.76 9.08

12.0 19.2 25.6 32.6

11.7 6.70 4.78 3.39

152 127 113 116

W 16 × 40 W 12 × 50 W 14 × 43 W 10 × 54

73.0 71.9 69.6 66.6

518 391 428 303

274 270 261 250

194 193 188 180

5.55 6.92 6.68 9.04

14.7 21.5 18.2 30.2

8.71 5.30 6.31 3.30

132 122 113 101

W 18 × 35 W 12 × 45 W 16 × 36 W 14 × 38 W 10 × 49 W 12 × 40 W 10 × 45

66.5 64.2 64.0 61.1 60.4 57.0 54.9

510 348 448 383 272 307 248

249 241 240 229 227 214 206

173 173 170 163 164 155 147

4.31 6.89 5.37 5.47 8.97 68.5 7.10

11.5 20.3 14.1 14.9 28.3 19.2 24.1

10.7 5.06 8.11 7.05 3.24 4.79 3.44

143 109 127 118 91.6 94.8 95.4

W 14 × 34

54.2

337

203

145

5.40

14.3

6.58

108

128

CIVIL ENGINEERING (continued)

129

CIVIL ENGINEERING (continued)

Table C – C.2.1. K VALUES FOR COLUMNS ♦

Theoretical K value

0.5

0.7

1.0

1.0

2.0

2.0

Recommended design value when ideal conditions are approximated

0.65

0.80

1.2

1.0

2.10

2.0

Figure C – C.2.2. ALIGNMENT CHART FOR EFFECTIVE LENGTH OF COLUMNS IN CONTINUOUS FRAMES ♦

GA

K

GB

50.0 10.0 5.0

1.0

50.0 10.0 5.0

3.0

0.9

3.0

GA

0.8 0.7 0.6

1.0

0.7

0.8 0.7 0.6

10.0 9.0 8.0 7.0 6.0 5.0

4.0

2.0

4.0

0.4

0.4

3.0

0.3

0.3

2.0

3.0 2.0 1.5

0.2 1.0

1.0 0.1

0.1

0

20.0

3.0

0.5

0.6

100.0 50.0 30.0

10.0 9.0 8.0 7.0 6.0 5.0

0.5

0.2

5.0 4.0

20.0

0.8 1.0

GB 20.0 10.0

100.0 50.0 30.0

2.0

2.0

K

0.5

0

1.0

0

SIDEWAY INHIBITED

0

SIDEWAY UNINHIBITED

The subscripts A and B refer to the joints at the two ends of the column section being considered. G is defined as Σ (I c /Lc ) G= Σ (I g /L g ) in which Σ indicates a summation of all members rigidly connected to that joint and lying on the plane in which buckling of the column is being considered. Ic is the moment of inertia and Lc the unsupported length of a column section, and Ig is the moment of inertia and Lg the unsupported length of a girder or other restraining member. Ic and Ig are taken about axes perpendicular to the plane of buckling being considered. For column ends supported by but not rigidly connected to a footing or foundation, G is theoretically infinity, but, unless actually designed as a true friction-free pin, may be taken as "10" for practical designs. If the column end is rigidly attached to a properly designed footing, G may be taken as 1.0. Smaller values may be used if justified by analysis. ♦ Manual of Steel Construction: Allowable Stress Design, American Institute of Steel Construction, 9th ed., 1989.

130

CIVIL ENGINEERING (continued)

Design Stress, φc Fcr, for Compression Members of 50 ksi Specified Yield Stress Steel, φc = 0.85 Kl r

φcFcr

φcFcr

ksi

Kl r

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

φcFcr

ksi

Kl r

42.50 42.49 42.47 42.45 42.42 42.39 42.35 42.30 42.25 42.19 42.13 42.05 41.98 41.90 41.81 41.71 41.61 41.51 41.39 41.28 41.15 41.02 40.89 40.75 40.60 40.45 40.29 40.13 39.97 39.79 39.62 39.43 39.25 39.06 38.86 38.66 38.45 38.24 38.03 37.81

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

φcFcr

ksi

Kl r

37.59 37.36 37.13 36.89 36.65 36.41 36.16 35.91 35.66 35.40 35.14 34.88 34.61 34.34 34.07 33.79 33.51 33.23 32.95 32.67 32.38 32.09 31.80 31.50 31.21 30.91 30.61 30.31 30.01 29.70 29.40 20.09 28.79 28.48 28.17 27.86 27.55 27.24 26.93 26.62

81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

26.31 26.00 25.68 25.37 25.06 24.75 24.44 24.13 23.82 23.51 23.20 22.89 22.58 22.28 21.97 21.67 21.36 21.06 20.76 20.46 20.16 19.86 19.57 19.28 18.98 18.69 18.40 18.12 17.83 17.55 17.27 16.99 16.71 16.42 16.13 15.86 15.59 15.32 15.07 14.82

121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160

131

φcFcr

ksi

Kl r

14.57 14.33 14.10 13.88 13.66 13.44 13.23 13.02 12.82 12.62 12.43 12.25 12.06 11.88 11.71 11.54 11.37 11.20 11.04 10.89 10.73 10.58 10.43 10.29 10.15 10.01 9.87 9.74 9.61 9.48 9.36 9.23 9.11 9.00 8.88 8.77 8.66 8.55 8.44 8.33

161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200

8.23 8.13 8.03 7.93 7.84 7.74 7.65 7.56 7.47 7.38 7.30 7.21 7.13 7.05 6.97 6.89 6.81 6.73 6.66 6.59 6.51 6.44 6.37 6.30 6.23 6.17 6.10 6.04 5.97 5.91 5.85 5.79 5.73 5.67 5.61 5.55 5.50 5.44 5.39 5.33

ksi

CIVIL ENGINEERING (continued)

ALLOWABLE STRESS DESIGN SELECTION TABLE

Sx

For shapes used as beams Fy = 50 ksi Lc

Lu

MR

Fy = 36 ksi

Sx

SHAPE 3

Lc

Lu

MR

Ft

Ft

Kip-ft

Ft

Ft

Kip-ft

In.

5.0 9.0 5.9 6.8 10.8 9.0

6.3 18.6 6.7 9.6 24.0 17.2

314 308 305 297 294 283

114 112 111 108 107 103

W 24 X 55 W 14 x 74 W 21 x 57 W 18 x 60 W 12 x 79 W 14 x 68

7.0 10.6 6.9 8.0 12.8 10.6

7.5 25.9 9.4 13.3 33.3 23.9

226 222 220 214 212 204

6.7 10.8

8.7 21.9

270 268

98.3 97.4

W 18 X 55 W 12 x 72

7.9 12.7

12.1 30.5

195 193

5.6 6.4 9.0

6.0 10.3 15.5

260 254 254

94.5 92.2 92.2

W 21 X 50 W 16 x 57 W 14 x 61

6.9 7.5 10.6

7.8 14.3 21.5

187 183 183

6.7 10.7

7.9 20.0

244 238

88.9 87.9

W 18 X 50 W 12 x 65

7.9 12.7

11.0 27.7

176 174

4.7 6.3 5.4 9.0 7.2 6.3 9.0 7.2

5.9 9.1 6.8 17.5 12.7 8.2 15.9 11.5

224 223 217 215 214 200 194 193

81.6 81.0 78.8 78.0 77.8 72.7 70.6 70.3

W 21 X 44 W 16 x 50 W 18 x 46 W 12 x 58 W 14 x 53 W 16 x 45 W 12 x 53 W 14 x 48

6.6 7.5 6.4 10.6 8.5 7.4 10.6 8.5

7.0 12.7 9.4 24.4 17.7 11.4 22.0 16.0

162 160 156 154 154 144 140 139

5.4 9.0

5.9 22.4

188 183

68.4 66.7

W 18 X 40 W 10 x 60

6.3 10.6

8.2 31.1

135 132

6.3 7.2 7.2 9.0 7.2

7.4 14.1 10.4 20.3 12.8

178 178 172 165 160

64.7 64.7 62.7 60.0 58.1

W 16 X 40 W 12 x 50 W 14 x 43 W 10 x 54 W 12 x 45

7.4 8.5 8.4 10.6 8.5

10.2 19.6 14.4 28.2 17.7

128 128 124 119 115

4.8 6.3 6.1 9.0 7.2 7.2

5.6 6.7 8.3 18.7 11.5 16.4

158 115 150 150 143 135

57.6 56.5 54.6 54.6 51.9 49.1

W 18 X 35 W 16 x 36 W 14 x 38 W 10 x 49 W 12 x 40 W 10 x 45

6.3 7.4 7.1 10.6 8.4 8.5

6.7 8.8 11.5 26.0 16.0 22.8

114 112 108 108 103 97

6.0

7.3

134

48.6

W 14 X 34

7.1

10.2

96

4.9 5.9 7.2

5.2 9.1 14.2

130 125 116

47.2 45.6 42.1

W 16 X 31 W 12 x 35 W 10 x 39

5.8 6.9 8.4

7.1 12.6 19.8

93 90 83

6.0

6.5

116

42.0

W 14 X 30

7.1

8.7

83

5.8

7.8

106

38.6

W 12 X 30

6.9

10.8

76

4.0

5.1

106

38.4

W 16 x 26

5.6

6.0

76

132

CIVIL ENGINEERING (continued)

133

CIVIL ENGINEERING (continued)

ASD Table C–50. Allowable Stress for Compression Members of 50-ksi Specified Yield Stress Steela,b Fa

Fa (ksi)

Kl r

41 42 43 44 45

25.69 25.55 25.40 25.26 25.11

29.58 29.50 29.42 29.34 29.26

46 47 48 49 50

11 12 13 14 15

29.17 29.08 28.99 28.90 28.80

16 17 18 19 20

Fa (ksi)

Kl r

81 82 83 84 85

18.81 18.61 18.41 18.20 17.99

24.96 24.81 24.66 24.51 24.35

86 87 88 89 90

51 52 53 54 55

24.19 24.04 23.88 23.72 23.55

28.71 28.61 28.51 28.40 28.30

56 57 58 59 60

21 22 23 24 25

28.19 28.08 27.97 27.86 27.75

26 27 28 29 30 31 32 33 34 35

Kl r

Fa

Fa

(ksi)

Kl r

(ksi)

121 122 123 124 125

10.20 10.03 9.87 9.71 9.56

161 162 163 164 165

5.76 5.69 5.62 5.55 5.49

17.79 17.58 17.37 17.15 16.94

126 127 128 129 130

9.41 9.26 9.11 8.97 8.84

166 167 168 169 170

5.42 5.35 5.29 5.23 5.17

91 92 93 94 95

16.72 16.50 16.29 16.06 15.84

131 132 133 134 135

8.70 8.57 8.44 8.32 8.19

171 172 173 174 175

5.11 5.05 4.99 4.93 4.88

23.39 23.22 23.06 22.89 22.72

96 97 98 99 100

15.62 15.39 15.17 14.94 14.71

136 137 138 139 140

8.07 7.96 7.84 7.73 7.62

176 177 178 179 180

4.82 4.77 4.71 4.66 4.61

61 62 63 64 65

22.55 22.37 22.20 22.02 21.85

101 102 103 104 105

14.47 14.24 14.00 13.77 13.53

141 142 143 144 145

7.51 7.41 7.30 7.20 7.10

181 182 183 184 185

4.56 4.51 4.46 4.41 4.36

27.63 27.52 27.40 27.28 27.15

66 67 68 69 70

21.67 21.49 21.31 21.12 20.94

106 107 108 109 110

13.29 13.04 12.80 12.57 12.34

146 147 148 149 150

7.01 6.91 6.82 6.73 6.64

186 187 188 189 190

4.32 4.27 4.23 4.18 4.14

27.03 26.90 26.77 26.64 26.51

71 72 73 74 75

20.75 20.56 20.38 20.10 19.99

111 112 113 114 115

12.12 11.90 11.69 11.49 11.29

151 152 153 154 155

6.55 6.46 6.38 6.30 6.22

191 192 193 194 195

4.09 4.05 4.01 3.97 3.93

(ksi)

Kl r

1 2 3 4 5

29.94 29.87 29.80 29.73 29.66

6 7 8 9 10

36 26.38 76 19.80 116 11.10 156 6.14 196 3.89 37 26.25 77 19.61 117 10.91 157 6.06 197 3.85 38 26.11 78 19.41 118 10.72 158 5.98 198 3.81 39 25.97 79 19.21 119 10.55 159 5.91 199 3.77 40 25.83 80 19.01 120 10.37 160 5.83 200 3.73 a When element width-to-thickness ratio exceeds noncompact section limits of Sect. B5.1, see Appendix B5. b

Values also applicable for steel of any yield stress ≥ 39 ksi.

Note: Cc = 107.0

134

CIVIL ENGINEERING (continued)

ENVIRONMENTAL ENGINEERING For information about environmental engineering refer to the ENVIRONMENTAL ENGINEERING section.

DARCY'S LAW Q = –KA(dh/dx), where Q = Discharge rate (ft3/s or m3/s), K = Hydraulic conductivity (ft/s or m/s), h = Hydraulic head (ft or m), and A = Cross-sectional area of flow (ft2 or m2). q = –K(dh/dx) q = specific discharge or Darcy velocity v = q/n = –K/n(dh/dx) v = average seepage velocity n = effective porosity

HYDROLOGY NRCS (SCS) Rainfall-Runoff Q=

(P − 0.2S )2 ,

P + 0.8S 1,000 − 10, S= CN 1,000 , CN = S + 10

P

=

precipitation (inches),

S

=

maximum basin retention (inches),

Q =

runoff (inches), and

CN =

curve number.

Rational Formula Q = CIA, where A = watershed area (acres), C = runoff coefficient, I = rainfall intensity (in/hr), and Q = peak discharge (cfs).

Unit hydrograph:

The direct runoff hydrograph that would result from one unit of effective rainfall occurring uniformly in space and time over a unit period of time.

Transmissivity, T,

is the product of hydraulic conductivity and thickness, b, of the aquifer (L2T –1).

Storativity or storage of an aquifer is the volume of water coefficient, S, taken into or released from storage per unit surface area per unit change in potentiometric (piezometric) head.

SEWAGE FLOW RATIO CURVES

Curve A2:

Curve B :

CurveG:

Population in Thousands (P)

135

P 0.2 5

14 4+ P

+1

18 + P 4+ P

CIVIL ENGINEERING (continued)

HYDRAULIC-ELEMENTS GRAPH FOR CIRCULAR SEWERS

For rectangular channels

Open-Channel Flow Specific Energy E =α E

=

Q =

13

⎛ q2 ⎞ yc = ⎜⎜ ⎟⎟ ⎝ g ⎠

αQ 2 V2 + y= + y , where 2g 2 gA 2

yc =

specific energy, discharge,

V

=

velocity,

y

=

depth of flow,

A

=

cross-sectional area of flow, and

α

=

kinetic energy correction factor, usually 1.0.

T

=

width of the water surface.

unit discharge = Q/B,

B

=

channel width, and

g

=

acceleration due to gravity.

=

yh =

where Q and A are as defined above, acceleration due to gravity, and

=

V

Q A = g T =

q

F=

3

g

critical depth,

Froude Number = ratio of inertial forces to gravity forces

Critical Depth = that depth in a channel at minimum specific energy 2

, where

136

V gy h

, where

velocity, and hydraulic depth = A/T

CIVIL ENGINEERING (continued)

Values of Hazen-Williams Coefficient C

Specific Energy Diagram

Pipe Material C Concrete (regardless of age) 130 Cast iron: New 130 5 yr old 120 20 yr old 100 Welded steel, new 120 Wood stave (regardless of age) 120 Vitrified clay 110 Riveted steel, new 110 Brick sewers 100 Asbestos-cement 140 Plastic 150 For additional fluids information, see the FLUID MECHANICS section.

y 1

1

αV 2 +y 2g Alternate depths: depths with the same specific energy. E=

Uniform flow: a flow condition where depth and velocity do not change along a channel.

Manning's Equation K Q = AR 2 3 S 1 2 n Q = discharge (m3/s or ft3/s), K = 1.486 for USCS units, 1.0 for SI units, A = cross-sectional area of flow (m2 or ft2), R = hydraulic radius = A/P (m or ft), P = wetted perimeter (m or ft), S = slope of hydraulic surface (m/m or ft/ft), and n = Manning’s roughness coefficient. Normal depth (uniform flow depth) Qn AR 2 3 = KS 1 2 Weir Formulas

TRANSPORTATION U.S. Customary Units

a = deceleration rate (ft/sec2) A = algebraic difference in grades (%) C = vertical clearance for overhead structure (overpass) located within 200 feet of the midpoint of the curve e

= superelevation (%)

f

= side friction factor

± G = percent grade divided by 100 (uphill grade"+")

h1 = height of driver's eyes above the roadway surface (ft)

Fully submerged with no side restrictions Q = CLH3/2 V-Notch Q = CH5/2, where Q = discharge (cfs or m3/s), C = 3.33 for submerged rectangular weir (USCS units), C = 1.84 for submerged rectangular weir (SI units), C = 2.54 for 90° V-notch weir (USCS units), C = 1.40 for 90° V-notch weir (SI units), L = weir length (ft or m), and H = head (depth of discharge over weir) ft or m.

h2 L Ls R S t V

= height of object above the roadway surface (ft) = length of curve (ft) = spiral transition length (ft) = radius of curve (ft) = stopping sight distance (ft) = driver reaction time (sec) = design speed (mph)

Stopping Sight Distance

Hazen-Williams Equation V = k1CR0.63S0.54, where C = roughness coefficient, k1 = 0.849 for SI units, and k1 = 1.318 for USCS units, R = hydraulic radius (ft or m), S = slope of energy grade line, = hf /L (ft/ft or m/m), and V = velocity (ft/s or m/s).

S =

137

V2 + 1.47Vt ⎛⎛ a ⎞ ⎞ ± G⎟ 30 ⎜ ⎜ ⎝ ⎝ 32.2 ⎟⎠ ⎠

CIVIL ENGINEERING (continued)

DENSITY k (veh/mi)

CAPACITY

DENSITY k (veh/mi)

CAPACITY

SPEED v (mph)

SPEED v (mph)

VOLUME q (veh/hr)

Transportation Models See INDUSTRIAL ENGINEERING for optimization models and methods, including queueing theory. Traffic Flow Relationships (q = kv)

VOLUME q (veh/hr)

Vertical Curves: Sight Distance Related to Curve Length

S ≤ L Crest Vertical Curve

L =

General equation:

For h1 = 3.50 ft and h2 = 2.0 ft :

L =

Sag Vertical Curve

AS2 100 ( 2h1 + 2h2 ) 2

L = 2S −

AS2 2,158

L = 2S −

L =

(based on standard headlight criteria)

S > L

L =

(based on riding comfort) Sag Vertical Curve

L =

(based on adequate sight distance under an overhead structure to see an object beyond a sag vertical curve)

AS2 h +h ⎞ ⎛ 800 ⎜ C − 1 2 ⎟ ⎝ 2 ⎠

(

h1 +

h2

)

2

A

2,158 A

⎛ 400 + 3.5 S ⎞ L = 2S − ⎜ ⎟⎠ ⎝ A

AS2 400 + 3.5 S

Sag Vertical Curve

200

AV 2 46.5 L = 2S −

h +h ⎞ 800 ⎛ C − 1 2⎟ ⎜ ⎝ A 2 ⎠

C = vertical clearance for overhead structure (overpass) located within 200 feet of the midpoint of the curve

Horizontal Curves 0.01e + f =

Side friction factor (based on superelevation)

Ls =

Spiral Transition Length

V2 15R

3.15V 3 RC

C = rate of increase of lateral acceleration [use 1 ft/sec3 unless otherwise stated] ⎡ ⎛ 28.65 S ⎞ ⎤ HSO = R ⎢1 − cos ⎜ ⎝ R ⎟⎠ ⎥⎦ ⎣

Sight Distance (to see around obstruction)

HSO = Horizontal sight line offset

138

CIVIL ENGINEERING (continued)

HORIZONTAL CURVE FORMULAS D

= Degree of Curve, Arc Definition

P.C. = Point of Curve (also called B.C.) P.T. = Point of Tangent (also called E.C.) P.I. = Point of Intersection = Intersection Angle (also called ∆)

I

Angle between two tangents L

= Length of Curve, from P.C. to P.T.

T

= Tangent Distance

E

= External Distance

R

= Radius

L.C. = Length of Long Chord M

= Length of Middle Ordinate

c

= Length of Sub-Chord

d

= Angle of Sub-Chord

R=

5729.58 D LATITUDES AND DEPARTURES

L.C. R= 2 sin ( I/2) T = R tan ( I/2) = L = RI

+ Latitude

L.C. 2 cos ( I/2)

π I = 100 180 D

- Departure

+ Departure

M = R [1 − cos (I/ 2 )]

R = cos ( I/ 2) E+R R−M = cos ( I/ 2) R

- Latitude

c = 2 R sin ( d/ 2)

⎡ ⎤ 1 − 1⎥ E=R⎢ ⎣ cos( I/2) ⎦

Deflection angle per 100 feet of arc length equals D

2

139

CIVIL ENGINEERING (continued)

VERTICAL CURVE FORMULAS

L

= Length of Curve (horizontal)

g2 = Grade of Forward Tangent

PVC = Point of Vertical Curvature

a = Parabola Constant

PVI = Point of Vertical Intersection

y = Tangent Offset

PVT = Point of Vertical Tangency

E = Tangent Offset at PVI

g1

= Grade of Back Tangent

r = Rate of Change of Grade

x

= Horizontal Distance from PVC to Point on Curve

xm = Horizontal Distance to Min/Max Elevation on Curve = −

Tangent Elevation = YPVC + g1x Curve Elevation

and

g1 g1 L = 2a g1 − g 2

= YPVI + g2 (x – L/2)

= YPVC + g1x + ax = YPVC + g1x + [(g2 – g1)/(2L)]x2

y = ax 2

2

a=

g 2 − g1 2L

⎛ L⎞ E=a ⎜ ⎟ ⎝ 2⎠

2

r=

g 2 _ g1 L

CONSTRUCTION Construction project scheduling and analysis questions may be based on either activity-on-node method or on activity-on-arrow method. CPM PRECEDENCE RELATIONSHIPS (ACTIVITY ON NODE) A

A

B

B

Start-to-start: start of B depends on the start of A

Finish-to-finish: finish of B depends on the finish of A

A

140

B

Finish-to-start: start of B depends on the finish of A

CIVIL ENGINEERING (continued)

HIGHWAY PAVEMENT DESIGN AASHTO Structural Number Equation SN = a1D1 + a2D2 +…+ anDn, where SN = structural number for the pavement ai

Gross Axle Load kN

lb

= layer coefficient and Di = thickness of layer (inches). Load Equivalency

Gross Axle Load

Factors Single

Tandem

Axles

Axles

kN

lb

Load Equivalency Factors Single

Tandem

Axles

Axles

4.45

1,000

0.00002

187.0

42,000

25.64

2.51

8.9

2,000

0.00018

195.7

44,000

31.00

3.00

17.8

4,000

0.00209

200.0

45,000

34.00

3.27

22.25

5,000

0.00500

204.5

46,000

37.24

3.55

26.7

6,000

0.01043

213.5

48,000

44.50

4.17

35.6

8,000

0.0343

222.4

50,000

52.88

4.86

44.5

10,000

0.0877

0.00688

231.3

52,000

5.63

53.4

12,000

0.189

0.0144

240.2

54,000

6.47

62.3

14,000

0.360

0.0270

244.6

55,000

6.93

66.7

15,000

0.478

0.0360

249.0

56,000

7.41

71.2

16,000

0.623

0.0472

258.0

58,000

8.45

80.0

18,000

1.000

0.0773

267.0

60,000

9.59

89.0

20,000

1.51

0.1206

275.8

62,000

10.84

97.8

22,000

2.18

0.180

284.5

64,000

12.22

106.8

24,000

3.03

0.260

289.0

65,000

12.96

111.2

25,000

3.53

0.308

293.5

66,000

13.73

115.6

26,000

4.09

0.364

302.5

68,000

15.38

124.5

28,000

5.39

0.495

311.5

70,000

17.19

133.5

30,000

6.97

0.658

320.0

72,000

19.16

142.3

32,000

8.88

0.857

329.0

74,000

21.32

151.2

34,000

11.18

1.095

333.5

75,000

22.47

155.7

35,000

12.50

1.23

338.0

76,000

23.66

160.0

36,000

13.93

1.38

347.0

78,000

26.22

169.0

38,000

17.20

1.70

356.0

80,000

28.99

178.0

40,000

21.08

2.08

Note: kN converted to lb are within 0.1 percent of lb shown.

141

CIVIL ENGINEERING (continued)

EARTHWORK FORMULAS Average End Area Formula, V = L(A1 + A2)/2 Prismoidal Formula, V = L (A1 + 4Am + A2)/6, where Am = area of mid-section Pyramid or Cone, V = h (Area of Base)/3 where L = distance between A1 and A2 AREA FORMULAS Area by Coordinates: Area = [XA (YB – YN) + XB (YC – YA) + XC (YD – YB) + ... + XN (YA – YN–1)] / 2 ⎛ h + hn ⎞ Trapezoidal Rule: Area = w ⎜ 1 + h2 + h3 + h4 + … + hn −1 ⎟ 2 ⎝ ⎠

w = common interval

⎡ ⎤ ⎛ n − 2 ⎞ ⎛ n −1 ⎞ Simpson's 1/3 Rule: Area = w ⎢h1 + 2⎜ ∑ hk ⎟ + 4⎜ ∑ hk ⎟ + hn ⎥ 3 ⎝ k =3,5 ,… ⎠ ⎝ k = 2 ,4 ,… ⎠ ⎣ ⎦

n must be odd number of measurements w = common interval

142