1. CIVIL ENGINEERING
GEOTECHNICAL Q = KH(Nf /Nd) (for flow nets, Q per unit width),
Definitions where
K = coefficient permeability,
c = cohesion
H = total hydraulic head (potential),
cc = coefficient of curvature or gradation
Nf = number of flow tubes, and
= (D30)2/[(D60)(D10)], where
Nd = number of potential drops.
D10, D30, D60 = particle diameter corresponding to 10%,
30%, and 60% finer on grain-size curve. γ = total unit weight of soil = W/V
cu = uniformity coefficient = D60 /D10 γd = dry unit weight of soil = Ws /V
e = void ratio = Vv /Vs, where = Gγw /(1 + e) = γ /(1 + w), where
Vv = volume of voids, and Gw = Se
Vs = volume of the solids. γs = unit weight of solid = Ws / Vs
K = coefficient of permeability = hydraulic conductivity n = porosity = Vv /V = e/(1 + e)
= Q/(iA) (from Darcy's equation), where τ = general shear strength = c + σtan φ, where
Q = discharge rate φ = angle of internal friction,
i = hydraulic gradient = dH/dx, σ = normal stress = P/A,
H = hydraulic head, P = force, and
A = cross-sectional area. A = area.
qu = unconfined compressive strength = 2c
Ka = coefficient of active earth pressure
w = water content (%) = (Ww /Ws) ×100, where
Ww = weight of water, and = tan2(45 – φ/2)
Ws = weight of solids. Kp = coefficient of passive earth pressure
= tan2(45 + φ/2)
Cc = compression index = ∆e/∆log p Pa = active resultant force = 0.5γH 2Ka, where
= (e1 – e2)/(log p2 – log p1), where H = height of wall.
e1 and e2 = void ratio, and
p1 and p2 = pressure. qult = bearing capacity equation
= cNc + γDf Nq + 0.5γBNγ , where
Dr = relative density (%) Nc, Nq, and Nγ = bearing capacity factors
= [(emax – e)/(emax – emin)] ×100 B = width of strip footing, and
= [(1/γmin – 1/γd) /(1/γmin – 1/γmax)] × 100, where Df = depth of footing below surface.
emax and emin = maximum and minimum void ratio, and
γmax and γmin = maximum and minimum unit dry weight. FS = factor of safety (slope stability)
cL + Wcosα tanφ
= , where
Gs = specific gravity = Ws /(Vsγw), where W sinα
γw = unit weight of water (62.4 lb/ft3 or 1,000 kg/m3). L = length of slip plane,
∆H = settlement = H [Cc /(1 + ei)] log [(pi + ∆p)/pi] α = slope of slip plane,
= H∆e/(1 + ei), where φ = angle of friction, and
H = thickness of soil layer W = total weight of soil above slip plane.
∆e = change in void ratio, and Cv = coefficient of consolidation = TH 2/t, where
p = pressure. T = time factor,
PI = plasticity index = LL – PL, where t = consolidation time.
LL = liquid limit, and Hdr = length of drainage path
PL = plasticity limit. n = number of drainage layers
Cc = compression index for normally consolidated clay
S = degree of saturation (%) = (Vw /Vv) × 100, where
Vw = volume of water, = 0.009 (LL – 10)
Vv = volume of voids. σ′ = effective stress = σ – u, where
σ = normal stress, and
u = pore water pressure.
93

2. CIVIL ENGINEERING (continued)
UNIFIED SOIL CLASSIFICATION SYSTEM (ASTM D-2487)
Group
Major Divisions Typical Names Laboratory Classification Criteria
Symbols
D 60
(More than half of coarse fraction is larger than No.
Depending on percentage of fines (fraction smaller than No. 200 sieve size), coarse-grained soils are
Clean gravels (Little or no
cu = greater than 4;
D10
Well-graded gravels, gravel-sand
GW
mixtures, little or no fines
(D ) 2
fines)
cc =
30
between 1 and 3
D 10 × D 60
Determine percentages of sand and gravel from grain-size curve.
4 sieve size)
(More than half of material is larger than No. 200 sieve size)
Gravels
Poorly-graded gravels, gravel-sand
GP Not meeting all gradiation requirements for GW
mixtures, little or no fines
d
Gravels with fines
GMa Above "A" line
amount of fines)
Silty gravels, gravel-sand-silt mixtures Atterberg limits below "A"
(Appreciable
u with PI between 4
line or PI less than 4
and 7 are
classified as follows:
Coarse-grained soils
borderline cases
5 to 12 percent: Borderline cases requiring dual symbolsb
Clayey gravels, gravel-sand-clay requiring use of
GC Atterberg limits above "A"
mixtures dual symbols
line with PI greater than 7
D 60
(More than half of coarse fraction is smaller
Clean sands (Little or no
cu = greater than 6;
More than 12 percent: GM, GC, SM, SC
D10
Well-graded sands, gravelly sands, little
Less than 5 percent: GW, GP, SW, SP
SW
or no fines
(D )2
fines)
than No. 4 sieve size)
cc =
30
between 1 and 3
D 10 × D 60
Sands
Poorly graded sand, gravelly sands, little
SP Not meeting all gradation requirements for SW
or no fines
d Limits plotting in
Atterberg limits below "A"
(Appreciable
SMa Silty sands, sand-silt mixtures hatched zone with
Sands with
amount of
u line or PI less than 4
fines)
PI between 4 and 7
fines
are borderline
Atterberg limits above "A" cases requiring use
SC Clayey sands, sand-clay mixtures
line with PI greater than 7 of dual symbols
Inorganic silts and very fine sands, rock
ML flour, silty or clayey fine sands, or
(Liquid limit less
Silts and clays
clayey silts with slight plasticity
(More than half material is smaller than No. 200 sieve)
than 50)
Inorganic clays of low to medium
CL plasticity, gravelly clays, sandy clays,
silty clays, lean clays
Organic silts and organic silty clays of
OL
low plasticity
Inorganic silts, micaceous or
Fine-grained soils
greater than 50)
MH diatomaceous fine sandy or silty soils,
Silts and clays
(Liquid limit
elastic silts
Inorganic clays of high plasticity, fat
CH
clays
Organic clays of medium to high
OH
plasticity, organic silts
Highly organic
Peat and other highly organic soils
soils
Pt
a
Division of GM and SM groups into subdivisions of d and u are for roads and airfields only. Subdivision is based on
Atterberg limits; suffix d used when LL is 28 or less and the PI is 6 or less; the suffix u used when LL is greater than 28.
b
Borderline classification, used for soils possessing characteristics of two groups, are designated by combinations of group
symbols. For example GW-GC, well-graded gravel-sand mixture with clay binder.
94

3. CIVIL ENGINEERING (continued)
STRUCTURAL ANALYSIS
Influence Lines α = coefficient of thermal expansion
An influence diagram shows the variation of a function L = member length
(reaction, shear, bending moment) as a single unit load Fp = member force due to external load
moves across the structure. An influence line is used to (1)
determine the position of load where a maximum quantity A = cross-sectional area of member
will occur and (2) determine the maximum value of the E = modulus of elasticity
quantity. ∆T = T–TO; T = final temperature, and TO = initial
Deflection of Trusses temperature
Principle of virtual work as applied to trusses
∆ = ΣfQδL Deflection of Frames
∆ = deflection at point of interest The principle of virtual work as applied to frames:
fQ = member force due to virtual unit load applied at ­ L mM ½
∆ = ¦ ®³O dx ¾
the point of interest ¯ EI ¿
δL = change in member length m = bending moment as a funtion of x due to virtual
unit load applied at the point of interest
= αL(∆T) for temperature
M = bending moment as a function of x due to external
= FpL/AE for external load
loads
BEAM FIXED-END MOMENT FORMULAS
Pab 2 Pa 2 b
FEM AB = FEM BA =
L2 L2
w o L2 w o L2
FEM AB = FEM BA =
12 12
w o L2 w o L2
FEM AB = FEM BA =
30 20
Live Load Reduction
The live load applied to a structure member can be reduced as the loaded area supported by the member is increased. A typical
reduction model (as used in ASCE 7 and in building codes) for a column supporting two or more floors is:
§ 15 ·
Lreduced = Lnominal ¨ 0.25 + ¸ ≥ 0.4 L Columns: kLL = 4
¨ k LL AT ¸ nominal
© ¹
Beams: kLL = 2
where Lnominal is the nominal live load (as given in a load standard or building code), AT is the floor tributary area(s) supported
by the member, and kLL is the ratio of the area of influence to the tributary area.
95

4. CIVIL ENGINEERING (continued)
REINFORCED CONCRETE DESIGN ACI 318-02
US Customary units ASTM STANDARD REINFORCING BARS
Definitions BAR SIZE DIAMETER, IN AREA, IN2 WEIGHT, LB/FT
a = depth of equivalent rectangular stress block, in
#3 0.375 0.11 0.376
Ag = gross area of column, in2
#4 0.500 0.20 0.668
As = area of tension reinforcement, in2 #5 0.625 0.31 1.043
As' = area of compression reinforcement, in2 #6 0.750 0.44 1.502
Ast = total area of longitudinal reinforcement, in2 #7 0.875 0.60 2.044
Av = area of shear reinforcment within a distance s, in #8 1.000 0.79 2.670
#9 1.128 1.00 3.400
b = width of compression face of member, in #10 1.270 1.27 4.303
be = effective compression flange width, in #11 1.410 1.56 5.313
bw = web width, in #14 1.693 2.25 7.650
#18 2.257 4.00 13.60
β1 = ratio of depth of rectangular stress block, a, to depth
to neutral axis, c
§ f c ' − 4,000 ·
= 0.85 ≥ 0.85 – 0.05 ¨
¨ ¸ ≥ 0.65
¸
© 1,000 ¹ LOAD FACTORS FOR REQUIRED STRENGTH
c = distance from extreme compression fiber to neutral U = 1.4 D
axis, in U = 1.2 D + 1.6 L
d = distance from extreme compression fiber to centroid
of nonprestressed tension reinforcement, in
SELECTED ACI MOMENT COEFFICIENTS
dt = distance from extreme tension fiber to extreme
Approximate moments in continuous beams of three or
tension steel, in
more spans, provided:
Ec = modulus of elasticity = 33 wc1.5 f c ' , psi 1. Span lengths approximately equal (length of
εt = net tensile strain in extreme tension steel at nominal longer adjacent span within 20% of shorter)
strength 2. Uniformly distributed load
fc' = compressive strength of concrete, psi 3. Live load not more than three times dead load
fy = yield strength of steel reinforcement, psi Mu = coefficient * wu * Ln2
hf = T-beam flange thickness, in wu = factored load per unit beam length
Mc = factored column moment, including slenderness Ln = clear span for positive moment; average
effect, in-lb adjacent clear spans for negative moment
Mn = nominal moment strength at section, in-lb
φMn = design moment strength at section, in-lb Column 1 1
+ +
Mu = factored moment at section, in-lb 14 16
Pn = nominal axial load strength at given eccentricity, lb
1 1 1 1
φPn = design axial load strength at given − − − −
eccentricity, lb 16 10 11 11
Pu = factored axial force at section, lb Ln
ρg = ratio of total reinforcement area to cross-sectional
area of column = Ast/Ag Spandrel
1 1
s = spacing of shear ties measured along longitudinal beam + +
14 16
axis of member, in
Vc = nominal shear strength provided by concrete, lb 1 1 1 1
− − − −
Vn = nominal shear strength at section, lb 24 10 11 11
φVn = design shear strength at section, lb
Vs = nominal shear strength provided by reinforcement, 1 1
lb Unrestrained + +
11 16
Vu = factored shear force at section, lb end
1 1 1
− − −
10 11 11
End span Interior span
96

5. CIVIL ENGINEERING (continued)
UNIFIED DESIGN PROVISIONS BEAMS − FLEXURE: φMN ≥ MU
Internal Forces and Strains For all beams
d' Comp.strain Net tensile strain: a = β1 c
Mu 0.003 ( dt − c ) 0.003 ( β1 dt − a )
ε's εt = =
A's C c C s' c c a
Pu d Design moment strength: φMn
dt
As where: φ = 0.9 [εt ≥ 0.005]
Ts
φ = 0.48 + 83εt [0.004 ≤ εt < 0.005]
Net tensile strain: εt Reinforcement limits:
AS, max εt = 0.004 @ Mn
Strain Conditions ­ ′
° 3 f c bw d 200 b d
w
0.003 0.003 0.003 AS ,min = larger ® or
f f
° y y
¯
c c As,min limits need not be applied if
A's c
As (provided ≥ 1.33 As (required)
dt
As
Singly-reinforced beams
0.85 f c ' β 1 b § 3 d t ·
As,max = ¨
¨ 7 ¸
¸
εt ≥ 0.005 0.005> εt >0.002 εt ≤ 0.002 fy © ¹
Tension- Transition Compression- As f y
a=
controlled section controlled 0.85 f c′ b
section: section:
c ≤ 0.375 dt a a
c ≥ 0.6 dt Mn = 0.85 fc' a b (d − ) = As fy (d − )
2 2
Doubly-reinforced beams
Balanced Strain: εt = εy Compression steel yields if:
0.003 0.85 β1 f c′ d' b § 87,000 ·
A s − A s' ≥ ¨ ¸
A's fy ¨ 87,000 + f y ¸
© ¹
dt If compression steel yields:
As 0.85 f c′ β1 b § 3 d t ·
As,max = ¨ ′
¸ − As
fy fy © 7 ¹
εt = εy = = 0.002
Es ′
( As − As ) f y
a =
0.85 f c ' b
ª § a· º
Mn = fy « ( As − As ) ¨ d − ¸ + As ( d − d ' ) »
′ ′
RESISTANCE FACTORS, φ ¬ © 2¹ ¼
If compression steel does not yield (four steps):
Tension-controlled sections ( εt ≥ 0.005 ): φ = 0.9
1. Solve for c:
Compression-controlled sections ( εt ≤ 0.002 ):
Members with spiral reinforcement φ = 0.70 § (87,000 − 0.85 f c ' ) As ' − As f y ·
c2 + ¨
¨
¸c
¸
Members with tied reinforcement φ = 0.65 © 0.85 f c ' β1 b ¹
Transition sections ( 0.002 < εt < 0.005 ): 87,000 As ' d '
Members w/ spiral reinforcement φ = 0.57 + 67εt − =0
0.85 f c ' β1 b
Members w/ tied reinforcement φ = 0.48 + 83εt
Shear and torsion φ = 0.75
Bearing on concrete φ = 0.65
97

6. CIVIL ENGINEERING (continued)
BEAMS − FLEXURE: φMN ≥ MU (CONTINUED) BEAMS − SHEAR: φVN ≥ Vu
Doubly-reinforced beams (continued) Beam width used in shear equations:
Compression steel does not yield (continued)
b (rectangular beams )
§ c − d' · bw =
2. fs'=87,000 ¨ ¸ bw (T−beams)
© c ¹
Nominal shear strength:
0.85 f c 'β1 b § 3 d t · § f '·
3. As,max= ¨ ¸ − A s' ¨ s ¸ Vn = Vc + Vs
fy © 7 ¹ ¨ fy ¸
© ¹ Vc = 2 bw d f c '
( As f y − As ' f s ' )
4. a = Av f y d
0.85 f c ' b Vs = [may not exceed 8 bw d f c' ]
s
ª § As f y ·§ a· º Required and maximum-permitted stirrup spacing, s
Mn = fs' « ¨ ¸
¨ f ' − As ' ¸ ¨ d − ¸ + As ' ( d − d ' ) »
«© s
¬ ¹© 2¹ »
¼ φVc
Vu ≤ : No stirrups required
2
T-beams − tension reinforcement in stem φVc
Vu > : Use the following table ( Av given ):
2
Effective flange width:
1/4 • span length
be = bw + 16 • hf φVc
< Vu ≤ φVc Vu > φVc
2
beam centerline spacing
smallest
Design moment strenth: Smaller of:
Vs = Vu − φVc :
As f y Av f y
a= s=
0.85 f c ' be 50b w
Required
φ Av f y d
If a ≤ hf : Av f y s=
spacing s= Vs
0.85 f c ' β 1 be § 3 dt · 0.75 bw fc '
As,max = ¨
¨ 7 ¸ ¸
fy © ¹
a
Mn = 0.85 fc' a be (d- )
2
Vs ≤ 4 b w d fc '
If a > hf : Smaller of:
Smaller of: d
0.85 f c 'β1 be § 3 d t · 0.85 f c ' (be − bw ) h f s= OR
As,max = ¨ ¸+ Maximum d 2
fy © 7 ¹ fy permitted
s=
2 s =24"
hf spacing
Mn = 0.85 fc' [hf (be − bw) (d − ) OR
2 Vs > 4 b w d fc '
s =24"
a
+ a bw (d − )] Smaller of:
2
d
s=
4
s =12"
98

7. CIVIL ENGINEERING (continued)
SHORT COLUMNS:
Reinforcement limits: Concentrically-loaded short columns: φPn ≥ Pu
A M1 = M2 = 0
ρ g = st
Ag KL
≤ 22
0.01 ≤ ρg ≤ 0.08 r
Design column strength, spiral columns: φ = 0.70
Definition of a short column: φPn = 0.85φ [ 0.85 fc' ( Ag − Ast ) + Ast fy ]
KL 12 M 1
≤ 34 −
r M2 Design column strength, tied columns: φ = 0.65
where: KL = Lcol clear height of column φPn = 0.80φ [ 0.85 fc' ( Ag − Ast ) + Ast fy ]
[assume K = 1.0]
Short columns with end moments:
r = 0.288h rectangular column, h is side length Mu = M2 or Mu = Pu e
perpendicular to buckling axis ( i.e., Use Load-moment strength interaction diagram to:
side length in the plane of buckling ) 1. Obtain φPn at applied moment Mu
r = 0.25h circular column, h = diameter 2. Obtain φPn at eccentricity e
3. Select As for Pu , Mu
M1 = smaller end moment
M2 = larger end moment
M1
M2
LONG COLUMNS − Braced (non-sway) frames
Definition of a long column: Long columns with end moments:
KL 12 M 1 M1 = smaller end moment
> 34 −
r M2 M2 = larger end moment
M1
Critical load: positive if M1 , M2 produce single curvature
M2
π2 E I π2 E I
Pc = = 0 .4 M 1
( KL ) 2 ( Lcol ) 2 C m = 0.6 + ≥ 0.4
M2
where: EI = 0.25 Ec Ig
Cm M 2
Mc = ≥ M2
Pu
Concentrically-loaded long columns: 1−
0.75 Pc
emin = (0.6 + 0.03h) minimum eccentricity
Use Load-moment strength interaction diagram
M1 = M2 = Pu emin (positive curvature)
to design/analyze column for Pu , Mu
KL
> 22
r
M2
Mc =
Pu
1−
0.75 Pc
Use Load-moment strength interaction diagram
to design/analyze column for Pu , Mu
99

8. 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 Edition (2004), Nilson, Darwin, Dolan
McGraw-Hill ISBN 0-07-248305-9 GRAPH A.11, Page 762
Used by permission
100

9. 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
Used by permission
101

10. CIVIL ENGINEERING (continued)
STEEL STRUCTURES References: AISC LRFD Manual, 3rd Edition
AISC ASD Manual, 9th Edition
LOAD COMBINATIONS (LRFD)
Floor systems: 1.4D Roof systems: 1.2D + 1.6(Lr or S or R) + 0.8W
1.2D + 1.6L 1.2D + 0.5(Lr or S or R) + 1.3W
0.9D ± 1.3W
where: D = dead load due to the weight of the structure and permanent features
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)
s2
Net area: An = (bg − ΣDh + ) t across critical chain of holes
4g
where: 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)
Ae = UAn U = 0.85 (angles with ≥ 3 bolts in line) Ae = UAg U = 0.87 (flat bars, 2w > L ≥ 1.5w)
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 ASD
Fracture: φTn = φf Ae Fu = 0.75 Ae Fu Yielding: Ta = Ag Ft = Ag (0.6 Fy)
Block shear rupture (bolted tension members): Fracture: Ta = Ae Ft = Ae (0.5 Fu)
Agt =gross tension area
Agv =gross shear area Block shear rupture (bolted tension members):
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 0.75 [0.6 Fu Anv + Fu Ant]
0
102

11. CIVIL ENGINEERING (continued)
BEAMS: homogeneous beams, flexure about x-axis
Flexure – local buckling:
bf 65 h 640
No local buckling if section is compact: ≤ and ≤
2t f Fy tw Fy
bf h
where: For rolled sections, use tabulated values of and
2t f 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
LRFD–compact rolled shapes ASD–compact rolled shapes
300 ry 76 b f 20,000
Lp = Lc = or use smaller
Fy Fy (d / A f ) Fy
Zx Table
ry X 1 Cb = 1.75 + 1.05(M1 /M2) + 0.3(M1 /M2)2 ≤ 2.3
Lr = 1 + 1 + X 2 FL2
FL M1 is smaller end moment
where: FL = Fy – 10 ksi M1 /M2 is positive for reverse curvature
π EGJA
X1 = Ma = S Fb
Sx 2 W-Shapes
Dimensions Lb ≤ Lc: Fb = 0.66 Fy
C § S ·2
X2 = 4 w ¨ x¸ and Properties Lb > Lc :
I y © GJ ¹ Table
ª2 F ( L / r )2 º
φ = 0.90 Fb = « − y b T » ≤ 0.6 Fy (F1-6)
φMp = φ Fy Zx «3
¬
1,530,000 Cb »
¼
Zx Table
φMr = φ FL Sx 170,000 Cb
Fb = ≤ 0.6 Fy (F1-7)
Cb =
12.5 M max ( Lb / rT )2
2.5 M max + 3M A + 4M B + 3MC
12 ,000 Cb
Lb ≤ Lp: φMn = φMp
Fb = ≤ 0.6 Fy (F1-8)
Lb d / A f
Lp < Lb ≤ Lr:
102 ,000 Cb L 510 ,000 Cb
ª § Lb − L p ·º For: < b ≤ :
φMn = Cb «φM p − ( φM p − φM r ) ¨ ¸» Fy rT Fy
« ¨ Lr − L p ¸»
¬ © ¹¼
Use larger of (F1-6) and (F1-8)
= Cb [φMp − BF (Lb − Lp)] ≤ φMp
Lb 510,000 Cb
See Zx Table for BF For: > :
rT Fy
Use larger of (F1-7) and (F1-8)
Lb > Lr :
2
φC b S x X 1 2 X1 X 2 See Allowable Moments in Beams curve
φM n = 1+ ≤ φMp
Lb /ry (
2 Lb /r y 2 )
See Beam Design Moments curve
103

12. CIVIL ENGINEERING (continued)
Shear – unstiffened beams
LRFD – E = 29,000 ksi ASD
h 380
For ≤ : Fv = 0.40 Fy
φ = 0.90 Aw = d t w tw Fy
h 417
≤ φVn = φ (0.6 Fy) Aw h 380 Fy
tw Fy For > : Fv = (Cv ) ≤ 0.4 Fy
tw Fy 2.89
417 h 523 where for unstiffened beams:
< ≤
Fy tw Fy kv = 5.34
ª 417 º 190 kv 439
φVn = φ (0.6 Fy) Aw « » Cv = =
« ( h/t w ) F y » h/t w Fy ( h/t w ) Fy
¬ ¼
523 h
< ≤ 260
Fy tw
ª 218,000 º
φVn = φ (0.6 Fy) Aw « 2
»
« ( h/t w ) F y »
¬ ¼
COLUMNS
Column effective length KL:
AISC Table C-C2.1 (LRFD and ASD)− Effective Length Factors (K) for Columns
AISC Figure C-C2.2 (LRFD and ASD)− Alignment Chart for Effective Length of Columns in Frames
Column capacities:
LRFD ASD
Column slenderness parameter: Column slenderness parameter:
§ KL · § 1 Fy · 2 π2 E
λc = ¨ ¸ ¨ ¸ Cc =
© r ¹ max ¨ π E ¸ Fy
© ¹
Nominal capacity of axially loaded columns (doubly Allowable stress for axially loaded columns (doubly
symmetric section, no local buckling): symmetric section, no local buckling):
φ = 0.85 § KL ·
When ¨ ¸ ≤ Cc
© r ¹ max
φFcr = φ § 0.658 λc · Fy
2
λc ≤ 1.5: ¨ ¸
© ¹
ª ( KL/r ) 2 º
ª 0.877 º «1 − » Fy
λc > 1.5: φFcr = φ « 2 » Fy «
¬ 2 Cc 2 » ¼
Fa =
« λc »
¬ ¼ 5 3 ( KL/r ) ( KL / r ) 3
+ −
See Table 3-50: Design Stress for Compression 3 8 Cc 8 Cc 3
Members (Fy = 50 ksi, φ = 0.85)
§ KL · 12 π 2 E
When ¨ ¸ > Cc: Fa =
© r ¹ max 23 ( KL / r ) 2
See Table C-50: Allowable Stress for Compression
Members (Fy = 50 ksi)
104

13. 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 Pu 8 Mu fa fa Cm f b
≥ 0.2 : + ≤ 1.0 > 0.15 : + ≤ 1 .0
φ Pn φ Pn 9 φ M n Fa Fa § fa ·
¨1−
¨ ¸ Fb
Pu Pu Mu © Fe′ ¸
¹
< 0.2 : + ≤ 1 .0
φ Pn 2 φ Pn φMn fa fa f
≤ 0.15 : + b ≤ 1 .0
where: Fa Fa Fb
Mu = B1 Mnt where:
Cm Cm = 1.0 for conditions stated above
B1 = ≥ 1.0
Pu
1− 12 π 2 E
Pex Fe′ = x-axis bending
23 ( KLx /rx ) 2
Cm = 1.0 for conditions stated above
§ π2 E I x ·
Pex = ¨ ¸ x-axis bending
¨ ( KL ) 2 ¸
© x ¹
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
1
Dh = bolt hole diameter = db + /16" [standard holes]
Bolt tension and shear strengths:
LRFD ASD
Design strength (kips / bolt):
Design strength ( kips / bolt ):
Tension: φRt = φ Ft Ab
Tension: Rt = Ft Ab
Shear: φRv = φ Fv Ab
Shear: Rv = Fv Ab
Design resistance to slip at factored loads
Design resistance to slip at service loads
( kips / bolt ): φRn (kips / bolt): Rv
Bolt size Bolt size
Bolt strength Bolt strength
3/4" 7/8" 1" 3/4" 7/8" 1"
φRt 29.8 40.6 53.0 Rt 19.4 26.5 34.6
φRv ( A325-N ) 15.9 21.6 28.3 Rv ( A325-N ) 9.3 12.6 16.5
φRn (A325-SC ) 10.4 14.5 19.0 Rv ( A325-SC ) 6.63 9.02 11.8
φRv and φRn values are single shear Rv values are single shear
105

14. CIVIL ENGINEERING (continued)
Bearing strength
LRFD
ASD
Design strength (kips/bolt/inch thickness): Design strength (kips/bolt/inch thickness):
φrn = φ 1.2 Lc Fu ≤ φ 2.4 db Fu
When s ≥ 3 db and Le ≥ 1.5 db
φ = 0.75
Lc = clear distance between edge of hole rb = 1.2 Fu db
and edge of adjacent hole, or edge of Le Fu
member, in direction of force When Le < 1.5 db : rb =
2
Lc = s – D h When s < 3 db :
D
Lc = Le – h § d ·
2 ¨ s − b ¸ Fu
¨ 2 ¸
Design bearing strength (kips/bolt/inch rb = © ¹
≤ 1.2 Fu db
2
thickness) for various bolt spacings, s,
and end distances, Le: Design bearing strength (kips/bolt/inch
thickness) for various bolt spacings, s,
and end distances, Le:
Bearing Bolt size
strength Bearing Bolt size
3/4" 7/8" 1"
φrn (k/bolt/in strength
rb(k/bolt/in) 3/4" 7/8" 1"
s = 2 2/3 db ( minimum permitted )
s ≥ 3 db and Le ≥ 1.5 db
Fu = 58 ksi 62.0 72.9 83.7
Fu = 58 ksi 52.2 60.9 69.6
Fu = 65 ksi 69.5 81.7 93.8
Fu = 65 ksi 58.5 68.3 78.0
s = 3"
s = 2 2/3 db (minimum permitted)
Fu = 58 ksi 78.3 91.3 101 47.1 55.0 62.8
Fu = 58 ksi
Fu = 65 ksi 87.7 102 113 Fu = 65 ksi 52.8 61.6 70.4
Le = 1 1/4" Le = 1 1/4"
Fu = 58 ksi 44.0 40.8 37.5 Fu = 58 ksi 36.3 [all bolt sizes]
Fu = 65 ksi 49.4 45.7 42.0 Fu = 65 ksi 40.6 [all bolt sizes]
Le = 2"
Fu = 58 ksi 78.3 79.9 76.7
Fu = 65 ksi 87.7 89.6 85.9
The bearing resistance of the connection shall be taken as
the sum of the bearing resistances of the individual bolts.
106

15. CIVIL ENGINEERING (continued)
Area Depth Web Flange Compact X1 X2 rT d/Af Axis X-X Axis Y-Y
Shape 6
A d tw bf tf section x 10 ** ** I S r Z I r
2 4 3 3 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
107

16. CIVIL ENGINEERING (continued)
Table 1-1: W-Shapes Dimensions and Properties (continued)
Area Depth Web Flange Compact X1 X2 rT d/Af Axis X-X Axis Y-Y
Shape A d section 6 I S r Z I r
tw bf tf x 10 ** **
2 4 3 3 4
in. in. in. in. in. bf/2tf h/tw ksi 1/ksi in. 1/in. in. in. in. in. in. in.
W18 × 86 25.3 18.4 0.48 11.1 0.77 7.20 33.4 2460 4060 2.97 2.15 1530 166 7.77 186 175 2.63
W18 × 76 22.3 18.2 0.43 11.0 0.68 8.11 37.8 2180 6520 2.95 2.43 1330 146 7.73 163 152 2.61
W18 × 71 20.8 18.5 0.50 7.64 0.81 4.71 32.4 2690 3290 1.98 2.99 1170 127 7.50 146 60.3 1.70
W18 × 65 19.1 18.4 0.45 7.59 0.75 5.06 35.7 2470 4540 1.97 3.22 1070 117 7.49 133 54.8 1.69
W18 × 60 17.6 18.2 0.42 7.56 0.70 5.44 38.7 2290 6080 1.96 3.47 984 108 7.47 123 50.1 1.68
W18 × 55 16.2 18.1 0.39 7.53 0.63 5.98 41.1 2110 8540 1.95 3.82 890 98.3 7.41 112 44.9 1.67
W18 × 50 14.7 18.0 0.36 7.50 0.57 6.57 45.2 1920 12400 1.94 4.21 800 88.9 7.38 101 40.1 1.65
W18 × 46 13.5 18.1 0.36 6.06 0.61 5.01 44.6 2060 10100 1.54 4.93 712 78.8 7.25 90.7 22.5 1.29
W18 × 40 11.8 17.9 0.32 6.02 0.53 5.73 50.9 1810 17200 1.52 5.67 612 68.4 7.21 78.4 19.1 1.27
W18 × 35 10.3 17.7 0.30 6.00 0.43 7.06 53.5 1590 30800 1.49 6.94 510 57.6 7.04 66.5 15.3 1.22
W16 × 89 26.4 16.8 0.53 10.4 0.88 5.92 25.9 3160 1460 2.79 1.85 1310 157 7.05 177 163 2.48
W16 × 77 22.9 16.5 0.46 10.3 0.76 6.77 29.9 2770 2460 2.77 2.11 1120 136 7.00 152 138 2.46
W16 × 67 20.0 16.3 0.40 10.2 0.67 7.70 34.4 2440 4040 2.75 2.40 970 119 6.97 132 119 2.44
W16 × 57 16.8 16.4 0.43 7.12 0.72 4.98 33.0 2650 3400 1.86 3.23 758 92.2 6.72 105 43.1 1.60
W16 × 50 14.7 16.3 0.38 7.07 0.63 5.61 37.4 2340 5530 1.84 3.65 659 81.0 6.68 92.0 37.2 1.59
W16 × 45 13.3 16.1 0.35 7.04 0.57 6.23 41.1 2120 8280 1.83 4.06 586 72.7 6.65 82.3 32.8 1.57
W16 × 40 11.8 16.0 0.31 7.00 0.51 6.93 46.5 1890 12700 1.82 4.53 518 64.7 6.63 73.0 28.9 1.57
W16 × 36 10.6 15.9 0.30 6.99 0.43 8.12 48.1 1700 20400 1.79 5.28 448 56.5 6.51 64.0 24.5 1.52
W16 × 31 9.1 15.9 0.28 5.53 0.44 6.28 51.6 1740 19900 1.39 6.53 375 47.2 6.41 54.0 12.4 1.17
W16 × 26 7.7 15.7 0.25 5.50 0.35 7.97 56.8 1480 40300 1.36 8.27 301 38.4 6.26 44.2 9.59 1.12
W14 × 120 35.3 14.5 0.59 14.7 0.94 7.80 19.3 3830 601 4.04 1.05 1380 190 6.24 212 495 3.74
W14 × 109 32.0 14.3 0.53 14.6 0.86 8.49 21.7 3490 853 4.02 1.14 1240 173 6.22 192 447 3.73
W14 × 99 29.1 14.2 0.49 14.6 0.78 9.34 23.5 3190 1220 4.00 1.25 1110 157 6.17 173 402 3.71
W14 × 90 26.5 14.0 0.44 14.5 0.71 10.2 25.9 2900 1750 3.99 1.36 999 143 6.14 157 362 3.70
W14 × 82 24.0 14.3 0.51 10.1 0.86 5.92 22.4 3590 849 2.74 1.65 881 123 6.05 139 148 2.48
W14 × 74 21.8 14.2 0.45 10.1 0.79 6.41 25.4 3280 1200 2.72 1.79 795 112 6.04 126 134 2.48
W14 × 68 20.0 14.0 0.42 10.0 0.72 6.97 27.5 3020 1660 2.71 1.94 722 103 6.01 115 121 2.46
W14 × 61 17.9 13.9 0.38 9.99 0.65 7.75 30.4 2720 2470 2.70 2.15 640 92.1 5.98 102 107 2.45
W14 × 53 15.6 13.9 0.37 8.06 0.66 6.11 30.9 2830 2250 2.15 2.62 541 77.8 5.89 87.1 57.7 1.92
W14 × 48 14.1 13.8 0.34 8.03 0.60 6.75 33.6 2580 3250 2.13 2.89 484 70.2 5.85 78.4 51.4 1.91
W12 × 106 31.2 12.9 0.61 12.2 0.99 6.17 15.9 4660 285 3.36 1.07 933 145 5.47 164 301 3.11
W12 × 96 28.2 12.7 0.55 12.2 0.90 6.76 17.7 4250 407 3.34 1.16 833 131 5.44 147 270 3.09
W12 × 87 25.6 12.5 0.52 12.1 0.81 7.48 18.9 3880 586 3.32 1.28 740 118 5.38 132 241 3.07
W12 × 79 23.2 12.4 0.47 12.1 0.74 8.22 20.7 3530 839 3.31 1.39 662 107 5.34 119 216 3.05
W12 × 72 21.1 12.3 0.43 12.0 0.67 8.99 22.6 3230 1180 3.29 1.52 597 97.4 5.31 108 195 3.04
W12 × 65 19.1 12.1 0.39 12.0 0.61 9.92 24.9 2940 1720 3.28 1.67 533 87.9 5.28 96.8 174 3.02
W12 × 58 17.0 12.2 0.36 10.0 0.64 7.82 27.0 3070 1470 2.72 1.90 475 78.0 5.28 86.4 107 2.51
W12 × 53 15.6 12.1 0.35 9.99 0.58 8.69 28.1 2820 2100 2.71 2.10 425 70.6 5.23 77.9 95.8 2.48
W12 × 50 14.6 12.2 0.37 8.08 0.64 6.31 26.8 3120 1500 2.17 2.36 391 64.2 5.18 71.9 56.3 1.96
W12 × 45 13.1 12.1 0.34 8.05 0.58 7.00 29.6 2820 2210 2.15 2.61 348 57.7 5.15 64.2 50.0 1.95
W12 × 40 11.7 11.9 0.30 8.01 0.52 7.77 33.6 2530 3360 2.14 2.90 307 51.5 5.13 57.0 44.1 1.94
** AISC ASD Manual, 9th Edition
108