CIVIL ENGINEERING
Q
GEOTECHNICAL
Definitions
c = cohesion
cc = coefficient of curvature or gradation
= (D30)2/[(D60)(D10)], where
D10, D30, D60 = particle diameter corresponding to 10%,
30%, and 60% finer on grain-size curve.
cu
e
Vv
Vs
K
Q
i
H
A
qu
w
Ww
Ws
=
=
=
=
=
=
=
=
=
=
=
=
=
=
K
H
Nf
Nd
γ
γd
uniformity coefficient = D60 /D10
void ratio = Vv /Vs, where
volume of voids, and
volume of the solids.
coefficient of permeability = hydraulic conductivity
Q/(iA) (from Darcy's equation), where
discharge rate
hydraulic gradient = dH/dx,
hydraulic head,
cross-sectional area.
unconfined compressive strength = 2c
water content (%) = (Ww /Ws) ×100, where
weight of water, and
weight of solids.
Gw
γs
n
τ
φ
σ
P
A
Ka
Kp
Cc = compression index = ∆e/∆log p
= (e1 – e2)/(log p2 – log p1), where
e1 and e2 = void ratio, and
p1 and p2 = pressure.
Pa
H
=
=
=
=
=
=
=
=
=
=
=
total unit weight of soil = W/V
dry unit weight of soil = Ws /V
Gγw /(1 + e) = γ /(1 + w), where
Se
unit weight of solid = Ws / Vs
porosity = Vv /V = e/(1 + e)
general shear strength = c + σtan φ, where
angle of internal friction,
normal stress = P/A,
force, and
area.
=
=
=
=
=
=
coefficient of active earth pressure
tan2(45 – φ/2)
coefficient of passive earth pressure
tan2(45 + φ/2)
active resultant force = 0.5γH 2Ka, where
height of wall.
= bearing capacity equation
= cNc + γDf Nq + 0.5γBNγ , where
Nc, Nq, and Nγ = bearing capacity factors
B
= width of strip footing, and
Df
= depth of footing below surface.
qult
Dr = relative density (%)
= [(emax – e)/(emax – emin)] ×100
= [(1/γmin – 1/γd) /(1/γmin – 1/γmax)] × 100, where
emax and emin = maximum and minimum void ratio, and
γmax and γmin = maximum and minimum unit dry weight.
FS
Gs = specific gravity = Ws /(Vsγw), where
γw = unit weight of water (62.4 lb/ft3 or 1,000 kg/m3).
∆H =
=
H =
∆e =
p =
= KH(Nf /Nd) (for flow nets, Q per unit width),
where
= coefficient permeability,
= total hydraulic head (potential),
= number of flow tubes, and
= number of potential drops.
settlement = H [Cc /(1 + ei)] log [(pi + ∆p)/pi]
H∆e/(1 + ei), where
thickness of soil layer
change in void ratio, and
pressure.
PI = plasticity index = LL – PL, where
LL = liquid limit, and
PL = plasticity limit.
S = degree of saturation (%) = (Vw /Vv) × 100, where
Vw = volume of water,
Vv = volume of voids.
L
α
φ
W
= factor of safety (slope stability)
cL + Wcosα tanφ
, where
=
W sinα
= length of slip plane,
= slope of slip plane,
= angle of friction, and
= total weight of soil above slip plane.
Cv
T
t
= coefficient of consolidation = TH 2/t, where
= time factor,
= consolidation time.
Hdr
n
Cc
=
=
=
=
=
=
=
σ′
σ
u
93
length of drainage path
number of drainage layers
compression index for normally consolidated clay
0.009 (LL – 10)
effective stress = σ – u, where
normal stress, and
pore water pressure.
CIVIL ENGINEERING (continued)
UNIFIED SOIL CLASSIFICATION SYSTEM (ASTM D-2487)
Clean gravels (Little or no
fines)
Gravels with fines
(Appreciable
amount of fines)
Clean sands (Little or no
fines)
d
Silty gravels, gravel-sand-silt mixtures
u
Clayey gravels, gravel-sand-clay
mixtures
GC
Well-graded sands, gravelly sands, little
or no fines
SW
Poorly graded sand, gravelly sands, little
or no fines
SP
d
SMa
u
Silts and clays
(Liquid limit less
than 50)
SC
ML
MH
Highly organic
soils
Poorly-graded gravels, gravel-sand
mixtures, little or no fines
GP
Silts and clays
(Liquid limit
greater than 50)
Sands with
fines
(Appreciable
amount of
fines)
Gravels
(More than half of coarse fraction is larger than No.
4 sieve size)
Sands
(More than half of coarse fraction is smaller
than No. 4 sieve size)
Coarse-grained soils
(More than half of material is larger than No. 200 sieve size)
Fine-grained soils
(More than half material is smaller than No. 200 sieve)
Laboratory Classification Criteria
Well-graded gravels, gravel-sand
mixtures, little or no fines
GW
GMa
Typical Names
CL
OL
CH
OH
Pt
Silty sands, sand-silt mixtures
Clayey sands, sand-clay mixtures
Determine percentages of sand and gravel from grain-size curve.
Depending on percentage of fines (fraction smaller than No. 200 sieve size), coarse-grained soils are
classified as follows:
Less than 5 percent: GW, GP, SW, SP
More than 12 percent: GM, GC, SM, SC
5 to 12 percent: Borderline cases requiring dual symbolsb
Group
Symbols
Major Divisions
cu =
D 60
greater than 4;
D10
(D )
2
cc =
30
D 10 × D 60
between 1 and 3
Not meeting all gradiation requirements for GW
Atterberg limits below "A"
line or PI less than 4
Atterberg limits above "A"
line with PI greater than 7
cu =
cc =
D 60
Above "A" line
with PI between 4
and 7 are
borderline cases
requiring use of
dual symbols
greater than 6;
D10
(D )2
30
D 10 × D 60
between 1 and 3
Not meeting all gradation requirements for SW
Atterberg limits below "A"
line or PI less than 4
Atterberg limits above "A"
line with PI greater than 7
Limits plotting in
hatched zone with
PI between 4 and 7
are borderline
cases requiring use
of dual symbols
Inorganic silts and very fine sands, rock
flour, silty or clayey fine sands, or
clayey silts with slight plasticity
Inorganic clays of low to medium
plasticity, gravelly clays, sandy clays,
silty clays, lean clays
Organic silts and organic silty clays of
low plasticity
Inorganic silts, micaceous or
diatomaceous fine sandy or silty soils,
elastic silts
Inorganic clays of high plasticity, fat
clays
Organic clays of medium to high
plasticity, organic silts
Peat and other highly organic soils
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
CIVIL ENGINEERING (continued)
STRUCTURAL ANALYSIS
Influence Lines
An influence diagram shows the variation of a function
(reaction, shear, bending moment) as a single unit load
moves across the structure. An influence line is used to (1)
determine the position of load where a maximum quantity
will occur and (2) determine the maximum value of the
quantity.
Deflection of Trusses
Principle of virtual work as applied to trusses
∆
= ΣfQδL
∆
= deflection at point of interest
α = coefficient of thermal expansion
L
= member length
Fp = member force due to external load
A
= cross-sectional area of member
E
= modulus of elasticity
∆T = T–TO; T = final temperature, and TO = initial
temperature
Deflection of Frames
The principle of virtual work as applied to frames:
­ mM ½
dx ¾
∆ = ¦ ®³OL
EI
¯
¿
fQ = member force due to virtual unit load applied at
the point of interest
δL = change in member length
= αL(∆T) for temperature
= FpL/AE for external load
m =
bending moment as a funtion of x due to virtual
unit load applied at the point of interest
M =
bending moment as a function of x due to external
loads
BEAM FIXED-END MOMENT FORMULAS
FEM AB =
Pab 2
L2
FEM BA =
Pa 2 b
L2
FEM AB =
w o L2
12
FEM BA =
w o L2
12
FEM AB =
w o L2
30
FEM BA =
w o L2
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 +
¨
k LL AT
©
·
¸ ≥ 0.4 L
nominal
¸
¹
Columns:
kLL = 4
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
CIVIL ENGINEERING (continued)
REINFORCED CONCRETE DESIGN
US Customary units
ACI 318-02
ASTM STANDARD REINFORCING BARS
Definitions
a
= depth of equivalent rectangular stress block, in
Ag = gross area of column, in2
As = area of tension reinforcement, in2
As' = area of compression reinforcement, in2
Ast = total area of longitudinal reinforcement, in2
Av = area of shear reinforcment within a distance s, in
b
= width of compression face of member, in
be = effective compression flange width, in
bw = web width, in
β1 = ratio of depth of rectangular stress block, a, to depth
to neutral axis, c
§ f c ' − 4,000 ·
¸¸ ≥ 0.65
© 1,000 ¹
=
0.85 ≥ 0.85 – 0.05 ¨¨
c
=
d
=
dt
=
distance from extreme compression fiber to neutral
axis, in
distance from extreme compression fiber to centroid
of nonprestressed tension reinforcement, in
distance from extreme tension fiber to extreme
tension steel, in
Ec
=
modulus of elasticity = 33 wc1.5
εt
=
fc'
fy
hf
Mc
=
=
=
=
Mn
φMn
Mu
Pn
φPn
=
=
=
=
=
Pu
ρg
=
=
s
=
Vc
Vn
φVn
Vs
lb
Vu
=
=
=
=
net tensile strain in extreme tension steel at nominal
strength
compressive strength of concrete, psi
yield strength of steel reinforcement, psi
T-beam flange thickness, in
factored column moment, including slenderness
effect, in-lb
nominal moment strength at section, in-lb
design moment strength at section, in-lb
factored moment at section, in-lb
nominal axial load strength at given eccentricity, lb
design axial load strength at given
eccentricity, lb
factored axial force at section, lb
ratio of total reinforcement area to cross-sectional
area of column = Ast/Ag
spacing of shear ties measured along longitudinal
axis of member, in
nominal shear strength provided by concrete, lb
nominal shear strength at section, lb
design shear strength at section, lb
nominal shear strength provided by reinforcement,
=
factored shear force at section, lb
BAR SIZE
DIAMETER, IN
#3
#4
#5
#6
#7
#8
#9
#10
#11
#14
#18
AREA, IN2
0.375
0.500
0.625
0.750
0.875
1.000
1.128
1.270
1.410
1.693
2.257
WEIGHT, LB/FT
0.11
0.20
0.31
0.44
0.60
0.79
1.00
1.27
1.56
2.25
4.00
0.376
0.668
1.043
1.502
2.044
2.670
3.400
4.303
5.313
7.650
13.60
LOAD FACTORS FOR REQUIRED STRENGTH
f c ' , psi
U = 1.4 D
U = 1.2 D + 1.6 L
SELECTED ACI MOMENT COEFFICIENTS
Approximate moments in continuous beams of three or
more spans, provided:
1.
2.
3.
Span lengths approximately equal (length of
longer adjacent span within 20% of shorter)
Uniformly distributed load
Live load not more than three times dead load
Mu = coefficient * wu * Ln2
wu = factored load per unit beam length
Ln = clear span for positive moment; average
adjacent clear spans for negative moment
Column
+
−
1
14
1
16
+
−
1
10
1
11
−
1
11
−
1
11
−
1
11
Ln
Spandrel
+
beam
−
1
14
1
24
+
−
1
10
−
1
Unrestrained +
11
end
1
10
End span
−
1
16
1
11
+
−
96
−
1
16
1
11
1
16
Interior span
CIVIL ENGINEERING (continued)
BEAMS − FLEXURE: φMN ≥ MU
UNIFIED DESIGN PROVISIONS
For all beams
Net tensile strain: a = β1 c
0.003 ( dt − c )
0.003 ( β1 dt − a )
εt =
=
c
a
Design moment strength: φMn
where: φ
= 0.9 [εt ≥ 0.005]
φ
= 0.48 + 83εt [0.004 ≤ εt < 0.005]
Reinforcement limits:
AS, max εt = 0.004 @ Mn
­
′
200 b d
° 3 f c bw d
w
AS ,min = larger ®
or
f
f
°
y
y
¯
As,min limits need not be applied if
As (provided ≥ 1.33 As (required)
Internal Forces and Strains
d'
Comp.strain
Mu
C c C s'
A's
Pu
As
ε's
c
d
dt
Ts
Net tensile strain:
Strain Conditions
0.003
0.003
A's
c
εt
0.003
c
c
dt
As
Singly-reinforced beams
εt ≥ 0.005
Tensioncontrolled
section:
c ≤ 0.375 dt
0.005> εt >0.002
As,max =
εt ≤ 0.002
Transition
section
·
¸¸
¹
As f y
Compressioncontrolled
section:
a=
c ≥ 0.6 dt
Mn = 0.85 fc' a b (d −
0.85 f c′ b
a
a
) = As fy (d − )
2
2
Doubly-reinforced beams
Compression steel yields if:
Balanced Strain: εt = εy
0.003
0.85 β1 f c′ d' b §¨ 87,000
¨ 87,000 + f y
fy
©
If compression steel yields:
A s − A s' ≥
A's
dt
As
εt = εy =
0.85 f c ' β 1 b § 3 d t
¨¨
fy
© 7
fy
Es
As,max =
= 0.002
a =
·
¸
¸
¹
0.85 f c′ β1 b § 3 d t ·
¨
¸ − As′
fy
© 7 ¹
( As − As′ ) f y
0.85 f c ' b
ª
º
a·
§
Mn = fy « ( As − As′ ) ¨ d − ¸ + As′ ( d − d ' ) »
2¹
©
¬
¼
If compression steel does not yield (four steps):
1. Solve for c:
§ (87,000 − 0.85 f c ' ) As ' − As f y ·
¸c
c2 + ¨¨
¸
0.85 f c ' β1 b
©
¹
87,000 As ' d '
−
=0
0.85 f c ' β1 b
RESISTANCE FACTORS, φ
Tension-controlled sections ( εt ≥ 0.005 ):
φ = 0.9
Compression-controlled sections ( εt ≤ 0.002 ):
Members with spiral reinforcement
φ = 0.70
Members with tied reinforcement
φ = 0.65
Transition sections ( 0.002 < εt < 0.005 ):
Members w/ spiral reinforcement
φ = 0.57 + 67εt
Members w/ tied reinforcement
φ = 0.48 + 83εt
Shear and torsion
φ = 0.75
Bearing on concrete
φ = 0.65
97
CIVIL ENGINEERING (continued)
BEAMS − SHEAR:
BEAMS − FLEXURE: φMN ≥ MU (CONTINUED)
Beam width used in shear equations:
Doubly-reinforced beams (continued)
Compression steel does not yield (continued)
2.
§ c − d' ·
fs'=87,000 ¨
¸
© c ¹
a =
Vn = Vc + Vs
Vc = 2 bw d f c '
Vs =
0.85 f c ' b
º
a·
¸ + As ' ( d − d ' ) »
2¹
¼»
φVc
< Vu ≤ φVc
2
beam centerline spacing
Design moment strenth:
smallest
As f y
0.85 f c ' be
Required
spacing
If a ≤ hf :
§ 3 dt ·
¨¨
¸¸
© 7 ¹
Mn = 0.85 fc' a be (d-
a
)
2
f c' ]
Smaller of:
Av f y
s=
50b w
s=
Vs = Vu − φVc :
s=
Av f y
0.75 bw
Vu > φVc
fc '
φ Av f y d
Vs
Vs ≤ 4 b w d
If a > hf :
As,max =
[may not exceed 8 bw d
φVc
: No stirrups required
2
φVc
Vu >
: Use the following table ( Av given ):
2
1/4 • span length
bw + 16 • hf
0.85 f c ' β 1 be
As,max =
fy
s
Vu ≤
Effective flange width:
a=
Av f y d
Required and maximum-permitted stirrup spacing, s
T-beams − tension reinforcement in stem
=
bw (T−beams)
Nominal shear strength:
§ f '·
0.85 f c 'β1 b § 3 d t ·
¨
¸ − A s' ¨ s ¸
¨ fy ¸
fy
© 7 ¹
©
¹
( As f y − As ' f s ' )
ª § As f y
·§
− As ' ¸¸ ¨ d −
Mn = fs' « ¨¨
¹©
¬« © f s '
be
b (rectangular beams )
bw =
3. As,max=
4.
φVN ≥ Vu
0.85 f c 'β1 be § 3 d t · 0.85 f c ' (be − bw ) h f
¨
¸+
fy
fy
© 7 ¹
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 b w d
Smaller of:
d
s=
4
s =12"
98
fc '
fc '
CIVIL ENGINEERING (continued)
SHORT COLUMNS:
Reinforcement limits:
A
ρ g = st
Ag
Concentrically-loaded short columns: φPn ≥ Pu
M1 = M2 = 0
KL
≤ 22
r
0.01 ≤ ρg ≤ 0.08
Design column strength, spiral columns: φ = 0.70
φPn = 0.85φ [ 0.85 fc' ( Ag − Ast ) + Ast fy ]
Definition of a short column:
12 M 1
KL
≤ 34 −
r
M2
where:
KL = Lcol
Design column strength, tied columns: φ = 0.65
φPn = 0.80φ [ 0.85 fc' ( Ag − Ast ) + Ast fy ]
clear height of column
[assume K = 1.0]
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
LONG COLUMNS − Braced (non-sway) frames
Definition of a long column:
Long columns with end moments:
M1 = smaller end moment
M2 = larger end moment
12 M 1
KL
> 34 −
r
M2
M1
positive if M1 , M2 produce single curvature
M2
Critical load:
Pc =
π2 E I
π2 E I
=
( KL ) 2
( Lcol ) 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)
Cm M 2
≥ M2
Pu
1−
0.75 Pc
Use Load-moment strength interaction diagram
to design/analyze column for Pu , Mu
KL
> 22
r
Mc =
0 .4 M 1
≥ 0.4
M2
M2
Pu
1−
0.75 Pc
Use Load-moment strength interaction diagram
to design/analyze column for Pu , Mu
99
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
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
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:
s2
) t across critical chain of holes
4g
bg = gross width
where:
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
0.75 [0.6 Fu Anv + Fu Ant]
0
102
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
LRFD–compact rolled shapes
ASD–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 ¹
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
Lb ≤ Lp:
Fb
ª2
Fy ( Lb / rT )2 º
«
» ≤ 0.6 Fy
−
=
1,530,000 Cb »
«¬ 3
¼
Fb
=
Fb
=
Zx Table
12.5 M max
+ 3M A + 4M B + 3MC
φMn = φMp
Lp < Lb ≤ Lr:
For:
ª
§ Lb − L p ·º
¸»
φMn = Cb «φM p − ( φM p − φM r ) ¨
¨ Lr − L p ¸»
«¬
©
¹¼
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 /ry
2 Lb /r y 2
(
See Allowable Moments in Beams curve
)
See Beam Design Moments curve
103
(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 t w
h
417
≤
tw
Fy
φVn = φ (0.6 Fy) Aw
417
Fy
<
ASD
h
523
≤
tw
Fy
Fy
<
h
380
:
≤
tw
Fy
Fv = 0.40 Fy
For
h
380
:
>
tw
Fy
Fv =
Fy
2.89
(Cv ) ≤ 0.4 Fy
where for unstiffened beams:
kv = 5.34
ª
º
417
»
φVn = φ (0.6 Fy) Aw «
« ( h/t w ) F y »
¬
¼
523
For
Cv =
190
h/t w
kv
Fy
=
439
( h/t w )
Fy
h
≤ 260
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
Column slenderness parameter:
§ KL ·
λc = ¨
¸
© r ¹ max
§ 1
¨
¨ π
©
ASD
Column slenderness parameter:
Fy ·¸
E ¸
¹
Cc =
Allowable stress for axially loaded columns (doubly
symmetric section, no local buckling):
Nominal capacity of axially loaded columns (doubly
symmetric section, no local buckling):
φ = 0.85
λc ≤ 1.5:
λc > 1.5:
2 π2 E
Fy
§ KL ·
When ¨
≤ Cc
¸
© r ¹ max
2
φFcr = φ §¨ 0.658 λc ·¸ Fy
©
¹
ª 0.877 º
φFcr = φ « 2 » Fy
«¬ λc »¼
Fa =
See Table 3-50: Design Stress for Compression
Members (Fy = 50 ksi, φ = 0.85)
ª
( KL/r ) 2 º
«1 −
» Fy
2 Cc 2 ¼»
¬«
5 3 ( KL/r ) ( KL / r ) 3
+
−
3
8 Cc
8 Cc 3
§ KL ·
When ¨
> Cc:
¸
© r ¹ max
Fa =
12 π 2 E
23 ( KL / r ) 2
See Table C-50: Allowable Stress for Compression
Members (Fy = 50 ksi)
104
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
> 0.15 :
Fa
fa
Cm f b
+
≤ 1 .0
Fa §
fa ·
¨¨ 1 −
¸ Fb
Fe′ ¸¹
©
fa
≤ 0.15 :
Fa
fa
f
+ b ≤ 1 .0
Fa
Fb
where:
Cm
≥ 1.0
Pu
1−
Pex
Cm = 1.0
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
©
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
Design strength (kips / bolt):
Tension:
φRt = φ Ft Ab
ASD
Design strength ( kips / bolt ):
Tension: Rt = Ft Ab
Shear:
Rv = Fv Ab
Design resistance to slip at service loads
(kips / bolt): Rv
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
105
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
member, in direction of force
Lc = s – D h
D
Lc = Le – h
2
thickness) for various bolt spacings, s,
and end distances, Le:
φrn (k/bolt/in
When
s < 3 db :
Le Fu
2
≤ 1.2 Fu db
Design bearing strength (kips/bolt/inch
thickness) for various bolt spacings, s,
and end distances, Le:
Bolt size
3/4"
Le < 1.5 db : rb =
d ·
§
¨¨ s − b ¸¸ Fu
2 ¹
rb = ©
2
Design bearing strength (kips/bolt/inch
Bearing
strength
When
7/8"
rb(k/bolt/in)
s = 2 2/3 db ( minimum permitted )
Fu = 58 ksi
62.0
72.9
83.7
Fu = 65 ksi
69.5
81.7
93.8
s ≥ 3 db
Fu = 58 ksi
Fu = 65 ksi
s = 3"
Fu = 58 ksi
78.3
91.3
101
Fu = 65 ksi
87.7
102
113
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
1"
The bearing resistance of the connection shall be taken as
the sum of the bearing resistances of the individual bolts.
106
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
X2
section
x 10
6
rT
d/Af
**
**
Axis X-X
I
S
4
in.
in.
in.
in.
bf/2tf
h/tw
ksi
1/ksi
in.
1/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
W24 × 94
27.7
24.3
0.52
9.07
0.88
5.18
41.9
2180
7800
2.33
3.06
W24 × 84
24.7
24.1
0.47
9.02
0.77
5.86
45.9
1950
12200
2.31
W24 × 76
22.4
23.9
0.44
8.99
0.68
6.61
49.0
1760
18600
W24 × 68
20.1
23.7
0.42
8.97
0.59
7.66
52.0
1590
W24 × 62
18.3
23.7
0.43
7.04
0.59
5.97
49.7
W24 × 55
16.3
23.6
0.40
7.01
0.51
6.94
W21 × 93
27.3
21.6
0.58
8.42
0.93
W21 × 83
24.3
21.4
0.52
8.36
W21 × 73
21.5
21.2
0.46
W21 × 68
20.0
21.1
W21 × 62
18.3
*
W21 × 55
*
r
3
Axis Y-Y
Z
3
I
r
4
in.
in.
245
9.96
280
119
1.99
2700
222
9.87
254
109
1.98
3.47
2370
196
9.79
224
94.4
1.95
2.29
3.91
2100
176
9.69
200
82.5
1.92
29000
2.26
4.52
1830
154
9.55
177
70.4
1.87
1730
23800
1.71
5.72
1560
132
9.24
154
34.5
1.37
54.1
1570
36500
1.68
6.66
1360
115
9.13
135
29.1
1.34
4.53
32.3
2680
3460
2.17
2.76
2070
192
8.70
221
92.9
1.84
0.84
5.00
36.4
2400
5250
2.15
3.07
1830
171
8.67
196
81.4
1.83
8.30
0.74
5.60
41.2
2140
8380
2.13
3.46
1600
151
8.64
172
70.6
1.81
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
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
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
in.
* LRFD Manual only
in.
** AISC ASD Manual, 9th Edition
107
in.
in.
CIVIL ENGINEERING (continued)
Table 1-1: W-Shapes Dimensions and Properties (continued)
Area Depth Web
Shape
A
in.
2
Flange
Compact
d
tw
bf
tf
in.
in.
in.
in.
bf/2tf
h/tw
X1
X2
rT
d/Af
Axis X-X
6
**
**
I
ksi
1/ksi
in.
1/in.
in.
section
x 10
S
4
in.
3
Axis Y-Y
r
Z
in.
in.
3
I
in.
r
4
in.
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
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
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
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
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
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
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
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
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
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
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
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
** AISC ASD Manual, 9th Edition
108
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
109
CIVIL ENGINEERING (continued)
110
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
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.
111
CIVIL ENGINEERING (continued)
112
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
113
CIVIL ENGINEERING (continued)
114
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
115
CIVIL ENGINEERING (continued)
ENVIRONMENTAL ENGINEERING
For information about environmental engineering refer
to the ENVIRONMENTAL ENGINEERING section.
DARCY'S EQUATION
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 .8 S
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.
Unit hydrograph:
Rational Formula
Q = CIA, where
A = watershed area (acres),
C = runoff coefficient,
I = rainfall intensity (in/hr), and
Q = peak discharge (cfs).
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
coefficient, S,
of an aquifer is the volume of water
taken into or released from storage per
unit surface area per unit change in
potentiometric (piezometric) head.
SEWAGE FLOW RATIO CURVES
CurveA 2:
(P)
116
5
P
0.167
Curve B :
14
+1
4+ P
Curve G:
18 + P
4+ P
CIVIL ENGINEERING (continued)
HYDRAULIC-ELEMENTS GRAPH FOR CIRCULAR SEWERS
Open-Channel Flow
Specific Energy
E =α
E
=
For rectangular channels
αQ 2
V2
+y=
+ y , where
2g
2 gA 2
13
§ q2 ·
yc = ¨¨ ¸¸
© g ¹
specific energy,
Q =
discharge,
V
=
velocity,
y
=
depth of flow,
A
=
cross-sectional area of flow, and
α
=
kinetic energy correction factor, usually 1.0.
yc =
critical depth,
q
=
unit discharge = Q/B,
B
=
channel width, and
g
=
acceleration due to gravity.
Froude Number = ratio of inertial forces to gravity forces
F=
Critical Depth = that depth in a channel at minimum
specific energy
V
Q 2 A3
=
g
T
=
acceleration due to gravity, and
T
=
width of the water surface.
=
yh =
where Q and A are as defined above,
g
, where
117
V
gy h
, where
velocity, and
hydraulic depth = A/T
CIVIL ENGINEERING (continued)
Specific Energy Diagram
Values of Hazen-Williams Coefficient C
y
1
1
Pipe Material
C
Concrete (regardless of age)
130
Cast iron:
New
130
5 yr old
120
20 yr old
100
Welded steel, new
120
Uniform flow: a flow condition where depth and velocity do
not change along a channel.
Wood stave (regardless of age)
120
Vitrified clay
110
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 – the uniform flow depth
Qn
AR 2 3 =
KS 1 2
Weir Formulas
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.
Riveted steel, new
110
Brick sewers
100
Asbestos-cement
140
αV 2
+y
2g
Alternate depths: depths with the same specific energy.
E=
Plastic
150
For additional fluids information, see the FLUID
MECHANICS section.
TRANSPORTATION
Stopping Sight Distance
U.S. Customary Units Equation
S=
V2
+ 1.47Vt
30[(a / 32.2 ) ± G ]
Metric Equation:
S=
V2
+ 0.278Vt ,
254[(a / 9.81) ± G ]
where (as appropriate):
S = stopping sight distance (ft or m),
G = percent grade divided by 100,
V = design speed (mph or km/h),
a = deceleration rate (ft/s2 or m/s2),
= 11.2 ft/s2 = 3.4 m/s2 and
t = driver reaction time (s).
Sight Distance Related to Curve Length
a. Crest Vertical Curve (general equations):
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 gradeline,
= hf /L (ft/ft or m/m), and
V = velocity (ft/s or m/s).
L=
(
200 h1 + h2
L = 2S −
where
L =
A =
S =
h1 =
AS 2
200
(
)
for S ≤ L
2
h1 + h2
A
)
2
for S > L
length of vertical curve (ft or m),
algebraic difference in grades (%),
sight distance for stopping or passing, (ft or m),
height of drivers' eyes above the roadway surface
(ft or m), and
h2 = height of object above the roadway surface
(ft or m).
118
CIVIL ENGINEERING (continued)
U.S. Customary Units:
where (as appropriate):
L =
V =
A =
When h1 = 3.50 ft and h2 = 2.0 ft,
2
AS
2,158
2,158
L = 2S −
A
L=
for S ≤ L
for S > L
e.
Horizontal curve (to see around obstruction):
ª
§ 28.65 S · º
M = R «1 − cos¨
¸»
© R ¹¼
¬
Metric Units:
When h1 = 1,080 mm and h2 = 600 mm,
AS 2
L=
658
L = 2S −
where
for S ≤ L
658
A
R = radius (ft or m)
M = middle ordinate (ft or m),
S = stopping sight distance (ft or m).
for S > L
b. Sag Vertical Curve (based on standard headlight
criteria):
Superelevation of Horizontal Curves
a. Highways:
U.S. Customary Units:
U.S. Customary Units
L=
AS 2
400 + 3.5 S
for S ≤ L
400 + 3.5 A
L = 2S −
A
e
V2
+f =
100
15R
Metric Units:
for S > L
e
V2
+f =
100
127 R
Metric Units
L=
AS 2
120 + 3.5 S
L = 2S −
c.
where (as appropriate):
for S ≤ L
120 + 3.5 A
A
e=
f =
V=
R=
for S > L
AS 2 §
h +h ·
¨C − 1 2 ¸
800 ©
2 ¹
L = 2S −
−1
800 §
h +h ·
¨C − 1 2 ¸
A ©
2 ¹
where
for S ≤ L
E=
G=
v=
g=
R=
for S > L
C = vertical clearance for overhead structure
(underpass) located within 200 ft (60 m) of the
midpoint of the curve (ft or m).
equilibrium elevation of outer rail (in.),
effective gage (center-to-center of rails) (in.),
train speed (ft/s),
acceleration of gravity (ft/s2), and
radius of curve (ft).
Spiral Transitions to Horizontal Curves
a. Highways:
U.S. Customary Units:
d. Sag Vertical Curve (based on riding comfort):
U.S. Customary Units
Ls =
2
AV
,
46.5
3.15V 3
RC
Metric Units:
Metric Units
L=
Gv 2
gR
E=
where
L=
superelevation (%),
side-friction factor,
vehicle speed (mph or km/hr), and
radius of curve (ft or m).
b. Railroads:
Sag Vertical Curve (based on adequate sight distance
under an overhead structure to see an object beyond a
sag vertical curve)
L=
length of vertical curve (ft or m),
design speed (mph or km/hr), and
algebraic difference in grades (%)
Ls =
AV 2
,
395
119
0.0214 V 3
RC
CIVIL ENGINEERING (continued)
where (as appropriate):
AREA Vertical Curve Criteria for Track Profile
Maximum Rate of Change of Gradient in Percent
Grade per Station
In
On
Line Rating
Sags Crests
High-speed Main Line Tracks
0.05
0.10
Secondary or Branch Line Tracks
0.10
0.20
Ls = length of spiral (ft or m),
V = design speed (mph or km/hr),
R = curve radius (ft or m),
C = rate of increase of lateral acceleration
(ft/s3 or m/s3) = 1 ft/s3 = 0.3m/s3
b. Railroads:
Ls = 62E
Transportation Models
Optimization models and methods, including queueing
theory, can be found in the INDUSTRIAL ENGINEERING
section.
Traffic Flow Relationships (q = kv)
E = 0.0007V 2D
where
Ls = length of spiral (ft),
E = equilibrium elevation of outer rail (in.),
V = speed (mph),
D = degree of curve.
Modified Davis Equation – Railroads
R = 0.6 + 20/W + 0.01V + KV 2/(WN)
where
K = air resistance coefficient,
N = number of axles,
R = level tangent resistance [lb/(ton of car weight)],
V = train or car speed (mph), and
W = average load per axle (tons).
Standard values of K
K = 0.0935, containers on flat car,
K = 0.16, trucks or trailers on flat car, and
K = 0.07, all other standard rail units.
Railroad curve resistance is 0.8 lb per ton of car weight per
degree of curvature.
TE = 375 (HP) e/V, where
e = efficiency of diesel-electric drive system (0.82 to
0.93),
HP = rated horsepower of a diesel-electric locomotive
unit,
TE = tractive effort (lb force of a locomotive unit), and
V = locomotive speed (mph).
DENSITY k (veh/mi)
VOLUME q (veh/hr)
120
DENSITY k (veh/mi)
CIVIL ENGINEERING (continued)
AIRPORT LAYOUT AND DESIGN
1. Cross-wind component of 12 mph maximum for aircraft of 12,500 lb or less weight and 15 mph maximum for aircraft
weighing more than 12,500 lb.
2. Cross-wind components maximum shall not be exceeded more than 5% of the time at an airport having a single runway.
3. A cross-wind runway is to be provided if a single runway does not provide 95% wind coverage with less than the maximum
cross-wind component.
LONGITUDINAL GRADE DESIGN CRITERIA FOR RUNWAYS
Item
Maximum longitudinal grade (percent)
Maximum grade change such as A or B (percent)
Maximum grade, first and last quarter of runway (percent)
Minimum distance (D, feet) between PI's for vertical curves
Minimum length of vertical curve (L, feet) per 1 percent grade change
a
Use absolute values of A and B (percent).
121
Transport Airports
Utility Airports
1.5
1.5
0.8
1,000 (A + B)a
1,000
2.0
2.0
-----250 (A + B)a
300
CIVIL ENGINEERING (continued)
AUTOMOBILE PAVEMENT DESIGN
AASHTO Structural Number Equation
SN = a1D1 + a2D2 +…+ anDn, where
SN = structural number for the pavement
ai
= layer coefficient and Di = thickness of layer (inches).
EARTHWORK FORMULAS
Distance between A1 and A2 = L
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,
AREA FORMULAS
Area by Coordinates: Area = [XA (YB – YN) + XB (YC – YA) + XC (YD – YB) + ... + XN (YA – YN–1)] / 2,
§h +h
·
Trapezoidal Rule: Area = w ¨ 1 n + h2 + h3 + h4 + + hn −1 ¸
© 2
¹
w = common interval,
ª
º
§ n − 2 · § n −1 ·
Simpson's 1/3 Rule: Area = w «h1 + 2¨ ¦ hk ¸ + 4¨ ¦ hk ¸ + hn » 3
© k =3,5, ¹ © k = 2 ,4 , ¹
¬
¼
n must be odd number of measurements,
w = common interval
122
CIVIL ENGINEERING (continued)
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
Finish-to-start: start of B
depends on the finish of A
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
(or point of tangency) to Point on Curve
xm =
Horizontal Distance to Min/Max Elevation on Curve = −
123
g1
g1 L
=
2a g1 − g 2
B
CIVIL ENGINEERING (continued)
Tangent Elevation
Curve Elevation
=
=
YPVC + g1x
and
= YPVI + g2 (x – L/2)
2
YPVC + g1x + ax = YPVC + g1x + [(g2 – g1)/(2L)]x2
y = ax 2 ;
a=
g 2 − g1
;
2L
r=
g 2 _ g1
L
2
§ L·
E=a ¨ ¸ ;
©2¹
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=
L.C.
;
2 sin (I/ 2 )
T = R tan (I/ 2) =
5729.58
;
R=
D
L.C.
2 cos(I/ 2 )
LATITUDES AND DEPARTURES
I
π
= 100
L = RI
180 D
+ Latitude
M = R [1 − cos(I/ 2 )]
R
R −M
= cos (I/ 2 );
= cos (I/ 2 )
E+R
R
- Departure
c = 2 R sin (d/ 2 );
0,0
ª
º
1
E=R«
−1»
¬ cos( I/2)
¼
Deflection angle per 100 feet of arc length equals D
2
- Latitude
124
+ Departure