ARCH 331

51 Pages·2016·7.6 MB·English

by  Anne Abell

ARCH 331 Note Set 18 Su2016abn
Steel Design

Notation:
a = name for width dimension d = nominal bolt diameter
b
A = name for area D = shorthand for dead load
A = area of a bolt DL = shorthand for dead load
b
A = effective net area found from the e = eccentricity
e
product of the net area A by the E = shorthand for earthquake load
n
shear lag factor U = modulus of elasticity
A = gross area, equal to the total area f = axial compressive stress
g c
ignoring any holes f = bending stress
b
A = gross area subjected to shear for f = bearing stress
gv p
block shear rupture f = shear stress
v
A = net area, equal to the gross area f = maximum shear stress
n v-max
subtracting any holes, as is A f = yield stress
net y
A = net area subjected to tension for F = shorthand for fluid load
nt
block shear rupture F = allowable stress
allow(able)
A = net area subjected to shear for block F = allowable axial (compressive) stress
nv a
shear rupture F = allowable bending stress
b
A = area of the web of a wide flange F = flexural buckling stress
w cr
section F = elastic critical buckling stress
e
AISC = American Institute of Steel F = yield strength of weld material
EXX
Construction F = nominal strength in LRFD
n
ASD = allowable stress design = nominal tension or shear strength of
b = name for a (base) width a bolt
= total width of material at a F = allowable bearing stress
p
horizontal section F = allowable tensile stress
t
= name for height dimension F = ultimate stress prior to failure
u
b = width of the flange of a steel beam F = allowable shear stress
f v
cross section F = yield strength
y
B = factor for determining M for F = yield strength of web material
1 u yw
combined bending and compression F.S. = factor of safety
c = largest distance from the neutral g = gage spacing of staggered bolt
axis to the top or bottom edge of a holes
beam G = relative stiffness of columns to
c = coefficient for shear stress for a beams in a rigid connection, as is 
1
rectangular bar in torsion h = name for a height
C = lateral torsional buckling h = height of the web of a wide flange
b c
modification factor for moment in steel section
ASD & LRFD steel beam design H = shorthand for lateral pressure load
C = column slenderness classification I = moment of inertia with respect to
c
constant for steel column design neutral axis bending
C = modification factor accounting for I = moment of inertia of trial section
m trial
combined stress in steel design I = moment of inertia required at
req’d
C = web shear coefficient limiting deflection
v
d = calculus symbol for differentiation I = moment of inertia about the y axis
y
= depth of a wide flange section J = polar moment of inertia
= nominal bolt diameter
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ARCH 331 Note Set 18 Su2016abn
k = distance from outer face of W N = bearing length on a wide flange
flange to the web toe of fillet steel section
= shape factor for plastic design of = bearing type connection with
steel beams threads included in shear plane
K = effective length factor for columns, p = bolt hole spacing (pitch)
as is k P = name for load or axial force vector
l = name for length P = allowable axial force
a
 = length of beam in rigid joint = required axial force (ASD)
b
P = allowable axial force
 = length of column in rigid joint allowable
c P = available axial strength
c
L = name for length or span length
P = Euler buckling strength
e1
= shorthand for live load
P = nominal column load capacity in
n
L = unbraced length of a steel beam
b LRFD steel design
L = clear distance between the edge of a
c P = required axial force
r
hole and edge of next hole or edge
P = factored column load calculated
u
of the connected steel plate in the
from load factors in LRFD steel
direction of the load
design
L = effective length that can buckle for
e Q = first moment area about a neutral
column design, as is 
e axis
Lr = shorthand for live roof load = generic axial load quantity for
= maximum unbraced length of a LRFD design
steel beam in LRFD design for r = radius of gyration
inelastic lateral-torsional buckling r = radius of gyration with respect to a
y
Lp = maximum unbraced length of a y-axis
steel beam in LRFD design for full R = generic load quantity (force, shear,
plastic flexural strength moment, etc.) for LRFD design
L’ = length of an angle in a connector = shorthand for rain or ice load
with staggered holes = radius of curvature of a deformed
LL = shorthand for live load beam
LRFD = load and resistance factor design R = required strength (ASD)
a
M = internal bending moment Rn = nominal value (capacity) to be
Ma = required bending moment (ASD) multiplied by  in LRFD and
M = nominal flexure strength with the
n divided by the safety factor  in
full section at the yield stress for
ASD
LRFD beam design R = factored design value for LRFD
u
M = maximum internal bending moment
max design
Mmax-adj = maximum bending moment s = longitudinal center-to-center
adjusted to include self weight
spacing of any two consecutive
Mp = internal bending moment when all holes
fibers in a cross section reach the
S = shorthand for snow load
yield stress = section modulus
M maximum moment from factored
u = = allowable strength per length of a
loads for LRFD beam design weld for a given size
M internal bending moment when the
y = S = section modulus required at
req’d
extreme fibers in a cross section allowable stress
reach the yield stress
S = section modulus required at
req’d-adj
n = number of bolts allowable stress when moment is
n.a. = shorthand for neutral axis
adjusted to include self weight
SC = slip critical bolted connection
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ARCH 331 Note Set 18 Su2016abn
t = thickness of the connected material y = vertical distance
t = thickness of flange of wide flange Z = plastic section modulus of a steel
f
t = thickness of web of wide flange beam
w
T = torque (axial moment) Z = plastic section modulus required
req’d
= shorthand for thermal load Z = plastic section modulus of a steel
x
= throat size of a weld beam with respect to the x axis
U = shear lag factor for steel tension  = method factor for B equation
1
member design  = actual beam deflection
actual
Ubs = reduction coefficient for block allowable = allowable beam deflection
shear rupture  = allowable beam deflection limit
limit
V = internal shear force  = maximum beam deflection
max
Va = required shear (ASD)  = yield strain (no units)
y
V = maximum internal shear force
max
 = resistance factor
V = maximum internal shear force
max-adj
= diameter symbol
adjusted to include self weight
 = resistance factor for bending for
V = nominal shear strength capacity for
n b
LRFD beam design LRFD
V = maximum shear from factored loads  = resistance factor for compression
u
c
for LRFD beam design
for LRFD
w = name for distributed load
 = resistance factor for tension for
w = adjusted distributed load for t
adjusted
LRFD
equivalent live load deflection limit
 = resistance factor for shear for
wequivalent = the equivalent distributed load v
derived from the maximum bending LRFD
moment  = load factor in LRFD design
w = name for distributed load from self
self wt  = pi (3.1415 radians or 180)
weight of member
 = slope of the beam deflection curve
W = shorthand for wind load
 = radial distance
x = horizontal distance
 = safety factor for ASD
X = bearing type connection with
 = symbol for integration
threads excluded from the shear
plane  = summation symbol

Steel Design

Structural design standards for steel are established
by the Manual of Steel Construction published by the
American Institute of Steel Construction, and uses
Allowable Stress Design and Load and Factor
Resistance Design. With the 13th edition, both
methods are combined in one volume which provides
common requirements for analyses and design and
requires the application of the same set of
specifications.

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ARCH 331 Note Set 18 Su2016abn
Materials

American Society for Testing Materials (ASTM) is the organization responsible for material and
other standards related to manufacturing. Materials meeting their standards are guaranteed to
have the published strength and material properties for a designation.

A36 – carbon steel used for plates, angles F = 36 ksi, F = 58 ksi, E = 29,000 ksi
y u
A572 – high strength low-alloy use for some beams F = 60 ksi, F = 75 ksi, E = 29,000 ksi
y u
A992 – for building framing used for most beams F = 50 ksi, F = 65 ksi, E = 29,000 ksi
y u
(A572 Grade 50 has the same properties as A992)

R
ASD R  n
a 

where R = required strength (dead or live; force, moment or stress)
a
R = nominal strength specified for ASD
n
 = safety factor

Factors of Safety are applied to the limit stresses for allowable stress values:

bending (braced, L < L )  = 1.67
b p
bending (unbraced, L < L and L > L)  = 1.67 (nominal moment reduces)
p b b r
shear (beams)  = 1.5 or 1.67
shear (bolts)  = 2.00 (tabular nominal strength)
shear (welds)  = 2.00

Lb is the unbraced length between bracing points, laterally
Lp is the limiting laterally unbraced length for the limit state of yielding
Lr is the limiting laterally unbraced length for the limit state of inelastic lateral-torsional
buckling

LRFD
R R whereR R
u n u i i
where  = resistance factor
 = load factor for the type of load
R = load (dead or live; force, moment or stress)
R = factored load (moment or stress)
u
R = nominal load (ultimate capacity; force, moment or stress)
n

Nominal strength is defined as the

capacity of a structure or component to resist the effects of loads, as determined by
computations using specified material strengths (such as yield strength, F , or ultimate
y
strength, F ) and dimensions and formulas derived from accepted principles of structural
u
mechanics or by field tests or laboratory tests of scaled models, allowing for modeling
effects and differences between laboratory and field conditions

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ARCH 331 Note Set 18 Su2016abn
Factored Load Combinations

R
The design strength, , of each structural element or structural assembly must equal or exceed
n
the design strength based on the ASCE-7 (2010) combinations of factored nominal loads:

1.4D
1.2D + 1.6L + 0.5(L or S or R)
r
1.2D + 1.6(L or S or R) + (L or 0.5W)
r
1.2D + 1.0W + L+ 0.5(L or S or R)
r
1.2D + 1.0E + L + 0.2S
0.9D + 1.0W
0.9D + 1.0E

Criteria for Design of Beams
Mc
F orF  f 
Allowable normal stress or normal stress from LRFD should not be
b n b I
exceeded:
(M  M /or M M )
a n u b n
Knowing M and F , the minimum plastic section modulus fitting the limit is:
y
M  M 
Z  a S  
req'd F   req'd F 
y b
Determining Maximum Bending Moment

Drawing V and M diagrams will show us the maximum values for design. Remember:

V (w)dx dV dM
w V
M (V)dx dx dx

Determining Maximum Bending Stress

For a prismatic member (constant cross section), the maximum normal stress will occur at the
maximum moment.

For a non-prismatic member, the stress varies with the cross section AND the moment.

Deflections

1 M(x)
If the bending moment changes, M(x) across a beam of constant material and cross 
R EI
section then the curvature will change:

1
The slope of the n.a. of a beam, , will be tangent to the radius of slope M(x)dx
EI
curvature, R:

1 1
The equation for deflection, y, along a beam is: y dx M(x)dx
EI EI

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ARCH 331 Note Set 18 Su2016abn
Elastic curve equations can be found in handbooks, textbooks, design manuals, etc...Computer
programs can be used as well. Elastic curve equations can be superimposed ONLY if the stresses
are in the elastic range.

The deflected shape is roughly the same shape flipped as the bending moment diagram but is
constrained by supports and geometry.

Allowable Deflection Limits

All building codes and design codes limit deflection for beam types and damage that could
happen based on service condition and severity.

L
y (x)  
max actual allowable value

Use LL only DL+LL*
Roof beams:
Industrial (no ceiling) L/180 L/120
Commercial
plaster ceiling L/240 L/180
no plaster L/360 L/240
Floor beams:
Ordinary Usage L/360 L/240
Roof or floor (damageable elements) L/480
* IBC 2012 states that DL for steel elements shall be taken as zero

Lateral Buckling

With compression stresses in the top of a beam, a sudden “popping” or buckling can happen
even at low stresses. In order to prevent it, we need to brace it along the top, or laterally brace it,
or provide a bigger I .
y

Local Buckling in Steel Wide-flange Beams– Web Crippling or Flange Buckling

Concentrated forces on a steel beam can cause the web to buckle (called web crippling). Web
stiffeners under the beam loads and bearing plates at the supports reduce that tendency. Web
stiffeners also prevent the web from shearing in plate girders.
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ARCH 331 Note Set 18 Su2016abn
The maximum support load and interior load can be
determined from:

P (2.5k N)F t
n (maxend) yw w

P (5k N)F t
n (interior) yw w
where t = thickness of the web
w
F = yield strength of the
yw
web
N = bearing length
k = dimension to fillet found in beam section tables
 = 1.00 (LRFD)  = 1.50 (ASD)

Beam Loads & Load Tracing

In order to determine the loads on a beam (or girder, joist, column, frame, foundation...) we can
start at the top of a structure and determine the tributary area that a load acts over and the beam
needs to support. Loads come from material weights, people, and the environment. This area is
assumed to be from half the distance to the next beam over to halfway to the next beam.

The reactions must be supported by the next lower structural element ad infinitum, to the ground.

LRFD - Bending or Flexure

M
For determining the flexural design strength, , for resistance to pure bending (no axial load)
b n
in most flexural members where the following conditions exist, a single calculation will suffice:

R  M M 0.9F Z

i i u b n y

where M = maximum moment from factored loads
u
 = resistance factor for bending = 0.9
b
M = nominal moment (ultimate capacity)
n
F = yield strength of the steel
y
Z = plastic section modulus

f
Plastic Section Modulus
f = 50ksi
y
Plastic behavior is characterized by a yield point and an
increase in strain with no increase in stress.
E

1

 = 0.001724
y
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ARCH 331 Note Set 18 Su2016abn
Internal Moments and Plastic Hinges

Plastic hinges can develop when all of the material in a cross
section sees the yield stress. Because all the material at that section
can strain without any additional load, the member segments on
either side of the hinge can rotate, possibly causing instability.

For a rectangular section:

I bh2 b2c2 2bc2
Elastic to f : M  f  f  f  f
y
y c y 6 y 6 y 3 y

Fully Plastic: M or M bc2 f  3 M
ult p y 2 y

For a non-rectangular section and internal equilibrium at  , the
y
n.a. will not necessarily be at the centroid. The n.a. occurs where
the A = A . The reactions occur at the centroids of the
tension compression
tension and compression areas.

Instability from Plastic Hinges

Atension = Acompression

Shape Factor:

The ratio of the plastic moment to the elastic moment at yield:
M
k  p
M k = 3/2 for a rectangle
y
k  1.1 for an I beam

Plastic Section Modulus
M
Z  p and k Z
f S
y
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ARCH 331 Note Set 18 Su2016abn
Design for Shear

V V / or V V
a n u v n
The nominal shear strength is dependent on the cross section shape. Case 1: With a thick or stiff
web, the shear stress is resisted by the web of a wide flange shape (with the exception of a
handful of W’s). Case 2: When the web is not stiff for doubly symmetric shapes, singly
symmetric shapes (like channels) (excluding round high strength steel shapes), inelastic web
buckling occurs. When the web is very slender, elastic web buckling occurs, reducing the
capacity even more:
E
Case 1) For h t 2.24 V 0.6F A  = 1.00 (LRFD)  = 1.50 (ASD)
w F n yw w v
y
where h equals the clear distance between flanges less the fillet or corner
radius for rolled shapes
V = nominal shear strength
n
F = yield strength of the steel in the web
yw
A = t d = area of the web
w w

E
Case 2) For h t 2.24 V 0.6F A C  = 0.9 (LRFD)  = 1.67 (ASD)
w F n yw w v v
y
where C is a reduction factor (1.0 or less by equation)
v

Design for Flexure

M M / or M M  = 0.90 (LRFD)  = 1.67 (ASD)
a n u b n b
The nominal flexural strength M is the lowest value obtained according to the limit states of
n
E
1. yielding, limited at length L 1.76r , where r is the radius of gyration in y
p y F y
y
2. lateral-torsional buckling limited at lengthL
r
3. flange local buckling
4. web local buckling

Beam design charts show available moment, M / and M , for unbraced length, L , of the
n b n b
compression flange in one-foot increments from 1 to 50 ft. for values of the bending coefficient
C = 1. For values of 1<C 2.3, the required flexural strength M can be reduced by dividing it
b b u
by C . (C = 1 when the bending moment at any point within an unbraced length is larger than
b b
that at both ends of the length. C of 1 is conservative and permitted to be used in any case.
b
When the free end is unbraced in a cantilever or overhang, C = 1. The full formula is provided
b
below.)

NOTE: the self weight is not included in determination of M / and M
n b n
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ARCH 331 Note Set 18 Su2016abn
Compact Sections
For a laterally braced compact section (one for which the plastic moment can be reached before
local buckling) only the limit state of yielding is applicable. For unbraced compact beams and
non-compact tees and double angles, only the limit states of yielding and lateral-torsional
buckling are applicable.
b E h E
Compact sections meet the following criteria: f 0.38 and c 3.76
2t F t F
f y w y
where:
b = flange width in inches
f
t = flange thickness in inches
f
E = modulus of elasticity in ksi
F = minimum yield stress in ksi
y
h = height of the web in inches
c
tw = web thickness in inches

With lateral-torsional buckling the nominal flexural strength is

 L L 
M C M (M 0.7F S ) b p   M
n b p p y x Lr Lp  p

where M = M = F Z
p n y x
and C is a modification factor for non-uniform moment diagrams where, when both ends of
b
the beam segment are braced:
12.5M
C  max
b 2.5M 3M 4M 3M
max A B C
M = absolute value of the maximum moment in the unbraced beam segment
max
M = absolute value of the moment at the quarter point of the unbraced beam segment
A
M = absolute value of the moment at the center point of the unbraced beam segment
B
M = absolute value of the moment at the three quarter point of the unbraced beam
C
segment length.

Available Flexural Strength Plots

Plots of the available moment for the unbraced length for wide flange sections are useful to find
sections to satisfy the design criteria of M M / or M M . The maximum moment
a n u b n
that can be applied on a beam (taking self weight into account), M or M , can be plotted against
a u
the unbraced length, L . The limiting length, L (fully plastic), is indicated by a solid dot (),
b p
while the limiting length, Lr (for lateral torsional buckling), is indicated by an open dot ().
Solid lines indicate the most economical, while dashed lines indicate there is a lighter section
that could be used. C , which is a lateral torsional buckling modification factor for non-zero
b
moments at the ends, is 1 for simply supported beams (0 moments at the ends). (see figure)

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