Design of Beams (Flexural Members) (Part 5 of AISC - User pages

15 Pages·2006·0.26 MB·English

53:134 Structural Design II
Design of Beams (Flexural Members)
(Part 5 of AISC/LRFD)

References

1. Part 5 of the AISC LRFD Manual
2. Chapter F and Appendix F of the AISC LRFD Specifications
(Part 16 of LRFD Manual)
3. Chapter F and Appendix F of the Commentary of the AISC
LRFD Specifications (Part 16 of LRFD Manual)

Basic Theory

If the axial load effects are negligible, it is a beam; otherwise it is a
beam-column.

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Shapes that are built up from plate elements are usually called plate
h
girders; the difference is the height-thickness ratio of the web.
t
w
⎧ h E
⎪ ≤ 5.70 beam
t F
⎪ w y
⎨

⎪ h E
> 5.70 plate girder
⎪
t F
⎩ w y

Bending

M = bending moment at the cross section under consideration
y = perpendicular distance from the neutral plane to the point of
interest
I = moment of inertia with respect to the neutral axis
x
S = elastic section modulus of the cross section
x

For elastic analysis, from the elementary mechanics of materials, the
bending stress at any point can be found
My
f =
b
I
x

The maximum stress

Mc M M
f = = =
max
I I /c S
x x x

This is valid as long as the loads are small and the material remains
linearly elastic. For steel, this means f must not exceed F and
max y
the bending moment must not exceed

M = F S
y y x

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M = the maximum moment that brings the beam to the point of
y
yielding

For plastic analysis, the bending stress everywhere in the section is
F , the plastic moment is
y
⎛ A⎞
M = F ⎜ ⎟a = F Z
p y y
⎝ 2 ⎠

M = plastic moment
p

A = total cross-sectional area

a = distance between the resultant tension and compression forces
on the cross-section

⎛ A⎞
Z = ⎜ ⎟a = plastic section modulus of the cross section
⎝ 2 ⎠

Shear

Shear stresses are usually not a controlling factor in the design of
beams, except for the following cases: 1) The beam is very short. 2)
There are holes in the web of the beam. 3) The beam is subjected to a
very heavy concentrated load near one of the supports. 4) The beam is
coped.
f = shear stress at the point of interest
v
V = vertical shear force at the section under consideration
Q = first moment, about the neutral axis, of the area of the cross
section between the point of interest and the top or bottom of
the cross section
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53:134 Structural Design II
I = moment of inertia with respect to the neutral axis
b= width of the cross section at the point of interest

From the elementary mechanics of materials, the shear stress at any
point can be found
VQ
f =
v
Ib

This equation is accurate for small b. Clearly the web will completely
yield long before the flange begins to yield. Therefore, yield of the
web represents one of the shear limit states. Take the shear yield
stress as 60% of the tensile yield stress, for the web at failure
V
f = n = 0.60F
v y
A
w
A = area of the web
w
The nominal strength corresponding to the limit state is
V = 0.60F A
n y w
This will be the nominal strength in shear provided that there is no
h
shear buckling of the web. This depends on , the width-thickness
t
w
ratio of the web. Three cases:

h E
No web instability: ≤2.45
t F
w y

V =0.60F A AISC Eq. (F2-1)
n y w

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E h E
Inelastic web buckling: 2.45 < ≤3.07
F t F
y w y

⎛2.45 E/F ⎞
⎜ y ⎟
V = 0.60F A AISC Eq. (F2-2)
n y w⎜ h/t ⎟
⎝ w ⎠

E h
Elastic web buckling: 3.07 < ≤ 260
F t
y w

⎡ ⎤
4.52E
V = A ⎢ ⎥ AISC Eq. (F2-3)
n w ( )2
⎢⎣ h/tw ⎥⎦

Failure Modes

Shear: A beam can fail due to violation of its shear design strength.
Flexure: Several possible failure modes must be considered. A beam
can fail by reaching M (fully plastic), or it can fail by
p
• Lateral torsional buckling (LTB), elastically or inelastically
• Flange local buckling (FLB), elastically or inelastically
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• Web local buckling (WLB), elastically or inelastically
If the maximum bending stress is less than the proportional limit when
buckling occurs, the failure is elastic. Otherwise, it is inelastic.

Lateral Torsional Buckling

The compressive flange of a beam behaves like an axially loaded
column. Thus, in beams covering long spans the compression flange
may tend to buckle. However, this tendency is resisted by the tensile
flange to certain extent. The overall effect is a phenomenon known as
lateral torsional buckling, in which the beam tends to twist and
displace laterally. Lateral torsional buckling may be prevented by: 1)
Using lateral supports at intermediate points. 2) Using torsionally
strong sections (e.g., box sections). 3) Using I-sections with relatively
wide flanges.

Local Buckling

The hot-rolled steel sections are thin-walled sections consisting of
a number of thin plates. When normal stresses due to bending and/or
direct axial forces are large, each plate (for example, flange or web
plate) may buckle locally in a plane perpendicular to its plane. In
order to prevent this undesirable phenomenon, the width-to-thickness
ratios of the thin flange and the web plates are limited by the code.
AISC classifies cross-sectional shapes as compact, noncompact
and slender ones, depending on the value of the width-thickness
ratios. (LRFD-Specification Table B5.1)
λ = width-thickness ratio
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λ = upper limit for compact category
p
λ = upper limit for noncompact category
r
Then the three cases are
λ≤λ and the flange is continuously connected to the
p
web, the shape is compact.
λ <λ≤λ the shape is noncompact
p r
λ>λ the shape is slender
r
The above conditions are based on the worst width-thickness ratio of
the elements of the cross section. The following table summarizes the
width-thickness limits for rolled I-, H- and C- sections (for C-
sections, λ=b /t . The web criterion is met by all standard I- and
f f
C- sections listed in the Manual. Built-up welded I- shapes (such as
plate girders can have noncompact or slender elements).

Element λ λ λ
p r
Flange b
E E
f 0.38 0.83

2t F F −10
f y y
Web h
E E
3.76 5.70

t
F F
w
y y

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Design Requirements

1. Design for flexure (LRFD SPEC F1)

L unbraced length, distance between points braced against lateral
b
displacement of the compression flange (in.)
L limiting laterally unbraced length for full plastic bending
p
capacity (in.) – a property of the section
L limiting laterally unbraced length for inelastic lateral-torsional
r
buckling (in.) – a property of the section
E modulus of elasticity for steel (29,000 ksi)
G shear modulus for steel (11,200 ksi)
J torsional constant (in.4)
C warping constant (in.6)
w
M limiting buckling moment (kip-in.)
r
M plastic moment, M = F Z ≤1.5M
p p y y
M moment corresponding to the onset of yielding at the extreme
y
fiber from an elastic stress distribution M = F S
y y x
M controlling combination of factored load moment
u
M nominal moment strength
n
φ resistance factor for beams (0.9)
b
The limit of 1.5M for M is to prevent excessive working-load
y p
deformation that is satisfied when

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Z
M = F Z ≤1.5M or F Z ≤1.5F S or ≤1.5
p y y y y S

Design equation
Applied factored moment ≤ moment capacity of the section
OR
Required moment strength ≤ design strength of the section
M ≤φM
u b n

In order to calculate the nominal moment strength M , first calculate
n
L , L , and M for I-shaped members including hybrid sections and
p r r
channels as

E
Lp =1.76ry - a section property AISC Eq. (F1-4)
F
y

r X
y 1 2
Lr = 1+ 1+ X2FL - a section property AISC Eq. (F1-6)
F
L

M = F S - section property AISC Eq. (F1-7)
r L x

F = F − F for nonhybrid member, otherwise it is the smaller
L y r
of F − F or F (subscripts f and w mean flange and
yf r yw
web)
F compressive residual stress in flange, 10 ksi for rolled shapes;
r
16.5 ksi for welded built-up shapes
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π EGJA
X = AISC Eq. (F1-8)
1
S 2
x
2
4C ⎛ S ⎞
X = w ⎜ x ⎟ AISC Eq. (F1-9)
2
I ⎝GJ ⎠
y
S section modulus about the major axis (in.3)
x
I moment of inertia about the minor y-axis (in.4)
y
r radius of gyration about the minor y-axis (in.4)
y

Nominal Bending Strength of Compact Shapes

( )
If the shape is compact λ≤λ , no need to check FLB (flange local
p
buckling) and WLB (web local buckling).
• Lateral torsional buckling (LTB)
If L ≤ L , no LTB:
b p
M = M ≤1.5M AISC Eq. (F1-1)
n p y
If L < L ≤ L , inelastic LTB:
p b r
⎡ ⎛ L −L ⎞⎤
( )
⎜ b p ⎟
M =C ⎢M − M −M ⎥ ≤ M AISC Eq. (F1-2)
n b⎢ p p r ⎜ L −L ⎟⎥ p
⎣ ⎝ r p ⎠⎦
Note that M is a linear function of L .
n b

If L > L (slender member), elastic LTB:
b r

M = M ≤ M AISC Eq. (F1-12)
n cr p
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