beams calculation - aisc summary
TRANSCRIPT
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Limit States
• Flexure•Elastic•Plastic•Stability (buckling)
• Shear• Deflection• Fatigue• Supports
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Flexure
unb MM
ElasticPlasticStability (buckling)
ab
n MM
LRFD ASD
90.0b 67.1b
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Flexure - Elastic
I
Myf
S=I/c : Section Modulus (Tabulated Value)
S
M
I
cMf max
maxmax
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Flexure - Plastic
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Flexure - Plastic
Z=(0.5A)a : Plastic Section Modulus (Tabulated Value)
Mp = Acfy = Atfy = fy (0.5A) a = Mp=Zfy
Mp/ My =Z/SFor shapes that are symmetrical about the axis of bending the plastic and elastic neutral axes are the same
C=TAcfy=Atfy
Ac=At
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Flexure - Stability
Mp is reached and section becomes fully plastic
Or
Flange Local Buckling (FLB) Elastically or InelasticallyWeb Local Buckling (WLB) Elastically or Inelastically
Lateral Torsional Buckling (LTB) Elastically or Inelastically
A beam has failed when:
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Flexure - Stability
Slenderness ParameterFLB
=bf/2tf
WLB
=h/tw
LTB
= Lb /ry
tf
bf
twh
Lb
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Flexure - Stability
FLB and WLB (Section B5 Table B4.1)Evaluate Moment Capacity for Different
FLB
=bf/2tf
WLB
=h/tw
CompactNonCompact
Slender
Mp
Mr
p r
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Slenderness Parameter - Limiting Values
AISC B5 Table B4.1 pp 16.1-16
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Slenderness Parameter - Limiting Values
AISC B5 Table B4.1 pp 16.1-17
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Slenderness Parameter - Limiting Values
AISC B5 Table B4.1 pp 16.1-18
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Flexure - Stability
FLB and WLB (Section B5 Table B4.1)
FLB
=bf/2tf
WLB
=h/tw
CompactNonCompact
Slender
Mp
Mr
p r
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Bending Strength of Compact Shapes
Lateral Torsional Buckling
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Bending Strength of Compact Shapes
yyp F
ErL 76.1
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Bending Strength of Compact Shapes
Laterally Supported Compact Beams
xypn ZFMM
yypb F
ErLL 76.1
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Bending Strength of Compact Shapes
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Bending Strength of Compact Shapes
Elastic Buckling
pxcrn MSFM
27.0
76.6117.0
95.1
EJc
hSF
hS
Jc
F
ErLL oxy
oxytsrb
2
2
2
078.01
ts
b
oxtsb
bcr r
L
hS
Jc
rL
ECF
xyr SFM 7.0
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Elastic Buckling
Cb = factor to account for non-uniform bending within the unbraced length
L/4 L/4 L/4 L/4
A B C
Mmax
0.33435.2
5.12
max
max
mCBA
b RMMMM
MC
See AISC table 3-1 p 3.10
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Elastic Buckling
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Elastic Buckling
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Elastic Buckling
Cb = factor to account for non-uniform bending within the unbraced length
0.33435.2
5.12
max
max
mCBA
b RMMMM
MC
Rm= 1 for doubly symmetric cross sections and singly symmetric subject to single curvature
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Elastic Buckling
Cb = factor to account for non-uniform bending within the unbraced length
2
2
2
078.01
ts
b
oxtsb
bcr r
L
hS
Jc
rL
ECF
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Elastic Buckling
Cb = factor to account for non-uniform bending within the unbraced length
x
wyts S
CIr 2
channelsfor
2
shapes I symmetricdoubly for 1
w
yo
C
Ihc
ho = distance between flange centroids = d-tf
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Bending Strength of Compact Shapes
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Bending Strength of Compact Shapes
Inelastic Buckling
ppr
pbrppbn M
LL
LLMMMCM
rbp LLL
xyr SFM 7.0
Linear variation between Mp and Mr
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Nominal Flexural Strength – Compact Shapes
2
2
2
078.01
ts
b
oxtsb
bcr r
L
hS
Jc
rL
ECF
rbp
brpxcr
ppr
pbrppb
pbp
n LLL
LLMSF
MLL
LLMMMC
LLM
M
for
for
for
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Nominal Flexural Strength – NON-Compact Shapes
Most W- M- S- and C- shapes are compact
A few are NON-compact
NONE is slender
Webs of ALL hot rolled shapes in the manual are compactFLB and LTB
Built-Up welded shapes can have non-compact or slender websFLB, WLB, LTB (AISC F4 and F5)
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Nominal Flexural Strength – NON-Compact Shapes
for Manualin shapes rolledfor /A
for
for
br
rpppr
prpp
pp
n
N
MMMM
M
M
WLB
t
bλ
f
f
2
F
Eλ
yp 38.0
F
Eλ
yr 0.1
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Design of Beams - Limit States
• Flexure•Elastic•Plastic•Stability (buckling)
• ShearShear• DeflectionDeflection
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Design for Shear
• Large concentrated loads placed near beam supports
• Rigid connection of beams and columns with webs on the same plane
• Notched or coped beams
• Heavily loaded short beams
• Thin webs in girders
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Design for Shear
V: Vertical shear at the section under considerationQ: First moment about of neutral axis of area of the
cross section between point of interest and top or bottom of section (depends on y)
I: Moment of inertia of sectionb: width of section at point of interest
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Design for Shear
Web fails before flanges
d/b=2 Error ~3%d/b=1 Error ~12%d/b=1/4 Error 100%
Small width bSmall width b
Nominal Strength if no buckling:
yw
nV F
A
Vf 6.0 wyn AFV 6.0
Average Shear Stress
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Design for Shear
• Yielding• Inelastic Buckling• Elastic Buckling
Failure of Web due to Shear:Failure of Web due to Shear: h/tw
h/tw>260 Stiffeners are requiredAppendix F2
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Design for ShearAISC Specs G pp 16.1-64
Shear Strength must be sufficient to satisfy
unV VV resistance factor for shear=0.9
nominal shear strengthdepends on failure mode
maximum shear based on the controlling combination for factored loads
LRFD
aV
n VV
Safety factormaximum shear based on the controlling combination for service loads
ASD
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AISC Spec requirements for Shear
vwyn CAFV 6.0
Cv depends on whether the limit state is web yielding, web inelastic
buckling or web elastic buckling
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AISC Spec requirements for Shear
yw F
E
t
h24.2Special Case for Hot Rolled I shapes with
5.1
1
1
V
V
VC
Most W shapes with ksi 50yF
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AISC Spec requirements for Shear Chapter G
All other doubly and singly symmetric shapes except round HSS
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DEFLECTIONSAISC Specs Chapter L
Serviceability Limit State
Use deflection formulas in AISC Part 3 Or standard analytical or numerical methods
Calculate due to UNFACTORED (service) loads
Governing Building Code, IBC etc
Deflections due to Service Loads Limiting Value<
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Design
Shear is rarely a problem in rolled steel beamsusual practice
Design for Flexure and Check for Shear and DeflectionsOr
Design for Deflections and Check for Flexure and Shear
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Design
• Compute Required Moment Strength Mu or Ma
– Weight of Beam can be assumed and verified or ignored and checked after member is selected
• Select shape that satisfies strength requirements
A) Assume shape, compute strength, compare with required, revise if necessary or
B) Use beam design aids in Part 3 of the Manual
• Check Shear and deflections