Beams Calculation - AISC Summary
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Transcript of Beams Calculation - AISC Summary
Limit States
• Flexure•Elastic•Plastic•Stability (buckling)
• Shear• Deflection• Fatigue• Supports
Flexure
unb MM
ElasticPlasticStability (buckling)
ab
n MM
LRFD ASD
90.0b 67.1b
Flexure - Elastic
I
Myf
S=I/c : Section Modulus (Tabulated Value)
S
M
I
cMf max
maxmax
Flexure - Plastic
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
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:
Flexure - Stability
Slenderness ParameterFLB
=bf/2tf
WLB
=h/tw
LTB
= Lb /ry
tf
bf
twh
Lb
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
Slenderness Parameter - Limiting Values
AISC B5 Table B4.1 pp 16.1-16
Slenderness Parameter - Limiting Values
AISC B5 Table B4.1 pp 16.1-17
Slenderness Parameter - Limiting Values
AISC B5 Table B4.1 pp 16.1-18
Flexure - Stability
FLB and WLB (Section B5 Table B4.1)
FLB
=bf/2tf
WLB
=h/tw
CompactNonCompact
Slender
Mp
Mr
p r
Bending Strength of Compact Shapes
Lateral Torsional Buckling
Bending Strength of Compact Shapes
yyp F
ErL 76.1
Bending Strength of Compact Shapes
Laterally Supported Compact Beams
xypn ZFMM
yypb F
ErLL 76.1
Bending Strength of Compact Shapes
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
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
Elastic Buckling
Elastic Buckling
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
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
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
Bending Strength of Compact Shapes
Bending Strength of Compact Shapes
Inelastic Buckling
ppr
pbrppbn M
LL
LLMMMCM
rbp LLL
xyr SFM 7.0
Linear variation between Mp and Mr
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
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)
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
Design of Beams - Limit States
• Flexure•Elastic•Plastic•Stability (buckling)
• ShearShear• DeflectionDeflection
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
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
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
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
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
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
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
AISC Spec requirements for Shear Chapter G
All other doubly and singly symmetric shapes except round HSS
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<
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
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