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

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


yyp F

ErL 76.1


Laterally Supported Compact Beams

xypn ZFMM

yypb F

ErLL 76.1


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


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


2

2

2

078.01

ts

b

oxtsb

bcr r

L

hS

Jc

rL

ECF

Elastic Buckling


x

wyts S

CIr 2

channelsfor

2

shapes I symmetricdoubly for 1

w

yo

C

Ihc

ho = distance between flange centroids = d-tf


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

beams calculation - aisc summary

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