Limit States

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# Limit States - PowerPoint PPT Presentation

Limit States. Flexure Elastic Plastic Stability (buckling) Shear Deflection Fatigue Supports. Flexure. LRFD. ASD. Elastic Plastic Stability (buckling). Flexure - Elastic. S=I/c : Section Modulus (Tabulated Value). Flexure - Plastic. Flexure - Plastic. C=T A c f y =A t f y

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## Limit States

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Limit States
• Flexure
• Elastic
• Plastic
• Stability (buckling)
• Shear
• Deflection
• Fatigue
• Supports
Flexure

LRFD

ASD

Elastic

Plastic

Stability (buckling)

Flexure - Elastic

S=I/c : Section Modulus (Tabulated Value)

Flexure - Plastic

C=T

Acfy=Atfy

Ac=At

Mp = Acfy = Atfy = fy (0.5A) a = Mp=Zfy

Mp/ My =Z/S

For shapes that are symmetrical about the axis of bending the plastic and elastic neutral axes are the same

Z=(0.5A)a : Plastic Section Modulus (Tabulated Value)

Flexure - Stability

A beam has failed when:

Mp is reached and section becomes fully plastic

Or

Flange Local Buckling (FLB) Elastically or Inelastically

Web Local Buckling (WLB) Elastically or Inelastically

Lateral Torsional Buckling (LTB) Elastically or Inelastically

bf

tf

h

tw

Lb

Flexure - Stability

Slenderness Parameter

FLB

l=bf/2tf

WLB

l=h/tw

LTB

l= Lb /ry

Mp

Mr

Non

Compact

Compact

Slender

lp

lr

Flexure - Stability

FLB and WLB (Section B5 Table B4.1)

Evaluate Moment Capacity for Different l

FLB

l=bf/2tf

WLB

l=h/tw

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

Mp

Mr

Non

Compact

Compact

Slender

lp

lr

Flexure - Stability

FLB and WLB (Section B5 Table B4.1)

FLB

l=bf/2tf

WLB

l=h/tw

Bending Strength of Compact Shapes

Lateral Torsional Buckling

Bending Strength of Compact Shapes

Laterally Supported Compact Beams

L/4

L/4

L/4

L/4

Elastic Buckling

Cb = factor to account for non-uniform bending within the unbraced length

See AISC table 3-1 p 3.10

Mmax

A

B

C

Elastic Buckling

Cb = factor to account for non-uniform bending within the unbraced length

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

Elastic Buckling

Cb = factor to account for non-uniform bending within the unbraced length

ho = distance between flange centroids = d-tf

Bending Strength of Compact Shapes

Inelastic Buckling

Linear variation between Mp and Mr

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 compact

FLB and LTB

Built-Up welded shapes can have non-compact or slender webs

FLB, WLB, LTB (AISC F4 and F5)

Design of Beams - Limit States
• Flexure
• Elastic
• Plastic
• Stability (buckling)
• Shear
• Deflection
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
Design for Shear

V: Vertical shear at the section under consideration

Q: 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 section

b: width of section at point of interest

Design for Shear

Small width b

d/b=2 Error ~3%

d/b=1 Error ~12%

d/b=1/4 Error 100%

Web fails before flanges

Average Shear Stress

Nominal Strength if no buckling:

Design for Shear

h/tw

Failure of Web due to Shear:

• Yielding
• Inelastic Buckling
• Elastic Buckling

h/tw>260 Stiffeners are required

Appendix F2

Design for ShearAISC Specs G pp 16.1-64

Shear Strength must be sufficient to satisfy

LRFD

resistance factor for shear=0.9

maximum shear based on the controlling combination for factored loads

nominal shear strength

depends on failure mode

ASD

maximum shear based on the controlling combination for service loads

Safety factor

AISC Spec requirements for Shear

Cv depends on whether the limit state is web yielding, web inelastic buckling or web elastic buckling

AISC Spec requirements for Shear

Special Case for Hot Rolled I shapes with

Most W shapes with

AISC Spec requirements for Shear Chapter G

All other doubly and singly symmetric shapes except round HSS

DEFLECTIONSAISC Specs Chapter L

Serviceability Limit State

Limiting Value

Deflections due to Service Loads

<

Governing Building Code, IBC etc

Use deflection formulas in AISC Part 3 Or standard analytical or numerical methods

Calculate due to UNFACTORED (service) loads

Design

Shear is rarely a problem in rolled steel beams

usual practice

Design for Flexure and Check for Shear and Deflections

Or

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
• Assume shape, compute strength, compare with required, revise if necessary or
• Use beam design aids in Part 3 of the Manual
• Check Shear and deflections