columns & buckling

1Elastic Buckling

Illustrations: Daniel L. Schodek: Structures, fifth edition; Pearson Prentice-Hall, 2004

Schodek fig. 7.1


Schodek fig. 7.1

Metastable


Schodek fig. 7.3

2Column Buckling Formula

Pcr = 2EIL2

Pcr = Critical force that initiates buckling failure

E = Modulus of elasticity of material (material stiffness)

I = Moment of inertia (geometric stiffness)

L = Unbraced column length

Notice it does NOT matter what the STENGTHof the material is!

Its all about STIFFNESS

(Euler Buckling Equation)


Schodek fig. 7.3, 7.1

Schodek fig. 7.5

EffectiveLengthFactors

Illustrations: Daniel L. Schodek: Structures, fifth edition; Pearson Prentice-Hall, 2004Schodek fig. 7.10

3Illustrations: Daniel L. Schodek: Structures, fifth edition; Pearson Prentice-Hall, 2004

Schodek fig. 7.7

Real Columns

Columns: Design Considerations

Have significant architectural impact due to how they affect spaceEstablish an organizing grid (structural bay)Define lengths of primary girders and beams framing into them


Most critical structural element Typically no redundancy Remove a column and all above it that it

supports will fall Controlled demolition by selected removal

of columns

Loads to columns typically computed by tributary area

4A B C D E F G H I J K L

23

1

45

67

89

10

30 9 30 9 30 8 11 8 16 16 8

21

25

7

7

21

Col Tributary Area = (9+30)/2+(21+7)/2=546 ft2(2 floors) = 1092 ft2


Layout of column grid is important for space planningCan be difficult to coordinate with varying needs if different floors, especially if parking is below Sometimes use transfer girders to carry

floors above Expensive option and requires much

deeper member than regular girder

Columns:Intermediate-Length

Intermediate-length columns are normally what is actually used in actual constructionFailure mode is a combination of crushing action and buckling action simultaneouslyComputing the allowable axial stress for these involves complex equations

Columns:Intermediate-Length

Fortunately, the equations can be simplified to a table lookup based on the column slenderness ration. (for steel, see table C-36)For pinned-ended columns, the capacity can be looked up directly in tables that relate member size to load capacity relative to the height of the column.

5Illustrations: Daniel L. Schodek: Structures, fifth edition; Pearson Prentice-Hall, 2004

Schodek fig. 7.3 Columns: Influences on Buckling Capacity

Ratio of length to cross-sectional area slenderness ratioEnd support conditionsIntermediate bracing 2% rule of thumb for bracing design: For

most cases bracing can be achieved with only 2% of the compressive force in the column!

Elastic buckling is not an issue of strength. It is rather one of stiffness.

Columns: Influences on Buckling Capacity

Eccentric loading dramatically reduces capacity Creates a moment at the top of column that causes

stress needing to be resisted in addition to axial load.

P- (P-Delta) Effect: As building moves laterally, this induces an

eccentric loading, even on columns concentricallyloaded.

For some materials (e.g. concrete), design for a minimal eccentricity is mandatory by code.


6Columns: Transfer Girder

Transfer Grider Column Variations

BCE Place, Toronto, Ontario, Canada Santiago Calatrava

7Bath House, Jewish Community Center, Trenton, New Jersey Louis Kahn

8Hurva Synagogue, Jerusalem, Israel (unbuilt project) Louis Kahn Dulles International Airport Terminal, Chantilly, Virginia Eero Saarinen

Illustration: Understanding Structures, Fuller Moore, WCB/ McGraw-Hill, 1999 Stuttgart Airport, Germany Von Gerkan Maarg

9Stuttgart Airport, Germany Von Gerkan Maarg

Stuttgart Airport, Germany Von Gerkan MaargStudent Model by Caroline Addis, Philadelphia University

Rose Center for Earth & Space, NYC Polsheck & Partners

Stanstead Airport, Essex, England Norman Foster

10



Foster & Partners web site

Stanstead Airport, Essex, England Sir Norman FosterStudent Model by James Fickes, Philadelphia University Lincoln School Library, Sprekles, CA

11

Hong Kong & Shanghai Bank, Hong Kong, China Norman Foster

Illustration: Understanding Structures, Fuller Moore, WCB/ McGraw-Hill, 1999

Illustration: Understanding Structures, Fuller Moore, WCB/ McGraw-Hill, 1999 Illustration: Understanding Structures, Fuller Moore, WCB/ McGraw-Hill, 1999

12

Alamillo Bridge, Near Barcelona, Spain Santiago Calatrava

Illustration: Understanding Structures, Fuller Moore, WCB/ McGraw-Hill, 1999

Library, Phillips Exeter Academy, Exeter, NU Louis I Kahn

13

Sendai Mediatheque, Sendai, Japan Toyo Ito

Tokyo International Forum, Tokyo, Japan Rafael Violy

columns & buckling

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