Elastic Buckling. Metastable. Schodek fig Schodek fig Schodek fig. 7.3

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1 Illustrations: Daniel L. Schodek: Structures, Schodek fig. 7.1 Elastic Buckling Illustrations: Daniel L. Schodek: Structures, Illustrations: Daniel L. Schodek: Structures, Schodek fig. 7.1 Schodek fig. 7.3 Metastable 1

2 Column Buckling Formula Illustrations: Daniel L. Schodek: Structures, Schodek fig. 7.3, 7.1 P cr = π 2 EI L 2 (Euler Buckling Equation) P cr = 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 STENGTH of the material is! It s all about STIFFNESS Effective Length Factors Schodek fig. 7.5 Schodek fig Illustrations: Daniel L. Schodek: Structures, 2

3 Illustrations: Daniel L. Schodek: Structures, Schodek fig. 7.7 Real Columns Columns: Design Considerations Have significant architectural impact due to how they affect space Establish an organizing grid (structural bay) Define lengths of primary girders and beams framing into them Columns: Design Considerations 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 3

4 A B C D E F G H I J K L Columns: Design Considerations Layout of column grid is important for space planning Can 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 Col Tributary Area = (9 +30 )/2+(21 +7 )/2=546 ft 2 (2 floors) = 1092 ft 2 Columns: Intermediate-Length Intermediate-length columns are normally what is actually used in actual construction Failure mode is a combination of crushing action and buckling action simultaneously Computing 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. 4

5 Illustrations: Daniel L. Schodek: Structures, Schodek fig. 7.3 Columns: Influences on Buckling Capacity Ratio of length to cross-sectional area slenderness ratio End support conditions Intermediate 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-Δ (P-Delta) Effect: As building moves laterally, this induces an eccentric loading, even on columns concentrically loaded. For some materials (e.g. concrete), design for a minimal eccentricity is mandatory by code. Illustrations: Daniel L. Schodek: Structures, 5

6 Columns: Transfer Girder Transfer Grider Column Variations BCE Place, Toronto, Ontario, Canada Santiago Calatrava 6

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

8 Hurva 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 8

9 Stuttgart Airport, Germany Von Gerkan Maarg Stuttgart Airport, Germany Von Gerkan Maarg Student Model by Caroline Addis, Philadelphia University Rose Center for Earth & Space, NYC Polsheck & Partners Stanstead Airport, Essex, England Norman Foster 9

10 Foster & Partners web site Stanstead Airport, Essex, England Norman Foster Stanstead Airport, Essex, England Norman Foster Stanstead Airport, Essex, England Sir Norman Foster Student Model by James Fickes, Philadelphia University Lincoln School Library, Sprekles, CA 10

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

12 Illustration: Understanding Structures, Fuller Moore, WCB/ McGraw-Hill, 1999 Alamillo Bridge, Near Barcelona, Spain Santiago Calatrava Library, Phillips Exeter Academy, Exeter, NU Louis I Kahn 12

13 Sendai Mediatheque, Sendai, Japan Toyo Ito Tokyo International Forum, Tokyo, Japan Rafael Viñoly 13

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