© Dr S R Satish Kumar, IIT Madras1 SECTION 7 DESIGN OF COMPRESSION MEMBERS

2 INTRODUCTION TO COLUMN BUCKLING Introduction Elastic buckling of an ideal column Strength curve for an ideal column Strength of practical column Concepts of effective lengths Torsional and torsional-flexural buckling Conclusions

3 INTRODUCTION Compression members: short or long Squashing of short column Buckling of long column Steel members more susceptible to buckling compared to RC and PSC members

4 ELASTIC BUCKLING OF EULER COLUMN Assumptions: Material of strut - homogenous and linearly elastic No imperfections (perfectly straight) No eccentricity of loading No residual stresss

5 The governing differential equation is ELASTIC BUCKLING OF EULER COLUMN Lowest value of the critical load

6 axially loaded initially straight pin-ended column B 1 f f y A c = /r Plastic yield defined by  f f=y Elastic buckling (  cr ) defined by   2 E /  2 A C B STRENGTH CURVE FOR AN IDEAL STRUT Column fails when the compressive stress is greater than or equal to the values defined by ACB. AC  Failure by yielding (Low slenderness ratios) CB  Failure by bucking (  c )

7  f /f y 1.0 = (f y /  cr ) 1/2 1.0 Elastic buckling Plastic yield Strength curve in a non-dimensional form STRENGTH CURVE FOR AN IDEAL STRUT

8 FACTORS AFFECTING STRENGTH OF A COLUMN IN PRACTICE: Effect of initial out of straightness Effect of eccentricity of applied loading Effect of residual stress Effect of a strain hardening and the absence of clearly defined yield point Effect of all features taken together

9 Residual Stresses

10 Effect of all features taken together

© Dr S R Satish Kumar, IIT Madras11 SECTION 7 DESIGN OF COMPRESSION MEMBERS 7.1Design Strength 7.2Effective Length of Compression Members 7.3Design Details 7.3.1Thickness of Plate Elements 7.3.2Effective Sectional Area Eccentricity for Stanchions and Columns Splices ]7.4Column Bases Gusseted Bases Slab Bases 7.5 Angle Struts Single Angle Struts Double Angle Struts Continuous Members Combined Stresses Cont...

© Dr S R Satish Kumar, IIT Madras12 SECTION 7 DESIGN OF COMPRESSION MEMBERS 7.6Laced Columns General Design of Lacings Width of Lacing Bars Thickness of Lacing Bars Angle of Inclination 7.6.6Spacing Attachment to Main Members End Tie Plates 7.7 Battened Columns 7.7.1General Design of Battens Spacing of Battens Attachment to Main Members 7.8 Compression Members Composed of Two Components Back-to-Back end

© Dr S R Satish Kumar, IIT Madras13 INTRODUCTION Typical column design curve cc fyfy Test data (x) from collapse tests on practical columns Euler curve Design curve Slenderness ( /r) x x x x x x x x x x

© Dr S R Satish Kumar, IIT Madras14 (a) Single Angle (b) Double Angle (c) Tee (d) Channel (e) Hollow Circular Section (CHS) (f) Rectangular Hollow Section (RHS) Cross Section Shapes for Rolled Steel Compression Members

© Dr S R Satish Kumar, IIT Madras15 (b) Box Section(c) Box Section (d) Plated I Section(e) Built - up I Section(f) Built-up Box Section (a) Box Section Cross Section Shapes for Built - up or fabricated Compression Members

© Dr S R Satish Kumar, IIT Madras The design compressive strength of a member is given by 7.1 DESIGN STRENGTH  = 0.5[1+  ( - 0.2)+ 2 ] f cd = the design compressive stress, λ = non-dimensional effective slenderness ratio, f cc = Euler buckling stress =  2 E/(KL/r) 2  = imperfection factor as in Table 7  = stress reduction factor as in Table 8

© Dr S R Satish Kumar, IIT Madras17 Cross SectionLimitsBuckling about axis Buckling Curve Rolled I-Sectionsh/b > 1.2 : t f 40 mm 40 < t f <100 z-z y-y z-z y-y a b c Welded I-Sectiont f <40 mm t f >40 mm z-z y-y z-z y-y b c d Hollow SectionHot rolled Cold formed Any abab Welded Box Section, built-up GenerallyAny bcbc Channel, Angle, T and Solid Sections Anyc Table 10 Buckling Class of Cross-sections

© Dr S R Satish Kumar, IIT Madras DESIGN STRENGTH a b c d

© Dr S R Satish Kumar, IIT Madras Effective Length of Compression Members (Table 11) Boundary Conditions Schematic represen -tation Effective Length At one endAt the other end TranslationRotationTranslationRotation Restrained Free 2.0L FreeRestrained Free RestrainedFree1.0L Restrained FreeRestrained 1.2L Restrained Free 0.8L Restrained 0.65 L Restrained

© Dr S R Satish Kumar, IIT Madras COLUMN BASES Gusseted Bases Slab Bases a b

© Dr S R Satish Kumar, IIT Madras21 STEPS IN THE DESIGN OF AXIALLY LOADED COLUMNS Design steps: Assume a trial section of area A = P/150 Make sure the section is at least semi-compact ! Arrive at the effective length of the column. Calculate the slenderness ratios. Calculate f cd values along both major and minor axes. Calculate design compressive strength P d = (f cd A). Check P < Pd

© Dr S R Satish Kumar, IIT Madras22 Angles under compression –Concentric loading - Axial force 1. Local buckling 2. Flexural buckling about v-v axis 3. Torsional - Flexural buckling about u-u axis –Eccentric loading - Axial force & bi-axial moments –Most practical case –May fail by bi-axial bending or FTB –(Equal 1, 2, 3 & Unequal 1, 3) BEHAVIOUR OF ANGLE COMPRESSION MEMBERS V V U U V V U U

© Dr S R Satish Kumar, IIT Madras ANGLE STRUTS Basic compressive strength curve Curve C of Eurocode 3 Slenderness Ratio: concentric loadingkL/r Single leg Connection(kl/r) eq Equivalent normalised slenderness ratio Where, k 1, k 2, k 3 are constants to account for different end conditions and type of angle.

© Dr S R Satish Kumar, IIT Madras24 Where L = laterally unsupported length of the member r vv = radius of gyration about the minor axis b 1, b 2 = width of the two legs of the angle t = thickness of the leg ε = yield stress ratio ( 250/f y ) 0.5

© Dr S R Satish Kumar, IIT Madras ANGLE STRUTS Loaded through one leg k 1, k 2, k 3 = constants depending upon the end condition (Table 12) No. of bolts at the each end connection Gusset/Connec -ting member Fixity † k1k1 k2k2 k3k3 > 2 Fixed Hinged Fixed Hinged Design ?

© Dr S R Satish Kumar, IIT Madras26 DESIGN CONSIDERATIONS FOR LACED AND BATTENED COLUMNS (a) Single Lacing (b) Double Lacing(c) Battens Built-up column members

© Dr S R Satish Kumar, IIT Madras The effective slenderness ratio, (KL/r) e = 1.05 (KL/r) 0, to account for shear deformation effects The effective slenderness ratio of battened column, shall be taken as 1.1 times the (KL/r) 0, where (KL/r) 0 is the maximum actual slenderness ratio of the column, to account for shear deformation effects. LACED AND BATTENED COLUMNS

© Dr S R Satish Kumar, IIT Madras28 Dr S R Satish Kumar Department of Civil Engineering IIT Madras Chennai