SECTION 7 DESIGN OF COMPRESSION MEMBERS IS 800: 2007 SECTION 7 DESIGN OF COMPRESSION MEMBERS Dr S R Satish Kumar Department of Civil Engineering IIT Madras Chennai 600 036 sr.satishkumar@gmail.com © Dr S R Satish Kumar, IIT Madras

INTRODUCTION TO COLUMN BUCKLING 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

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

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

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

STRENGTH CURVE FOR AN IDEAL STRUT axially loaded initially straight pin-ended column B 1 f y A l c = /r Plastic yield defined by s Elastic buckling ( cr ) defined by p 2 E / ¢ C 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 )

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

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

Residual Stresses

Effect of all features taken together

SECTION 7 DESIGN OF COMPRESSION MEMBERS 7.1 Design Strength 7.2 Effective Length of Compression Members 7.3 Design Details 7.3.1 Thickness of Plate Elements 7.3.2 Effective Sectional Area 7.3.3 Eccentricity for Stanchions and Columns 7.3.4 Splices ]7.4 Column Bases 7.4.1 Gusseted Bases 7.4.2 Slab Bases 7.5 Angle Struts 7.5.1 Single Angle Struts 7.5.2 Double Angle Struts 7.5.3 Continuous Members 7.5.4 Combined Stresses Cont... © Dr S R Satish Kumar, IIT Madras

SECTION 7 DESIGN OF COMPRESSION MEMBERS 7.6 Laced Columns 7.6.1 General 7.6.2 Design of Lacings 7.6.3 Width of Lacing Bars 7.6.4 Thickness of Lacing Bars 7.6.5 Angle of Inclination 7.6.6 Spacing 7.6.7 Attachment to Main Members 7.6.8 End Tie Plates 7.7 Battened Columns 7.7.1 General 7.7.2 Design of Battens 7.7.3 Spacing of Battens 7.7.4 Attachment to Main Members 7.8 Compression Members Composed of Two Components Back-to-Back end © Dr S R Satish Kumar, IIT Madras

INTRODUCTION Dominant factors affecting ultimate strength of columns subjected to axial compressive loading: Slenderness ratio (/r) Material yield stress (fy) Dominant factors affecting ultimate strength of practical columns: Initial imperfection Eccentricity of loading Residual stresses Lack of defined yield point Strain hardening © Dr S R Satish Kumar, IIT Madras

Typical column design curve INTRODUCTION c fy Test data (x) from collapse tests on practical columns Euler curve Design curve Slenderness  (/r) x x x x x x x x 200 100 50 100 150 Typical column design curve © Dr S R Satish Kumar, IIT Madras

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

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

7.1 DESIGN STRENGTH 7.1.2 The design compressive strength of a member is given by  = 0.5[1+ ( - 0.2)+ 2] fcd = the design compressive stress, λ = non-dimensional effective slenderness ratio, fcc = Euler buckling stress = 2E/(KL/r)2 = imperfection factor as in Table 7  = stress reduction factor as in Table 8 © Dr S R Satish Kumar, IIT Madras

Table 10 Buckling Class of Cross-sections Limits Buckling about axis Buckling Curve Rolled I-Sections h/b > 1.2 : tf 40 mm  40 < tf <100 z-z y-y a b  b c Welded I-Section tf <40 mm tf >40 mm  z-z  c d Hollow Section Hot rolled Cold formed Any Welded Box Section, built-up Generally Channel, Angle, T and Solid Sections © Dr S R Satish Kumar, IIT Madras

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

7.2 Effective Length of Compression Members (Table 11) Boundary Conditions Schematic represen -tation   Effective Length At one end At the other end Translation Rotation Restrained Free 2.0L 1.0L 1.2L 0.8L 0.65 L © Dr S R Satish Kumar, IIT Madras

7.4 COLUMN BASES 7.4.2 Gusseted Bases 7.4.3 Slab Bases b a 2c+tf 2c+tw © Dr S R Satish Kumar, IIT Madras

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 fcd values along both major and minor axes. Calculate design compressive strength Pd = (fcd A). Check P < Pd © Dr S R Satish Kumar, IIT Madras

BEHAVIOUR OF ANGLE COMPRESSION MEMBERS 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) V U © Dr S R Satish Kumar, IIT Madras

Basic compressive strength curve 7.5 ANGLE STRUTS Basic compressive strength curve Curve C of Eurocode 3 Slenderness Ratio: concentric loading kL/r Single leg Connection (kl/r)eq Equivalent normalised slenderness ratio Where, k1, k2, k3 are constants to account for different end conditions and type of angle. © Dr S R Satish Kumar, IIT Madras

L = laterally unsupported length of the member Where L = laterally unsupported length of the member rvv = radius of gyration about the minor axis b1, b2 = width of the two legs of the angle t = thickness of the leg ε = yield stress ratio ( 250/fy)0.5 © Dr S R Satish Kumar, IIT Madras

No. of bolts at the each end connection 7.5 ANGLE STRUTS 7.5.1.2 Loaded through one leg k1, k2, k3 = constants depending upon the end condition (Table 12) No. of bolts at the each end connection Gusset/Connec -ting member Fixity† k1 k2 k3 > 2 Fixed 0.20 0.35 20 Hinged 0.70 0.60 5 1 0.75 1.25 0.50 60 Design ? © Dr S R Satish Kumar, IIT Madras

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

LACED AND BATTENED COLUMNS 7.6.1.5 The effective slenderness ratio, (KL/r)e = 1.05 (KL/r)0, to account for shear deformation effects. 7.7.1.4 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. © Dr S R Satish Kumar, IIT Madras

Thank You SUMMARY 7.1 Design Strength 7.2 Effective Length of Compression Members 7.3 Design Details 7.4 Column Bases 7.5 Angle Struts 7.6 Laced Columns 7.7 Battened Columns 7.8 Compression Members Composed of Two Components Back-to-Back Thank You © Dr S R Satish Kumar, IIT Madras