SECTION 7 DESIGN OF COMPRESSION MEMBERS 1
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
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 3
ELASTIC BUCKLING OF EULER Assumptions: COLUMN Material of strut - homogenous and linearly elastic No imperfections (perfectly straight) No eccentricity of loading No residual stresss 4
ELASTIC BUCKLING OF EULER COLUMN Pcr The governing differential equation is y x l P + y EI d y cr. dx Lowest value of the critical load Pcr π E I σ cr = = A Al π E r π E σ cr = = l = π E ( l / r) λ = 0 P cr π EI = l 5
STRENGTH CURVE FOR AN IDEAL STRUT axially loaded initially straight pin-ended column 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 (λ λ c ) f f y A B 1 C Plastic yield defined by σ f = f y A Elastic buckling ( σ cr ) defined by π E / λ B λ c λ = l /r 6
STRENGTH CURVE FOR AN IDEAL STRUT σ f / f y Plastic yield 1.0 Elastic buckling 1.0 λ = ( f y / σ cr ) 1/ Strength curve in a non-dimensional form 7
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 8
Residual Stresses Residual stresses in web Residual stresses in flanges Residual stresses distribution (no applied load) 9
Effect of all features taken together σ a Data from collapse tests f y Theoretical elastic bucklin Lower bound curve π (E/f y ) 1/ l/r 10
SECTION 7 DESIGN OF COMPRESSION MEMBERS 7.1 Design Strength 7. Effective Length of Compression Members 7.3 Design Details 7.3.1 Thickness of Plate Elements 7.3. 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. Slab Bases 7.5 Angle Struts 7.5.1 Single Angle Struts 7.5. Double Angle Struts 7.5.3 Continuous Members 7.5.4 Combined Stresses Cont... 11
SECTION 7 DESIGN OF COMPRESSION MEMBERS 7.6 Laced Columns 7.6.1 General 7.6. 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. 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 1
INTRODUCTION σ c f y 00 x x x x x x x x x x x Test data (x) from collapse tests on practical columns Euler curve 100 x x x x x x x x Design curve 50 100 150 Slenderness λ (l/r) Typical column design curve 13
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) 14
Cross Section Shapes for Built - up or fabricated Compression Members (a) Box Section (b) Box Section (c) Box Section (d) Plated I Section (e) Built - up I Section (f) Built-up Box Section 15
7.1 DESIGN STRENGTH 7.1. The design compressive strength of a member is given by P d = A e f cd f cd = f y / γ m0 0.5 φ + φ λ = χ f y / γ m0 f y / γ m0 φ = 0.5[1+α (λ - 0.)+ λ ] f cd = the design compressive stress, λ = non-dimensional effective slenderness ratio, f cc = Euler buckling stress = π E/(KL/r) α = imperfection factor as in Table 7 χ = stress reduction factor as in Table 8 f ( KL = ) π E y f cc f y r 16
Table 10 Buckling Class of Cross-sections Cross Section Limits Buckling about axis Buckling Curve Rolled I-Sections h/b > 1. : t f 40 mm 40 < t f <100 z-z y-y z-z y-y a b b c Welded I-Section t f <40 mm z-z y-y b c t f >40 mm z-z c y-y d Hollow Section Hot rolled Cold formed Any Any a b Welded Box Section, built-up Generally Any Any b c Channel, Angle, T and Solid Sections Any c 17
7.1 DESIGN STRENGTH 1 Buckling Curves fcd/fy 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0.1 0 a b c d 0 0.5 1 1.5.5 3 Lamda TABLE 7.1 IMPERFECTION FACTOR, α Buckling Class a b c d α 0.1 0.34 0.49 0.76 18
7. Effective Length of Compression Members (Table 11) Boundary Conditions At one end At the other end Translation Rotation Translation Rotation Schematic represen -tation Effective Length Restrained Restrained Free Free Free Restrained Restrained Free.0L Restrained Free Restrained Free 1.0L Restrained Restrained Free Restrained 1.L Restrained Restrained Restrained Free 0.8L Restrained Restrained Restrained Restrained 0.65 L 19
7.4. Gusseted Bases 7.4.3 Slab Bases 7.4 COLUMN BASES t = γ / f > s.5w( a 0.3b ) m0 y t f a b 0
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 1
BEHAVIOUR OF ANGLE COMPRESSION MEMBERS Angles under compression Concentric loading - Axial force 1. Local buckling. Flexural buckling about v-v axis V 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,, 3 & Unequal 1, 3) U U V U V V U
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 λ = k + k λ + e 1 vv k λ 3 φ Where, k 1, k, k 3 are constants to account for different end conditions and type of angle. 3
λ vv = ε KL rvv π E 50 λ φ = ε ( b + b ) 1 E 50 π t Where L = laterally unsupported length of the member r vv = radius of gyration about the minor axis b 1, b = width of the two legs of the angle t = thickness of the leg ε = yield stress ratio ( 50/f y ) 0.5 4
7.5 ANGLE STRUTS 7.5.1. Loaded through one leg k 1, k, k 3 = constants depending upon the end condition (Table 1) λ e = k1 + kλvv + k3λφ No. of bolts at the each end connection Gusset/Connec -ting member Fixity k 1 k k 3 > 1 Fixed 0.0 0.35 0 Hinged 0.70 0.60 5 Fixed 0.75 0.35 0 Hinged 1.5 0.50 60 5
DESIGN CONSIDERATIONS FOR LACED AND BATTENED COLUMNS (a) Single Lacing (b) Double Lacing (c) Battens Built-up column members 6
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. 7