Basic principles of steel structures. Dr. Xianzhong ZHAO

Basic principles of steel structures Dr. Xianzhong ZHAO.zhao@mail.tongji.edu.cn www.sals.org.cn 1

Introduction Resistance of cross-section Compression members Outlines Overall stabilit of uniform (solid web) compression members Local buckling of plate element in solid-web compression members Overall stabilit and chord stabilit of built-up compression members Rigidit of compression members Design of aiall loaded compression members

Compression members in structures - truss members -bracing - pinned columns Compression members introduction Structural shapes used in compression members - doubl smmetric cross-section - singl smmetric cross-section - unsmmetric cross-section 3

Compression members possible buckling mode fleural buckling torsional buckling fleural torsional buckling 4

Compression members global buckling of members 5

Compression members local buckling of plates 6

Compression members local buckling of plates 7

Compression members resistance of cross section Resistance of cross section for compression members - same as tension members, but no fracture - no need to check if no large hole eistence = u A n f Design equation in design code A, = γ n fd fd f / R, or f d = f / K σ = A n f d 8

Overall stabilit of solid-web compression members concept of ideal compression member Ideal compression member perfectl centrall loaded straight member Definition (or assumption) of ideal compression member - center of figure (centroid) alwas coincides with its barcenter - the member ais is perfectl straight - perfectl centrall loaded member (ais of force alwas coincides with member ais) 9

Overall stabilit of solid-web compression members differential equation of elastic buckling for ideal member (1) EI EI IV v + v θ = IV u + u θ = IV ωθ GI θ v EI t + u + ( r R) θ = coordinates, principal ais shear center, coordinate of shear center z centroid shear center L 1

Overall stabilit of solid-web compression members differential equation of elastic buckling for ideal member () simultaneous differential equation EI EI IV v + v θ = IV u + u θ = IV ωθ GI θ v EI t + u + ( r R ) θ = z effect of actions reaction or resistance global buckling mode (deformation) - involves fleural and torsional buckling mode - small deformation, but must be considered - fied peripheral shape - second-order nonlinear analsis: equilibrium in a deflected position 11

Overall stabilit of solid-web compression members differential equation of elastic buckling for ideal member (3) EI EI IV v + v θ = IV u + u θ = IV ωθ GI θ v EI t + u + ( r R ) θ = z z Equilibrium in fleural mode - effect of action (1): deflection in -direction equilibrium while bending: M - effect of action (): rotation of centroid moment generated due to offset of aial force after the centroid rotating about shear center: - equilibrium in fleural mode EI v = ( M + M ) M 1 = v θ 1 II I θ II v I v is negative in this sstem M M 1 1

Overall stabilit of solid-web compression members differential equation of elastic buckling for ideal member (4) EI EI IV v + v θ = IV u + u θ = IV ωθ GI θ v EI t + t f L b θ u h u + ( r R ) θ = restrained torsion and warping u =. 5hθ φ = u =. 5hθ = EI u =. 5 = dm dz =. 5 M V M I ω EI hθ / EI hθ ω = Vh =. 5 =.5I M = EI ω h ω θ M + M k = EI h θ ' b ω M T Eample: cantilevered I-shape beam under end torsional moment 3 t f h / 4 ' ' M k = GI t θ ' 13

Overall stabilit of solid-web compression members differential equation of elastic buckling for ideal member (5) EI EI IV v + v θ = IV u + u θ = IV ωθ GI θ v - effect of action (1): bending of ais moment generated: - effect of action (): longitudinal stress and residual stress ' ' ' σ daas ( ) θ as ( ) θ as ( ) σda rθ EI t + Equilibrium in torsional mode I + I r = + + A ' ' θ as ( ) σ da θ R r - equilibrium in torsional mode M + M k = ω M T u + ( r R ) θ = ' ' v, u σ daa ' θ dθ a ' v v z σ da aθ γ = a M ' v θ = v ' z1 ' σ daaθ as () θ adθ / dz σ daa( s) ' θ 14

Overall stabilit of solid-web compression members critical load of ideal member with doubl smmetric section critical buckling load IV = = EI v + v θ EI v + v similarl = π EI / L E o = π EI / L E o = π EA( I / A)/ L = π EA/ λ o =π EA/λ E θ = ( π EIω / Lo θ + GIt + R)/ r =π EA/ λθ Each equation onl has one unknown variable and can be solved separatel, indicating that those three buckling modes are independent from each other. λ = l i o = l I o A λ = l o i = I l o A λ = θ I Ar w o l l + π oθ oθ GIt + R EAr o Which critical load among these three will be the dominant one? 15

Overall stabilit of solid-web compression members critical load of ideal member with doubl smmetric section Effective length factor for different end constraint - End constraint: bending and torsional deformation - Constraint tpes: free end, pinned, fied end - Effective length factor: Table 5-4 in page 11 How about the critical stress? σ E = E A = π E / / λ 16

Overall stabilit of solid-web compression members relationships: critical load, critical stress and slenderness E / E A σ cr p E = π EA λ / p A f Wh modif? Modif due to elasto-plastic λe How to determine? critical buckling load vs. slenderness λ λ e critical buckling stress vs. slenderness λ 17

Overall stabilit of solid-web compression members critical load of ideal member with singl smmetric section decouple one of the simultaneous differential equation (-ais is ais of smmetr) EI EI IV v + v θ = IV u + u θ = IV two buckling mode: fleural buckling and fleural-torsional buckling (FTB) fleural buckling about the unsmmetric ais and FTB about smmetric ais Critical buckling load EIωθ GIt θ v + u = π EI / L E o 1 1 ( + ) (1 ) = 1 r Eω Eω E Eθ E Eθ EI EI u + u = IV v + v θ = + ( r R) θ = EIωθ ( GI t + R r ) θ v = λ ω = 1 ( λ + λ ) + θ 1 Eω = π EA λ ω ( λ + λ ) θ 4(1 r o o ) λ λ θ 18

Overall stabilit of solid-web compression members buckling capacit for compression member with imperfection Imperfection in compression members - Mechanical imperfection residual stress, variable ield strength at each point of a plane - Geometrical imperfection initial out-of-straightness (crookedness), initial non-concentric loading Deformation of perfect column without an initial geometrical imperfection EI v + v= v v m : w E E = π EI / L E o v 19

Overall stabilit of solid-web compression members buckling capacit for compression member with imperfection Effect of initial crookedness on buckling capacit If initial crookedness about major ais eists for doubl smmetric cross-section = Fleural equilibrium equation EI v + v v v m : w v o v om E v if EI v v v or ( o) + = v = v sin( π z/ L) o om v m = vom/( 1 / E) = ( 1 v o / vm) ult < E E

Overall stabilit of solid-web compression members buckling capacit for compression member with imperfection Effect of distribution and amplitude of residual stress on buckling capacit Ideal stress-strain curve and buckling curve residual stress and plastic area after compression stress-strain curve with residual stress and corresponding buckling curve

Overall stabilit of solid-web compression members buckling curve ( 柱子曲线 ) E ( σ cr ) σ cr / f ϕ = σ cr / f stabilit coefficient for aiall loaded compression members p ( f ) λ e Modif due to elastoplastic (including effect of residual stress) λ 1. effect of imperfection λ Critical buckling load (stress) vs. slenderness Buckling curve (stabilit coefficient) for aiall load compression member 3

Overall stabilit of solid-web compression members buckling curve how to get the stabilit coefficient (1) fleural buckling capacit (1) criteria of ield at etra fibre A + Δ W m = f Δ m Δo = 1 E σ A f + (1 + ε ) σ + (1 + ε ) σ o E o E cr = = [ ] f f σ E ε o = AΔ W o (eccentric ratio) σ cr 1 1 1 4 ϕ = = 1 + (1 + ε ) [1 + (1 + )] o ε o f λ λ λ λ λ = π f E (non-dimensional slenderness) applied in Technical code of cold-formed thin-wall steel structures (GB518-) Feasibilit used in thin-wall structures: (a) should not consider plastic development for thin-wall element (b) less effect of residual stress (c) just consider initial crookedness and non-concentric loading 4

Overall stabilit of solid-web compression members buckling curve how to get the stabilit coefficient () fleural buckling capacit () criteria of ultimate buckling capacit numerical simulation method, considering: shape and dimension of section, mechanical properties of steel, distribution and amplitude of residual stress, initial crookedness and torsion of member, non-concentric loading and buckling direction etc. applied in code for design of steel structures (GB517-3) four buckling curves: curve a, b, c, d if if λ.15, λ.15, σ cr ϕ = f σ cr ϕ = f = 1 α λ 1 1 = [( α + α3λ + λ ) λ ( α + α3λ + λ ) 4λ ] 1. ϕ = σ cr / f buckling curve λ 5

Overall stabilit of solid-web compression members buckling curve ( 柱子曲线 ) buckling curves used in GB517 4 buckling curves: curve a, b, c, d select relevant curve b: section tpes, buckling direction, plate thickness and manufacture method refer to Table5-4 in page 15 1. ϕ = σ cr / f calculation of stabilit coefficient Equation method calculate from former equations Chart method slenderness section tpe Anne Table 4-3 to 4-6 in page 371-374 λ buckling curve (reduction factor for relevant buckling mode) 6

Overall stabilit of solid-web compression members buckling resistance (design) of a member practical equation ote: ϕaf ult = cr A = σ cr / f ) d σ ( Af = ϕaf - adopt gross section for buckling resistance - adopt stabilit coefficient specified in GB517-3 procedure of buckling resistance design - ascertain the design value of aial load - calculate the slenderness about two principal ais separatel, or the equivalent slenderness if needed - ascertain the stabilit coefficient - check the buckling resistance of the member 7

Local buckling of plate elements introduction u.c.e: unstiffened compression element local buckling of plate element s.c.e: stiffened or partiall stiffened compression element 8

Local buckling of plate elements differential equation of local buckling of plates a square plate subjected to a uniform compression stress in one direction flat and straight plate with equal thickness ratio of width to thickness greater than 1 uniform compression stress in mid-surface perform as compression members? t b a w t 9

Local buckling of plate elements differential equation of local buckling of plates comparison with compression members perform as compression members! but, fleural buckling 3

Local buckling of plate elements differential equation of local buckling of plates differential equations of local buckling for plate 4 4 w w D( + 4 4 plate rigidit per width + 4 w w ) + = 4 3 Et D = 1(1 μ ) EI IV v + v = EI = Ebh 1 3 uniform compression stress per width 31

Local buckling of plate elements critical local buckling of a simpl supported square plate a simpl supported square plate: boundar conditions deflection at four edges equals zero bending moment at four edges equals zero w = m= 1 n= 1 A mn mπ nπ sin sin a b 4 4 w w D( + 4 4 4 w + ) + 4 w = critical local buckling of a simpl supported square plate D mb n a = π + b a mb cr ( ) numbers of half sine waves m a = ( + ) = k cr n= 1 cr π D mb a π D b a mb b π D = 4 b π E t = 4 1(1 μ ) b critical buckling while n=1 3 a/b 3

Local buckling of plate elements critical local buckling stress & its boundar conditions critical local buckling stress for a simpl supported square plate σ cr π E cr = = 4 / t σe = π E /λ 1(1 μ ) boundar conditions of a plate simpl-supported, fied, free end combination of different constraint at edges constraint in a real plate b t b/t: ratio of width to thickness ( 宽厚比 ) critical local buckling stress for a square plate σ cr? π E b = 4 / 1(1 μ ) t π E t k 1(1 μ ) b buckling coefficient for a plate, pertain to load distribution and boundar conditions 33

Local buckling of plate elements critical local buckling stress & its boundar conditions buckling coefficient for different boundar conditions σ cr π E b = 4 / 1(1 μ ) t π E t k 1(1 μ ) b 1.35 if free end changing to stiffeners The stronger the constraint, the larger the local buckling coefficient, and the bigger the critical local buckling resistance 34

Local buckling of plate elements constraint actions between plate elements & effects of elasto-plastic propert constraint actions between plate elements effects of constraint of adjacent plate elements σ cr π E t = k 1(1 μ ) b σ constraint coefficient between plate elements for a designated plate cr = π E t χ k 1(1 μ ) b modification due to the elasto-plastic propert σ cr π E t = k 1(1 μ ) b σ cr = ψ t ψ t = π E t k 1(1 μ ) b E / t E 35

Local buckling of plate elements post-buckling strength of thin plate elements structural performance of thin plate after local buckling 36

Local buckling of plate elements post-buckling strength of thin plate elements structural performance of thin plate after local buckling Mechanism (phsics): stress redistribution Mechanism (mathematics): large deflection theor, pp13-15 Post-buckling strength will be larger than ield strength? 37

Local buckling of plate elements effective width concept and post-buckling strength Effective width concept for simpl supported plates be b = 1 1 (1. ) λ λ e e increase the bending rigidit of member increase the buckling resistance of member decrease the cross-section resistance become less development of plasticit deteriorate the hsteretic performance λe = 1. 5 b t σ e ke 38

Local buckling of plate elements design criteria of preventing local buckling of plates Criteria 1: o local buckling allowed in an plate σ σ cr cr f ϕ f σ π E 1(1 υ ) t b cr = ψ t χk ( ) ϕf b t 1 ψ t χkπ E 1(1 ) υ ϕf Use ratio of width to thickness of a plate to guarantee the local buckling, pp17 Criteria : local buckling allowed, use post-buckling strength 1. effective width and effective cross-section be b 1 1 = (1. ) λ λ e e λe = 1. 5. cross-section resistance and buckling resistance using effective width fd ϕ A e b t σ e ke 39

Overall stabilit of built-up compression member concept of built-up compression members Wh use built-up compression members? pursuing identical buckling resistance in, direction Tpes of built-up members Chord: built-up section / shaped section two-, three-, four-chord Bracing: lacing form triangular shape battening form rectangular shape Ais: actual ais / virtual ais Overall stabilit about actual ais The same as solid-web compression members chord lacing virtual ais actual ais even for channel shape? 4

Overall stabilit of built-up compression member concept of built-up compression members 41

Overall stabilit of built-up compression member differential equation considering shear deformation differential equation for buckling resistance considering the effect of shear deformation v = v 1 + v v1 = M / EI = v / EI dv dm ' = γ = γ 1V = γ 1 = 1v dz dz γ v = γ 1v γ 1 shear stain under unit shear force 1 v = v + v = v / EI + γ 1v v + v = EI 1 γ ) ( 1 solution of the differential equation π EI π EA (1 γ1cr ) cr = cr = L λ + π EAγ 1 v dv1 dv dz γ 4

Overall stabilit of built-up compression member buckling resistance of built-up members considering shear deformation buckling resistance of built-up members π EA π EA cr = = λ + π EAγ λ Equivalent slenderness λ = λ + π EAγ 1 Shear stain decrease the bending rigidit of member, thus decrease the fleural buckling resistance calculation of equivalent slenderness 1 λ λ How to get the shear strain γ 1? Table 5-5, pp111 λ = λ + A / = λ λ1 λ + 7 A 1 laced compression members battened compression members 43

Local buckling of laced compression members local buckling of plate, chord and lacing Local buckling of compressive plate in chords Design as that of solid-web compression members. buckling of each chord λ.7λ =.7 ma{ λ, λ } 1 ma o buckling of lacing Af d 35 V ma = 85 f V1 Vma t = = n cos α n cos α t γ fd ϕ A t t V 1 V ma 44

Local buckling of battened compression member local buckling of plate, chord and battening Local buckling of compressive plate in chords Design as that of solid-web compression members. buckling of each chord λ1 min{ 4,.5λ ma buckling of battening } V 1 a Structural behavior of battened compression members is similar to that of multi-store rigid frame. Internal force of battening: a T = V1 c a M = V1 Local buckling and strength like deep beam V 1 T c a 45

Overall and local buckling of built-up compression member design procedures distinguish the actual ais and virtual ais overall stabilit design about actual ais design as that of solid-web compression members. overall stabilit design about virtual ais how about channel section? compute the slenderness about virtual ais (note: just consider the chord) compute the equivalent slenderness (note: tpes of built-up members) Table 5-5 or λ π 1 o = λ + (1 + ) 1 kb compute the buckling coefficient of member about virtual ais compute the buckling resistance about virtual ais local buckling resistance local buckling of plates, buckling of chord and bracing k λ 1 46

Rigidit of compression members allowable slenderness concept of rigidit of compression members is the same as that of tension members allowable slenderness more rigorous than allowable slenderness of tension members λ ma [ λ ] [λ ] is about 15~ 47

Overall and local buckling of compression member design procedures Selection of member section Section: requirement of overall stabilit, local buckling and ease to connect Strength: cross-section resistance Overall buckling resistance Solid-web compression members Laced or battened compression members: actual and virtual ais/ chord and bracing local buckling resistance Allowable ratio of width to thickness Effective width and effective area rigidit of compression members 48