Fundamentals of Structural Design Part of Steel Structures Civil Engineering for Bachelors 133FSTD Teacher: Zdeněk Sokol Office number: B619 1 Syllabus of lectures 1. Introduction, history of steel structures, the applications and some representative structures, production of steel 2. Steel products, material properties and testing, steel grades 3. anufacturing of steel structures, welding, mechanical fasteners 4. Safety of structures, limit state design, codes and specifications for the design 5. Tension, compression, buckling 6. Classification of cross sections, bending, shear, serviceability limit states 7. Buckling of webs, lateral-torsional stability, torsion, combination of internal forces 8. Fatigue 9. Design of bolted and welded connections 10. Steel-concrete composite structures 11. Fire and corrosion resistance, protection of steel structures, life cycle assessment 2 1
Scope of the lecture Stability of webs Web stiffeners Shear loads Local concentrated loads Torsion Combined actions 3 Stability (buckling) of webs Webs are loaded By compression This is introduced as effective sections properties of Class 4 cross-sections, but not explained in details in FSTD By shear By concentrated local load 4 2
Beam web stiffeners Beam web stiffeners Beam web stiffeners I section beam with stiffeners welded by fillet welds Stiffeners of composite box girder for steel-concrete composite bridge 5 Beam web stiffeners Typical shapes of beam web stiffeners 6 3
Scope of the lecture Stability of webs Web stiffeners Shear loads Local concentrated loads Torsion Combined actions 7 Shear buckling of beam web Shear buckling of beam web 8 4
Principle of tension field Fictive truss is used as model for the beam just before the collapse when the shear is resisted by tensile fields (now replaced by diagonals of the truss) and the stiffeners of the beam web are loaded in compression (now replaced by vertical elements of the truss) Tension field in the beam web Fictive truss with diagonals loaded in tension 9 Web loaded by shear Buckling of the web is in principle similar to buckling of elements in compression Two cases need to be considered: Perfect plate critical stress cr can be obtained Real plate imperfections play important role and should be considered However, behaviour of web after buckling occurs is significantly different The effect of these parameters needs to be considered - Imperfections lead to web buckling and reduce the shear resistance + Post-critical reserve caused by tensile fields which reduce the effect of imperfections (actually reduce the amplitude of bow-shaped deformation of the web) The tensile fields reduce the effect of buckling and increase the resistance of the web, therefore higher Critical stress of the web cr resistance can be observed than resistance based on critical stress Real stress in the web cr Real and critical stress of the beam web 10 5
Shear buckling of webs when to check It is necessary to consider web buckling in shear and therefore to reduce the shear resistance in the following cases: web without stiffeners d t w web with transverse stiffeners only d tw 69 30 k for more complicated stiffeners pattern, the method is given in Eurocode Beam web stiffened by longitudinal and transverse stiffeners 12 Shear resistance of slender web The shear resistance with respect to web buckling is given by d tw ba Vba,Rd 1 where ba is function of the web slenderness λ w w f y cr 3 d t 37, 4 t w w The strength ba is equal to ba f y 3 when w 0, 8 ba f y 1 0, 425 w 0, 8 when 3 0, 8 w 1, 2 ba f y 0, 9 3 when 1, 2 w w ba k 13 6
Scope of the lecture Stability of webs Web stiffeners Shear loads Local concentrated loads Torsion Combined actions 14 Local concentrated loads (transverse forces) ormally, web stiffener is designed at locations where local concentrated load is presented (column, connected beam, etc.) In some situations, the load can not transferred into the stiffener (wheel of bridge crane) The column delivers a concentrated load to the beam and a bearing stiffener is used on the web 15 7
Local concentrated loads (transverse forces) Web crushing at a support point of thin-walled cold formed C section, no stiffeners can be used here 16 Local concentrated loads Resistance check to concentrated loads include: Buckling of the web below the concentrated load Combination of stresses in web at the vicinity of load 2 2 2 y x z x z 3 1 where x, z are axial stresses at perpendicular directions (sign included) x is the stress from bending moment z is the stress from concentrated load distributed to length s s is the shear stress f Distribution length s s 17 8
Scope of the lecture Stability of webs Web stiffeners Shear loads Local concentrated loads Torsion Combined actions 18 Torsion Only torsion is quite rare but combination of torsion and bending is more frequent Torsion occurs when the load plane does not pass through the shear centre Shear center of typical steel sections It is always better to avoid torsion when possible When torsion can not be avoided, hollow sections should be preferably used e F Ed F Ed Bending and torsion of I-sections beam 19 9
Torsion cross-sections Open cross-sections They have low torsional stiffness, therefore are not suitable for high torsion moments Significant stresses (both shear and axial) are created which must be considered for the resistance check St.Venant + warping torsion occurs, resulting in shear t + w and axial stress w appear ( x = T t + T w ) Exception: when all parts intersect at single point (L, T sections), only St. Venant torsion and only t occurs ( x = T t ) Hollow cross-sections They have high torsional stiffness and are suitable to transfer torsion St.Venant occurs, only shear stress t appears 20 Open cross-sections, St. Venant torsion Shear stress Tt t,i t I t where T t torsion moment I t torsion constant t thickness of the element The torsion constant is 1 3 It bi ti 3 b i width t i thickness of that part of the element cross-section shape coefficient ( 1) 21 10
Open cross-sections, warping torsion Axial stress w B I w w where I w warping moment of inertia [m 6 ] B bimoment [km 2 ] w warping coordinate [m 2 ] Axial stress patterns for I and channel sections 22 Bending and torsion of U cross-section Bending Torsion 23 11
Bi-axial bending and torsion of I cross-section Bending y Bending z Torsion 24 Hollow cross-sections Only shear stress occurs t,i 1 ti Q Tt t i where T t torsion moment t i thickness of the element area enclosed by the cross-section A s 2 A s A s 25 12
Scope of the lecture Stability of webs Shear loads Local concentrated loads Torsion Combined actions 27 Combined actions Bi-axial bending Bending + tension Bending + compression 28 13
Bi-axial bending - review y,ed c,y,rd z,ed c,z,rd 1 y,ed, z,ed c,y,rd, c,z,rd bending moments acting about y and z axes bending moment resistances It is possible to take into account = = 1 (conservative approach) Accurate method for various cross section shapes (i.e. the values of and ) is given in Eurocode c,z,rd z c,y,rd y Interaction diagram for bi-axial bending 29 Bending + tension Example: Tension chord of truss with inter-nodal load Class 1,2 sections plastic behaviour can be considered 2 Ed t,ed 1 pl,rd pl,rd Class 3 sections elastic behaviour is be considered, the stresses from bending and axial force are combined A t,ed W Ed el,y f y 0 f A t,ed y 0 Ed f Wel,y y 0 1 30 14
Bending + tension Bi-axial bending + tension t,ed pl,rd y,ed pl,y,rd z,ed pl,z,rd 1 31 Bending + compression Typical cases: columns with lateral load columns with eccentric load frames compression chord of truss with inter-nodal load 32 15
Bending + compression 33 Bending + compression 34 16
Bending + compression Second-order effect should be included Derivation as for buckling resistance of elements loaded in compression initial curvature primary moments Ed secondary moments Ed e omenty: L e 0 e e 35 Bending + compression For element pinned at both ends, the maximum bending moment on the element (including the second order effect) is equal to: 1 max Ed Ed e Ed Ed 1 cr when Ed cr, the moment max aximum stress in the element is or max A Ed max W Ed Ed A f y 1 Ed W f cr f y y 1 factor k yy introducing the second order effect 36 17
Effect of moment pattern Effects of + add k yy > 1 Effects of + eliminate k yy < 1 37 Check according to E 1993-1-1 2 conditions are considered: 1. In plane buckling y,ed Δ Ed k yy χ y Rk χlt γ γ 1 1 y,rk y,ed 1 2. Out of plane buckling Ed χ z γ 1 Rk k zy χ y,ed LT Δ γ 1 y,rk y,ed 1 Both formulas have to be fulfilled W (and y,rk ) should be taken according to class of the cross-section 38 18
Factors k ij second order effect There is influence of some parameters on the k ij factors Effect of applied axial force Ed e Effect of applied moment pattern Ed along L cr,y (as seen recently) when Ed is small or 1 then k yy = 1,0, k zy = 1,0 (no second order effect is presented) 39 Thank you for your attention 40 19