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21-11solution:Known:maintruss,bottomchord,H-shapecrosssection,wedding,geometricallengthofbottomchordisl=8m.Q235steel,high-strengthbolts,d0=24mm.Variableaxialforce,N1=3319.4kN(tension),N2=772kN(tension).FatigueloadingmodelI:N1f=1399kN(tension),N2f=425kN(tension)Unknown:checkthestrengthandfatigue.Solve:bylookinguptable,thedesignvalueofstrengthofsteel:fd=275kN.1.Determinethedimensionofcrosssectionofbottomchord.1).TherequiredgrossareaAmandnetareaAn𝐴𝑚=1.15𝑁𝑚𝑎𝑥𝑓𝑑=1.15×3319.4×103275=13881𝑚𝑚2𝐴𝑛≥𝑁𝑚𝑎𝑥𝑓𝑑=3319.4×103275=12071𝑚𝑚22).Determinethedimensionofthicknessofweb.Selectingtw=16mm,thicknessofflangetf=20mm,widthofwebbw=440mm,widthofflangeh=460mm.thegrossareaactuallyis𝐴𝑚=𝑏𝑤𝑡𝑤+2ℎ𝑡𝑓=440×16+2×20×460=25440𝑚𝑚213881𝑚𝑚2OK.Singlefacefrictionnf=1,µ=0.35,pre-tensionPf=190MPa.Shearingcapacityperhigh-strengthbolt:𝑁𝑣𝑢𝑏=0.9𝑛𝑓µ𝑃𝑓=0.9×1×0.35×190×103=59.85𝑘𝑁Numberofbolts.𝑛=𝑁𝑚𝑎𝑥𝑁𝑣𝑢𝑏=3319.4×10359850=55.46𝑚𝑚2Selectingn=64,arrange32boltslinebylineperflange,4lines.Thenetareaactuallyis𝐴𝑛=𝐴𝑚−2×4×24×20=25440−3840=21600𝑚𝑚212071𝑚𝑚2OK2.Checkthestrength.N′=𝑁max(1−0.5𝑛1𝑛)=3319.4×(1−0.5×432)=3112𝑘𝑁σ′=N𝐴𝑛=3112×10321600=144.1𝑀𝑃𝑎𝑓𝑑=275𝑀𝑃𝑎OK3.Checkthestiffness.Inertiamomentofx-axisandy-axis𝐼𝑥=2×112×20×4603=324453333.3𝑚𝑚4𝐼𝑦=112×16×4403+2×460×20×2302=1086938667𝑚𝑚4Thecorrespondinggyrationradius𝑖𝑥=√𝐼𝑥𝐴𝑚=√324453333.325440=35.7𝑚𝑚4𝑖𝑦=√𝐼𝑦𝐴𝑚=√108693866725440=206.7𝑚𝑚4Thecorrespondingslendernessratio𝜆𝑥=𝑙𝑜𝑥𝑖𝑥=800035.7=224[𝜆]=130Dissatisfy𝜆𝑦=𝑙𝑜𝑦𝑖𝑦=8000206.7=38.7[𝜆]=130OK4.Checkthefatiguelimitstate.Δσ𝐷=0.737Δσ𝐶=0.737×90=66.33𝑀𝑃𝑎Δσ𝐷𝛾𝑀𝑓=66.331.35=49.13𝑀𝑃𝑎σ𝑝𝑚𝑎𝑥=𝑁𝑃𝑚𝑎𝑥𝐴𝑚=1399×10325440=54.99𝑀𝑃𝑎σ𝑝𝑚𝑖𝑛=𝑁𝑃𝑚𝑖𝑛𝐴𝑚=425×10325440=16.71𝑀𝑃𝑎NormalstressamplitudeΔσ𝑝=𝛾𝑀𝑓(1+Δϕ)(σ𝑝𝑚𝑎𝑥−σ𝑝𝑚𝑖𝑛)=1×1×(54.99−16.71)=38.28𝑀𝑃𝑎𝛾𝐹𝑓Δσ𝑝=38.28𝑀𝑃𝑎Δσ𝐷𝛾𝑀𝑓=49.13𝑀𝑃𝑎OKConclusion:21-12solution:a)Known:H-shapecrosssection,wedding,axialcompression,lox=loy=8m.Q345steel,N=2650kN(compression).Unknown:checkthestrength,stiffnessandstabilityofsteelmember.Solve:bylookinguptable,thedesignvalueofstrengthofsteel:fd=275kN.1.Checkthestrength.1).flange:flangewith3simply-supportededgesandonefreeedge,elasticbucklingcoefficientKf=0.425.𝜆𝑝𝑓̅̅̅̅̅=1.05(ℎ2𝑡𝑓)√𝑓𝑦𝐸𝐾𝑓=1.05×460220×√3452.06×105×0.425=0.7580.4Theimpactoflocalbucklingneedconsidering.𝜀𝑜𝑓=0.8(𝜆𝑝𝑓̅̅̅̅̅−0.4)=0.8×(0.758−0.4)=0.2864𝜌𝑓=12[1+1+𝜀𝑜𝑓𝜆𝑝𝑓̅̅̅̅̅2−√(1+1+𝜀𝑜𝑓𝜆𝑝𝑓̅̅̅̅̅2)2−4𝜆𝑝𝑓̅̅̅̅̅2]=12×[1+1+0.28640.7582−√(1+1+0.28640.7582)2−40.7582]=0.68Theeffectivewidthofflange𝑏𝑒𝑓=𝜌𝑓ℎ=0.68×460=312.8𝑚𝑚2).web:webwith4simply-supportededges,elasticbucklingcoefficientKf=4.𝜆𝑝𝑤̅̅̅̅̅=1.05(𝑏𝑡𝑤)√𝑓𝑦𝐸𝐾𝑤=1.05×42014×√3452.06×105×0.425=0.7060.4Theimpactoflocalbucklingneedconsidering.𝜀𝑜𝑤=0.8(𝜆𝑝𝑤̅̅̅̅̅−0.4)=0.8×(0.706−0.4)=0.2448𝜌𝑤=12[1+1+𝜀𝑜𝑤𝜆𝑝𝑤̅̅̅̅̅2−√(1+1+𝜀𝑜𝑤𝜆𝑝𝑤̅̅̅̅̅2)2−4𝜆𝑝𝑤̅̅̅̅̅2]=12×[1+1+0.24480.7062−√(1+1+0.24480.7062)2−40.7062]=0.723Theeffectivewidthofweb𝑏𝑒𝑤=𝜌𝑤𝑏=0.723×420=303.66𝑚𝑚Theeffectivearea𝐴𝑒𝑓𝑓=2𝑏𝑒𝑓𝑡𝑓+𝑏𝑒𝑤𝑡𝑤=2×312.8×20+303.66×14=16763.24𝑚𝑚2σ=N𝐴𝑒𝑓𝑓=2650×10316763.24=158.08𝑀𝑃𝑎𝑓𝑑=270𝑀𝑃𝑎OK2.Checktheglobalstability.Inertiamomentofx-axisandy-axis𝐼𝑥=2×112×20×4603=324453333.3𝑚𝑚4𝐼𝑦=112×14×4203+2×460×20×2202=976996000𝑚𝑚4𝐼𝑥𝐼𝑦𝑖𝑚𝑖𝑛=𝑖𝑥=√𝐼𝑥𝐴𝑚=√324453333.324280=115.6𝑚𝑚4𝜆𝑚𝑎𝑥=𝑙𝑜𝑥𝑖𝑥=8000115.6=69.2𝜆̅=𝜆𝑚𝑎𝑥𝜋√𝑓𝑦𝐸=69.2𝜋√3452.06×105=0.90.2𝜀𝑜=𝛼(𝜆̅−0.2)=0.5×(0.9−0.2)=0.35Thereductionfactorofglobalstabilityχ=12[1+1+𝜀𝑜𝜆̅2−√(1+1+𝜀𝑜𝜆̅2)2−4𝜆̅2]=12×[1+1+0.350.92−√(1+1+0.350.92)2−40.92]=0.596𝑁χ𝐴𝑒𝑓𝑓=2650×1030.596×16763.24=265.24𝑀𝑃𝑎𝑓𝑑=270𝑀𝑃𝑎OK3.Checkthelocalstability.Forflange:𝑏𝑓𝑡𝑓=46020=2312√345𝑓𝑦=12DissatisfyForweb:𝑏𝑤𝑡𝑤=42014=30=30√345𝑓𝑦=30OK4.Checkthestiffness𝜆𝑚𝑎𝑥=69.2[𝜆]=100OKb)Known:H-shapecrosssection,wedding,axialcompression,lox=loy=8m.Q345steel,N=2650kN(compression).Unknown:checkthestrength,stiffnessandstabilityofsteelmember.Solve:bylookinguptable,thedesignvalueofstrengthofsteel:fd=275kN.1.Checkthestrength.1).flange:flangewith3simply-supportededgesandonefreeedge,elasticbucklingcoefficientKf=0.425.𝜆𝑝𝑓̅̅̅̅̅=1.05(ℎ2𝑡𝑓)√𝑓𝑦𝐸𝐾𝑓=1.05×520218×√3452.06×105×0.425=0.9520.4Theimpactoflocalbucklingneedconsidering.𝜀𝑜𝑓=0.8(𝜆𝑝𝑓̅̅̅̅̅−0.4)=0.8×(0.758−0.4)=0.4416𝜌𝑓=12[1+1+𝜀𝑜𝑓𝜆𝑝𝑓̅̅̅̅̅2−√(1+1+𝜀𝑜𝑓𝜆𝑝𝑓̅̅̅̅̅2)2−4𝜆𝑝𝑓̅̅̅̅̅2]=12×[1+1+0.44160.9522−√(1+1+0.44160.9522)2−40.9522]=0.537Theeffectivewidthofflange𝑏𝑒𝑓=𝜌𝑓ℎ=0.537×460=279.24𝑚𝑚2).web:webwith4simply-supportededges,elasticbucklingcoefficientKf=4.𝜆𝑝𝑤̅̅̅̅̅=1.05(𝑏𝑡𝑤)√𝑓𝑦𝐸𝐾𝑤=1.05×46412×√3452.06×105×0.425=0.830.4Theimpactoflocalbucklingneedconsidering.𝜀𝑜𝑤=0.8(𝜆𝑝𝑤̅̅̅̅̅−0.4)=0.8×(0.83−0.4)=0.344𝜌𝑤=12[1+1+𝜀𝑜𝑤𝜆𝑝𝑤̅̅̅̅̅2−√(1+1+𝜀𝑜𝑤𝜆𝑝𝑤̅̅̅̅̅2)2−4𝜆𝑝𝑤̅̅̅̅̅2]=12×[1+1+0.3440.832−√(1+1+0.3440.832)2−40.832]=0.62Theeffectivewidthofweb𝑏𝑒𝑤=𝜌𝑤𝑏=0.62×464=287.68𝑚𝑚Theeffectivearea𝐴𝑒𝑓𝑓=2𝑏𝑒𝑓𝑡𝑓+𝑏𝑒𝑤𝑡𝑤=2×279.24×18+287.68×12=13504.8𝑚𝑚2σ=N𝐴𝑒𝑓𝑓=265