图示悬臂圆截面杆件受三个载荷作用,已知 F=4.2kN, Me=1.5kN·m,l=0.5m,d=100mm,[s]=80MPa,试按第三强度理论校核杆的强度。
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Ïx=F*L*y/(Ï/64*D^4)+ F*1/2*L*z/(Ï/64*D^4)=64/Ï*F*L/D^4*[y+1/2*z]
注æå°è¡¨é¢æy^2+z^2=D^2/4ï¼æ
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Ïx=64/Ï*F*L/D^4*[y+1/2*sqrt(D^2/4-y^2)]
令dÏx/dy=0ï¼å³64/Ï*F*L/D^4*{1+1/2*1/[2*sqrt(D^2/4-y^2)]*(-2*y)}=0
解åºy=D/sqrt(5) ï¼z=D/sqrt(20)
Ïx max =64/Ï*F*L/D^4*[D/sqrt(5)+1/2*D/sqrt(20)]
=64/Ï*F*L/D^4*Sqrt(5)/4*D
=16*sqrt(5)/Ï*F*L/D^3
=16*sqrt(5)/Ï*4200*500/100^3 N/mm^2â23.92 N/mm^2=23.92 MPa
æ转åªååºåÏ=Me/[Ï/16*d^3]=1500000/[Ï/16*1000000]=7.64 MPa
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Ïæ»=sqrt{Ïx^2+4*Ï^2}=sqrt{23.92^2+4*7.64^2}=28.38 MPaâ¤[s]=80MPa
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