Question The couple M is applied to a beam of the cross section shown in a plane forming an angle b with the vertical. Determine the stress at (a) point A, (b) point B, (c) point D.  

Draw the cross section with couple:Area of the part 1 ,{:[A_(1)=(80mm)(90mm)],[=7200mm^(2)]:}Area of the part 2 ,{:[A_(2)=(80mm)(30mm)],[=2400mm^(2)]:}Calculate the Total area of the cross section,{:[A=A_(1)+A_(2)],[=(7200mm^(2))+(2400mm^(2))],[=9600mm^(2)]:}Calculate the Moment of inertia of a rectangular part about any point,I=(bd^(3))/(12)+Ay_(i)^(2)Calculate the Moment of inertia of the part 1 about the y axis,{:[I_(y1)=(bd^(3))/(12)+A1y_(1)^(2)],[I_(y1)=((80)(90)^(3))/(12)+(7200)(0)^(2)],[=4.86 xx10^(6)mm^(4)]:}Calculate the Moment of inertia of the part 2 about the y axis,{:[I_(y2)=(bd^(3))/(12)+A_(2)y_(2)^(2)],[I_(y2)=((80)(30)^(3))/(12)+(2400)(0)^(2)],[=0.18 xx10^(6)mm^(4)]:}Calculate the Moment of inertia of the total cross section about the y axis,{:[I_(y)=I_(y1)+I_(y2)],[=(4.86 xx10^(6))+(0.18 xx10^(6))],[=5.04 xx10^(6)mm^(4)]:}Calculate the moment of inertia of the part 1 about the z axis,{:[I_(z1)=(bd^(3))/(12)+A_(1)z_(1)^(2)],[I_(z1)=((90)(80)^(3))/(12)+(7200)(20)^(2)],[=6.72 xx10^(6)mm^(4)]:}Moment of inertia of the part 2 about the z axis,{:[I_(z2)=(bd^(3))/(12)+A_(2)z_(2)^(2)],[I_(z2)=((30)(80)^(3))/(12)+(2400)(60)^(2)],[=9.92 xx10^(6)mm^(4)]:}Calculate the Moment of inertia of the total cross section about the y axis,{:[I_(z)=I_(z1)+I_(z2)],[=(6.72 xx10^(6))+(9.92 xx10^(6))],[=16.64 xx10^(6)mm^(4)]:}Component of the moment on the y axis,{:[M_(y)=M sin beta],[M_(y)=(25)sin 15^(@)],[=6.47kN*m],[=6.47 xx10^(6)N*mm]:}Component of the moment on the z axis,{:[M_(z)=M cos beta],[M_(z)=(25)cos 15^(@)],[=24.15kN*m],[=24.15 xx10^(6)N*mm]:}Calculate the Normal stress acting on any point due to the moment applied,sigma_(x)=-(M_(z)y)/(I_(z))+(M_(y)z)/(I_(y))(a)Calculate the Normal stress acting on the point A,{:[sigma_(A)=-(M_(z)y_(A))/(I_(z))+(M_(y)z_(A))/(I_(y))],[sigma_(A)=-((24.15 xx10^(6))(60))/(16.64 xx10^(6))+((6.47 xx10^(6))(45))/(5.04 xx10^(6))],[=-29.3N//mm^(2)],[sigma_(A)=-29.3MPa]:}(b)Calculate the Normal stress acting on the point B,{:[sigma_(B)=-(M_(z)y_(B))/(I_(z))+(M_(y)z_(B))/(I_(y))],[=-((24.15 xx10^(6))(60))/(16.64 xx10^(6))+((6.47 xx10^(6))(-45))/(5.04 xx10^(6))],[=-144.8N//mm^(2)],[sigma_(B)=-144.8MPa]:}(c)Calculate the Normal stress acting on the point D,{:[sigma_(D)=-(M_(z)y_(D))/(I_(z))+(M_(y)z_(D))/(I_(y))sigma_(D)],[=-((24.15 xx10^(6))(-100))/(16.64 xx10^(6))+((6.47 xx10^(6))(-15))/(5.04 xx10^(6))],[=125.9N//mm^(2)],[sigma_(D)=125.9MPa]:} ...