The cross section in Question 2 is replaced by a T-shape as shown above. What ar

Question

The cross section in Question 2 is replaced by a T-shape as shown above. What are the maximum tensile and compressive stresses of this bean) if bent about (a) the x axis, (b) the y axis? (83MPa (C) 198MPa (T), 356 MPa (C&T;)) What are the maximum tensile and compressive strains for each case (a) and (b). if the beam is made of Aluminium ?(Eal=75GPa) (1107 mustrain (C) 2640 mustrain (T), 4750 mustrain C&T;) What maximum loads for each case (a) and (b). could the beam take if the allowable stress is 150MPa in tension or lOOMPa in compression? (2.27kN tension. 0.84kN compression govern)

Explanation / Answer

The maximum bending moment is not given. I use the answer given to guess that M = 6000 N.m

The centroid of the cross section from the top surface is

y = yiAi/Ai = [5*(10*100)+60*(10*100)]/(10*100+10*100) = 32.5 mm

Therefore

Ix = [Ixi + Aidi2] =(Ix1 + A1d12 ) + (Ix2 + A2d22 )

   = [(0.1)(0.013)/12 + (0.01*0.1)(0.0325-0.005)2]

    + [(0.01)(0.13)/12 + (0.01*0.1)(0.06 - 0.0325)2] = 2.35417 x 10-6 m4

Iy = (0.01)(0.13)/12 + (0.1)(0.013)/12 = 0.841667 x 10-6 m4

(A)

(a) The maximum Tensile stress occurs at the lower surface and

         the maximum compressive stress occurs at the top surface

         max_comp = Mctop/Ixx = 6000(0.0325)/[2.35417 x 10-6] = 83 x 106 Pa =83 MPa

         max_tens = Mcbot/Ixx = 6000(0.0775)/[2.35417 x 10-6] = 198 x 106 Pa =198 MPa

(b) Due to symmetry, the maximum tensile and compressive stresses are the same

         max_yy = Mc/Iyy = 6000(0.05)/[0.841667x 10-6] = 356x 106 Pa =356 MPa

(B)

(a) max_comp = max_comp /E = 83 x 106/(75 x 109 ) =1.107x10-3 =1107

        max_tens = max_tens /E = 198 x 106/(75 x 109 ) =2.640x10-3 =2640

(b) max_yy = max_yy /E = 356 x 106 Pa/(75 x 109 Pa) = 4.747x10-3 = 4747

(C)

(a) Pmax_comp = (allow_comp/max_comp ) (3kN) = (100/83 ) (3kN) = 3.61 kN

        Pmax_tens = (allow_tens/max_tens ) (3kN) = (150/198) (3kN) = 2.27 kN

       So tension governs, P_max = 2.27 kN

(b)

        Pmax_comp = (allow_comp/max_comp ) (3kN) = (100/356 ) (3kN) = 0.84 kN

        Pmax_tens = (allow_tens/max_tens ) (3kN) = (150/356) (3kN) = 1.26 kN

     So compression governs, P_max = 0.84 kN

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