INFORMATIO

# 10 48 18 10 6 3 3 mpa 61 10 6 3 3 mpa pa made from

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10 48 . 18 10 6 3 3 MPa 71054 . 89 10 46874 . 10 52 . 61 10 6 3 3 MPa Pa made from different materials in order etermined from =E 1 ε etermined from =E 2 ε dA of the beam is ormed into material 2 -6 m 4 10 -6 m 4 r to efficiently carry a

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A d F d =( E 2 ε)ndydz dF= F d ( E 1 ε)dydz=( E 2 ε)ndyd n= 2 1 E E n: transformation factor (modula If the material 2 is being transfo b 1 = n b where n = 1 2 E E For the transformed m =n 2 z dz ar ratio). ormed into material 1 material
3 Example 46: A composite beam is made of wood and reinforced with a steel strap located on its bottom side. It has the cross sectional area shown below. If the beam is subjected to a bending moment of M=2 KN.m determine the normal stress at point B and C . Take E w =12 GPa and E st =200 GPa . st w E E n 06 . 0 200 12 n w st b n b mm b st 9 150 06 . 0 mm 150 mm 150 mm 20 B C mm 150 C mm 20 mm 150 mm 9 B 1 2 mm 150 mm 150 mm 9 mm 20

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4 A A y y c mm m y c 379 . 36 10 379 . 36 10 4350 10 158250 3 3 9 I=I 1 +I 2 2 3 1 12 Ad bh I 2 3 6 3 3 3 1 ) 10 379 . 26 ( 10 3000 12 ) 10 20 ( 10 150 I I 1 =2.187554×10 -6 m 4 2 3 2 12 Ad bh I 2 3 6 3 3 3 2 ) 10 621 . 58 ( 10 1350 12 ) 10 150 ( 10 9 I I 2 =7.170419×10 -6 m 4 I=9.35797×10 -6 m 4 I My B MPa B 557689 . 28 10 35797 . 9 10 621 . 133 10 2 6 3 3 B B n MPa B 71346134 . 1 ) 557689 . 28 ( 06 . 0 I My B MPa B 774976 . 7 10 35797 . 9 10 379 . 36 10 2 6 3 3 No. of Area A(m 2 ) y (m) A y (m 3 ) 1 3000×10 -6 10×10 -3 30000×10 -9 2 1350×10 -6 95×10 -3 128250×10 -9 A =4350×10 -6 A y =158250×10 -9 mm 150 mm 150 mm 9 mm 20 mm c 621 . 133 1 mm c 379 . 36 2 A N .
5 Shear Stresses in Beams A ydA It V A y ydA A = Q It VQ :- the shear stress in the member at the point located a distance y from the neutral axis. V:-the internal resultant shear force. I:-the moment of inertia of the entire cross sectional area computed about the neutral axis. t:-the width of the members cross sectional area, measured at the point where is to be determined. Q A y , where A is the top (or bottom) portion of the members cross sectional area, defined from the section where t is measured, and y is the distance to the centroid of A , measured from the neutral axis.

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6 Example 47: A metal beam with span L=3 ft is simply supported at points A and B . The uniform load on the beam is q=160 lb/ in. The cross section of the beam is rectangular with width b=1 in and height h=4 in . Determine the normal stress and shear stress at point C , which is located 1 in below the top of the beam and 8 in from the right hand support. 0 A M B y ×3×12-5760×1.5×12=0 B y =2880 lb 0 y F A y +2880-5760=0 A y =2880 lb A B ft 3 in lb q / 160 C lb 5760 y A y B lb 2880 in lb q / 160 lb 2880 in 18 in k M . 92 . 25 max lb 2880 lb 2880 in 18 C in 8 V 18 2880 10 V V=1600 lb
7 At point C x=28 in from left end from shear force diagram N.A 4 3 3 3333 . 5 12 ) 4 ( 1 12 in bh I A =1×1=1 in 2 y =1.5 in Q A y =1.5×1=1.5 in 3 ksi I My C

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• Winter '15
• MAhmoudali

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