# hw2_sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT...

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± MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 2.002 MECHANICS AND MATERIALS II SOLUTIONS FOR HOMEWORK NO. 2 Problem 1 (15 points) The axial stiﬀness of the structure is: F x σ xx A A k axial = = = E (1) δ x ± xx L L where A is the area of the cross section and L is the length of the beam. The bending stiﬀness is deﬁned as: F y δ k bending = , (2) y where F y is the concentrated tip load, and δ y is the related tip deﬂection. According to the beam theory, we know that δ y = F y L 3 . Thus, the bending stiﬀness is: 3 EI F y = F y 3 EI k bending = δ y F y L 3 = L 3 . (3) 3 EI Combining Eq.1 and Eq.3, the ratio of a slender cantilever’s bending stiﬀness to its axial stiﬀness is: 3 EI k bending = L 3 3 I/A =3( l 2 ) 2 = (4) E A L 2 L k axial L where l 2 I/A .S in c e L ± l 2 , we ﬁnd the above ratio is extremely small. For a solid circular cross-section beam with diameter d ,we have : πd 4 I = (5) 64 and πd 2 A = (6) 4 Substitution of the A and I into Eq. 4, we have k bending 3 I/A 3 d ) 2 = = ( (7) k axial L 2 16 L Problem 2 (45 points) Part A: No forces and moments are applied to the beam. And at any x ,0 <x <l ,wehave : N = F x = σ xx

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and M = σ xx ( y ) dA =0 (9) The stress is a function of y , σ xx = σ xx ( y
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hw2_sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT...

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