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HW6+Solutions - HOMEWORK PROBLEMS Week 6 Due Friday 21...

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HOMEWORK PROBLEMS Week 6: Due Friday 21 October 1. The rectangular plate ABCD is homogeneously deformed into the shape A B C D as shown in Figure 1. Determine the strain components ǫ x , ǫ y and γ xy . x y C A B D A B C D 0.007 0.006 0.004 0.002 0.002 0.002 0.001 0.004 15 20 all dimensions in inches [not to scale]. Figure 1 Solution The displacements of the corners are u A x = 0 . 001 ; u A y = 0 . 002 ; u B x = 0 . 006 ; u B y = 0 . 004 , 1
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u C x = 0 . 007 ; u C y = 0 . 002 ; u D x = 0 . 002 ; u D y = 0 . 004 . so the strains are ǫ x = u D x u A x L AD = 0 . 002 + 0 . 001 15 = 66 . 7 × 10 6 . ǫ y = u B y u A y L AB = 0 . 004 0 . 002 20 = 300 × 10 6 . γ xy = u D y u A y L AD + u B x u A x L AB = 0 . 004 0 . 002 15 + 0 . 006 + 0 . 001 20 = 116 . 7 × 10 6 . 2
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2. Figure 2 shows a rigid rectangular block which is attached to a rigid foundation through two (deformable) rubber blocks. The 5 kN force shown causes the rigid block to move downwards a distance 1.8 mm. Find the average shear strain γ and the average shear stress τ in each of the rubber blocks and hence estimate the shear modulus G for the rubber. rigid rigid rigid rubber 30 30 100 70 5 kN all dimensions in mm Figure 2 Solution Figure 2.1(a) shows the deformation of the rubber, from which we deduce that the shear strain is γ = 1 . 8 30 = 0 . 06 . 30 1.8 5 kN V V (a) (b) Figure 2.1 3
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To find the stress in the rubber, we must cut through the rubber ( and only the rubber ), so as to leave an object that is not connected to the rest of the world. This means we have to cut through the rubber on both sides, leaving the free-body diagram of Figure 2.1(b), from which
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