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Aircraft Structures 3E Episode 2

Tham khảo tài liệu 'aircraft structures 3e episode 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 26 Basic elasticity For the case of plane stress they simplify to x x- y 1 y ơy vơx Zz - x- ry Txy lxy G 1-47 It may be seen from the third of Eqs 1.47 that the conditions of plane stress and plane strain do not necessarily describe identical situations. Changes in the linear dimensions of a strained body may lead to a change in volume. Suppose that a small element of a body has dimensions 6x Sy and Sz. When subjected to a three-dimensional stress system the element will sustain a volumetric strain e change in volume unit volume equal to 1 Ex ổx l e 5y l Ez fe SxSySz e SxSySz Neglecting products of small quantities in the expansion of the right-hand side of the above equation yields r Ey z 1.48 Substituting for ex Ey and z from Eqs 1.42 we find for a linearly elastic isotropic body e E ơy ơz 2v ơx ơy or l-2iz . _ x e - o ơy Tz In the case of a uniform hydrostatic pressure ơx ơy ơz p and - P 1.49 The constant 3 1 2v is known as the bulk modulus or modulus of volume expansion and is often given the symbol K. An examination of Eq. 1.49 shows that V 0.5 since a body cannot increase in volume under pressure. Also the lateral dimensions of a body subjected to uniaxial tension cannot increase so that V 0. Therefore for an isotropic material 0 V sg 0.5 and for most isotropic materials V is in the range 0.25 to 0.33 below the elastic limit. Above the limit of proportionality V increases and approaches 0.5. Example 1.2 A rectangular element in a linearly elastic isotropic material is subjected to tensile stresses of 83 N mm2 and 65 N mm2 on mutually perpendicular planes. Determine the strain in the direction of each stress and in the direction perpendicular to both 1.15 Stress-strain relationships 27 stresses. Find also the principal strains the maximum shear stress the maximum shear strain and their directions at the point. Take E 200000N mm2 and V 0.3. If we assume that ơx 83N mm2 and ơy 65N mm2 then from Eqs 1.47 2Õ0ÕÕÕ 83- -3 x 65 3-175 x 10 200W 65- -3x83 M05xI 4 ez