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Aircraft structures for engineering students - part 3

Szat 60 = (4.33) h Bây giờ chúng ta có thể áp dụng các nguyên tắc của giá trị văn phòng của tổng năng lượng bổ sung kết hợp với phương pháp tải đơn vị để xác định sự lệch A, do nhiệt độ của bất kỳ điểm nào của chùm tia được hiển thị trong hình. 4,28. Chúng ta đã thấy rằng các nguyên tắc trên là tương đương với việc áp dụng nguyên tắc làm việc ảo, | 108 Energy methods of structural analysis Fig. 4.29 a Linear temperature gradient applied to beam element b bending of beam element due to temperature gradient. the element will increase in length to fe l at where a is the coefficient of linear expansion of the material of the beam. Thus from Fig. 4.29 b R R h Sz ôz ỉ at giving R h at 4.32 Also ÔQ Sz R so that from Eq. 4.32 66 4.33 We may now apply the principle of the stationary value of the total complementary energy in conjunction with the unit load method to determine the deflection ATe due to the temperature of any point of the beam shown in Fig. 4.28. We have seen that the above principle is equivalent to the application of the principle of virtual work where virtual forces act through real displacements. Therefore we may specify that the displacements are those produced by the temperature gradient while the virtual force system is the unit load. Thus the deflection ATeiB of the tip of the beam is found by writing down the increment in total complementary energy caused by the application of a virtual unit load at B and equating the resulting expression to zero see Eqs 4.13 and 4.18 . Thus 6C i Ml dớ - lATe.B 0 JL References 109 or ATe.B Mị dớ 4.34 J . where Mị is the bending moment at any section due to the unit load. Substituting for dớ from Eq. 4.33 we have f_ Oct . . d 4.35 L where t can vary arbitrarily along the span of the beam but only linearly with depth. For a beam supporting some form of external loading the total deflection is given by the superposition of the temperature deflection from Eq. 4.35 and the bending deflection from Eqs 4.27 thus A i 4.36 . L El nJ Example 4.11 Determine the deflection of the tip of the cantilever in Fig. 4.30 with the temperature gradient shown. Applying a unit load vertically downwards at B M I X z. Also the temperature t at a section z is 0 z l. Substituting in Eq. 4.35 gives ATc.b f -. de i Jo n I Integrating Eq. i gives ATcB i.c. downwards 6 7 References 1 Charlton