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Course of Spacecraft Structures Written test, September 11 th, 2017 Exercise 1 The structure in the gure is composed of three beams of length l, which are charaterized by the same elastic properties. In particular, the bending and torsional stinesses are denoted with EJ and GJ, respectively. The structure is elastically supported by a torsional spring of stiness kt at one end. A linearly varying force per unit length is applied along one of the beams, as illustrated in the sketch. The force per unit length is equal to zero at one end, and is equal to q at the other end. Determine the reaction forces and plot the internal actions. Data EJ = 72e6 Nmm 2 GJ = EJ/2 kt = 72 Nmm l = 1000 mm q = 100 N/mm Exercise 2 The beam in the gure is made of an isotropic material with modulus E. It is characterized by a square cross section of dimension a, where a(x) = γ 4 √ a0 + (a1−a0) x l , and x∈ [0, l]. The rst end of the beam is xed; the second one is free to translate along the transverse direction, while the rotation is prevented. A concentrated load of magnitude P is applied at x =l. Assume a Euler-Bernoulli beam model, and determine: the transverse displacement at x =l; the maxi- mum stress at x =l/4. To this aim, apply the method of Ritz and, using polynomial functions, solve the problem by considering one single shape function. Discuss how the problem can be formulated when two shape functions are adopted (report the relevant equations; no calculations are required). Data E = 72000 Nmm 2 l = 1000 mm γ = 10 mm a0 = 1 a1 = 16 P = 1 kN Question 1 Illustrate the relation between the stress measures according to Cauchy, Piola-Kirchho I and Piola- Kirchho II. 1