辅导案例-ECMM108

  • August 10, 2020

ECMM108 UNIVERSITY OF EXETER COLLEGE OF ENGINEERING, MATHEMATICS AND PHYSICAL SCIENCES ENGINEERING Advanced Structural Engineering Module Convenor: M. K. Wadee You have 24 hours to complete this paper from the time of its release. Intended Duration: TWO HOURS May 2020 Answer any THREE questions out of FOUR Materials to be supplied: Graph paper Approved calculators are permitted. This is an OPEN BOOK examination. ECMM108 1 TURN OVER Question 1 (20 marks) (a) Consider a linear translational spring with stiffness k, where k > 0. Using the concept of total potential energy, show that, under the action of an axial force, P , the spring is always stable. Comment on the validity of this result. (8 marks) (b) A vertical telecommunication mast of height h = 10 m has been designed to be fixed at its base with its top end stabilized by four cables splayed out in the north, east, south and west directions (see Figure Q1(a)). The mast is made of a square hollow tubular steel section with outer edges of dimension 200 mm and thickness of 10 mm (Figure Q1(b)). The plane faces of the tubular section are parallel to the four cables. (i) Calculate the buckling capacity of the mast for axial loading (you may neglect the axial compression arising from the cables). (8 marks) (ii) During construction, it is found that ground conditions are poorer than expected and only a pin connection is possible at the base of the mast to the ground. Calculate the percentage reduction in the carrying capacity of the structure due to the weaker foundation. (4 marks) (a) (b) h t = 10 mm 200 mm 200 mm Figure Q1: (a) North-south elevation of a telecommunication mast. (b) Cross-section of mast. ECMM108 2 CONTINUE Question 2 (20 marks) The rectangular frame shown in Figure Q2(a) is being modelled as an undamped single- degree-of-freedom (SDOF) system and is acted upon by a time-varying horizontal load, P (t), as depicted. The height of the frame is L and the two columns may be considered to be massless and made of the same section, with bending stiffness EI . However, they have different connections at their bases as indicated. The rigid beam has mass m and full-moment connections have been used to fix it to each column. (a) Calculate the undamped natural frequency, ω, of the the structure in terms of m, EI and L for small horizontal vibrations. (6 marks) (b) The numerical values of the above quantities are L = 2.5 m, m = 250 kg and EI = 8× 1011 Nmm2. The horizontal force is an impulse load of the form P (t) = P0f(t), where P0 = 5 kN and f(t) has the form shown in Figure Q2(b) and t1 = 4 s. Find the maximum deflection of the structure under this loading. (8 marks) (c) Calculate the maximum bending moments at the top of each column under the given loading. (6 marks) (a) (b) L m u(t) P (t) f(t) 1 t t1 Figure Q2: (a) Rectangular frame under dynamic loading. (b) Impulse loading function. ECMM108 3 TURN OVER Question 3 (20 marks) (a) Describe the assumption on shear behaviour in Kirchhoff’s theory for plate bend- ing. Explain using relevant equations, how the assumption enables rotations (ψx, ψy) to be derived from the deflection field (w). (3 marks) (b) Explain using relevant equations, how bending moments (Mx, My and Mxy) can be calculated from the deflection field (w) in a Kirchhoff plate. (2 marks) (c) The floor slab given in Figure Q3 is to be analysed using Mindlin–Reissner plate elements. A uniformly distributed load of magnitude 6 kNm−2 is applied over the shaded portion of the slab. Edges AB, BC and CD are simply supported. Edge DA is unsupported. (i) Define the displacement boundary conditions that need to be prescribed along the three supported edges. (2 marks) (ii) Along the four edges, state the known force boundary conditions with respect toMx,My,Mxy, Qx and Qy. (3 marks) (iii) How can symmetry be used to model only a symmetric portion of the floor slab given in Figure Q3? Clearly state the boundary conditions that need to be prescribed when modelling only a portion of the slab. Also state the known force boundary conditions along the planes of symmetry. (4 marks) (iv) Label on a diagram drawn in your answer book where you expect to see maximum sagging and hogging bending moments forMx andMy. Also give a qualitative sketch of the distribution of the reactions along edge AB.(3 marks) (d) State the governing equations for equilibrium for a plate under bending. Do so- lutions from finite element analysis satisfy equilibrium across the plate? Explain your answer. (3 marks) B A x y 6 m 1 m D C 3 m 6 kNm−2 Figure Q3 ECMM108 4 CONTINUE Question 4 (20 marks) (a) State the master safe theorem. Explain the assumptions made when using the master safe theorem for limit analysis of slabs. (3 marks) (b) In the context of limit analysis of slabs, explain why upper bound methods are considered unsafe and lower bound methods are considered safe. (2 marks) (c) Figure Q4 shows the yield line pattern used to compute the yield bending moment for a slab. Hogging and sagging yield lines are shown using thick dashed and thick solid lines respectively. The slab is simply supported on edges BC and AD. A uniformly distributed load of magnitude 8 kNm−2 acts over the entire plate. Com- plete the following tasks towards calculating the yield bending moment for the slab based on the given yield line pattern. i. Parametrize the yield line pattern and illustrate this using a diagram.(3 marks) ii. Write down the equations for internal and external work. (6 marks) iii. Derive the equation for the yield bending moment by relating the equations obtained for internal and external work. (4 marks) iv. Describe how the yield bending moment can be obtained from the equation derived in part iii through the use of optimization. (2 marks) D B C A 4 m 4 m 4 m Figure Q4 END OF QUESTION PAPER ECMM108 5

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