ECMM108 UNIVERSITY OF EXETER COLLEGE OF ENGINEERING, MATHEMATICS AND PHYSICAL SCIENCES ENGINEERING May 2021 Advanced Structural Engineering Module Convenor: M. K. Wadee Duration: TWO HOURS + 30 minutes upload time Answer THREE out of FOUR questions This is an OPEN BOOK examination. ECMM108 1 CONTINUE Question 1 (20 marks) Consider the structural model given in Figure Q1. It shows a structure comprising two identical links of length L joined together at a pin connection. The joint is supported both by a horizontal translational spring with linear stiffness k and a rotational spring of stiffness C. The model is acted upon by a conservative vertical force of magnitude P . Displacement is measured by the angle, θ, that the bottom link makes with the vertical. The structure is perfectly vertical when the springs are unstressed. P C θ k L L Figure Q1: Rigid link model. (a) For the case when C = 1 4 kL2, determine the equilibrium paths and sketch them. Demonstrate that the critical point is unstable. (14 marks) (b) If the rotational spring can be replaced, determine the value of C for which the critical point will switch from being unstable to stable. (6 marks) ECMM108 2 CONTINUE Question 2 (20 marks) Figure Q2 depicts a simply-supported beam with uniform bending stiffness EI and length L, modelled as a single-degree-of-freedom structure with its mass, m, concen- trated at its centre. It is further supported by a linear vertical translational spring of stiffness k as shown. The damping ratio of the structure is ξ = 0.1. P (t) m k L/2L/2 Figure Q2: A beam-spring structural system. (a) Calculate the structure’s stiffness to central vertical deflection and hence determine its natural period, T , given thatm = 1000 kg, EI = 6× 108 kNmm2, L = 3 m and k = 250 Nmm 1 for the spring. (10 marks) (b) The structure is acted upon by a time-varying force of the form P (t) = P0(sin t+ cos t), where P0 = 4 kN and = 45 rad s 1. Estimate the maximum dynamic deflection of the structure to this action. (10 marks) ECMM108 3 CONTINUE Question 3 (20 marks) (a) The collapse load for the plate in Figure Q3 is to be determined using yield line techniques. The plate is fixed along edge AB to prevent rotation and vertical trans- lation. It is simply supported along edge BC. Using diagrams, illustrate two poten- tial yield line patterns that may be considered for yield line analysis. (4 marks) 4 m B A 6 m D C Figure Q3: Elastic plate (b) A floor slab is analysed using 4-noded Mindlin-Reissner plate elements. The slab is made of a linear, isotropic material with elastic modulus and Poisson’s ratio of 25 kNmm 2 and 0.2 respectively. The thickness of the slab is 0.2 m. Table Q3 shows the nodal coordinates and the computed nodal displacements for one rectangular element. Answer the following questions on the element. i. Compute the bending moments,Mx,My andMxy at Node 1. (8 marks) ii. Evaluate the shear strain γxz at the centre of the element, and then compute the corresponding shear force Qx. (6 marks) iii. Would the shear force Qx derived from the bending moment fields using the following equation: Qx = Mx x + Mxy y equal the shear force computed from the shear strain γxz, as done in part ii Explain why. (2 marks) Node Coordinates (m) w (×10 3) m ψx (×10 4) radians ψy (×10 4) radians 1 (-1,-1) 4 0 80 2 (1,-1) 4 0 80 3 (1,1) 2 100 0 4 (-1,1) 2 100 0 Table Q3: Nodal data. ECMM108 4 CONTINUE Question 4 (20 marks) Use two families of beam strips, one set each along x and y directions, to evaluate the maximum design moments for the homogeneous and isotropic slab in Figure Q4. The slab is simply-supported on two edges AB and DC, and carries a uniformly distributed load of magnitude 8 kNm 2 as illustrated in the figure. (a) Briefly describe the load routes adopted for the slab. Indicate the loads taken by the beam strips along the x- and y-directions and how the strips interact. (6 marks) (b) Plot reaction forces along the boundaries taking care to convert any twist moments into equivalent shear forces. (4 marks) (c) Derive the equations for shear and bending moments for one beam strip along the x-direction that has the maximum bending moments. Also give qualitative sketches of the shear and bending moment diagrams. (10 marks) x 4 m A B CD 2 m u.d.l. applied here 6 m y Figure Q4: Partially loaded plate. END OF QUESTION PAPER ECMM108 5
