ECMM149 Linear Systems and Structural Analysis 2019
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ECMM149
Linear Systems and Structural Analysis
2019
Question 1 (40 marks)
(a) Given the linear system of equations:
i. Write the system in matrix form and find the det(A).
(4 marks)
ii. For which value of β is there a solution to the system Ax = b?
(6 marks)
iii. For the value of β calculated above in (ii), find the general solution of the system Ax = b . (6 marks)
(b) Consider a system whose response d(t) to a force y(t) is described by the differential equation:
i. For = 30 kg, = 3000 Nm-1 , and = 6 Nms-1 , calculate the damped natural frequency ௗ in rad s-1 . (2 marks)
ii. If the force applied on the system is () = 20 sin(), calculate and sketch in a graph the magnitude of the receptance Frequency Response Function (FRF) for the values of = [7, 8, 9, 10, 11, 12] . Ensure you add the correct units in your graph.
(12 marks)
iii. Similar to (ii), calculate the magnitude of the FRF for a force () =
60 sin() for the same values of = [7, 8, 9, 10, 11, 12] . (3 marks)
(c) Assume that you are required to find the velocity on a fixed location of a structure, however you cannot measure it directly. You can either record its displacement or its acceleration and derive the velocity from either of them. Explain both approaches, their associated problems, and state which one of the two (displacement or acceleration) would you prefer and justify your answers.
(7 marks)
Question 2 (30 marks)
(a) Determine whether the three plane frame structures shown in Figure Q2 are statically determinate, statically indeterminate or a mechanism. If a structure is statically indeterminate, state its degree of indeterminacy.
(6 marks)
(b) Assuming that the structures in Figures Q2(b) and Q2(c) are loaded as shown, draw clear and qualitative sketches of their deformed shapes, indicating any points of contraflexure.
(8 marks)
(c) For the structures in Figures Q2(b) and Q2(c), provide clear and qualitative sketches of the bending moment diagrams, drawn on the tension side of members. (8 marks)
(d) By removing an appropriate number of internal and/or external forces from the statically indeterminate structure shown in Figure Q2(c), obtain a new structure that will be statically determinate. Offer at least two solutions, and then sketch the bending moment diagrams for both solutions.
(8 marks)
Question 3 (30 marks)
A bridge structure shown in Figure Q3 is exposed to a horizontal point force of 200 kN acting at node 7. A computer program based on the stiffness matrix method is used to find unknown displacements and internal forces. Nodes, shown in circles, elements shown in squares, and cross section properties required for calculation, as shown in Figure Q3 and Table Q3, were used as input data to the program. All elements of the structure are made of the same material.
Table Q3: Input data
(a) Sketch the local coordinate system for every element as adopted by the program. (3 marks)
(b) Label all degrees of freedom in the global coordinate system for each joint and
decide which displacements are unknown. (5 marks)
(c) Write the stiffness matrix for element 5, and element 2 in the global coordinate system . Label the rows and columns according to the degrees of freedom they relate to.
(12 marks)
(d) The program output lists the force vectors for all elements in their local coordinate systems as follows:
Draw the bending moment and shear force diagrams for the structure.
(10 marks)
2022-01-15