结构分析代写-ECMM149 1
时间:2022-01-13
ECMM149 1 TURN OVER
ECMM149
UNIVERSITY OF EXETER
COLLEGE OF ENGINEERING, MATHEMATICS
AND PHYSICAL SCIENCES
ENGINEERING

Linear Systems and Structural Analysis

Module Convenor: Dr E Papatheou
Duration: TWO HOURS
January 2019



Answer ALL questions.

Approved calculators are permitted.

This is an OPEN BOOK/NOTE examination.




ECMM149 2 CONTINUE
Answer ALL questions.

Question 1 (40 marks)

(a) Given the linear system of equations:

+ 2 − 3 = 1
3 − + 2 = 2
5 + 3 − 4 =
i. Write the system in matrix form = where = ቈ



቉ and find the det(A).
(4 marks)
ii. For which value of is there a solution to the system = ?
(6 marks)
iii. For the value of calculated above in (ii), find the general solution of the
system = . (6 marks)


(b) Consider a system whose response () to a force () 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)
ECMM149 3 TURN OVER
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)

(a) (b)


(c)
Figure Q2: Structures for Q2.

(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)







a b
c
d e

P
P
ECMM149 4 CONTINUE
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.

Figure Q3
Table Q3: Input data
Element
number
Element
type
First
node
Second
node
Cross
section area
Second moment
of area
1 Beam 1 2
2 Beam 2 3
3 Beam 3 4
4 Beam 4 5
5 Beam 4 6 3 5
6 Beam 4 7 3 5
7 Bar 2 7 2 -
8 Bar 3 7 2 -
9 Bar 7 5 2 -
=10ହ kNm2
=10଺ kN

(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)
1 2 3 4
5
6 .
1
4 3
5
6
7
20.0m 20.0m
1 5
. 0 m

1 5
. 0 m



200kN
10.0m 10.0m
ECMM149 5
(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:

ଵ =





−195.0
−2.9
−28.0
195.0
2.9
−29.2 ⎭




ଶ =





−186.0
3.9
29.2
186.0
−3.9
9.5 ⎭




ଷ =





−189.3
−1.0
−9.5
189.3
1.0
−0.5 ⎭




ସ =





−194.7
−0.5
−9.2
194.7
0.5
0.0 ⎭





ହ =





−143.6
5.0
15.8
143.6
−5.0
58.9 ⎭




଺ =





−144.2
−0.4
−6.2
144.2
0.4
0.0 ⎭




଻ = ቄ
−11.2
11.2 ቅ = ቄ
5.8
−5.8ቅ

ଽ = ቄ
243.4
−243.4ቅ

Draw the bending moment and shear force diagrams for the structure.
(10 marks)


END OF QUESTION PAPER
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