Structures Exam City 2005.pdf

CITY UNIVERSITY LONDON

BEng (Hons)/MEng (Hons) Degree in Civil Engineering BEng (Hons)/MEng (Hons) Degree in Civil Engineering with Surveying BEng (Hons)/MEng (Hons) Degree in Civil Engineering with Architecture

PART 2 EXAMINATION STRUCTURAL ANALYSIS CV2303

Date: 6 May 2005

Time: 10:00 – 13:00

Answer Question 1 in Section A and FOUR out of the SIX questions from Section B

Calculators are permitted Dictionaries are NOT permitted Students should be provided with an Answer Book with graph paper

Section A Question 1 (a)

Derive, from first principles, the element stiffness matrix for a pin-ended bar element in the local axes. (8 marks)

(b)

For a beam of length L and rigidity EI, the stiffness matrix excluding axial effects is:

6 L − 12 6 L   12  6 L 4 L2 − 6 L 2 L2  EI  [k] = 3  L  − 12 − 6 L 12 − 6 L   2 − 6 L 4 L2   6L 2 L Using the stiffness matrix derived in Q1(a), write down the stiffness matrix for a plane rigidjointed member including axial effects. (4 marks) The pin-jointed truss shown in Figure Q1 has been analysed by the stiffness matrix method. 1 50kN

2

[500]

[400] 3

[400]

3000

[300]

4

[500]

[400] 5

[400]

3000

[300]

6

[500] [300]

[400] 7

[400] [500]

3000

(c)

8

4000

Figure Q1 The figures within square brackets are areas of cross-section in mm2 and the dimensions given are in mm. Assume that E for the structural material is 210 kN/mm2. The node numbers are shown encircled. The positive directions of Global axes are: x-axis towards the right and y-axis towards the top. (Continued on the next page)

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The global displacements in metres are obtained from the analysis as: Node 1 2 3 4 5 6 7 8

δx

0.043400 0.041496 0.026255 0.024350 0.011119 0.009214 0.000000 0.000000

δy

0.004018 -0.008036 0.004018 -0.006696 0.002679 -0.004018 0.000000 0.000000

Determine the element stiffness matrix for element 3-5 and then calculate the internal force for Member 3-5. (8 marks)

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SECTION B Question 2 Figure Q2 shows the cross-section of a beam which is subject to a 50 kN shear force acting in a plane parallel to that of its web. The thickness of the section is 16 mm throughout. All other dimensions are shown in mm in Figure Q2. (a)

Calculate the position of the shear centre. (17 marks)

(b)

Calculate the torque, Τ, that would act on the section if the 50kN load is applied uniformly across the top flange. (3 marks)

150 50

t = 16 t = 16 200

t = 16

t = 16

t = 16

all dimensions in mm

Figure Q2

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50

Question 3 Using the Moment Distribution Method:

(a)

determine the moments in the three span continuous beam ABC shown in Figure Q3 assuming that there are no settlements of the supports,

(b)

draw the bending moment diagram, indicating all principal values,

(c)

calculate the additional moments in the beam if support C settles by 10mm.

(12 marks) (4 marks) (4 marks) 16kN 6kN/m

6kN/m

6m

4m

A

EI = 104kNm2 for all members

4m

B

C

Figure Q3

Question 4 The rigid-jointed frame ABCD shown in Figure Q4 is built-in at A and D, and simply supported at C. Using the Slope Deflection Method: (a)

calculate the moments in the frame for the loading shown,

(b)

draw the bending moment diagram indicating all principal values.

(16 marks) (4 marks)

3kN A

3kN/m

B

C 2m

5m

2m

3m D

Figure Q4

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EI is constant

Question 5 (a)

For the continuous beam shown in Figure Q5, determine the reactions at A B and C arising out of the applied loading and a differential downward settlement of 5mm at A. Use the release shown in the figure. Tables of area integrals are attached. Assume EI = 1010 kNmm2. (16 marks)

(b)

Without doing any calculations, state whether you expect the reaction at A arising out of a settlement of 10mm at A would be twice of that obtained in Q5(a). Give any qualitative reason(s) to justify your statement. (4 marks)

5m

5m Figure Q5

Question 6 (a)

Explain how the influence line for the reaction at the middle support of a two span beam may be determined experimentally. Illustrate your answer with sketches. Indicate what measurements will be required. What would be the most important feature of the test arrangement? (4 marks)

(b)

Using the Müller-Breslau principle, sketch the influence lines for the structure shown in Figure Q6 for the following quantities: (i) (ii) (iii) (iv)

Reaction at A Bending Moment at B Bending Moment at E Shear force at E (16 marks)

2L

L

L

Figure Q6

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2L

Question 7 The cantilever shown in Figure Q7 has a cross section where the depth varies as shown below:

d x = d o [ 1 + ( x / L) 2 ] where, do is the depth of the cantilever at the tip, and dx is the depth of the cantilever at a distance x from the origin. The origin is shown to be at the tip of the cantilever. Assume that the cross-section is rectangular with a width of b, and that the Modulus of Elasticity is E. Calculate the deflection of the tip of the cantilever using the numerical summation method. Divide the whole length of the beam into 5 segments. (20 marks)

do L Figure Q7

Internal Examiners:

Dr C D’Mello Professor K S Virdi

External Examiner:

Professor M R Barnes

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