Exam January 2015, questions PDF

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CIVE5959M/5970M

This question paper consists of 5 printed pages each of which is identified by the Code Number CIVE5959M/5970M

© UNIVERSITY OF LEEDS School of Civil Engineering

January 2015

CIVE5959M/5970M Advanced Structural Analysis

Time allowed: 2 hours

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Please Answer FOUR Questions

Page 1 of 5

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CIVE5959M/5970M

1. A 6 m long steel column of a channel section is pinned at top and fixed at the bottom, as shown in Figure 1. The section properties are as follows, A=12.71 cm2; Ix=312 cm4; Iy=19.9 cm4; x0=1.23cm. The Young’s modulus of the steel is 200GPa and the elastic limit of the material is 200MPa. (a) What are the radius of gyration and slenderness ratio in the x and y directions [8 marks] (b) Determine the possible failure mode of the column,

[8 marks]

(c) Find the ultimate load that can be applied

[9 marks]

x 6m x0 y

Figure 1

2. As shown in Figure 2, the two-bar assembly is fixed at both ends and connected by a pin joint at the mid span. It is subjected to a load P at the mid-point joint 2. Using the 1D Finite Element method (two bar elements), determine the stresses in the bar element A and B, given: P=6.0×104 N, E=2.0×104 N/mm2, AA (area of cross section for element A) =500mm2, AB (area of cross section for element B) =250mm2, L=150mm [25 marks]

A

B x

1

2 L

P

3

L

Figure 2

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3. A square element as shown in Figure3 (co-ordinates are in meters) is used in a plane stress finite element analysis. It has been assumed that the element has following shape functions:

N1  (1  x)(1  y ) / 4, N2  (1  x )(1  y ) / 4 N3  (1  x)(1  y ) / 4, N 4  (1  x )(1  y ) / 4 (a) If there is distributed loads of 100 kN/m on the edge 1-2, as shown in Figure 3, find the equivalent nodal forces at node 1 and 2, the thickness of element is 0.1m. [10 marks] (b) If the following nodal displacements are obtained from a finite element analysis, calculate the stresses at the centre of the element. Modulus of elasticity and Poisson’s ratio of the element are 200 kN/mm2 and 0.3 respectively. [15 marks]

u 1  0.1mm v1  0.1mm u 2  0.3mm v 2  0.3mm u 3  0.6mm v 3  0.7mm u 4  0.1mm v 4  0.5mm

100kN/m

Y

1(-1,1)

2(1,1) X

4(-1,-1)

3(1,-1) Figure 3

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4. For the un-damped fixed-footed portal frame shown n Figure 4 the mass of the columns can be neglected.

Figure 4 (a) Calculate the un-damped natural frequency and period of the sway vibration. [8 marks] (b) If the beam 2-4-6 is given an initial horizontal velocity of 1m/s (at time t=0) what is the position of point 2 relative to its starting position after 12 seconds [9 marks] (c) A dashpot damping system is added to the structure shown and it is found, experimentally, that after each cycles the amplitude of the motion has decreased by 10%. What is the percentage of critical damping? How much has the frequency changed? [8 marks] It is given that the stiffness of a fixed ended column is, k 

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12 EI l3

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5. Consider the sway vibrations of the frame structure of Figure 5. The floors are rigid and have centrally concentrated masses with values as shown. Each floor also carries an additional rigidly mounted weight with the value of 200t and 100t for the first and second floors respectively. The value of EI / L is 20MNm for a single column in all floors. The column mass can be ignored and the stiffness of each individual column can be calculated from the formula, k 

12 EI . l3

Figure 5

(a) Determine the stiffness and mass matrices of the frame [8 marks] (b) Calculate the circular frequencies and the vibration modes of the frame [13 marks] (c) Sketch all the sway modes. [4 marks]

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