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Vietnam National University – HCMC FINAL EXAM Ho Chi Minh City University of Technology Subject: Reliability of Structures (085085) Faculty of Civil Engineering Materials: allowed; Computer & mobile phone: NOT allowed; Duration: 120 minutes; Date: 03/12/2016 Prepared by: Dr. Cao Văn Vui Signature: ............................................... Approved by: Assoc. Prof. Dr. Lương Văn Hải Signature: ............................................... PART I: MULTIPLE CHOICE QUESTIONS (first 60 minutes) PROBLEM 1 A normally distributed variable X with its mean  X and standard deviation  X : Question 1) What is the value of P (  x   x  X   x   x ) a. 0.68 b. 0.87 c. 0.86 d. 0.78 e. Other: … Question 2) What is the value of P (  x  1.5 x  X   x  1.5 x ) a. 0.75 b. 0.78 c. 0.57 d. 0.87 e. Other: … Question 3) What is the value of P (  x  2 x  X   x  2 x ) a. 0.59 b. 0.95 c. 0.57 d. 0.75 e. Other: … Question 4) Given  X  10;  X  5 , what is the correct value of P(00) > 0 b. P(X<0) > 0 c. P(X>0) < 0 d. P(X<0) < 0 e. Other: … PROBLEM 2 If Z is a standard normal random variable Question 7) and z=-1.25, what are the CDF and PDF values, respectively? a. 1.056 and 1.826 b. 1.860 and 1.562 c. 0.1056 and 0.1826 d. 0.0154 and 0.0387 e. Other: … Question 8) and z=+2.5, what is the CDF value? a. 0.9938 b. -0.9938 c. 0.9398 d. -0.3998 e. Other: … Page 1/9 Question 9) and given  ( z )  6.55.10 2 , what is the corresponding value of z? a. -1.15 b. -5.41 c. -4.15 d. -1.51 e. Other: … PROBLEM 3 The tension load carrying capacity (kN) of 225 steel bar samples are presented in the normal probability paper as shown in Figure 1. The “best-fit” line for the data show that the data can be modelled as normal distribution. Figure 1. Question 10) What is the mean tension load carrying capacity (kN)? a. 15 b. 20 c. 25 d. 30 e. Other: … Question 11) What is the standard deviation? a. 5 b. 0 c. 10 d. 15 e. Other: … PROBLEM 4 An analytical result is obtained from unknown source as shown in Figure 2. No other information is found. 0.3 0.25 0.25 0.2 0.2 0.175 0.15 0.125 0.1 0.1 0.05 0.05 0.05 0 Figure 2. Page 2/9 Question 12) It can be a(an) … a. cumulative distribution function b. lognormal distribution c. probability density function d. probability mass function e. Other: … Question 13) The analytical result is a. reliable b. correct incorrect c. d. good e. Other: … PROBLEM 5 The applied load (Y) and displacement (X) of a structural component are determined from an experiment. The data are shown in Table 1. Table 1. Experimental applied load (Y) and displacement (X) of a structural component. Y 1015 2020 4999 3970 4005 9002 960 4940 4020 7025 X 1 2 5 4 4 9 1 5 4 7 Question 14) Calculate the sample standard deviation of X a. 3.15 b. 2.53 c. 0.15 d. 3.51 e. Other: … Question 15) Calculate the sample standard deviation of Y a. 2533 b. 2353 c. 253.3 d. 23.53 e. Other: … Question 16) Calculate the correlation coefficient of X and Y a. 0.099 b. 0.899 c. 0.999 d. 0.989 e. Other: … PROBLEM 6 Assume that the load carrying capacity X1 and the demand X2 of a structure are normal random variables with their distribution parameters are shown in Table 2. Table 2. Mean and standard deviation. Variable Mean standard deviation X1 100 10 X2 40 5 Consider the performance function Y = 3X1 − 5X2 Question 17) Calculate the mean of Y, assuming that X1 and X2 are uncorrelated. a. 50 b. 100 c. 10 d. 290 e. Other: … Question 18) Calculate the standard deviation of Y, assuming that X1 and X2 are uncorrelated. a. 39.05 b. 26 25 c. d. 15.8 e. Other: … Question 19) If two variables X1 and X2 are correlated with the correlation coefficient of X1 and X2 is 0.5. What is the standard deviation of Y? a. 14.7 b. 20.3 c. 5.2 d. 27.84 e. Other: … Question 20) The failure of the structure is defined as Y < 0. Determine the probability of failure, assuming that X1 and X2 are uncorrelated. 2.83.10-4 a. b. 2.53.10-4 c. 1.64.10-4 d. 3.28.10-4 e. Other: … Page 3/9 PROBLEM 7 The moment-carrying capacity of a beam is calculated as [M]=FyS where, S is the plastic section modulus. Fy is the yield stress. M is the maximum applied moment (demand) on the beam. Assume that Fy, S, and Q are statistically independent lognormal random variables with the parameters as shown in Table 3. Table 3. Distribution parameters. Mean Coefficient of variation Variable Fy (kN/cm2) 20 0.15 3 S (cm ) 0.5 0.10 M (kNcm) 10 0.25 capacity The performance function is defined as Y  and the failure of the beam is defined as Y<1. demand Question 21) The variance and mean values of lnFy are …, respectively: a. 2.022 and 3.033 b. 0.0223 and 2.985 c. 3.022 and 0.033 d. 0.0223 and 2.859 e. Other: … Question 22) The variance and mean values of lnS are …, respectively: a. 0.01 and 0.698 b. -0.01 and -0.698 c. 0.01 and -0.698 d. -0.01 and 0.698 e. Other: … Question 23) The variance and mean values of lnM are …, respectively: a. 0.0606 and 2.2723 b. 0.0606 and 1.2723 c. 1.0606 and 2.2723 d. 0.1606 and 2.2723 e. Other: … Question 24) The variance and mean values of lnY are …, respectively: a. 0.0298 and 0.0142 b. 0.0928 and 0.1042 c. 0.0928 and 0.0412 d. 0.0928 and 0.0142 e. Other: … Question 25) The failure probability of the performance function (variable Y) is a. 4.8x10-4 b. 4.8x10-3 c. 4.8x10-2 d. 4.8x10-1 e. Other: … PROBLEM 8 Using Latin hypercube sampling, estimate the mean value of the load combination C including dead load (D), live load (L) and wind load (W) acting on a structure: C = 1.1D + 0.9L + 0.9W Question 26) What is the value of K in the Latin hypercube sampling a. 1 b. 2 c. 3 d. 4 e. Other: … Question 27) In Latin hypercube, which of the following interval characteristics must have? a. Equality b. Equal probability c. The same length d. Different probability e. Other: … Question 28) Let N is the number of intervals. What is a total of possible combinations of these representative values. a. KxN b. N-K K c. N d. K-N e. Other: … PROBLEM 9 The load carrying capacity or resistance R and the demand or load effect Q of a structural member are assumed to be random variables and uncorrelated. Their distribution parameters are given as shown in Table 4. The performance function is defined as Y=R+Q. Page 4/9 Table 4. Distribution parameters. Mean Standard deviation Variable R R R Q Q Q Question 29) What is the mean of Y: a.  R + Q c.  R x Q b. d. e. Other: … Question 30) What is the variance of Y: 2 2 a.  R +  Q 2 2 c.  R x  Q   R / Q 2 2 b.  R -  Q 2 2 d.  R /  Q e. Other: … Question 31) The reliability index is a.   Q  R 2 2  R  Q c.  R - Q b. d.  R  Q 2 2  R  Q e. Other: … Question 32) The probability of failure is a.      b. c.      e. Other: …    R  Q 2 2  R  Q  R / Q 2 2  R  Q   d.     PROBLEM 10 Consider a 4-meter cantilever beam subjected to a distributed load w. The moment capacity MR and the distributed load are uncorrelated normal random variables. The distribution parameters for the random variables are shown in Table 5: Table 5. Distribution parameters.  Variable V (%) w 10 kN/m 10 MR 85 kNm 12 The performance function is defined as Y  capacity  demand  M R  M w Question 33) The standard deviations of w and MR are …, respectively: a. 1 kN/m and 1.2kNm b. 1.5 kN/m and 10.2kNm c. 2 kN/m and 10.2kNm d. 1 kN/m and 10.2kNm e. Other: … Question 34) The mean value and variance of the maximum moment (kNm) due to the distributed load w are …, respectively: a. 80 and 6.4 b. 80 and 4.6 80 and 64 c. d. 8 and 64 e. Other: … Question 35) The distribution of the performance function (variable Y) is a. lognormal b. uniform c. normal d. extreme type I e. Other: … Question 36) The mean value (kNm) of the performance function (variable Y) is a. 0 b. 5 c. 10 d. 15 e. Other: … Page 5/9 Question 37) The variance of the performance function (variable Y) is a. 186 b. 861 c. 618 d. 168 e. Other: … Question 38) The coefficient of variation of the performance function (variable Y) is a. 5.29 b. 5.29 c. 9.25 d. 2.59 Other: … e. Question 39) The reliability index of the performance function (variable Y) is a. 0.3857 b. 0.4857 c. 0.5738 d. 0.3251 e. Other: … Question 40) The failure probability of the performance function (variable Y) is a. 3.499x10-3 b. 3.499x10-3 -1 c. 3.499x10 d. 2.899x10-4 e. Other: … PROBLEM 11 The system shown in Figure 3 including element BC and CD. Element BC is a beam with a cross section is 9cm width and 20cm height, and the allowable stress []=4.2 kN/cm2. Member CD is a steel bar with cross sectional area A=1.5 cm2 and ultimate strength of fu=20 kN/cm2. Assume that the distributed load q=8kN/m and the length L=5m are deterministic. The stress caused by the axial force in element BC is neglected. The following statistical parameters are assumed for the resistance R of the members: Table 6. Parameters of variables. Coefficient of Variable Distribution Mean variation Lognormal RBC 2700 (kNm) 0.15 Normal RCD 80 (kN) 0.10 D q A C B L Figure 3. Question 41) The standard deviation and mean values of lnRBC are …, respectively: a. 1.492 and 79 b. 1.492 and 7.9 c. 14.92 and 9.7 d. 1.249 and 0.79 e. Other: … Question 42) The standard deviation value of RCD is: a. 1 b. 2 c. 3 d. 4 e. Other: … Question 43) The failure probability of the element BC is: a. 0. 9532 b. 0.3925 c. 0. 2935 d. 0.3295 e. Other: … Question 44) The reliability index of the element BC is: a. 0.44 b. 1.4 c. 0.04 d. 4.4 e. Other: … Page 6/9 Question 45) The failure probability of the element CD is: a. 0.3728 b. 2.837 c. 0.8327 d. 0.2837 e. Other: … Question 46) The reliability index of the element CD is: a. 2.75 b. 0.572 c. 1.33 d. 0.725 Other: … e. Question 47) The failure probability of the system is: a. 0.1957 b. 0.1597 c. 0.5197 d. 5.197 e. Other: … Question 48) The reliability index of the system is: a. 0.049 b. 0.009 c. 1.14 d. 0.49 e. Other: … PROBLEM 12 A system including 2 of the 10-element chains is used to slowly lift the mass m at a construction site, as shown in Figure 4. The failure probability of each chain element is 0.005. Force chain m Figure 4. Question 49) The failure probability of each 10-element chain is: a. 0.0498 b. 0.4980 c. 0.0489 d. 0.4098 e. Other: … Question 50) The failure probability of the system is: a. 0.24 b. 0.024 c. 0.42 d. 0.0024 e. Other: … Page 7/9 PART II: WRITING PRESENTATION (60 minutes) 2P P q C a B D E a a a Figure 5. PROBLEM 13 Consider elements BD and BE of the structure shown in Figure 5. a) P=2qa. Calculate the internal force N. b) Establish the performance function in the form: Y=N-[N], where, N is the applied tension load (demand) on the elements [N]= []A is the allowable buckling compression load-carrying capacity of the elements.  is the coefficient of stress reduction. A=28.27 cm2 is the cross-section area. []=20 kN/cm2 is the allowable stress. Given that a=1m is deterministic. Other variables are given in Table 7. Table 7. Parameters of variables. c) d) e) f) g) Variable Mean Coefficient of variation q (kN/m) 88 0.05 0.65 0.10  Calculate the mean of the performance function Y. Calculate the standard deviation of the performance function Y. Calculate the reliability index. Calculate the failure probability. Calculate the failure probability for the system including elements BD and BE. PROBLEM 14 The cross section of the beam BC in Figure 5 is rectangular: the width b and the depth d. Its performance function can be defined as follows: M Y  Fb S where, S=bd2/6 is the elastic section modulus, M is the applied bending moment, and Fb is the allowable bending stress. Assume that Fb, M, b and d are all random variables with the following parameters: Page 8/9 Table 8. Parameters of variables. Variable Mean Coefficient of variation Determined from Figure 5 and parameters given in 0.05 M (kNm) Problem 13. 45 0.15 Fb (MPa) 200 0.05 b (mm) 300 0.05 d (mm) Calculate the mean, variation of Y and the reliability index. Page 9/9