GATE-CE-2000 - 1.1 1.2 The following integral 1.3 1.4...

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Unformatted text preview: 1.1. 1.2. The following integral 1.3. 1.4. GATE-2000 CIVIL ENGINEERING M Maximum marks : 151] SECTION A. (75 Marks) This question consists of 2? (Twenty Seven) multiple—choice tycp sub-questions each carryiing one mark. The answers to the multiple choice questions MUST be written only in the boxes corresponding to the question number by writing A, B, C or D in the answer book. (27 x i=2?) If, A, B, C are square matrices of the sa me order, (ABCT1 is equal to (a) C'1 A" B" (b) C" B'1 A'1 (C) A-l B-l C—l A-l CA] B-l a 132?, xfildx I 1 (b) Converges to — (a) Diverges 3 1 (c) Converges to — g; (d) Converges to 0 A function with a period 211: is shown below f{ x) t -I an 0 5'2 ‘3 The Fourier series for this function is given by l i. e (a) “)0: 2+“; “I am 2 I” 2 , mt (b) f(x)= “HEW” cosnx 1 °° 2 .E (c) ftx)= 3+ 3“” w 2 . mt (d) f(x)= zES”'_‘-g'sinnx. Consider the following two statements : I. The maximum number of linearly independent column vectors of a matrix A is called the rank of A. II. if A is an n x 11 square matrix, it will be nonsingular is rank A = n. 1.5. 1.6. 1.7. With reference to the above statements, which of the following applies ? (rt) Both the statements are false (1:) Both the statements are true. (r) l is true but I] is false. (if) I is false but 11 is true. The dimensions for the I'iexural rigidity of a beam element in mass (M), length (L) and time (T) is given by (a) M’i“2 (c) NIL-1 T1 (b) ML3 "r1 (d) ML" 'r2 A two span beam with an internal hinge is shown below. Hinge i— .1 c a I: Conjugate beam corresponding to this beam is a I l] C d (a) a l) c d (b) m a b c d (U) Pick the incorrect statement from the following four statements : (a) On the plane which carries maximum normal stress, the shear stress is zero. (b) Principal planes are mutually orthogonal. {c} On the plane which carries maximum shear stress, the normal stress is zero. The principal stress axes and principal strain axes coincide for an isotropic material. (if) 1.8. .-\fmr1w Mil '1.) is sup 1.9. ported by a roller at A and is on a hinge at (T as shown below: ' U2 i. The reaction at the roller end A is given by (at P (ti) 2]) P (c) E [(1) Zero. The following two statements are made with reference to a simply supported under - reinforced RCC beam .- |. Failure takes place by crushing of concrete before the steel has yielded. ll. The neutral axis moves tip as the load is increased. With reference to the above statements, which of the following applies ? (r1) Both the statements are false. [11) | is true but ll is false. ((1 Both the statements are ture. (d) I is false but II is true. 1.1!]. The stress-strain diagram for two materials A and I3 is shown below : Material. A Material B The following statements are made based on this diagram : I (l) Material A is more brittle than material I}. (11) The ultimate strength of material B is more than 7.“- i thatol A. 1.11. 1.12. 1.13. 1.14. 1.15. With reference to the above statements, which or the following applies ? (a) Both the statements are false. (1)) Both the statements are true. in?) (d) I is false but [I is true. | is true but I] is false. Four column of the same material and having identical geometric properties are supported in different ways as shown below : l . II III 111’ It is required to order these four beams in the increasing order of their respective first buckling loads. The correct order is given by (a) 1, II, 111, W (12) Ill, IV, i, 11 (c) Ii, 1, 11!, Ill (at I, [1, IV. In A soil sample has a void ratio of 0.5 and its porosity will be close to (a) 50% (c) 100% (b) 66% (d) 33% A borrow pit soil has a dry density of 1? kN/m3. l' low many cubic meters of this soil will be required to censtruet an embankment of 100 in3 volume with a dry density of 16 kN/m3. (a) 94 m3 (b) 106 m3 (c) 100 m3 (a) 90 m3 The group efficiency of pile group (a) (b) (C) (It) will be always less than 100% will be always greater than 100% may be less than 100% or more than 100% will be more than 100% for pile groups in cohesionless soils and less than 100% for these in cohensive soils. The two criteria for the determination of allowable bearing capacity of a foundation are (rt) tensile failure and compression failure. (0) tensile failure and settlement. (c) bond failure and shear failure. (if) shear failure and settlement. 1.16. lfduty (D) is 1428 hectares/enmec and base period {8) is l20 days for an irrigated crop, then delta (A) in meters is given by (a) 102.8 (c) 1.38 (b) 0.73 {d} Utli 1.17. The relation that must hold for the. flow to be irrotational is _ _ ( dtl _' 81' (n) 31’ 8x _ (b) _' 65- 9%.22.0 ec_a (c) at“? (2v: (‘1) dy (11' 1.13. Cavitation is caused by (rt) high velocity (c) high pressure (it) low pressure ((1') high temperature 1.19. If the pump head is 75m, discharge is 0.464 m3/ s 1.20. 1.21. 1.22. 1.23. and the motor speed is 1440 rpm at rated condition, the specific—speed of the pump is about (a) 4 (b) 26 (c) 33 (d) 1440 The BOD removal efficiency, in percentage, during primary treatment, under normal conditions is about (a) 65% (c) 30% (b) 35% (a) Zen:- Critical factors for the activated sludge treatment process are (H) (D) (C) maximum hourly flow rate. maximum and minimum flow rate. maximum hourly flow rate and maximum daily organic load. (‘1) minimum hourly flow rate and minimum daily organic load. Use of coagulants such as alum (it) results in reduction of pH of the treated water. (11) (C) (11') results in increase of pH of the treated water. results in nochange in pH of the treated water. may cause and increase or decrease of pH of the treated water. The disinfection efficiency of chlorine in water treatment (I?) is not dependent on pH value, (b) is increase by increased pH value. (c) remains constant at all pint value. (it) is reduced by increased pH value. 1.24. 1.25. 1.26. 1.27. 2.1 2.2. The standard plate size in a plate hearing test for finding modulus of subgrade reaction (k) value is {It} 100 cm diameter (c) .75 cm diameter (ii) 50 cm diameter ((1) '25 cm diameter Width of carriage way for a single lane is recommended to be (rt) 5'5 m (c) 3.75 m (b) 7.0 m id) 55 m Stopping sight distance is the minimum distance available on a highway which is the (I?) distance of sufficient length to stop the vehicle without collision. (b) (c) (:1) distance visible to a d river during night driving height of the object above the road surface. distance equal to the height of the driver's eye above the road surface. Bituminous materials are commonly use in highway censtruction because of their good {a} tensile and compression properties. (it) binding and water proofing propertie's. (c) shear strength and tensile properties. (it) bond and tensile properties. . This question consists of 24 (Twenty Four) multiple—choice type sub—questions, each carrying TWO marks. (24 x 2 = 48) 1 lfftx,y,z)={x2+y2+zzl 1 air air 33; Ed—F‘l-EZTl-Sequflllo (b) 1 _; (c) 2 (d) —3 [x2+y2+zz) The Taylor expansion of sin at about x = 11/ 6 is given x3 x5 ? wt*a*a‘a .1 5 1 [A] [rt] tx-a ‘ (C) [x—KJ— 6 + - " I l s 3! 5! 7 l (d) 2 2.3. Let [5(5) = £ [ll [0] denote [he Laplace translorm of the function Fill. Which of the follow ing statements is correct ? | rdf l . - ._ ‘ _ (a) JELT}:;H5]; i! l(‘r)dt —5t‘{5) [(0) Fur ' - dF =Sl‘t:,i-F(U); Elbow}??? l £[fflfidf]: F(5_a) D . I 1 sh finer] = E as) D .23. The limit of the function f (x) = {1-34 /X"] as x a no is given by (rt) 1 (Cl 00 (33) EXP l‘a'll (d) Zero 2.5. The maxima and minima of the function fix) = 2x3 - 15):2 + 36x + 10 occur, respectively, at (a) x=3andx=2 (ii) x=1andx=3 (c) x=2amdx=3 (d) x=3andx=4 2.6. The four cross sections shown below are required to be ordered in the increasing order of their respective shape factOrs. 99$ El Which of the following order is correct ? (a) ill,l,iv,11 (b) I, 11, m, w (c) 11[,[v,1,11 (d) I11,1v,11,1 '23. A simply supported beam with an overhang is traversed by a unit concentrated moment from the left to the right as shown below : g—b Direction of Motion M :1 1 A B H—slw—bL— 1. N The influence line for reaction at B is given by 2.8. 2.9. (6!) zero everywhere The following two statements are made with reference to the planar truss shown below : I. The truss is statically determinate [I.I The truss is kinematically determinate With reference to die above statements, which of the following applies ? (it) Both statements are true. (l3) Both statements are false. (:3) [I is true but [ false. (d) l is true but [I is false. A cantilever beam of length L and a cross section with shape fact0r f supports a concentrated load P as shewn below : I L —+ The length LP of the plastic zone, when the maximum bending moment, equals the 13135th moment Mp, given by L l. W Tp=i “0 TP=L(1'B LP_ LI: 1 ‘0 T‘lmii (‘0 T=1‘? 2.10. For the structure shown below, the vertical deflection at point A is given by m.“ (a) SIEI 2171.3 (b) 8113i (L) Zero PL3 (d) 72E| 2.11. The ultimate bearing capacity of a soil is 300 kN/m2.The depth of foundation islm and unit weight of soil is 20 kN/m“. Choosing a factor of safety of 2.5, the net safe bearing capacity is (a) 100 kN/m3 (b) 112 lav/mi (c) 80 kN/m2 (d) 100.5 kN/m2 2.12. A deep cut of 7m has to be made in a clay with unit weight 16 kN/l'n" and a cohesion of 25 kN/mz. What will be the factor of safety if one has to have a slope angle of 30D ? Stability number is given to be 0.128 (from Taylor’s chart} for a depth factor of 3. (a) 0.80 (b) 1.1 (C) 1.25 (r!) 1.0 2.13. [n a drained triaxial compression test, a saturated specimen of a cohesionless sand fails under a deviatoric stress of 3 kgf/cm2 when the cell pressure is 1 kgf/cmz. The effective angle of shearing resistance of sand is about (a) 37° 0;) 45° (c) 53° (in 20° 2.14. Two footings, one circular and the other square, and founded on the surface of a purer cohesionless soil. The diameter of the clrcu Iar footing is same as that of the side of the square footing. The ratio of their ultimate bearing capacities is (a) 3/ 4 (b) 4/ 3 (C) 1-0 (ii) 1.3 2.15. To have zero active pressure intensity at the tip of a wall in cohesive soil, one shOuId apply a uniform surcharge intensity of (b) 2ccoto. (d) —2ctanot (fl) 2 c tan a (c) ~ 2 c tan [1 2.16. Water flows at a depth of 0.1m with a velocity of 5111/ s in a rectangular channel. The alternate depth is (a) 0.30m (0) 0.40m (C) 0.86 m (d) 0.81 m 2.17. 2.18. 2.19. 2.20. 2.21. 2.22. 2.23. 2.24. In an area of 200 hectare, water table drops by 4 m. If the porosity is 0.35 and the specific retention is 0.15, change in volume of storage in the aquifer is (ti) 160m“ (0) 1.6 x 10" m (if) SXIObm3 (d) Loatlflflm3 A tube well havinga capacity of4rn3/hour operates for 20 hours each day during the irrigation season. How much area can be commanded if the irrigation interval is 20 days and depth of irrigation is 1? cm ? (n) 1.71 x 10‘1 m2 (b) 1.14 ><104 m2 (c) 22.9 5:104 m2 (d) 2.29 x 10“ rn2 The parameters in Horton’s infiltration equation [_f(t) = fc + (fo — fc)‘“k'] are given as, to -.= 7.62 cm/ hour, fr = 1.34 cm/hour and k = 4.182/h0ur. For assumed continious pending the cumulative infiltration at the end of 2 hours is (a) 2.68 cm (b) 1.50 cm (:2) 1.34 cm (d) 4.18 cm Water flows at a rate of 10m3/s in a rectangular channel 3 m wide. The critical depth of flow is (n) 1.13111 (0} 2m (c) 1.45 m (cl) 1.04 m A circular sewer 2m diameter has to carry a discharge of 2m3/s when flowing nearly full. What is the minimum required slope to initiate the flow? Assume Manning’s n = 0.015. (.11) 0.00023 (1:) 0.000036 (t‘) 0.000091 (:1) 0.000014 The following characteristics pertain to the sand filters used in water industry. I. Filtration rate is 1 to 4 1'n3/(rn2 day). If. Typical duration of operation in one run is 24 to 72 hours. III. Operating cost is low. Which of the above characteristics pertain to slow sand filters ? (a) i, It and [11 (c) [land “I (b) l and 11 (d) [ and 111 The ruling minimum radius of horizontal curve of a national highway in plain terrain for a fuling design speed of 100 km/ hour with e = 0.07 and f: 0.15 is close to (n) 250 m (c) 36m Design rate of super elevation for horizontal highway curve of radius 450 m for a mixed traffic condition, having a speed of125 km/hour is (a) 1.0 (b) 0-05 (c) 0.07 (d) 0-15‘1 (b) 360 m (d) 300 m SECTION B. {75 Marks) This section consists ol'TWENTY questions of l-IVI; mirks each. ANY Fit-TEEN out of them have to be L ‘ s to each questions must be it more number of questions the answers not to be {irst fifteen unscored {IS x 7'1 :: 75) kinsu'ered. Answer started on a fresh page. are attempted, score ol'f evaluated. else only: the answer will be considered strictly. 3 Find the solution of the differential equation 321+ Ale .-= cos tut 4- k with initial conditions at2 ' 0) = 0, (fl) = 0.1Iere it, (u and k are constants. V l ‘ dt _ I _ _ Use either the method 0! undetermined coeffictents d or the operator [-0 = based method. 4. A is a square matrix is given by 2 G A = —t 2 0 —t 2 One of the eigenvalues of A is given to be equal to 2. Determine the other eigenvalues. Express the eigenvector corresponding to the lowest eigenvalue in the form [ 1 a b]T and determine a and b. 5. A cantilever beam AB carries a concentrated force F and a moment M = l’L/3 at its tip as shown below. Show using Castigliano’s theorem that, if the angle of inclination at of the line of action of the force P is such that sin or = 1/2, then the displacement of the point [5, due to bending will be in the direction of force F. —-——-———-l El = Constant H B ‘L . beam AB is suspended from a wire CB as shown in figure below. The beam carries a central concentrated load 1". it may be assumed that E. = 7 x IOION/mz, I = 800 mm“, A =12 mm2 and P 300 N. Determine, using stiffness matrix approach, we deflection of point B. What would be this defleetion if the load P were to be applied upwards, instead of downwards ? The axial deformation of the beam AB may be ignored C N? WireT P : 300N 600mm gal The stiffness matrix for a beam element is given to he as follows. 2 s '2 AIL Symmetric 0 '12 K_EL 0 6L at} _F 2 2 is 0 —12—6L0 12 ; D 6L—2L10 ~6L 4L2 7. A simply supported rectangular prestressed concrete beam 200 mm wide and 400 mm deep has an effective span of 12 m. The prestressing cable has a triangular profile with zero eccentricity at ends and 70 mm at the midspan as shown in the figure below. The effective prestress is 800 kN after all losses. Determine the value of a point load, the beam can support at the midspan if the pressure line passes through the upper kern of the secti0n- The weight density of the material of the beam can ' be taken to be 25 kN/m3. T '— 400mm S. A two hinged parabolic arch carries two t'oncentratt-‘d moments as shown in the figure below. The moment of inertia of the arch at any particular cross section is equal to the moment of interia at the crown multiplied by the secant of the angle 6. where 0 is the angle between the horizontal and the tangent to the arch axis at that particular sectiorg. Determine the support reactions. 2 M M It H—Lri-bt l-d—Lr 4—1»! n—L—~——>t 9. A continuous beam 250 mm x «150 mm carries 6 numbers of 12 mm diameter longitudinal bars as shown. The factored shear force at the point of inflection is 200 kN. Check if the beam is safe in bond. Assume M15 mix with ock = 15 N/mm2 and mild steel with or = 250 N/mnz. A clear cover of 25 mm can be assumed. The design bend stress for mild steel bars in M15 concrete is specified to be 1.0 N/mn13_ Section of point / of inflection 6 ms. of 12mm die. 10. A 12mm bracket plate is connected to a column flange as shown in the figure below. The bracket transmits a load of W = 200 kN to the column flange. A 10mm fillet weld is provided along AB, BC and CD. If e = 350 mm, b = 200 mm and d : 650 mm, verify if the size of the weld provided is adequate. Allowable shearing stress in the fillet weld can be taken to be 108 Ml’a. lat—— b—H4———e-——+| W=200kN ? I I I I’ll/l. '1 D 12mm thick plate 11. 12. 13. 15. A 5m thich clay layer lies between two layers of sand each 4m thick, the top of the upper layer of sand being at ground level. The water table is int below the piezometric surface being 4m above ground level. The saturated unit weight of clay is 20 kNXm3 and that of sand is 19 kN/mi. Above, the water table unit weight ofsand is 16.5 kam-‘. Calculate the effective vertical stresses at the top and bottom of the layer. Also draw the total vertical stress diagram in the given soil layers. In an oedometer test. a specimen of saturated clay 19 mm thick reaches 50% consolidation in 20 minutes. How long it would take a layer of thisclay 5m thick to reach the same degree of consolidation under the same stress and drainage conditions ? l-tow lond would it take for the clay layer to reach 30% consolidation? A footing 3m square carries a gross pressure of 350 kN/m2 at a depth of 1.2 m in sand. The saturated unit weight of sand is 20 RN] re3 and the unitweight above the water table is i7kN/m3.The shear strength parameters are c' = t} and ¢‘ = 30°. (For 41‘ = 30°, N‘‘ = 22 and N3 = 20). Determine the factor of safety with respect of shear failure for the following cases: (a) water table is 5m below ground level. (b)water table is at 1.2 m below ground level. mm 14.For the retaining wall shown in the figure below assume that the wall can yield sufficiently to develop active stage. Use Rankine's active earth pressure theory and determine (a) active force per meter of the wall, and (b) the location of the resultant line of action. T A pump is to deliver water at a rate of 1.4 m3/s from a sump to a reservoir which is 10 km away through a steel pipe 1200 mm in diameter for a length of 5 km and the remaining through a PSC pipe of diameter 1000 mm. The elevation difference between the full tank level of the delivery reservmr and the low water level of the sump is 50 or What is the Break Horse Power ( in kW) requirement of the motor ? Assume a pump house loss of 3.75 m and minor losses to be 10% of the friction lessee. Use Darcy-Weisbach friction factor (fluof 0.018 for steel pipe and 0.02 for PSC pipe with a pump efficiency of 80%. ufa proposed dam is used to conditions. The design flood is 15000 mfils. Establish 16. .-\ l : ~19 scale model predict prototype flow discharge near thespillway the rela ‘ prototvpe. What flow rate should be established In the model to simulate this flow .7 if a velocity of I .2 rm’s is measured at a point in the model what is the velocity at a corresponding point in the prototype ? 17. An area of 1 hectare is irrigated through a stream of 0.03 nia/s. Depth of the root zone is 1 m and available moisture holding capacity is 16 cm/m. Irrigation water is supplied when 50% of the available moisture is depleted. Water application efficiency is 60% Determine the storage efficiency. 18. A 6 hour unit hydrograph of a watershed is given below. Calculate 18 hour unit hydrograph using 5— curve method and tabulate the results. 6hr unit hydrograph '19. Consider a glucose solution (CfiHIZOG) of molarity , 1.75 x 10'?l that is completely oxidized to CD2 and HEO. Find the amount of oxygen required for this reaction.'Tl'ie chemical mass balance equation for the above reaction is given as CfiHuO6 602 —) 6C02 + 6H20 {Atomic weights are : C = 12; H = 1; 0 =16}. 20. The dissolved oxygen (D0) in an unseeded sample of dlluted waste having an initial D0 of 9.5 mg/l is if'neasured to be 3.5 rug/l after 5 days. The dilution actor is 0.03 and the reaction rate constant k = 0.22/day (to the b . ,. 356 e in the d WT.) Estimate any curve at lion between velocities in the model and _ 21. 22. (a) The 5 day BOD of the waste at 20° C. (b) Ultimate Carbonaceous BOD. (c) Remaining oxygen demand alter 5 days (d) The 5 day BUD of this waste at 25° C assuming a temperature coefficient of l .047. The load penetration date from a California Bearing Ratio (CBR) test is provided in the tollowingtable. indicate whether any correction is required for the calcula...
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