DEPARTEMENT SIVIELE EN BIOSISTEEM-INGENIEURSWESE DEPARTMENT OF CIVIL AND BIOSYSTEMS ENGINEERING MEGANIKA SWK 122 EKSAMEN MECHANICS SWK 122 EXAMINATION

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1 UNIVERSITEIT VAN PRETORIA UNIVERSITY OF PRETORIA Kopiereg voorbehou / Copyright reserved DEPARTEMENT SIVIELE EN BIOSISTEEM-INGENIEURSWESE DEPARTMENT OF CIVIL AND BIOSYSTEMS ENGINEERING MEGANIKA SWK 122 EKSAMEN MECHANICS SWK 122 EAMINATION 8 November 2005 / November 8, 2005 Dosente/Lecturers: Prof L Maree ; Ms B Ssamula ; Prof A van der Walt ; Mnr F J van Graan Eksterne Eksaminator / External Examiner: Prof WMG Burdzik Volpunte/Full Marks : 80 Tyd/Time : 2 ure/hours Van/Surname: Studentenommer/Student number Studierigting/Study Discipline Voorletters / Initials: AFDELING A / SECTION A: 16 Punte / Marks AFDELING B / SECTION B: Vraag Question Punt Mark Vraag Question Punt Mark

2 AFDELING A / SECTION A Gebruik asseblief kant 1 van die antwoordblad en volg die instruksies noukeurig. Die laaste bladsy van die vraestel is blanko en kan vir berekenings gebruik word. Please use side 1 of the answer sheet and follow the instructions carefully. The last page of this paper is blank and may be used for calculations. VRAAG / QUESTION Die krag F= i j + 3k kn gryp aan in die punt A(2;3;4) m en die krag F gryp aan in die punt B(1;1;2) m. The force F= i j + 3k kn acts at the point A(2;3;4) m and the force F acts at the point B(1;1;2) m. Antwoord 1 / Answer 1 (2 punte / marks) Die vektormoment van die koppel is: The vector moment of the couple is: (A) M = (2,67 ; -0,33, -1) kn.m (B) M = (-2,67 ; 0,33, 1) kn.m (C) M = (8 ; -1 ; -3) kn.m (D) M = (-8 ; 1 ; 3) kn.m VRAAG / QUESTION Die krag F = 9i +18j 18k kn gryp aan in die punt A(5 ; 4 ; 3) m. Die punte B(-1 ; 1 ; 5) m en C(1 ; 3 ; 6) m bepaal die lyn BC waarvan die rigting beskryf 1 2 ; 2 ;1. word deur die eenheidsvektor µ BC = ( ) The force F = 9i +18j 18k kn acts at the point A(5 ; 4 ; 3) m. The points B(-1 ; 1 ; 5) m and C(1 ; 3 ; 6) m determine the line BC with its direction 1 2 ; 2 ;1. described by the unit vector µ BC = ( ) Antwoord 2 / Answer 2 (3 punte / 3 marks) Die vektormoment van die krag F om die lyn BC is: The vector moment of the force F about the line BC is: 3 3 (A) M BC = ( 50 ; 50 ; 25) kn.m (B) M BC = (50 ; 50 ; 25) kn.m (C) M BC = (18 ; 15 ; 6) kn.m (D) M BC = ( 18 ; 15 ; 6) kn.m 2

3 VRAAG / QUESTION A 5 kn B C Die gladde balk ABC word binne in n pyp gesteun soos getoon in die skets. The smooth beam ABC is supported inside a pipe as shown in the figure. Antwoord 3 / Answer 3 (2 punte / 2 marks) Die korrekte vryliggaamskets van die balk ABC is: The correct free body diagram for the beam ABC is: (A) A R A 5 kn B R B C R A (B) A 5 kn R A B R B C A 5 kn (C) R B B C 3

4 VRAAG / QUESTION b 6 m c b 2 m c a a Beskou die ingekleurde deel van die skets. Die lyn c-c gaan deur die sentroïede van die ingekleurde deel. Die tweede moment van area van die ingekleurde deel van die figuur om die lyn b-b, is 450 m 4, en die oppervlakte is 20 m 2. Consider the shaded part of the figure. The line c-c passes through the centroid of the shaded area. The second moment of area of the shaded part about the line b-b is 450 m 4 and its area is 20 m 2. Antwoord 4 / Answer 4 (3 punte / 3 marks) Die tweede moment van area van die ingekleurde deel om die lyn a-a is: The second moment of area of the shaded area about the line a-a is: (A) I b-b = 690 m 4 (B) I b-b = 210 m 4 (C) I b-b = 1170 m 4 VRAAG / QUESTION 12 kn 1 m 2 m 8 kn 2 m A 4 kn Antwoord 5/ Answer 5 (2 punte / 2 marks) Die skuifkrag by punt A van die balk in die skets is: The shear force at point A of the beam in the figure is: (A) 4 kn opwaarts / upwards (B) + 4 kn (C) - 4 kn 4

5 Antwoord 6 / Answer 6 (2 punte / 2 marks) Die buigmoment by die punt A van die balk is: The bending moment at the point A of the beam is: (A) 4 kn.m Antikloksgewys / Anti clockwise (B) + 4 kn.m (C) - 4 kn.m VRAAG / QUESTION M=3 N.m x=1 x=2 x=3 Antwoord 7 / Answer 7 (2 punte / 2 marks) Die gedeelte van die buigmomentdiagram vir die balk in die skets tussen die punte x=1,9 m en x=2,1 m is: The part of the bending moment diagram for the beam in the figure between the points x=1,9 m and x=2,1 m is: (A) M(x) (B) x=1,9 x=2 x=2,1 M(x) x=1,9 x=2 x=2,1 5

6 AFDELING B / SECTION B VRAAG 1 / QUESTION 1 [5] Die kabels BC en BD kan n maksimum trekkrag van 20 kn weerstaan. Bepaal die gewig van die swaarste beton-elmboog wat, soos in die skets getoon, opgetel kan word. Die swaartepunt van die elmboog is by G. The cables BC and BD can each sustain a maximum tensile force of 20 kn. Determine the weight of the heaviest concrete elbow that can be lifted in the way shown in the figure. The centre of gravity of the elbow is located at G. 6

7 VRAAG 2 / QUESTION 2 [10] Y a) Bepaal die grootte en die hoek ten opsigte van die positiewe -as van die resultant van die kragte en koppelmoment wat op die raamwerk aangryp. (5) Determine the magnitude and angle with respect to the positive -axis of the resultant of the forces and couple moment acting on the frame. b) Bepaal die vergelyking van die werklyn van die resultant. Determine the equation of the line of action of the resultant force. (5) 7

8 VRAAG 3 / QUESTION 3 [10] n Homogene staal paal AB weeg 8 kn. Dit word as ankerpaal gebruik vir die elektriese paal CE, en AB is vasgebout aan n voetstuk by A(-4 ; 3 ; 0) m, soos in die skets getoon. Die kabel is vas by die punt B(-3 ; 2 ; 2) m en by C(-5 ; 4 ; 3) m. Die krag in die kabel BC is 6 kn. Teken n vryliggaamskets en bereken die groottes van al die reaksies by die voetstuk A. A homogeneous pole AB weighs 8 kn and is used to anchor an electric pole CE. The anchor pole AB is bolted to a base at A(-4 ; 3 ; 0) m, as shown in the figure. The anchor cable is fixed at the point B(-3 ; 2 ; 2) m and at the point C(-5 ; 4 ; 3) m. The force in the cable is 6 kn. Draw a free body diagram and determine the magnitudes of all the reactions at A. Z C B O A E Y 8

9 VRAAG 4 / QUESTION 4 [5] Bepaal die kragte in die stange CD, CF en FG van die vakwerk in die skets. Vul u antwoorde in op die gegewe tabel. Determine the forces in members CD, CF and FG of the given truss. Note your answers on the given table. Stang/Member CD Krag/Force Druk/Comp Trek/Tens CF FG 6 kn 6 kn 9

10 VRAAG 5 / QUESTION 5 [10] Die apparaat in die skets word gebruik om n balk wat 4 kn weeg op te tel. Bereken die horisontale en vertikale komponente A x en A y van die krag wat op die pen by A uitgeoefen word en die horisontale en vertikale komponente B x en B y van die krag wat die balk by B op die apparaat uitgeoefen. The apparatus in the figure is used to lift a beam that weighs 4 kn. Determine the horizontal and vertical components A x and A y of the force exerted on the pin at A, and the horizontal and vertical components B x and B y of the force exerted by the beam on the apparatus at B. 10

11 VRAAG 6 / QUESTION 6 [10] Y 10 m 6 m 2 m O 1 m 4 m 7 m 15 m a) Bereken die x-koördinaat van die sentroïede van die ingekleurde area in die skets. (3) Calculate the x-co-ordinate of the centroid of the shaded area of the figure. 11

12 b) Bereken die tweede moment van area om die Y-as, I y, vir die ingekleurde area in die skets. Die tweede moment van area van n driehoek en n reghoek, om n lyn ewewydig aan hulle basisse en wat deur hulle 3 sentroïedes gaan, is 1 bh en 1 3 bh respektiewelik, waar b die lengte van hulle basisse is en h die loodregte hoogte. (3) Determine the second moment of area about the Y-axis, I Y, for the shaded area in the figure, The second moment of area of a triangle and a rectangle about a line parallel to their basis passing through their centroids 3 are 1 bh and 1 3 bh respectively, where b is the length of the base and h the perpendicular height. c) Bereken die tweede moment van area om die -as, I x, vir die ingekleurde area in die skets. (4) Determine the second moment of area about the -axis, I, for the shaded area in the figure. 12

13 VRAAG 7 / QUESTION 7 [14] a) Voltooi die onderstaande lasdiagram vir die balk in die skets. Die balk is in die muur ingebou by A, het n skarnier by B en n roller by C (6) Complete the load diagram below for the beam in the figure. The beam is built into the wall at A, has a hinge at B and a roller at C. 700 kn 150 kn/m 800 kn.m Load Diagram Lasdiagram b) Teken die skuifkrag- en buigmoment diagramme vir die veranderde lasdiagram op die volgende blad. Dui alle tersaaklike waardes, wat insluit die maksimum waardes van die skuifkrag en die buigmoment en die waardes van x waarby hulle voorkom, op u grafiek aan. (8) Draw the shear force and bending moment diagrams for the altered load diagram given on the next page. Show all relevant values, including the maximum values of the shear force and the bending moment as well as the values of x at which they occur, on your graph. 13

14 1000 kn kn.m 80 kn/m 360 kn.m Load Diagram Lasdiagram 1180 kn 300 kn x=0 m x=8 m x=12 m x=18 m Shear Force Skuifkrag Bending Moment Buigmoment 14

15 15

16 K I A Q1 2D EQ particle 3-12 a la Q2 2D resultant 10 Q3 3D Equilibrium 10 Q4 Trusses P Q5 Frames&Machines 10 Q6 (a) 1 st Moment of area P (b) 2 nd Moment of area P Q7 SF and BM diagrams

WTW 158 : CALCULUS EKSAMEN / EXAMINATION Eksterne eksaminator / External examiner: Prof NFJ van Rensburg

WTW 158 : CALCULUS EKSAMEN / EXAMINATION Eksterne eksaminator / External examiner: Prof NFJ van Rensburg Outeursreg voorbehou UNIVERSITEIT VAN PRETORIA Departement Wiskunde en Toegepaste Wiskunde Copyright reserved UNIVERSITY OF PRETORIA Department of Mathematics and Applied Maths 4 Junie / June 00 Punte