3.2 Secure many-to-one communications for resource-constrained
4.1.2 System architecture
Common Data For Q. 1 and 2
Consider the loading of cantilever beam AB shown in figure.
SM 7.1 The equation of the elastic curve for the cantilever beam AB, is (A) y 3PxEI (3L x)
-SM 7.2 The deflection and slope at the free end respectively, are (A) PLEI
The deflection curve for a simple beam AB shown in figure, is given by the following equation:
=-SM 7.3 The load acting on the beam is expressed as (A) w w cos xL
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SM 7.5 The maximum bending moment Mmax, is
Consider the cantilever beam and loading shown in figure below.
SM 7.6 What will be the deflection at B ?
SM 7.8 A uniformly loaded steel wide-flange beam with simple supports is shown in figure. It has a downward deflection of 10 mm at the midpoint and angles of rotation equal to 0.01 radians at the ends. If the maximum bending stress is 90 MPa and the modulus of elasticity is 200 GPa, what will be the height h of the beam.
(A) 96 mm (B) 19 2. mm
(C) 9 6. mm (D) 48mm
SM 7.9 A gold-alloy microbeam attached to a silicon wafer behaves like a cantilever beam, subjected to a uniform load is shown in figure. The beam has length L=27.5 mm and rectangular cross section of width b=4.0 mm and thickness t=0.88 mm . The total load on the beam is 17.2 Nm . If the deflection at the end of the beam is 2.46 mm , what is the modulus of elasticity Eg of the gold alloy?
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(A) 75 GPa (B) 80 GPa
(C) 60 GPa (D) 46 GPa
SM 7.10 In the figure shown, uniformly distributed loads are applied to beam AE . If the beam is a W200#35.9 rolled shape and that L =3 m, w =5kN m/ , I=34.4#10 mm6 4 and E =200 GPa, the distance of the center of the beam from the x axis is
(A) 7 94. mm (B) 3 97. mm
(C) 0 397. mm (D) 39 7. mm
SM 7.11 What will be the reaction at the roller support for the beam and loading shown in figure ?
SM 7.12 For the beam and loading shown in figure, the reaction and bending moment at A, are as shown, where w0 is the maximum intensity of the load. What will be the
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deflection ymax at the midpoint of the beam using the fourth-order differential equation of the deflection curve (the load equation) ?
(A) EI
SM 7.14 In the figure shown, a simple beam with a uniform load is pin supported at one end and spring supported at the other. The spring has stiffness k=48EI L/ 3. What will be the angle of rotation qA at support A ?
(Hint: Use third-order differential equation (the shear-force equation)).
(A) wLEI
SM 7.15 In the figure shown, the two wooden meter sticks are separated at their centers by a smooth rigid cylinder having a diameter of 50 mm. Each stick has a width of 20 mm, a thickness of 5 mm and Ew=11 GPa. What will be the force F that must be applied at each end in order to just make their ends touch ?
(A) 0.6875 N (B) 2 75. N
(C) 1 375. N (D) 2.06N
SM 7.16 Beam ACB hangs from two springs is shown in figure. The springs have stiffnesses k1 and k2 and the beam has flexural rigidity EI. What is the downward displacement of the midpoint C of the beam when the load P is applied ?
(Use EI=216 kN-m2 )
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(A) 25 mm (B) 2 5. mm
(C) 20 5. mm (D) 4 5. mm
Common Data For Q. 17 and 18
Consider the beam and loading shown in figure.
SM 7.17 The slope at end A is
SM 7.19 For the beam and loading shown in figure, the reaction at A and the slope at C are
SM 7.20 A cantilever beam AB supporting a parabolic load defined by the equation /
w=w x L0 2 2 is shown in figure. What will be the angle of rotation qB and deflection yB at the free end of the beam ?
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(A) w LEI zero and EI is constant, the magnitude of force F that must be applied at the end of the overhang C, is
Common Data For Q. 22 and 23
Rigid bars are welded to the steel rod AD as shown in figure.
(Take E =200 GPa).
SM 7.22 For the loading shown, what will be the deflection at point B ?
(A) 0.30 mm (B) 15mm
(C) 1 5. mm (D) 0 15. mm
SM 7.23 What will be the slope at end A ?
(A) 5.65#10 rad-3 (B) 4 67. #10 rad-3 (C) 9 35. #10 rad-3 (D) 7 50. #10 rad-3
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SM 7.24 Consider a W360#39 beam and loading is shown in figure. If E =200 GPa and
SM 7.25 For the cantilever beam shown in figure, the slope and deflection at point B respectively, are
(Take E =200 GPa and I=2.53#10 mm6 4)
(A) 5.56#10-3rad, 25 mm (B) 11.12#10-3rad, 5mm (C) 5.56#10-3rad, 5mm (D) 5.56#10-3rad, 2. mm5
Common Data For Q. 26 and 27
In figure shown, a 50 mm diameter A-36 steel shaft is supported by the bearings at A and B. The bearings exert only vertical reactions on the shaft and take
200 GPa
E = for the steel shaft.
SM 7.26 What will be the slope of the shaft at A and B, respectively ?
(A) q =A 0.00968 rad, q =B 0.0105rad (B) q =A 0.00968 rad, q =B 0. 0105 rad0 (C) q =A 0.0968 rad, q =B 0.0 105 rad0 (D) q =A 0.0105rad, q =B 0.00968rad
SM 7.27 What will be the maximum deflection of the shaft?
(A) 8.16 mm (B) 16 32. mm
(C) 0 816. mm (D) 81 6. mm
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SM 7.28 The frame ABCD shown in figure is squeezed by two collinear forces P, acting at points A and D. The flexural rigidity EI is constant throughout the frame and disregard the effects of axial deformations. Considering the effects of bending due the loads P, what is the decrease y in the distance between points A and D ?
SM 7.29 A simple beam AB supports two concentrated loads P at the positions shown in the figure. A support C at the midpoint of the beam is positioned at distance d below the beam before the loads are applied. The magnitude of the loads P so that the beam just touches the support at C, is
(Data: d =10 mm, L =6 m, E =200 GPa and I=198#10 mm6 4)
(A) 64 kN (B) 80kN
(C) 48kN (D) 32kN
SM 7.30 A central beam BD is joined at hinges to two cantilever beams AB and DE. All beams have the cross section shown in figure. For the loading shown, if the deflection at C is not to exceed 3 mm and E =200 GPa, the largest allowable . For the loading shown in figure, the deflection at point B and at point D, are
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(A) yB=5.76 mm, yD=23 1. mm (B) .yB=10 38mm, yD =23 1. mm (C) yB=4 61. mm, yD =23 1. mm (D) .yB=10 38mm, yD =9 2. mm
SM 7.32 A steel shaft is used to support a rotor that exerts a uniform load of 5kN m / within the region CD of the shaft as shown in figure. The bearings exert only vertical reactions on the shaft. What will be the slope of the shaft at the bearings A and B? (Take E =200 GPa)
(A) q =A 0c, q =B 0 175. c (B) q =A 0 175. c, q =B 0c (C) q =A 0 175. c, q =B 0 350. c (D) qA=qB=0 175. c
Common Data For Linked Answer Q. 33 and 34
The cantilever beam ACB shown in figure, has moments of inertia I2 and I1 in parts AC and CB, respectively.
SM 7.33 Using the method of superposition, the deflection yB at the free end due to the load P, is of a prismatic cantilever with moment of inertia I1 carrying the same load ? (A) II A by the strain energy of the beam and using Castigliano’s theorem?
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(A) M LEI
SM 7.36 For the cantilever beam and loading shown in figure, what will be the deflection and slope at end D caused by the couple M0 ? the deflection at midpoint C ?
(Use data P =8 kN, I=16.6#10 mm6 4 and E =200 GPa )
(A) qA=0.3125#10 rad-3 , yC =1.016 mm (B) qA=31.25#10 rad-3 , yC =10.16 mm (C) qA=3.125#10 rad-3 , yC =1.016 mm (D) qA=3.125#10 rad-3 , yC =10 16. mm
Common Data For Q. 43 and 44
Consider the beam and loading as shown in figure with I=556#10 mm6 4 and
SM 7.40 The cantilever beam shown in figure, supports a triangularly distributed load of
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maximum intensity w0. The deflection dB at the free end B by using Castigliano’s theorem, is
Common Data For Q. 46 and 47
A W 103 #74 beam is loaded as shown in figure. It has I=1 56 #10 mm6 4 and
SM 7.43 The cantilever beam ACB shown in figure, is subjected to a uniform load of intensity w acting between points A and C. The angle of rotation qA at the free end A by using the modified form of Castigliano’s theorem, is
(A) wLEI deflection at point B and at the midpoint C of the beam, are
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(A) yB=0.175 mm, yC=0.882 mm (B) yB=1 75. mm, yC =0.882 mm (C) yB=0 882. mm, yC=0 175. mm (D) yB=0.175 mm, yC =8 82. mm
SM 7.45 In figure shown, an overhanging beam ABC is subjected to a couple MA at the free end. The lengths of the overhang and the main span are a and L, respectively. What will be the deflection yA at end A by using the modified form of Castigliano’s theorem ?
(A) M a L6EIA ^2 -3ah (B) M a L6EIA ^2 +3ah (C) M a a6EIA ^3 -2Lh (D) M a L6EIA ^3 +2ah
SM 7.46 The steel strip AB shown in figure, is 5 mm wide and 10 mm high. What will be the deflection at the end B of the clamped steel strip ? (Take E =200 GPa)
(A) 0.015 mm (B) 15 0. mm
(C) 0 15. mm (D) 1 50. mm
Common Data For Q. 46 and 47
Beam DE rests on the cantilever beam AC is shown in figure. A square rod of side 10 mm is used for each beam with E =200 GPa.
SM 7.47 What will be the deflection at end C, if the 25 N m- couple is applied to end E
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of beam DE ?
(A) 59.4 mm (B) 11 88. mm
(C) 29 7. mm (D) 5 94. mm
SM 7.48 What will be the deflection at end C, if the 25 N-m couple is applied to end C of beam AC ?
(A) 0.675 mm (B) 67 5. mm
(C) 6 75. mm (D) 1 35. mm
SM 7.49 A weight W =20 kN falls through a height h =1.0 mm onto the midpoint of a simple beam of length L =3 m as shown in figure. The beam is made of wood with square cross section (dimension d on each side) and E =12 GPa. If the allowable bending stress in the wood is sallow=10 MPa, what will be the minimum required dimension d?
(A) 140 mm (B) 2 81. mm
(C) 28 1. mm (D) 281mm
***********
SM 8
COLUMN
SM 8.1 A rigid bar AB is shown in figure. If the spring at A has a stiffness of k, the critical load Pcr will be
(A) kL 2 / (B) kL (C) kL2/2 (D) kL2
SM 8.2 What will be the critical buckling load for the column shown in figure when the material assumed to be rigid ?
(A) kL 2 / (B) kL 4/
(C) kL 6 (D) / kL 3/
SM 8.3 A rigid bar AD is attached to two springs of stiffness k each and is in equilibrium in the position as shown in figure. If the equal and opposite loads P and Pl remain horizontal, the magnitude of the critical load for the system, will be
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(A) ka2/2l (B) ka l/2
(C) ka2/4l (D) ka l2/
SM 8.4 In the figure given, each spring can act in either tension or compression. If 125 kg
m = , h = 700 mm and the stiffness of each spring is k =2.8kN m/ , which of the following is the value of distance d for which the equilibrium of the rigid rod AB is stable in the position shown ?
(A) 245 mm (B) 294 mm
(C) 392 mm (D) 490 mm
SM 8.5 In figure shown, the steel pipe is fixed supported at its ends for which Est =200 GPa and s =Y 250 MPa. If it is 4 m long with outer diameter of 50 mm and support an axial load of P =100 kN without buckling, what will be its required thickness ?
(A) 4.44 mm (B) 5 92. mm
(C) 7.4 mm (D) 3 7. mm
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SM 8.6 A horizontal beam AB is pin-supported at end A and carries a load Q at end B as shown in the figure. The beam is supported at C by a pinned-end column. The column is a solid steel bar of square cross section and have a length L =1.8 m, side dimension b =60 mm and E =200 GPa. Based upon the critical load of the column, if the factor of safety with respect to buckling is 2.0, the allowable load Q is
(A) 767.7 kN (B) 657 97. kN
(C) 109.7 kN (D) 548 27. kN
SM 8.7 A rectangular column with cross-sectional dimensions b and h is pin-supported at ends A and C as shown in figure. At mid-height, the column is restrained in the plane of the figure but is free to deflect perpendicular to the plane of the figure.
What will be the ratio h b/ such that the critical load is the same for buckling in the two principal planes of the column ?
(A) 1 (B) 4
(C) 3 (D) 2
SM 8.8 An A-36 steel column has a length of 4 m and is pinned at both ends as shown in figure. If the cross sectional area has the dimensions shown with E =200 GPa and s =Y 250 MPa, the critical load is
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(A) 28.38 kN (B) 17.02 kN
(C) 14.19 kN (D) 22.7 kN
SM 8.9 A steel tube (E =200 GPa) shown in figure is 5.0 m long. It has 100 mm outer diameter and 16 mm wall thickness. The critical load of tube is
(A) 305 kN (B) 244kN
(C) 183kN (D) 366kN
SM 8.10 A load P can be applied to the structure shown in figure. Consider only bucking in the plane of the structure and E =200 GPa. If a factor of safety of 2.6 is required, what will be the largest value of P ?
(A) 6 4. kN (B) 4kN
(C) 2 4. kN (D) 5 2. kN
SM 8.11 In figure shown, the steel bar AB of the frame is pin-connected at its ends. It has Est=200 GPa and s =Y 360 MPa. For buckling about the y-y axis due to the applied load of P =18 kN, the factor of safety is
(A) 4 76 (B) . 2 38.
(C) 6 (D) 3
SM 8.12 A strut AB is compressed by the action of tensile forces F in a cable that makes an angle a = 75c with the strut as shown in figure. The strut is a circular tube of aluminium with E =72 GPa It has outer diameter d2=50 mm and inner
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diameter d1=40 mm. The strut is 1.0 m long and is pin-connected at both ends.
If factor of safety is 2.5 with respect to the critical load, the allowable force F in the cable will be
(A) 128.7 kN (B) 99 5. kN
(C) 29 2. kN (D) 51 49. kN
SM 8.13 A compression member of 7 m effective length is made by welding together two 152 102 12.7
L # # angles as shown in figure. Data for angles are 3060 mm
A= 2, Ix 7.20 10 mm6 4
= # , Iy 2.64 10 mm6 4
= # , y =50.3 mm, 25.3 mm
x = .
If E =200 GPa and a factor of safety is 2.2, the allowable centric load for the member will be
(A) 370.5 kN (B) 168 4. kN
(C) 185 25. kN (D) 3 . kN36 8
SM 8.14 A distributed loading is supported by two pin-connected columns, each having a solid circular cross section as shown in figure. If AB is made of aluminium with
70 GPa
Eal = , ( )sY al =100 MPa and CD is made of steel with Est=200 GPa, ( )sY st =250 MPa, what will be the required diameter of each column so that both are on the point of buckling at the same time ?
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(A) dAB=40.2 mm, dCD=68.8 mm (B) dAB=34.4 mm, dCD=80.4 mm (C) dAB=34.4 mm, dCD=40.2 mm (D) dAB=68.8 mm, dCD=40.2 mm
SM 8.15 The truss ABC shown in the figure supports a vertical load W at joint B. The distance between supports is 7.0 m and joint B is restrained against displacement perpendicular to the plane of the truss. If each member is a slender circular steel pipe (E =200 GPa) with outside diameter 100 mm and wall thickness 6.0 mm, what will be the critical value Wcr of the load ? axis buckling and fixed at its ends for y y- axis buckling. The factor of safety with respect to buckling is 3. What will be the maximum allowable load P that can be applied to member BC without causing member AB to buckle ?
(A) 26 3. kN (B) 13 15. kN (C) 29 6. kN (D) 14 8. kN
SM 8.17 In the assembly shown in figure, force P is applied to the handle so that the steel control rod BC does not buckle. The rod has a diameter of 25 mm. Take
200 GPa
Est= and s =Y 250MPa. What will be the maximum value of force P that can be applied ?
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(A) 29 9. kN (B) 14 95. kN
(C) 59 8. kN (D) 37 38. kN
SM 8.18 Consider only buckling in the plane of the structure shown in figure. If the load 5.2 kN
P = and E =200 GPa, the factor of safety for the structure is
(A) 1 135 (B) . 3 13.
(C) 2 27. (D) 6.2
SM 8.19 A vertical post AB is embedded in a concrete foundation and held at the top by two cables. The cables are tightened equally by turnbuckles as shown in figure.
The post is a hollow steel tube with modulus of elasticity of 200 GPa. It has a outer diameter of 40 mm and thickness of 5 mm. If factor of safety against Euler buckling in the plane of the figure is 3.0, what will be the maximum allowable tensile force Tallow in the cables ?
(A) 39.34 kN (B) 26.22 kN
(C) 18 1. kN (D) 78 67. kN
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SM 8.20 Column AB carries a centric load P of magnitude 72 kN. Cables BC and BD are taut and prevent motion of point B in the xz plane as shown in figure. Moments of inertia for column are Ix 48.9 10 mm6 4
= # and Iy 4.73 10 mm6 4
= # . Neglect the tension in the cables. If factor of safety is 2.3 and E =200 GPa, what will be the maximum allowable length L using Euler’s formula ?
(A) 9.06 m (B) 15 1. m
(C) 7 55. m (D) 12.08 m
SM 8.21 The brace shown in figure is an A-36 steel rod of 50 mm diameter. For the brace CD is not to be buckled, the maximum distributed loading that can be applied to the wide-flange beam will be
(Use: Est =200 GPa and s =Y 360 MPa)
(A) 11.82kN m (B) / 9 46. kN m/ (C) 5 91. kN m/ (D) 7 1. kN m/
Common Data For Q. 22 and 23
The roof beams of a warehouse are supported by pipe columns as shown in figure.
Each column has outer diameter d2=100 mm, inner diameter d1=90 mm and length L = 4.0 m. The columns have fixed supports at the base with modulus
210 GPa
E = .
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SM 8.22 If we assume:
(1) the upper end is pinned and the beam prevents horizontal displacement, (2) the upper end is fixed against rotation and the beam prevents horizontal
displacement.
What will be the critical load Pcr of one of the columns ?
(A) ^ hPcr 1=219kN, ^ hPcr 2=54 7. kN (B) ^ hPcr 1=447kN, ^ hPcr 2=875kN (C) ^ hPcr 1=875kN, ^ hPcr 2=447kN (D) ^ hPcr 1=54 7. kN, ^ hPcr 2=219 kN
SM 8.23 If we assume:
(1) the upper end is pinned but the beam is free to move horizontally,
(2) the upper end is fixed against rotation but the beam is free to move horizontally.
What will be the critical load Pcr of one of the columns ?
(A) ^ hPcr 1=447kN, ^ hPcr 2=875kN (B) ^ hPcr 1=219 kN, ^ hPcr 2=54.7 kN (C) ^ hPcr 1=54 7. kN, ^ hPcr 2=219 kN (D) ^ hPcr 1=875kN, ^ hPcr 2=447 kN
SM 8.24 An aluminium (E =72 GPa) tube AB of circular cross section is fixed at the base and pinned at the top to a horizontal beam as shown in figure. The horizontal beam supports a load Q =200 kN. The tube has outside diameter d =100 mm. If the desired factor of safety with respect to Euler buckling is 3.0, the required thickness t of the tube will be
(A) 9.15 mm (B) 12.2 mm
(C) 15.25 mm (D) 1.22 mm
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