Performance analysis - Secure many-to-one communications for resource-constrained

3.2 Secure many-to-one communications for resource-constrained

3.2.5 Performance analysis

SM 6.1 A 1.5 m long cantilever beam AB is loaded by a couple M0 at its free end as shown in figure. The distance from the top surface of the beam to the neutral surface is 75 mm and the longitudinal normal strain at the top surface is 0 001. . What will be the curvature k and the vertical deflection d at the end of the beam ?

(A) k =0.1333 m^-¹, d =15.0 mm (B) k =0 1. 333 m^-¹, d =1.5 mm (C) k =0. 1333 m0 ^-¹, d =1. mm5 (D) k =0. 1333 m0 ^-¹, d =15.0 mm

SM 6.2 Wide-flange beam as shown in figure, is made of a high-strength, low-alloy steel for which s =Y 345 MPa and s =U 450 MPa. If a factor of safety of 3 0 is used . and the effect of fillets is neglected, what will be the largest couple that can be applied to the beam when it is bent about the y-axis ?

(A) 42 3. kN m^- (B) 35 15. kN m

-(C) 70 3. kN m^- (D) 56.3 kN m

-SM 6.3 A nylon spacing bar has the cross section as shown in figure. If the allowable stress for the grade of nylon used is 24 MPa, the largest couple Mz that can be applied to the bar, is

(A) 2 38. kN m^- (B) 1 785. kN m -(C) 2 975. kN m^- (D) 1 487. kN m

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SM 6.4 In the figure shown, if the couple acts in a vertical plane, the stress at point A and point B are

(A) s =A 116.4 MPa, s =-B 87 3. MPa (B) s =-A 116.4 MPa, s =-B 87.3 MPa (C) s =-A 116.4 MPa, s =B 87.3 MPa (D) s =A 116.4 MPa, s =B 87.3 MPa

SM 6.5 Each girder of a highway bridge has a cantilever length of 46 m and an I-shaped cross section with dimensions as shown in the figure. The load on each girder is 11.0kN m, which includes the weight of the girder. What will be the maximum / bending stress in a girder due to this load?

(A) 96 75. MPa (B) 129MPa

SM 6.6 For the extruded beam shown in figure, the allowable stress is 120 MPa in tension and 150 MPa in compression. What will be the largest couple M that can be applied ?

(A) 1 89. kN m^- (B) 5 65. kN m -(C) 7 58. kN m^- (D) 3 79. kN m

-SM 6.7 The box beam as shown in figure has an allowable bending stress of sallow=10 MPa and an allowable shear stress of tallow=775 kPa. What will be the maximum

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intensity w of the distributed loading that it can safely support ?

(A) 1.51 kN m^- (B) 6 04. kN m -(C) 2 01. kN m^- (D) 3 02. kN m

-Common Data For Q. 8 and 9

Consider the beam as shown in figure which has an allowable stress of 120 MPa.

SM 6.8 What will be the largest couple M that can be applied to the beam ? (A) 18 34. kN m^- (B) 9 17. kN m -(C) 4 585. kN m^- (D) 6 875. kN m

-SM 6.9 If the cross section of the beam is a square of 80 mm side, what will be the largest couple M ?

(A) 7 68. kN m^- (B) 10 24. kN m -(C) 15 36. kN m^- (D) 5 12. kN m

-SM 6.10 Vertical wood beams AB of thickness t =120 mm is used to construct a small dam of height h =2.0 m as shown in the figure. If the beams are simply supported at the top and bottom and the weight density of water is g =9.81kN m/ ³, the maximum bending stress smax in the beams will be

(A) 1.4 MPa (B) 3 15. MPa

SM 6.11 In figure shown, the tapered beam supports a concentrated force P at its center.

If it is made from a plate that has a constant width b, what will be the absolute

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maximum bending stress in the beam ?

(A) PLbh

Common Data For Q. 12 and 13

A beam of rectangular cross section is to be sawed from a log of circular cross section as shown in figure. A couple M is applied to the rectangular beam.

SM 6.12 For the smallest possible maximum stress sm, the ratio d b/ will be

SM 6.14 For the simple beam AB and loading shown in figure, consider the following data: P =5.4 kN, L =3.0 m, d =1.2 m, b =75 mm t =25 mm, h =100 mm and h1=75 mm. What will be the maximum tensile stress st and maximum compressive stress sc due to the load P ?

(A) s =t 73.2 MPa, s =c 43 9. MPa (B) s =t 43 9. MPa, s =c 73.2 MPa (C) s =t 36 6. MPa, s =c 43 9. MPa (D) s =t 36 6. MPa, s =c 87 8. MPa

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SM 6.15 The reinforced concrete beam as shown in figure, is subjected to a positive bending moment of 175 kN m^- . The modulus of elasticity is 25 GPa for the concrete and 200 GPa for the steel. What will be the stress in the steel and the maximum stress in the concrete ?

A cantilever beam AB of triangular cross section has length L =0.8 m, width 80 mm

b = and height h =120 mm as shown in figure. The beam is made of brass weighing 85kN m/ ³.

SM 6.16 What will be the maximum tensile stress st and maximum compressive stress sc

due to the beam’s own weight ? (A) s =t 1 36. MPa, s =c 2 72. MPa

SM 6.18 If the height h is doubled, the stresses will be

(A) half (B) double

SM 6.19 Consider the design of a reinforced concrete beam as shown in figure. The design is said to be balanced when the maximum stresses in the steel ss and concrete sc are equal to the allowable stresses. If Ec and Es are the moduli of elasticity of concrete and steel, respectively and d is the distance from the top of the beam to the reinforcing steel, the distance x from the top of the beam to the neutral axis to achieve a balanced design, must be

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(A) x

Common Data For Q. 20 and 21

Consider the two bars as shown in figure (a) and (b). The allowable stress used in the design of a steel bar is 80 MPa.

SM 6.20 If the bar is designed with grooves having semicircular portions of radius r =15 mm and K=1 92. as shown in figure (a), what will be the largest couple M that can be applied to the bar ?

(A) 2 5. kN m^- (B) 3.125kN m

-(C) 0 625. kN m^- (D) 1 25. kN m

-SM 6.21 If the bar is redesigned by removing of material above the grooves (K=1 57. ) as shown in figure (b), the largest couple M that applied to the bar can be

(A) 1 53. kN m^- (B) 3 825. kN m -(C) 2 04. kN m^- (D) 3 06. kN m

-SM 6.22 A 450 mm long cantilever beam AB of circular cross section supports a load 400 N

P = acting at the free end as shown in figure. The beam is made of steel with weight density of 77kN m/ ³ and an allowable bending stress of 60 MPa. If the effect of the beam’s own weight is considered, the required diameter dmin of the beam is

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(A) 39 51. mm (B) 23 71. mm

Common Data For Q. 23 and 24

A tapered cantilever beam AB of length L has square cross sections and supports a concentrated load P at the free end as shown in figure. The width and height of the beam vary linearly from hA at the free end to hB at the fixed end.

SM 6.23 If hB=3hA, what will be the distance x from the free end A to the cross section of maximum bending stress ?

(A) L

2 (B) 2L

SM 6.24 What will be the ratio of the maximum stress smax to the largest stress sB at the support ?

(A) 1 (B) 4

SM 6.25 In figure shown, a bar of rectangular cross section is made of steel which is assumed to be elastoplastic with s =Y 320 MPa and subjected to a couple M parallel to the y-axis. What will be the moments Ma of the couple for which yield first occurs and Mb for the 5 mm thick plastic zones at the top and bottom of the bar ?

(A) Ma=115 2. N m^- , Mb=171 2. N m^- (B) Ma=115.2 N m^- , Mb=214N m -(C) Ma=143 75. N m^- , Mb=171 2. N m^- (D) Ma=143 75. N m^- , Mb=214 N m

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SM 6.26 Consider a steel beam of the cross section as shown in figure. If the steel is elastoplastic with a yield strength of 240 MPa, the plastic moment MP of the beam is

(A) 14 26. kN m^- (B) 11 88. kN m -(C) 23 76. kN m^- (D) 19 01. kN m

-SM 6.27 A beam of width t has a depth that varies as shown in figure. If it supports a concentrated force P at its end, the absolute maximum bending stress in the beam and its location x, are

(A) x b b b L

Common Data For Q. 30 and 31

Consider a solid bar of elastoplastic material of rectangular cross section. The bending moment and radius of curvature at the onset of yield are, MY and rY, SM 6.29 What will be the radius of curvature after the couple is removed ?

(A) rR=0.70711rY (B) rR=6 09. rY

SM 6.30 A tall signboard supported by two vertical beams consisting of thin walled, tapered circular tubes is shown in figure. Each beam is represented as a cantilever AB of length L =8.0 m subjected to a lateral load P =2.4 kN at the free end. The tubes have constant thickness t =10.0 mm and average diameters dA=90 mm and dB=270 mm at ends A and B, respectively. The moment of inertia at any cross section is I=pd t 8³ / and the section modulus is S=pd t 4² / . At what distance x from the free end does the maximum bending stress occur and what is

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the magnitude smax of the maximum bending stress ?

(A) 5.0 m, 47.13 MPa (B) 4.0 m, 37 73. MPa (C) 2.5 m, 23 56. MPa (D) 3.0 m, 28 33. MPa

SM 6.31 A shaft is supported on journal bearings as shown in figure that do not offer resistance to axial load. The allowable normal stress for the shaft is sallow=80 MPa . Using the maximum distortion energy theory of failure, the smallest diameter of the shaft that support the loading will be

(A) 15mm (B) 20mm

SM 6.32 Consider a cantilever beam AB having rectangular cross sections with constant width b and varying height hx. The beam is subjected to a uniform load of intensity w as shown in figure. If the beam is fully stressed, what will be the height hx ?

(A) h 2h xL

x = B (B) h h xL

x = B

x B 2

= (D) h h xL

x = B

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SM 6.33 For the beam shown in figure, if the magnitude of the horizontal force P is 8 kN , the stresses at point A and B will be

(A) s =-A 102.8 MPa, s =B 80 6. MPa (B) s =A 102.8 MPa, s =-B 80.6 MPa (C) s =A 80 6. MPa, s =B 102 8. MPa (D) s =-A 80 6. MPa, s =B 80.6 MPa

Common Data For Q. 35 and 36

A laminated plastic beam of square cross section is built up by gluing together three strips of 10mm#30mm in cross section each as shown in figure. The beam has a total weight of 3.2 N and is simply supported with span length of

320 mm

L = . Consider the weight of the beam.

SM 6.34 If the allowable shear stress in the glued joints is 0.3 MPa, the maximum permissible load P that placed at the midpoint, will be

(A) 402 N (B) 488 N

SM 6.35 If the allowable bending stress in the plastic is 8 MPa, the maximum permissible load P that placed at the midpoint, will be

(A) 450N (B) 488N

SM 6.36 What will be the stress at point B, (a) for the loading shown in figure, (b) if the 60 kN loads applied at points 2 and 3 are removed ?

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(A) ^ hs_{B a}=8.33 MPa, ^ hs_{B b}=-7 64. MPa (B) ^ hs_{B a}=-8.33 MPa, ^ hs_{B b}=7.64 MPa (C) ^ hs_{B a}=6 25. MPa, ^ hs_{B b}=8 23. MPa (D) ^ hs_{B a}=7 64. MPa, ^ hs_{B b}=6 25. MPa

SM 6.37 The machine element shown in figure is formed by welding a steel rod to a steel plate. Centroid of the cross section is at C and Iz =4195 mm4. If the allowable stress is 135 MPa, what will be the largest force P that can be applied to the element ?

(A) 3 175. kN (B) 1. kN9

SM 6.38 A C-shaped steel (E =200 GPa) bar of square cross section of side 40 mm is shown in figure. The strain on the inner edge of the bar is found to be 450 m. What will be the magnitude of the forces P ?

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(A) 9 kN (B) 11 25. kN

SM 6.39 A 60 kN eccentric axial force P acts at point D, which is located 25 mm below the top surface of the steel bar as shown in figure. What will be the depth d of the bar for which the compressive stress at point B is maximum and the corresponding stress at point B ?

(A) d =75 mm, s =B 10 MPa (B) d =75 mm, s =-B 20MPa (C) d =75 mm, s =B 20MPa (D) d =150mm, s =-B 10 MPa

SM 6.40 An aluminium pole for a street light of weight 4600 N which supports an arm of weight 660 N is shown in figure. The outside diameter of the pole at its base is 225 mm and its thickness is 18 mm. If the center of gravity of the arm is 1.2 m from the axis of the pole, what will be the maximum tensile and compressive stresses st and sc, respectively, in the pole at its base due to the weights ?

(A) s =t 961 kPa, s =-c 1860 kPa (B) s =t 1860kPa, s =-c 961kPa (C) s =t 1410kPa, s =c 1860 kPa (D) s =t 450kPa, s =c 1410kPa

SM 6.41 A couple M=300 N m^- is applied to a beam of rectangular cross section as shown in figure, in a plane forming an angle b =60c with the vertical. What will be the stresses at point A, B and D ?

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(A) s =-A 3.37 MPa, s =-B 18 6. MPa, s =D 3 37. MPa (B) s =A 3.37 MPa, s =B 18.6 MPa, s =D 3.37 MPa (C) s =A 3.37 MPa, s =B 18.6 MPa, s =-D 3.37 MPa (D) s =-A 3.37 MPa, s =B 18.6 MPa, s =D 3.37 MPa

SM 6.42 A couple M acts in a vertical plane and is applied to a beam oriented as shown in figure. What will be the angle that the neutral axis forms with the horizontal plane and the maximum tensile stress in the beam, respectively ?

(A) 19 52c. , 1 09 MPa (B) 19.52c, 95 MPa (C) 5 475c. , 190MPa (D) 5 475c. , 95 MPa

SM 6.43 An axial load P =30 kN is applied to a short section of a C 150#12 2. rolled steel channel as shown in figure. Sectional properties for C 150#12 2. rolled steel channel are as below:

What will be the largest distance a for which the maximum compressive stress is 60 MPa?

(A) 46.65 mm (B) 93 3. mm

SM 6.44 The Z section shown in figure is subjected to a couple M0 acting in a horizontal plane. The Imax 2.28 10 mm6 4

= # , Imin 0.23 10 mm6 4

= # and the principal axes are 25 7c. and 64 3c. . If the maximum stress is not to exceed 80 MPa, the largest permissible value of the moment M0 of the couple is

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(A) 7 938. kN m^- (B) 3 969. kN m -(C) 1 323. kN m^- (D) 2.646kN m

-SM 6.45 In figure shown, a box beam of wood is constructed of two 260mm#50mm and two 260mm#25mm boards. The boards are nailed at a longitudinal spacing of

100 mm

s = . If each nail has an allowable shear force F =1200 N, the maximum allowable shear force Vmax will be

(A) 16.05 kN (B) 21 4. kN

SM 6.46 In figure shown, a is the distance from the line of action of the 3 kN force to the vertical plane containing the center of curvature of the bar. If the allowable stress in the bar is 150 MPa, the largest permissible value of a is

(A) 60.9 mm (B) 32 5. mm

SM 6.47 For the curved beam and loading shown in figure, the stresses at point A and point B are

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(A) s =A 63 9. MPa, s =-B 52 6. MPa (B) s =-A 63.9 MPa, s =-B 52 6. MPa (C) s =A 63 9. MPa, s =B 52.6 MPa (D) s =-A 63.9 MPa, s =B 52.6 MPa

SM 6.48 A beam of the cross section shown in figure is extruded from an aluminium alloy for which E =72 GPa. If the couple shown acts in a vertical plane, what will be the maximum stress in the beam and the corresponding radius of curvature ?

(A) smax=112.5 MPa, r =12 80. m (B) smax=56 25. MPa, r =6.4 m (C) smax=112.5 MPa, r =6.4 m (D) smax=56 25. MPa, r =12 8. m

SM 6.49 Consider the built-up wooden beam as shown in figure. It is subjected to a vertical shear of 8 kN and the nails are spaced longitudinally every 60 mm at A and every 25 mm at B and Ix 1.504 10 mm9 4

= # . What will be the shearing force in the nails at B?

(A) 549 N (B) 823N

SM 6.50 A vertical shear V causes a maximum shearing stress of 75 MPa in the hat-shaped extrusion as shown in figure. The corresponding shearing stress at point a and point b, are

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(A) ta=tb=20.7 MPa (B) t =a 41 4. MPa, t =b 20.7 MPa (C) t =a 20.7 MPa, t =b 41 4. MPa (D) ta=tb=41 4. MPa

SM 6.51 The section shown in figure is formed by welding three plates together of 12 mm thickness each. For a vertical shear of 100 kN, the shear flow through the welded surfaces is

(A) 199 5. kN m^- (B) 332 5. kN m -(C) 266kN m^- (D) 166 25. kN m

-SM 6.52 A thin-walled beam of uniform thickness has the cross section as shown in figure.

What will the dimension h for which the shear center O of the cross section is located at a distance e =25 mm from the center of the flange AB?

(A) 92 75. mm (B) 55 65. mm

SM 6.53 An extruded beam of 3 mm wall thickness has the cross section as shown in figure and subjected to a 10 kN vertical shear. What will be the shearing stress at point A and the maximum shearing stress in the beam ?

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(A) t =A 31.3 MPa, tmax=50.7 MPa (B) t =A 25 35. MPa, tmax=62 61. MPa (C) t =A 25 35. MPa, tmax=31 3. MPa (D) t =A 50 7. MPa, tmax=62 6. MPa

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