PURWANCHAL UNIVERSITY III SEMESTER FINAL EXAMINATION-2002 LEVEL : B. E. (Civil) SUBJECT: BEG256CI, Strength of Material Full Marks: 80 TIME: 03:00 hrs Pass marks: 32 Candidates are required to give their answers in their own words as far as possible.. The figure in the margin indicates full marks. Attempt any FIVE questions. Q. [1] A girder is composed of four 150 150 13 mm angles composed to a web plate 600 mm by 20 mm, plus two flange plates each 460 20 mm, as shown in the given figure. The properties of the angle are lx=l=8.05 10 6 mm 4, the area=3730 mm 2 and x=y=32.3 mm. Compute the moment of inertia with respect to the centroidal x 0 axis. [16] Q. [3] [a] Prove the torsional equation: [8] 4 4 π D d T = τ for hollow circular shaft. 16 D Where T = Torque, D = Outer diameter of the shaft, d = Inner diameter of the shaft, and τ = Maximum shear stress developed in the outer most layer of the shaft material. [b] A solid shaft of 150 mm diameter is to be replaced by a hollow shaft of the same material with internal diameter equal to 60% of the external diameter. Find the saving in material, if maximum allowable shear stress is the same for both the shafts. [8] Q. [2] The block of weight W in the figure given below hangs from the pin at A. the bars AB and AC are pinned to the support at B and C. The areas are 600 mm 2 for AB and 300 mm 2 for AC. Neglecting the weights of the bars, determine the maximum safe value of W if the stress in AB is limited to 100 MPa and that in AC to 110 MPa. [16] Q. [4] Draw bending moment and shear force diagram of the figure given below. Also draw axial force diagram. [16] / 1
Q. [5] A rectangular black of material is subjected to a tensile stress of 120 N/mm 2 on one plane and a tensile stress of 50 N/mm 2 on a plane at right angles. Together with sbear stresses of 65N/mm 2 on the same planes. Find: [16] (a) The direction of the principal planes, (b) The magnitude of the principal stress, and (c) The magnitude of the greatest shear stress. Q. [6] Determine the equation of the elastic curve of a cantilever beam supporting a uniformly distributed load of intensity w 0 over part of its length as shown below. [16] Q. [7] For a gas cylinder of diameter 50 mm and 7 mm thick, if the tensile stress in the material is not being exceeded 30 MPa, Find the maximum pressure which can be allowed in the cylinder. [16] / 2
PURWANCHAL UNIVERSITY III SEMESTER FINAL EXAMINATION-2003 LEVEL : B. E. (Civil) SUBJECT: BEG256C1, Strength of Material Full Marks: 80 TIME: 03:00 hrs Pass marks: 32 Candidates are required to give their answers in their own words as far as Practicable. The figure in the margin indicates full marks. Attempt any FIVE questions. Q. [2] [a] A bar of copper an steel form a composite system and they are heated to a certain temperature. If the coefficient of linear expansion of copper is more than that of steel, what type of stress will be in copper bar, tensile or compressive? Why? [6] [b] A cast iron block of 5cm 2 cross-section carries an axial compressive load of 50kN. Calculate the magnitude of the normal and shear stresses on a plane whose normal is inclined at 30º to the axis of the block. Also determine the maximum shear stress in the block. [10] Q. [1] [a] Find out the degree of static indeterminacy of the beam shown in Fig. 1 [a] [4] Q. [3] [a] For a body under the action of biaxial stress G x and G y, on an oblique plane, what will be the maximum shear stress and normal stress? Derive the expression. [6] [b] Two rods L meters long and 90 mm 2 cross-sectional area are fastened rigidly to a level support at distance of 1.2m from each other A horizontal cross-bar is provided at lower ends as shown in Fig. 3[b]. The bar has to remain horizontal after loading. Calculate the stresses in the two rods and also the loads on each bar. [Take E s =200 GN/m 2 and E b =90GN/m 2.] [10] [b] Draw the Bending Moment diagram for the frame shown in Fig. 1[b]. The frame consists of hinge at joint C. [12] Fig. 1 [b] Q. [4] [a] A sort iron column is of hollow section 300mm external diameter and 60mm thick. A vertical compressive load of 300kN acts at an eccentricity of 90mm from the axis. Determine the maximum tensile and compressive stresses. Plot the variation of stress in the column. [8] / 3
[b] A hollow shaft is to transmit 338kW at 100rmp. If the shear stress is not to exceed 65N/mm 2 and internal diameter is 0.6 of the external diameter, find the external and internal diameters, assuming that the maximum torque is 1.3 times the mean. [8] Q. [5] [a] Calculate the second moment of area of an L-section shown in Fig. 5[a] about its centroidal axis. [8] Q. [7] [a] Explain Haigh's theory of elastic failure of material. [6] [b] A rectangular beam 100mm wide 200mm deep and 4 meters long is simply supported at ends. It carries a uniformity distributed load of 5kN/m run over the entire span. If this load is removed and two loads W kn each are placed at one metre from each end. Calculate the greatest value which may be assigned to W so that the maximum bending stress remains same as before. [10] Fig. 5 [a] [b] The air vessel of a torpedo is 530mm external diameter and 10mm thick, the length being 1830mm. Find the change in the external diameter and length when charged to 10.5N/mm 2 internal pressure. Take E=2.1 10 5 N/mm 2 and Poisson's Ratio=0.3. [8] Q. [6] [a] Derive the expression for shear stress at a section in a uniformly distributed loaded rectangular beam. Ay i.e. τ = V, with their usual meanings. 1 b [b] Determine the ratio of the strengths in terms of crippling loads of a solids steel column to that of a hollow column of the same material and having the same cross-sectional area. The internal diameter of hollow column is ½ of the external diameter. Both the columns are of the same length and are pinned at both ends. [10] / 4
PURWANCHAL UNIVERSITY III SEMESTER BACK-PAPER EXAMINATION-2003 LEVEL : B. E. (Civil) SUBJECT: BEG256CI, Strength of Materials Full Marks: 80 TIME: 03:00 hrs Pass marks: 32 Candidates are required to give their answers in their own words as far as possible. The figures in the margin indicate full marks. Attempt any FIVE questions. Q. [2] [a] Derive the relationship between Young's Modulus of Elasticity and Modulus of Rigidity. [6] [b] A Plane element is subjected to the stresses as shown in Fig. 2[b]. Determine the following stresses: [i] The principal stress and their directions, and [ii] the maximum shearing stresses and the directions of the planes on which they occur. [10] Q. [1] [a] Find out the degree of static indeterminacy of the beam shown in Fig. 1[a]. [4] [b] Draw the Bending Moment diagram for the frame shown in Fig. 1[b]. The frame consists of hinge at Joint C. [12] Q. [3] [a] What are the rules for constructing Mohr's circle in the analysis of principal stresses? [6] [b] A component of a power generator consists of a torus supported by six tie rods from an overhead central point as shown. The weight of the torus is 3000N per meter of circumferential length. The point of attachment A is 1.25m above the plans of the torus. The radius of the middle line of the torus is 0.5m. Each tie rod has a cross sectional area of 25mm 2. Determine the vertical displacement of the torus duc to its own weight. [Take E=200 GN/m 2 ] [10] Fig. 1[b] / 5
Q. [4] [a] A short iron column is of hollow 300mm external diameter and 50mm thick. A vertical Compressive load of 200kN acts at an eccentricity of 90mm from the axis. Determine the maximum tensile and compressive stresses. Plot the variation of stress in the column. [8] [b] A solid aluminium shaft 1m long and 60mm diameter is to be replaced by a tubular steel shaft of the same length and same outside diameter such that each of the two shafts could have the same angle of twist per unit torsional moment over the total length. What must be the inner diameter of the rubular steel shaft? Let the modulus of rigidity of steel be three times that of aluminium. [8] Q. [5] [a] Find the moment of Inertia about the centroidal axis X-X for the lamina shown in Fig. 5[a[. [8] Fig. 5[a] [b] Calculate the change in diameter and length of an air vessel 400mm in diameter and 10mm thick when subjected to internal pressure of 16MPa. [Take modulus of elasticity of material as 200kN/mm 2 and Poisson's Ratio=0.3. The vessel is 1.5 in long] [8] Q. [6] [a] Prove that in simple bending of beam, the neuter axis passes through the center of gravity (C.G.) or the section. [6] [b] A steel bar of rectangular section 30mm 50mm is used as a column with both ends hinged and subjected to axial compression. If the critical stress developed is 250 MPa and modulus of elasticity is 200 GN/m 2, determine the minimum length for which Eluer's equation may be used. Again, if the length of the column is 2m, determine the safe load with the factor of safety of 4. [10] Q. [7] [a] Explain Saint Venant's theory of elastic failure of materials. [6] [b] A cast iron 540mm internal diameter and 30mm wall thickness is running full of water and supported over a length of 8m. Determine the maximum stress intensity in the metal if the density of the cast iron is 72 KN/m 3 and that of water is 10KN/m 3. [10] / 6
PURWANCHAL UNIVERSITY III SEMESTER FINAL EXAMINATION-2004 LEVEL : B. E. (Civil) SUBJECT: BEG256C1, Strength of Materials Full Marks: 80 TIME: 03:00 hrs Pass marks: 32 Candidates are required to give their answers in their own words as far as possible.. All questions carry equal marks. The marks allotted for each subquestion is specified along its side. Q. [2] [a] What is a bulk modulus? Derive an expression for young's modulus in terms of bulk modulus and Poisson's ratio. [6] [b] The normal stress in two mutually perpendicular directions are 600 N/mm 2 and 300 N/mm 2 both tensile. The complimentary shear stress in these directions is of intensity 450 N/mm 2. Find the normal and tangential stress on two planes, which are equally inclined to the planes carrying the normal stress, mentioned above. [10] Q. [3] [a] Write a note on Mohr's circle. [6] Attempt any FIVE questions. Q. [1] [a] What do you understand by the term "Indeterminate structure"? Determine the degree of indeterminacy of the given beam. [4] [b] A rod is 2 m long at a temperature of 10ºC. Find the expansion of the rod, when the temperature is raised to 80ºC. If this expansion is prevented, find the stress induced in the material of the rod. Take E=1.0 10 5 MN/m 2 and Coefficient of thermal expansion=12 10-6 /ºC. [10] [b] Analyze and draw bending moment, shear force diagram of the given frame. [12] Q. [4] [a] A circular column of diameter 200 mm carries a point load of 240 kn at an eccentricity of 10mm to the right of the center. Determine the maximum and minimum stress on the section. [8] [b] Ram is twisting a shaft of diameter 5m and length 40m by power 200 KW at 20 rev/min. Does the shaft transmit the power safely if the permissible stress is 50N/mm 2? [8] / 7 Q. [5] [a] Determine the principal moment of inertia and direction of principal axes of a circular section of radius 6 centimeters. [8] [b] A cylindrical pipe of diameter 1.5 cm and thickness 1.5cm is subjected to a fluid pressure of 1.2 N/mm 2. Determine longitudinal and hoop stress, and also
determine the percentage change in volume [1/m=0.3]. Take E=0.204 10 4 N/mm 2 and length of cylindrical pipe as 2m. [8] Q. [6] [a] Define pure bending, elastic bending and plastic bending. [6] [b] A solid round bar 2 m long and 5cm in diameter is used as a strut. Determine the crippling load for the condition of both ends hinged and both ends fixed. Also determine safe load for safety factor 3. Take E=2 10 5 N/mm 2. [10] Q. [7] [a] Discuss about theory of failure. [6] [b] Three equal loads of 5kN have been supported by a simply supported beam of 12m dividing the beam in four equal parts. Does the beam support the load if the allowable bending stress is 50N/mm 2 and section is 100mm 50mm?s [10] / 8
PURWANCHAL UNIVERSITY III SEMESTER BACK-PAPER EXAMINATION-2004 LEVEL : B. E. (Civil) SUBJECT: BEG256C1, Strength of Materials Full Marks: 80 TIME: 03:00 hrs Pass marks: 32 Candidates are required to give their answers in their own words as far as possible.. The figures in the margin indicate full marks. The marks allotted for each sub-question is specified along its side. Attempt any FIVE questions. Q. [1] [a] Check the statical determinacy of the beam in Fig. 1[b]. If the condition were roller instead of hinge, what would be the effect? [4] [b] Find and draw shear force and bending moment diagrams for the following loaded beam shown in Fig. 1[b]. [12] planes are inclined to one another at 30º and a plane AC at right angles to plane AB carries a direct stress of unknown magnitude and nature, determine; [i] the value of shear stress ' τ ' on BC, [ii] the magnitude and nature of direct stress on AC, [iii] principal stresses. [10] Q. [3] [a] Explain the law of complementary shear in a rectangular element in a strained material. [6] [b] Two copper rods and a steel rod support a load of 20KN as shown in Fig. 3[b]. The diameter of steel rod and copper rod are 15mm and 20mm. Take 'E' for steel and copper as 2.1 10 5 N/mm 2 and 1 10 5 N/mm 2 respectively. Find the load carried by each rod. [10] Fig. 1[b] Q. [2] [a] What do you understand by the term thermal stress? [6] [b] In a certain material under load, plane AB carries a tensile direct stress of 30 N/mm 2 and a shear stress of 20N/mm 2, while another plane BC carries a tensile direct stress of 20N/mm 2 and a shear stress ' τ '. If the / 9 Fig. 3[b] Q. [4] [a] A cost iron short column of 200mm diameter a vertical load of 400 KN at a distance of 40mm from the center. Determine the maximum and minimum intensities of stress developed in the section along the diameter passing through the point of loading. [8] [b] A hollow shaft 30mm outer diameter runs at 400 rpm against a power of 50KW. Find the inner diameter of the shaft so that the shear strain does not exceed 1/1000. Take 'C'=8 10 6 N/cm 2. [8]
Q. [5] [a] Find the moment of Inertia of the section about the horizontal centroidal axis as shown in Fig. 5[a]. Define principal moment of inertia of a section. [8] Q. [7] [a] Discuss on maximum principal stress theory of failure criteria. Also mention what are its limitations. [6] [b] A timber beam 150mm wide and 210mm deep is to be reinforced by bolting on two steel flitches each 150mm 10mm is section. Find the moment of resistance when: [i] Flitches are attached symmetrically at top and bottom [ii] The flitches are attached symmetrically at sides. Fig. 5[a] [b] A cylindrical thin shell 1.5m long internal diameter 300mm and wall thickness 100mm is filled up with a fluid at atmospheric pressure. If additional fluid of 350 10 3 mm 3 is pumped in the shell, find the pressure exerted by the fluid on the shell. Take Young's modulus of Elasticity of shell materials as 2 10 5 N/mm 2 and Poisson's Ratio as 0.3. Also find the hoop stress incuced. [8] Take the allowable stress in timber as 6N/mm 2 and E steel =20 E timbe. What is maximum stress in steel in each case? [10] Q. [6] [a] Prove that the bending stress at any layer in a beam depends upon the distance of the layer from the Neutral axis. [6] [b] A column having T-section with a flange 120mm 16mm and web 150mm 16mm is 4m long. Assuming the column to be hinged at one end and fixed at the other end, find the buckling load by using Euler's formula. Take the Young's modulus of elasticity of column material as 2 10 5 N/mm 2. [10 / 10
PURWANCHAL UNIVERSITY III SEMESTER FINAL EXAMINATION-2005 LEVEL : B. E. (Civil) SUBJECT: BEG256C1, Strength of Materials Full Marks: 80 TIME: 03:00 hrs Pass marks: 32 Candidates are required to give their answers in their own words as far as practicable. All questions carry equal marks. The marks allotted for each subquestion is specified along its side. Attempt any FIVE questions. Fig. [1.b] Q. [2] [a] Find an expression force total elongation of a bar due to its own weight, when the bar is fixed at its upper end and hanging freely at the lower end. [6] [b] Direct stress of 150N/mm 2 tensile and 100N/mm 2 compression exist on two perpendicular planes at a certained point in a body. They are accompanied by a shear stress on the planes. The greatest principal stress at the point due to these is 150N/mm 2. What must be the magnitude of the shearing stress on the two plane and what will be the maximum shearing stress at the point. [10] Q. [1] [a] Define statically determinate and indeterminate structures. Find out the degree of static indeterminacy of the given beam shown in fig. [1] [a]. [4] Q. [3] [a] A rectangular bar is subjected to a simple shear stress as shown in fig. [3.a] Determine the normal stress and tangential stress on an oblique plane AB which is inclined at an angle θ with the axis of minor stress. [6] [b] Draw blending moment of the given frame shown in fig.[1] [b] [12] [b] A rod is composed of three segments as shown in Fig. [3.b]. The rod is held between rigid supports A and B. Find the stresses developed in each material when the temperature of the system is raised by 55ºC. [10] E S = 200 KN/mm 2 α S = 12*10-6 /ºC E A = 70 KN/mm 2 α A = 23*10-6 /ºC E C = 100 KN/mm 2 α C = 17*10-6 /ºC / 11
Q. [4] [a] A square column 300 mm*300 mm carries vertical load of 100KN as shown in fig. [4.a]. Find the stress developed at each corner of column. [8] [b] A cylindrical shell is 4 m long, 2m internal diameter and 30mm metal thickness. Calculate the intensity of maximum shear stress induced and also change in dimensions of the shell if it is subjected to an internal pressure of 3 N/mm 2. Take Modulus of elasticity E=0.2* 10 6 N/mm 2 and Poisson's ratio 1/m=0.3. [8] Q. [6] [a] Explain about the theory of simple bending in term of polar moment of inertia. [6] [b] A composite shaft consists of copper rod of 30mm diameter enclosed in a steel tube of external diameter 50mm and 10mm thick. The shaft is required to transmit a torque of 1000Nm... Determine the shear stresses developed in copper and steel, if both the shafts have equal length and welded to a plate at each end, so that their twists are equal. Take modulus of rigidity for steel as twice that of copper. [8] [b] Compare the strength in term of crippling load of a solid circular column of diameter of 300mm and hollow circular column of same cross-sectional area and thickness 30mm. The other parameters are same for both sections. Assuming both the ends are fixed. [10] Q. [7] [a] A short column of rectangular section subjected to an eccentric load. Draw clearly the stress diagram if (i) Direct stress is greater than bending stress (ii) direct stress is less that bending stress and (iii) direct stress is equal to bending stress. [6] Q. [5] [a] Find the moment of inertia of Channel section about an axis pass through the centroid. All the dimensions shown in fig. [5.a.] are in Centimeter. [8] / 12 [b] A beam of size 25mm * 25mm is carrying the maximum central load of 330 KN on a simply supported span of 600mm. The beam of same material but of size 25mm wide and 100mm deep is used as a
cantilever of span 1200mm. Determine the maximum value of point load W that can be placed at the free and of this cantilever. [10] / 13
PURWANCHAL UNIVERSITY III SEMESTER BACK-PAPER EXAMINATION-2005 LEVEL : B. E. (Civil) SUBJECT: BEG256C1, Strength of Materials Full Marks: 80 TIME: 03:00 hrs Pass marks: 32 Candidates are required to give their answers in their own words as far as practicable. All questions carry equal marks. The marks allotted for each subquestion is specified along its side. Attempt any FIVE questions. Q. [1] [a] Define statically determinate and indeterminate structures. Find out the degree of static indeterminacy of the given beam shown in fig. [1] [a]. [4] Fig. [1.b] Q. [2] [a] Find an expression for total elongation of a bar due to its own weight, when the bar is fixed at its upper end and hanging freely at the lower end. [6] [b] i) Determine resultant stress on given plane. ii) Find out principal stresses. iii) Determine maximum shear stress. [10] Fig.[2.b] Q. [3] [a] At a point in a two dimensional system, the normal stresses on two mutually perpendicular planes are σ 1 and σ 2 (both tensile) with a shear stress τ. Show that one of the principal stress is zero if τ 2 = σ1σ 2 [6] [b] Draw blending moment of the given frame shown in fig.[1] [b] [12] / 14 [b] An aluminum rod 22mm diameter passes through a steel tube of 31mm external diameter. The rod and tube are fixed at a temperature 180ºC. Find the stress in the aluminum rod and steel tube, when the temperature falls to 60ºC. [10] Es=200KN/mm 2 E A =70KN/mm 2 σ =12 10-6 /ºC s σ A =23 10-6 /ºC Q. [4] [a] A square column 300mm 300mm carries a vehicles load of 100 KN at a distance 80 cm from the axis as shown in fig[4.a]. Find the maximum and minimum stress induced in the section. Plot the stress variation diagram. [8]
[i] Maximum permissible diamet'er of the shell for an internal pressure of 2 MN/m 2. [ii] Permissible intensity of internal pressure when the shell diameter is 1.5m. Q. [6] [a] What do you mean by pure bending. Explain with an example? What are the assumptions made in the theory of simple bending? [6] [b] A hollow circular shaft of external diameter 150 mm transmits 200KW power at 100 r.p.m. determine the maximum stress in the shaft is not to exceed 60N/mm 2.[8] Q. [5] [a] Find the moment of inertia of T-section about an axis pass through the centroid and parallel to the top face AB. All the dimensions shown in fig are in Centimeter. Angle size: 5 5 1 cm Flange size: 20 1cm, Web size: 20 1cm [8] [b] Compare the strength in term of crippling load of a solid circular column of diameter of 200mm and hollow circular column of same cross-sectional area and thickness 30mm. the other parameters are same for both section. Assuming both the ends are hinged. [10] Q. [7] [a] Write a short note on theory of elastic failure. [6] [b] A timber beam 110mm wide and 220mm deep is strengthened by two steel plate 110mm wide and 20mm thick at the top and bottom o the timber section. Find the moment of resistance of the section. Allowable stress in the timber is 7N/mm 2. Take E s =20E t [10] [b] A boiler shell is to be made of 15 mm thick plate having tensile stress of 120 MN/m 2. If the efficiencies of the longitudinal and circumferential joints are 70% and 30% respectively. Determine. [8] / 15
PURWANCHAL UNIVERSITY III SEMESTER FINAL EXAMINATION-2006 LEVEL : B. E. (Civil) SUBJECT: BEG256C1, Strength of Materials Full Marks: 80 TIME: 03:00 hrs Pass marks: 32 Candidates are required to give their answers in their own words as far as practicable. All questions carry equal marks. The marks allotted for each subquestion is specified along its side. Attempt any FIVE questions. Q. [1] [a] Find out the degree of indeterminacy of the beam shown in Fig.-[1.a] [4] Q. [2] [a] Find an expression for the total elongation of a uniformly tapering rectangular bar when it is subject to an axial load P. [6] [b] Direct stress of 160N/mm 2 tensile and 120N/mm 2 compression exist on two perpendicular planes at a certain point in a body. They are accompanied by a shear stress on the planes. The greatest principal stress at the point due to these is 200N/mm 2. (i) What must be the magnitude of the shearing stresses on the two planes? (ii) What will be the maximum shearing stress at the point? [10] Q. [3] [a] Describe Mohr's circle and its method of construction.[6] [b] Three rods of diameter 2m each, support a beam as shown in figure. Where should a load P be placed rom the left rod to remain the beam horizontal? Neglect the weight of the beam. Take E s =2 10 6 Kg/cm 2, E h =1 10 6 Kg/cm 2, E a =6 10 6 Kg/cm 2 [10] [b] Draw bending moment diagram for the frame shown in Fig.-[1.b] [12] Q. [4] [a] A solid circular shaft and a hollow circular shaft whose inside diamter is 3 4 of the outside diameter, are of same material, of equal lengths and are required to transmit a given torque. Compare the weights of these two shafts if the maximum shear stress developed in the two shafts are equal. [8] [b] A hollow rectangular column of external and internal dimensions are 2.4m 1.8m and 1.2m 1.2m respectively. Calculate the safe load that can be placed / 16
at an eccentricity of 50cm on a plane bisecting the longer side, if the maximum compressive stress is not to exceed 50kg/cm 2. [8] Q. [5] [a] Find the product of inertia for the plane hatched area about the axes X-X and Y-Y shown in figure. [8] [b] A timber cantilever 200mm wide and 300mm deep is 3m long. It is loaded with a UDL of 3KN/m over the entire length. A point load of 2.7KN is placed at the free end of cantilever. Find the maximum bending stress produced. [8] [b] A cylindrical shell is 90cm long, 20cm internal diameter having thickness of metal as 8mm is filled with fluid at atmospheric pressure. If an additional 20cm 3 of fluid is pumped into the cylinder, find (i) the pressure exerted by the fluid in the cylinder and (ii) the hoops stress induced. Take E=2 10 5 N/mm 2 and poisson's ratio=0.3. Q. [6] [a] What do you understand by deflection of a beam. Show the relationship between the slope, deflection and radius of curvature. [6] [b] A filtched beam consists of a timber joist 150mm wide and 300mm deep is strengthened by steel section 20mm thick and 300mm deep at on left side of the timber joist. Determine the moment of resistance of the beam if allowable stress in timber=7n/mm 2. Take Es=20E 1 [4] Q. [7] [a] Determine the ratio of buckling strength of two columns one hollow and other solid. Both are made of the same material and have the same length, crosssectional area and end conditions. The internal diameter of hollow column is 2/3 of its external diameter. [8] / 17