CHATER 9 Coumns and Struts robem. Compare the ratio of the strength of soid stee coumn to that of the hoow stee coumn of the same cross-sectiona area. The interna diameter of the hoow coumn is /th of the externa diameter. The coumns have the same ength and are pinned at both ends. Use Euer s theory. Soution: et d be the diameter of the soid strut and D be the outer diameter of the hoow strut. As the cross-sectiona areas are same, et d D D d 9 7 D D D 6 6 crh bucking oad of the hoow coumn crs bucking oad of soid coumn h east moment of inertia of hoow coumn D D 6 6 56 s east moment of the soid coumn Now crh Thus crh crs crh crs 6 d E h h s D 75 and crs 75 D 6 56 d 6 E s 75 D 75 D 56 d 7 56 D 6 5 7.57. Ans. robem. A soid round bar of 60 mm diameter and.5 m ong is used as a strut. Find the safe compressive oad for the strut using Euer s formua if (a) both ends are hinged (b) both ends are fixed. Take E 0 5 N/mm and factor of safety. Soution:.5 m 500 mm d 60 mm, E 0 5 N/mm d 60 667.5 mm 6 6 80
Factor of safety. (a) Both ends are hinged: cr 5 E 0 667.5 0090 N (500) 00.9 kn 00.9 Safe oad cr 66.97 kn. Factor of safety (b) Both ends are fixed: cr Safe oad 5 E 0 667.5 (500) 8068 N 80.68 kn c r 80.68 6.89 kn. Ans. Factor of safety robem. What is the ratio of the strength of a soid stee coumn of 50 mm diameter to that of a hoow circuar stee coumn of the same cross-sectiona area and a wa thickness of 5 mm? The two coumns have the same ength and simiar end conditions. Soution: Diameter of circuar coumn d 50 mm C.S. Area 50 et the thickness of circuar hoow coumn be t 5 mm et externa diameter of hoow circuar coumn be D mm ts interna diameter D t D 5 (D 0) mm C.S. area {D (D 0) } This area is same as that of soid coumn { ( 0) } D D 50 D {D 60D + 900} 50 60D 500 + 900 00 D 90 mm nterna diameter of hoow coumn 90 0 60 mm. east moment of inertia: s h crh crs 85088.7 mm 6 6 50 (90 60 ) 89.5 mm Eh e Es e 8
crh crs h s 89.5.5. Ans. 85088.7 robem. Find the Euer s crushing oad for a hoow cyindrica cast iron coumn 0 mm externa diameter and 0 mm thick, if it is. m ong and is hinged at both ends. Take E 80 kn/mm. Compare this oad with the crushing oad as given by Rankine s formua using constants f c 550 N/mm and a /600. For what ength of strut does the Euer s formua cease to appy? Soution: Externa diameter 0 mm Thickness 0 mm nterna diameter 0 0 80 mm east moment of inertia (0 80 ) 6 8680.89 mm Coumn is hinged at both ends. e. m 00 mm Euer s bucking oad E 80 0 8680.89 (00) 65606.89 N. Ans. A (0 80 ) 68.8 mm K 8680.89 00 A 68.8 K 6.05 mm fc A 550 68.8 Rankine s critica oad R 00 + a + K 600 6.05 Now, E Equating it to crushing oad, we get E R E E K 80 0 00 65.6 N 65606. 89.00. Ans. 65. 6 E f c A f c 550 80 000 00 550 66.08 mm. Ans. 8
robem 5. An SB 00 section is provided with a fange pate 00 mm mm for each fange. The composite member is used as a coumn with one end fixed and the other end hinged. Cacuate the ength of the coumn for which, cripping oads given by Rankine s formua and Euer s formua wi be the same. Take E 0 kn/mm, f c 0 N/mm, a /7500 roperties of SB 00 section are: Overa width 50 mm, Overa depth 00 mm, Thickness of fange 9. mm, Thickness of web 6.7 mm xx 7.9 0 6 mm yy.76 0 6 mm A 808 mm. Soution: f c 0 N/mm, a, E 0 0 N/mm 7500 Area A 808 mm xx 7.9 0 6 mm, yy.76 0 6 mm Sectiona area of SB 00 coumn. A 808 + (00 ) 9608 mm Moment of inertia about x x axis. 00 xx 7.9 0 6 + 909906 mm Moment of inertia about y-y axis. yy Since yy < xx, the coumn buckes about y-y axis. min 976000 mm 976000 east radius of gyration K A 9608 5.5 mm. et effective ength E + (00 ) 56 6 00.76 0 + 976000 mm f A c + a K 0 0 976000 0 9608 + 7500 056.8 989.65 8 + 6.85 0 989.65 + 0.87 89 mm 8
Therefore, required actua ength for one end hinged and other end fixed coumn for which critica oad by Rankine s formua and Euer s formua wi be equa is 89 606 mm.606 m. Ans. robem 6. A buit up stee coumn, 8 m ong and ends firmy fixed is having cross-section as shown in Fig.. The properties of -section are Area 900 mm, xx 0 6 min, yy 8. 0 6 mm. Determine: (i) The safe axia oad the coumn can carry with a factor of safety of.5 using (a) Euer s Formua, (b) Rankine s Formua. (ii) The ength of the coumn for which both formuae give the same cripping oad. (iii) The ength of the coumn for which the Euer s formua ceases to appy. Take E 0 5 N/mm, f c 0 N/mm, a /7500 Soution: (i) ength of the coumn 8 m 8000 mm Factor of safety.5, f c 0 N/mm, a Fig. 7500, E 05 N/mm A (900 + 50 5) 600 mm Moment of inertia of coumn section about x-x axis: xx 0 6 + 50 5 99008.5 mm Moment of inertia of the coumn section about y-y axis: yy 8. 0 6 + 858. mm yy < xx 5 50 min 858. mm K A 0.79 858. 600 + 50 5 7.5 + 900 00 8
Since coumn is fixed at both ends, E 5 E 0 858. (8000) 705899. N E 705899. Safe oad 5.0 0 N.5.5 5. kn. Ans. fc A 8000 (b) R where 000 mm + a K 0 600 000 + 7500 0.79 99.806 0 N 99.806 kn. 99.806 Safe oad 8.95 kn. Ans..5 (ii) et be the effective ength or or 5 E f A c + a K 0 858. 0.797 + 60550.5 0 600 + 7500 (0.79) 7670 mm 7.67 m. Ans. (iii) et be the ength of coumn for which Euer s formua ceases to appy. Then E f c A E E 5 EK 0 (0.79) 0 5850. 5900 mm 5.9 m. Ans. robem 7. Find the safe oad carrying capacity of coumn given in robem 6 by ndian Standard code procedure. Given f c 50 N/mm. 85
Soution: 858. mm A 600 mm K Senderness ratio λ 0.79 mm 000 k 0.79 8.9 From.S. tabe, λ 0 and f c 50 N/mm σ ac 5 N/mm and for λ 0, f c 50 N/mm σ ac 9 N/mm ineary interpoating between these two vaues of λ, 8.9 σ ac 5 (5 9) 0 9.65 N/mm Therefore, safe oad carrying capacity of the coumn is, σ ac A 9.65 600 50 N 50. kn. Ans. 86