ch11-3 - SECTION 11.5 Columns with Eccentric Axial Loads...

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SECTION 11.5 Columns with Eccentric Axial Loads 697 Problem 11.5-13 A frame ABCD is constructed of steel wide-flange members ( W 8 21; E 30 10 6 psi) and subjected to triangularly distributed loads of maximum intensity q 0 acting along the vertical members (see figure). The distance between supports is L 20 ft and the height of the frame is h 4 ft. The members are rigidly connected at B and C . (a) Calculate the intensity of load q 0 required to produce a maximum bending moment of 80 k-in. in the horizontal member BC . (b) If the load q 0 is reduced to one-half of the value calculated in part (a), what is the maximum bending moment in member BC ? What is the ratio of this moment to the moment of 80 k-in. in part (a)? Solution 11.5-13 Frame with triangular loads Section E - E E E B A h L D C q 0 q 0 P resultant force e eccentricity M AXIMUM BENDING MOMENT IN BEAM BC From Eq. (11-56): (1) N UMERICAL DATA W 8 21 I I 2 9.77 in. 4 (from Table E-1) E 30 10 6 psi L 20 ft 240 in. h 4 ft 48 in. e h 3 16 in. k B P EI M max Pe sec B PL 2 4 EI M max Pe sec kL 2 P q 0 h 2 e h 3 (a) L OAD q 0 TO PRODUCE M max 80 k-in. Substitute numerical values into Eq. (1). Units: pounds and inches (radians) (2) S OLVE E Q . (2) NUMERICALLY P 4461.9 lb (b) L OAD q 0 IS REDUCED TO ONE - HALF ITS VALUE P is reduced to one-half its value. Substitute numerical values into Eq. (1) and solve for M max . M max 37.75 k-in. This result shows that the bending moment varies nonlinearly with the load. Ratio: M max 80 k-in. 5 37.7 80 5 0.47 P 1 2 (4461.9 lb) 2231.0 lb q 0 2 P h 186 lb in. 2230 lb ft P 5,000[cos(0.0070093 P ) ] 0 5,000 P sec(0.0070093 P ) 80,000 P (16 in.) [sec(0.0070093 P ) ] 0.0070093 P M max 80,000 lb-in. B PL 2 4 EI B A h L D C h 3 h 3 P P
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Problem 11.6-2 A brass bar ( E 100 GPa) with a square cross section is subjected to axial forces having a resultant P acting at distance e from the center (see figure). The bar is pin supported at the ends and is 0.6 m in length. The side dimension b of the bar is 30 mm and the eccentricity e of the load is 10 mm. If the allowable stress in the brass is 150 MPa, what is the allowable axial force P allow ? Solution 11.6-2 Bar with square cross section 698 CHAPTER 11 Columns The Secant Formula When solving the problems for Section 11.6, assume that bending occurs in the principal plane containing the eccentric axial load. Problem 11.6-1 A steel bar has a square cross section of width b 2.0 in. (see figure). The bar has pinned supports at the ends and is 3.0 ft long. The axial forces acting at the end of the bar have a resultant P 20 k located at distance e 0.75 in. from the center of the cross section. Also, the modulus of elasticity of the steel is 29,000 ksi. (a) Determine the maximum compressive stress max in the bar. (b) If the allowable stress in the steel is 18,000 psi, what is the maximum permissible length L max of the bar? Solution 11.6-1 Bar with square cross section P e b b Pinned supports. D ATA b 2.0 in. L 3.0 ft 36 in. P 20 k e 0.75 in. E 29,000 ksi (a) M AXIMUM COMPRESSIVE STRESS Secant formula (Eq. 11-59): (1) L r 62.354 P EA 0.00017241 ec r 2 2.25 r 2 I A 0.3333 in.
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