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Unformatted text preview: 3.54. CHAPTER 3, PROBLEM 54 403 3.54 Chapter 3, Problem 54 Problem: A triangular access port, hinged as shown below, is provided in the side of a form containing liquid concrete. Determine the hydrostatic force exerted on the access port by the concrete, which has density ρ. Also, compute the force needed at the upper tip of the access port to prevent it from opening. Ar ............................................................................................. .......................... .... .............. . . . . .. . .. . .. . . . . . . . . . ............................................................................................. ............................................................................................. ................................................................................................. ............................................................................................ .... . . . . . . . . . . . . . . . . . . ................................................................................................................ ............................................................................................. . . . . . . .. . .. . .. . .. . .. . . . . ........... .............................................................................................. .............................................................................................. ................ ....... ....... ........................................... . .. . .. . . . .. ............................................................................................... ............................................................................................. ... ....... ... . . ... .. .... ... .. ....... ... .. ............................................................................................. ................................................................................................................. . . . ............................................................................................................ ................................................................................ ................................................................................................. .... . . .. .... .. ..... . .... . . 1 3h zcp h ρ F Ft p Ar H nge •............................................................................................................................................................................................................................................................................................................................................................................................ ............................................................................................. . . . . . . . . . . . ... .. . . . . ....... ............................................................................................. ............................................................................................. . . . . . . . . . . . ... . . . . . . . .................................................................................................. .............................................................................................. ............................................................................................... ............................................................................................ ...... .. .. .. ......... .. .. .. .. ... .. ... .. ... .. ....... . . . . . . . .. . .. . .......................................................................................................................................................................................................................... .................... .. .. .. .. .. .. .. .. ... .. ... .. ......... ...................................................................................................................................................................................................................... ...................................................................................................................................................................................................................... . .. .. . . C ¤ ¤C ¤Ch ¤C ¤ C 1 3h Solution: For the triangular gate, the centroid, area and moment of inertia are 1 1 11 14 2 11 z = h + h = h, A= h h = h2 , I= h h3 = h 3 3 23 6 36 3 108 Hence, the hydrostatic force on the access port is 1 1 F = ρg zA = ρgh h2 = ρgh3 6 6 To determine the force at the upper tip of the access port that is required to prevent it from opening, we require that the net moment about the hinge vanish. Thus, we require that 4 F h − zcp = Ft p h 3 The center of pressure is given by zcp = z + 1 h4 I 1 19 = h + 108 =h+ h= h zA 18 18 h 1 h2 6 So, balancing moments, we have 1 24 − 19 1 4 19 =⇒ Ft p = ρgh3 ρgh3 h − h = Ft p h 6 3 18 6 18 Therefore, the force at the tip of the access port is 5 Ft p = ρgh3 108 ...
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