HYDRAULICS CORRE 1 SAMPLE PROBLEM A reservoir of glycerin has a mass of 1,200 kg and a volume of 0.952 cu. m. Find its: (a.) Gravity Force (b.) Unit Weight (c.) Mass density (d.) Specific gravity SAMPLE PROBLEM Assuming specific weight of air to be constant at 12 N/ ^3, what is the approximate height of Mount Banahaw if a mercury barometer at the base of the mountain reads 654 mm and at the same instant, another barometer at the top of the mountain reads 480 mm. SAMPLE PROBLEM Determine the value of y in the manometer shown in the figure. TOTAL HYDROSTATIC PRESSURE TOTAL HYDROSTATIC FORCE ON PLANE SURFACES PROPERTIES OF COMMON PLANE SECTIONS TRIANGLE RECTANGLE CIRCLE QUARTER CIRCLE TOTAL HYDROSTATIC FORCE ON PLANE SURFACES PROPERTIES OF COMMON PLANE SECTIONS SEMI CIRCLE ELLIPSE HALF ELLIPSE QUARTER ELLIPSE TOTAL HYDROSTATIC FORCE ON PLANE SURFACES PROPERTIES OF COMMON PLANE SECTIONS SECTOR OF A CIRCLE PARABOLIC SEGMENT SPANDREL SEGMENT OF ARC SAMPLE PROBLEM The gate in the figure shown is 1.5 m wide, hinged at point A, and rests against a smooth wall at B. Compute (a) the total force on the gate due to seawater, (b) the reaction at B, and (c) the reaction at hinge A. Neglect the weight of the gate. TOTAL HYDROSTATIC PRESSURE SAMPLE PROBLEM The submerged curve AB is one quarter of a circle of radius 2 m and is located on the lower corner of a tank as shown. The length of the tank perpendicular to the sketch is 4 m. Find the magnitude and location of the horizontal and vertical components of the total force acting on AB. SAMPLE PROBLEM The gate shown is a quarter circle 2.5 m wide. Find the force F just sufficient to prevent rotation about hinge B. Neglect the weight of the gate. BUOYANCY where: πΎ = unit weight of the fluid π = volume displaced. Volume of the body below the liquid surface. BUOYANCY If the body of height H has a constant horizontal cross-sectional area If the body is of uniform constant vertical cross-sectional area SAMPLE PROBLEM 1.) An iceberg (sg = 0.917) floats in ocean water with 3000 m^3 of the iceberg protruding above the free surface. What is the volume of the iceberg below the free surface? 2.) A hollow cylinder 1 m in diameter and 2 m high weighs 3825 N. (a) How many kN of lead weighing 110 kN/m^3. must be fastened to the outside bottom of the cylinder to make it float with 1.5 m submerged in water? (b) How many kN of lead if it is placed inside the cylinder? ANALYSIS OF GRAVITY DAM UPSTREAM SIDE OBJECTIVES DOWNSTREAM SIDE 1. Determine the forces acting on the dam. ππ πΎπ ππ πΎπ h π· 2. Check if the dam will not slide or overturn (stability analysis). ππ πΎπ πΎπ ππ 3. Compute the pressure on the foundation of the dam. y 1m Toe Heel πΉπ UPLIFT PRESSURE DIAGRAM ππ πΌπ πΌπ π πΉπ π π R STEPS IN ANALYSIS 1. Consider 1 unit (1 m) length of dam ANALYSIS OF GRAVITY DAM UPSTREAM SIDE 2. Determine all the forces acting: A. Vertical Forces DOWNSTREAM SIDE • Weight of the Dam πΎπ π = πΎ π; πΎπ h π = πΎπ • Weight of permanent structures on the dam • Hydrostatic Uplift 1m π = πΎ π • Weight of water in the upstream side (if any) πΎπ πΎπ π = πΎ π; Toe Heel UPLIFT PRESSURE DIAGRAM πΌπ πΌπ π = πΎπ ; π = πΎπ STEPS IN ANALYSIS B. Horizontal Forces ANALYSIS OF GRAVITY DAM UPSTREAM SIDE • Total Hydrostatic Force acting at the vertical projection of the submerged portion of the dam DOWNSTREAM SIDE π = πΎβπ΄ • Wind pressure, wave action, floating bodies and earthquake load πΎπ πΎπ h π· 3. Solve for the reaction A. Vertical Reaction, Ry πΎπ πΎπ π = 1m A. Horizontal Reaction, Rx Toe Heel πΉπ UPLIFT PRESSURE DIAGRAM πΌπ πΌπ πΉπ πΉπ£ = π +π +π +π −π −π R π = πΉβ = π STEPS IN ANALYSIS 4. Moment about the Toe ANALYSIS OF GRAVITY DAM A. Righting Moment, Rm π = π π₯ +π π₯ +π π₯ +π π₯ UPSTREAM SIDE DOWNSTREAM SIDE B. Overturning Moment, Om ππ πΎπ π = ππ¦ + π π§ + π π§ ππ πΎπ h π· 5. Location of Ry, ππ πΎπ πΎπ π₯Μ = ππ y 1m Toe Heel πΉπ UPLIFT PRESSURE DIAGRAM ππ πΌπ πΌπ π πΉπ π π R π −π π ANALYSIS OF GRAVITY DAM Factors of Safety Factor of Safety against Sliding, πΉπ = ππ > 1 π Factor of Safety against Overturning, πΉπ = π > 1 π ANALYSIS OF GRAVITY DAM Foundation Pressure For π= − π 6π (1 ± ) π΅ π΅ For π= 2π 3π₯Μ SAMPLE PROBLEM A gravity dam of trapezoidal cross-section with one face vertical and horizontal base is 22 m high and has a thickness of 4 m at the top. Water upstream stands 2m below the crest of the dam. The specific gravity of masonry is 2.4. Considering hydrostatic uplift pressure to vary uniformly from full hydrostatic pressure at the heel to zero. a.)Find the base width B of the dam so that the resultant force will act at the extremity of the middle third near the toe. b.) Compute the maximum and minimum compressive stresses acting against the base of the dam. c.) Compute the factors of safety against sliding and overturning. Use = 0.5 MOVING VESSEL SAMPLE PROBLEM 1.) An open rectangular tank mounted on a truck is 5 m long, 2 m wide and 2.5 m high is filled with water to a depth of 2 m. (a) What maximum horizontal acceleration can be imposed on the tank without spilling any water and (b) determine the accelerating force on the liquid mass. (c) If the acceleration is increased to 6 m/s^2, how much water is spilled out. 2.) A closed horizontal cylindrical tank 1.5 m in diameter and 4 m long is completely filled with gasoline (sp.gr. = 0.82) and accelerated horizontally at 3 m/s^2. Find the total force acting at the rear wall and at the front wall of the tank. Find also the accelerating force on the fluid mass. SAMPLE PROBLEM 3.) A vessel containing oil is accelerated on a plane inclined 15° with the horizontal at 1.2 m/s^2. Determine the inclination of the oil surface when the motion is (a) upwards, and (b) downwards. 4.) A vessel 3 m in diameter containing 2.4 m of water is being raised. (a) Find the pressure at the bottom of the vessel in kPa when the velocity is constant, and (b) find the pressure at the bottom of the vessel when it is accelerating 0.6 m/s^2 upwards. ROTATING VESSELS where: w = angular speed in radians per second NOTE: 1 rpm = π/30 rad/sec ROTATING VESSELS SQUARED PROPERTY OF PARABOLA VOLUME OF PARABOLOID OF REVOLUTION ROTATING VESSELS LIQUID SURFACE CONDITIONS For open cylindrical containers more than half-full of liquid, rotated about its vertical axis ( h > H/2) ROTATING VESSELS LIQUID SURFACE CONDITIONS For open cylindrical containers more than half-full of liquid, rotated about its vertical axis ( h > H/2) ROTATING VESSELS LIQUID SURFACE CONDITIONS For open cylindrical containers more than half-full of liquid, rotated about its vertical axis ( h > H/2) SAMPLE PROBLEM 1.) An open cylindrical tank 2 m in diameter and 4 m high contains water to a depth of 3 m. It is rotated about its own vertical axis with a constant angular speed w. a) If w = 3 rad/s, is there any liquid spilled? b) What maximum value of w (in rpm) can be imposed without spilling any liquid? c) If w = 8 rad/s, how much water is spilled out and to what depth will the water stand when brought to rest? d) What angular speed w (in rpm) will just zero the depth of water at the center of the tank? e) If w = 100 rpm, how much area at the bottom of the tank is uncovered? ROTATING VESSELS LIQUID SURFACE CONDITIONS For closed cylindrical containers more than half-full of liquid, rotated about its vertical axis ( h > H/2) ROTATING VESSELS LIQUID SURFACE CONDITIONS For closed cylindrical containers more than half-full of liquid, rotated about its vertical axis ( h > H/2) ROTATING VESSELS LIQUID SURFACE CONDITIONS For closed cylindrical containers more than half-full of liquid, rotated about its vertical axis ( h > H/2) ROTATING VESSELS LIQUID SURFACE CONDITIONS For closed cylindrical containers completely filled with liquid: SAMPLE PROBLEM 1.) A closed cylindrical vessel, 2m in diameter and 4 m high is filled with water to a depth of 3 m and rotated about its own vertical axis at a constant angular speed, w. The air inside the vessel is under a pressure of 120 kPa. a) If w = 12 rad/sec, what is the pressure at the center and circumference at the bottom of the tank. b) What angular speed w will just zero the depth of water at the center? c) If w =20 rad/sec, how much area at the bottom is uncovered. 2.) A 1.90 m diameter closed cylinder, 2.75 m high is completely filled with oil having sp. gr. of 0.8 under a pressure of 5 kg/cm^2 at the top. a) What angular speed can be imposed on the cylinder so that the maximum pressure at the bottom of the tank is 14 kg/cm^2? b) Compute the pressure force exerted by oil on the side of the tank in kg.
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