A 4-m-long quarter-circular gate of radius 3 m and of negligible weight is hinged about its uppe

A 4-m-long quarter-circular gate of radius 3 m and of...

Key Concepts

Lever Arm

The lever arm is the perpendicular distance from the pivot point to the line of action of a force. In moment calculations, it is a critical parameter because the torque produced by a force is the product of the force magnitude and its lever arm. Accurate identification of lever arms is vital in evaluating the contributions of different forces in a system.

Statics of Fluid-Structure Interaction

This concept integrates principles of fluid statics and structural mechanics by analyzing how fluid pressures affect structural elements. It is essential for solving problems where the effects of fluid loading must be countered by structural components, ensuring that the structure remains in equilibrium under the applied loads.

Center of Pressure

The center of pressure is the point on a submerged surface through which the resultant hydrostatic force acts. It is typically located below the centroid of the area due to the increase of pressure with depth, and its determination is essential for correctly setting up the moment balance in static equilibrium problems.

Hydrostatic Pressure Distribution

This concept refers to the variation of pressure in a fluid at rest, which increases linearly with depth. In problems involving submerged surfaces, understanding how pressure changes with depth is crucial because it determines the magnitude and distribution of the forces exerted by the fluid on the surface.

Moment Equilibrium

Moment equilibrium involves balancing the torques (moments) about a pivot or hinge in a system. In force analysis problems, such as those involving gates or beams, ensuring that the sum of moments is zero is key to determining unknown forces such as spring forces or reaction forces at supports.

Transcript

00:01 So we are given with a gate or quarter cylinder shape and it's set let's be given as a diagram which is at bottom of 4 .5 meters.

00:08 So we have some formula.

00:09 So for horizontal force acting on this gate we can use formula pca where pc is the pressure acting at the centroid of this gate and it's a projected area of the gate and not the actual shape.

00:24 So it is a rectangle width of three meters and height of three meters.

00:30 Since the radius is 3 meters.

00:33 So now we have our area.

00:36 So to calculate pc, we have formula row g, hc, where hc is the height to the centroid.

00:42 So row is 1 ,000, g is 9 .81, and hc is r by 2 plus d minus r.

00:52 So we can substitute the values, 1 .5 plus 4 .5 minus 3.

00:57 On calculating this, we get our answer is 29430.

01:01 Pascal and now horizontal force will be equals to multiplication of this with the area that is 264870 newton so that is 265 kilo newton approximately now to calculate the vertical force we need to just calculate the volume of water above the gate and that can be calculated using the method that is now we have d minus r as the rectangle strip length then the depth is w or width is w now volume will be some of these two volumes w1 and w2 so first volume is w times r times d minus r similarly the second volume which is the part remaining after subtracting the quarter circle from the square so that is r square minus pi r square by four into w so now we can substitute the values d is 4 .5 r is 3 and similarly rest of the values on calculating this we get our answer as 19 .27 so our vertical force is row gv that is thousand times 9 .81 times 19 .27 and that gives us 189234 .9 newton that is equals to 189 kilo newton approximately.

02:26 Now to calculate the force position or the line of action of the force, we have formula y -dashes y -c plus i -xx over a -c.

02:37 So we have already know y -c is nothing but our hc value, which is 3.

02:42 So we can directly substitute that here.

02:44 Plus i xx we know its formula is 1 by 12 l w l cube where l is the length or height so w is 3 that is the depth or width and l is also 3 so substituting all other values we can calculate this as 3 .25 so this is our y -dash now for x dash or sorry x -dash we need to find the centroid of the liquid above the gate so we have two areas basically that is one bigger rectangle and out of that we have to remove the quarter circle.

03:26 So we just take the bigger rectangle as a 1 and smaller quarter circle as a2 and calculate ax bar dash is equal to a1 x1 minus a2 x2 where a1 is the area of first rectangle, a2 is the area of quarter circle.

03:43 X1 is the centroid of a1 and x2 is center of a 2...

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