Water exerts pressure on the upstream face of a dam as shown

Water exerts pressure on the upstream face of a dam as shown in Fig. P19.9. The pressure can be characterized by p(z) = rho g(D - z) where p(z) = pressure in pascals (or N/m^2) exerted at an elevation z meters above the reservoir bottom; p = density of water, which for this problem is assumed to be a constant 10^3; kg/m^3; g = acceleration due to gravity (9.81 m/s^2); and D = elevation (in m) of the water surface above the reservoir bottom. According to Eq. (P19.9), pressure increases linearly with depth, as depicted in Fig. P19.9a. Omitting atmospheric pressure (because it works against both sides of the dam face and essentially cancels out), the total force f(can be determined by multiplying pressure times the area of the dam face (as shown in Fig. P19.9b). Because both pressure and area vary with elevation, the total force is obtained by evaluating f_c = integral_0^D rho g w(z) (D z) dz where w(z) = width of the dam face (m) at elevation z(Fig. P19.9b). The line of action can also be obtained by evaluating d = integral_0^D rho gzw(z) (D - z)dz/integral_0^D rho gw (z) (D - z) dz Use trapezoidal rule to compute f_t and d.

Solution

The pressure can be characterized by

p(z)=pg(D-z)

where p(z) is pressure in pascals exerted an elivation z meters above the reservoir bottom and p=density of water which for this problem is assumed to be constant 10^3 k/m^3 , g=acceleration due to gravity 9.8 m/s^2 D= elivation (in m)of the water surface above the reservior bottom.Accordin to the above equation the pressure increases linearly with depth as depicited in the given figure.omitting atmospheric pressure because it works against the both sides of the dam face and essentially cancels out where as the total force f can be determined by multiplying pressure times the area of dam face because both pressure and area vary with elivation the total force obtained by evaluting

f(t)= integration 0 to D of pgw(z)(D-z)dz

where w(z)=width of dam face(m) at elivation z and the line of action obtained by

d=integration 0 to D of pgzw(z)(D-z)dz/integration 0 to D of pgw(z)(D-z)dz