# Chapter 3 Hydrostatics - forces exerted by water bodies

When water is motionless its weight exerts a pressure on surfaces with which it is in contact. The force is function of the density of the fluid and the depth.

## 3.1 Pressure and force

A consideration of all of the forces acting on a particle in a fluid in equilibrium produces Equation (3.1). $$$\frac{dp}{dz}=-{\gamma} \tag{3.1}$$$ where $$p$$ is pressure ($$p=F/A$$), $$z$$ is height measured upward from a datum, and $${\gamma}$$ is the specific weight of the fluid ($$\gamma={\rho}g$$). Rewritten using depth (downward from the water surface), $$h$$, produces Equation (3.2). $$$h=\frac{p}{\gamma} \tag{3.2}$$$

Example 3.1 Find the force on the bottom of a 0.4 m diameter barrel filled with (20 $$^\circ$$C) water for barrel heights from 0.5 m to 1.5 m.

area <- pi/4*0.4^2
gamma <- hydraulics::specwt(T = 20, units = 'SI')
heights <- seq(from=0.5, to=1.5, by=0.05)
pressures <-  gamma * heights
forces <- pressures * area
plot(forces,heights, xlab="Total force on barrel bottom, N", ylab="Depth of water, m", type="l")
grid()

The linear relationship is what was expected.

## 3.2 Force on a plane area

For a submerged flat surface, the magnitude of the hydrostatic force can be found using Equation (3.3). $$$F={\gamma}y_c\sin{\theta}A={\gamma}h_cA \tag{3.3}$$$

The force is located as defined by Equation (3.4). $$$y_p=y_c+\frac{I_c}{y_cA} \tag{3.4}$$$

The variables correspond to the definitions in Figure 3.3. The location of the centroid and the moment of inertia, $$I_c$$ for some common shapes are shown in Figure 3.4 . The variables correspond to the definitions in Figure 3.4.

Example 3.2 A 6 m long hinged gate with a width of 1 m (into the paper) is at an angle of 60o and is held in place by a horizontal cable. Plot the tension in the cable, $$T$$, as the water depth, $$h$$, varies from 0.1 to 4 m in depth. Ignore the weight of the gate.

The surface area of the gate that is wetted is $$A=L{\cdot}w=\frac{h{\cdot}w}{\sin(60)}$$. The wetted area is rectangular, so $$h_c=\frac{h}{2}$$. The magnitude of the force uses (3.3): $F={\gamma}h_cA={\gamma}\frac{h}{2}\frac{h{\cdot}w}{\sin(60)}$

The distance along the plane from the water surface to the centroid of the wetted area is $$y_c=\frac{1}{2}\frac{h}{\sin(60)}$$. The moment of inertia for the rectangular wetted area is $$I_c=\frac{1}{12}w\left(\frac{h}{\sin(60)}\right)^3$$.

Taking moments about the hinge at the bottom of the gate yields $$T{\cdot}6\sin(60)-F{\cdot}\left(\frac{h}{\sin(60)}-y_p\right)=0$$ or $$T=\frac{F}{6\cdot\sin(60)}\left(\frac{h}{\sin(60)}-y_p\right)$$

These equations can be used in R to create the desired plot.

gate_length <- 6.0
w <- 1.0
theta <- 60*pi/180  #convert angle to radians
h <- seq(from=0.1, to=4.1, by=0.25)
gamma <- hydraulics::specwt(T = 20, units = 'SI')
area <- h*w/sin(theta)
hc <- h/2
Force <- gamma*hc*area
yc <- (1/2)*h/(sin(theta))
Ic <- (1/12)*w*(h/sin(theta))^3
yp <- yc + (Ic/(yc*area))
Tension <- Force/(gate_length*sin(theta)) * (h/sin(theta) - yp)
plot(Tension,h, xlab="Cable tension, N", ylab="Depth of water, m", type="l")
grid()

## 3.3 Forces on curved surfaces

For forces on curved surfaces, the procedure is often to calculate the vertical, $$F_V$$, and horizontal, $$F_H$$, hydrostatic forces separately. $$F_H$$ is simpler, since it is the horizontal force on a (plane) vertical projection of the submerged surface, so the methods of Section 3.2 apply.

The vertical component, $$F_V$$, for a submerged surface with water above has a magnitude of the weight of the water above it, which acts through the center of volume. For a curved surface with water below it the magnitude of $$F_V$$ is the volume of the ‘mising’ water that would be above it, and the force acts upward.

A classic example of a curved surface in civil engineering hydraulics is a radial (or Tainter) gate, as in Figure 3.7.

To simplify the geometry, a problem is presented in Example 3.3 where the gate meets the base at a horizontal angle.

Example 3.3 A radial gate with radius R=6 m and a width of 1 m (into the paper) controls water. Find the horizontal and vertical hydrostatic forces for depths, $$h$$, from 0 to 6 m.

The horizontal hydrostatic force is that acting on a rectangle of height $$h$$ and width $$w$$: $F_H=\frac{1}{2}{\gamma}h^2w$ which acts at a height of $$y_c=\frac{h}{3}$$ from the bottom of the gate.

The vertical component has a magnitude equal to the weight of the ‘missing’ water indicated on the sketch. The calculation of its volume requires the area of a circular sector minus the area of a triangle above it. The angle, $$\theta$$ is found using geometry to be $${\theta}=cos^{-1}\left(\frac{R-h}{R}\right)$$. Using the equations for areas of these two components as in Figure 3.4, the following is obtained: $F_V={\gamma}w\left(\frac{R^2\theta}{2}-\frac{R-h}{2}R\sin{\theta}\right)$

The line of action of $$F_V$$ can be determined by combining the components for centroids of the composite shapes, again following Figure 3.4. Because the line of action of the resultant force on a curcular gate must pass through the center of the circle (since hydrostatic forces always act normal to the gate), the moments about the hinge of $$F_H$$ and $$F_V$$ must equal zero. $\sum{M}_{hinge}=0=F_H\left(R-h/3\right)-F_V{\cdot}x_c$ This produces the equation: $x_c=\frac{F_H\left(R-h/3\right)}{F_V}$ These equations can be solved in many ways, such as the following.

R <- units::set_units(6.0, m)
w <- units::set_units(1.0, m)
gamma <- hydraulics::specwt(T = 20, units = 'SI', ret_units = TRUE)
h <- units::set_units(seq(from=0, to=6, by=1), m)
#angle in radians throughout, units not needed
theta <- units::drop_units(acos((R-h)/R))
area <- h*w/sin(theta)
Fh <- (1/2)*gamma*h^2*w
yc <- h/3
Fv <- gamma*w*((R^2*theta)/2 - ((R-h)/2) * R*sin(theta))
xc <- Fh*(R-h/3)/Fv
Ftotal <- sqrt(Fh^2+Fv^2)
tibble::tibble(h=h, Fh=Fh, yc=yc, Fv=Fv, xc=xc, Ftotal=Ftotal)
#> # A tibble: 7 × 6
#>     h      Fh    yc      Fv     xc  Ftotal
#>   [m]     [N]   [m]     [N]    [m]     [N]
#> 1   0      0  0          0  NaN         0
#> 2   1   4896. 0.333  22041.   1.26  22578.
#> 3   2  19585. 0.667  60665.   1.72  63748.
#> 4   3  44067. 1     108261.   2.04 116886.
#> 5   4  78341. 1.33  161583.   2.26 179573.
#> 6   5 122408. 1.67  218398.   2.43 250363.
#> 7   6 176268. 2     276881.   2.55 328228.