---
title: "Resultant force on curved surface ABCDE (S = 1.1)"
subject: "Fluid Mechanics"
chapter: 3
key: "semester_3/fluid_mechanics:curved-surface-abcde-s11-20m"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/curved-surface-abcde-s11-20m
site: BCT Engineering
---

# Resultant force on curved surface ABCDE (S = 1.1)

Fluid Mechanics · Chapter 3: Hydrostatics

- Asked in: 2073 Shrawan, 8 marks
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/curved-surface-abcde-s11-20m
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-3

## Question

Find the resultant pressure force on curved surface ABCDE due to liquid with specific gravity S = 1.1, take length of the curved surface (normal to the paper) as 20 m. [Figure: free surface at A; surface goes from A down to B (4 m horizontal), curved portions with 2 m marked near B-C and C-D, then straight to E on the ground; 4 m vertical and 4 m horizontal dimensions at the bottom]

## Short answer

With γ = 10.79 kN/m³, horizontal force on 6 m vertical projection is 3884.8 kN; vertical force (water weight above) is 7584.3 kN. Resultant 8521 kN at 62.9° below horizontal.

## Answer

The horizontal component of the force equals the force on the vertical projection of the surface; the vertical component equals the weight of liquid vertically above the surface up to the free surface.

**Assumed geometry** (the figure is not fully clear): A is at the free surface, AB is a vertical wall 4 m high, BC is a quarter circle of radius 2 m (concave upward), and CDE is a horizontal bed 4 m long at 6 m depth. $S = 1.1$, $L = 20$ m, $\gamma = 9.81\times 1.1 = 10.79$ kN/m³.

```text
 A |~~~~~~~~~~~~~~~~~~~~~~  free surface
   |
   | 4 m
 B |
    \   R = 2 m
     '-.
 C ------ D ------ E   (depth 6 m)
   |<---- 4 m ---->|
```

**Horizontal component** (vertical projection = 6 m, free surface to the bed):

$$
\begin{aligned}
F_H &= \gamma\,\frac{h^2}{2}\,L = 10.79\times\frac{6^2}{2}\times 20\\
&= 3884.8\ \text{kN}
\end{aligned}
$$

**Vertical component** (area of liquid above the surface, per metre length):

- Above arc BC: the depth of the arc below the free surface is $4 + \sqrt{R^2-(x-2)^2}$, so the area is a $2\times 4$ rectangle plus the quarter disc of area $\pi R^2/4 = \pi$: $8 + \pi = 11.14$ m²
- Above bed CE: $4\times 6 = 24$ m²

$$
\begin{aligned}
A_{total} &= 8 + \pi + 24 = 35.14\ \text{m}^2\\
F_V &= \gamma A_{total} L = 10.79\times 35.14\times 20 = 7584.3\ \text{kN}
\end{aligned}
$$

**Resultant**

$$
\begin{aligned}
F &= \sqrt{3884.8^2 + 7584.3^2} = 8521.3\ \text{kN}\\
\theta &= \tan^{-1}\frac{7584.3}{3884.8} = 62.9^\circ \text{ below the horizontal}
\end{aligned}
$$

**Answer: $F \approx 8521$ kN (8.52 MN), at $62.9^\circ$ to the horizontal; $F_H = 3884.8$ kN, $F_V = 7584.3$ kN (downward).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which depth is used for F_H?
   A. 4 m (wall only)
   B. The vertical projection, 6 m (free surface to bed)
   C. 2 m (arc radius)
   D. 10 m (total length)
   Answer: B. F_H = γ h^2/2 L = 10.79 x 18 x 20 = 3885 kN. (Key point: Horizontal component)

2. What is γ for S = 1.1?
   A. 10.79 kN/m^3
   B. 9.81 kN/m^3
   C. 11.0 kN/m^3
   D. 8.92 kN/m^3
   Answer: A. γ = 1.1 x 9.81 = 10.79 kN/m^3. (Key point: Specific weight)

3. What area gives F_V per metre length?
   A. Only the quarter disc, π m^2
   B. 6 x 4 = 24 m^2 only
   C. Liquid above the surface: 8 + π (above arc BC) + 24 (above bed) = 35.14 m^2
   D. 8 + 24 = 32 m^2
   Answer: C. F_V = weight of liquid vertically above the surface up to the free surface, here 10.79 x 35.14 x 20 = 7584 kN. (Key point: Vertical component)

4. What is the resultant force and direction?
   A. About 3885 kN horizontal
   B. About 11,469 kN
   C. About 8521 kN at 27.1° to the horizontal
   D. About 8521 kN at 62.9° to the horizontal
   Answer: D. F = √(3884.8^2 + 7584.3^2); θ = tan^-1(F_V/F_H) = 62.9°. (Key point: Resultant)

## Related in Fluid Mechanics

- [Resultant force on curved surface ABCDE (S = 1.25)](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Acurved-surface-abcde-s125-10m) — Fluid Mechanics, chapter 3
- [Resultant force on curved surface BCDEF](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Acurved-surface-bcdef-force) — Fluid Mechanics, chapter 3