Circular arch 40 m: sketch, equation, forces at 10 m

Data: three-hinged circular arch, span $L = 40$ m, rise $f = 5$ m. Radius $R = \dfrac{L^2/4+f^2}{2f} = \dfrac{400+25}{10} = 42.5$ m; the centre lies $R - f = 37.5$ m below the springing line, at mid-span.

Sketch and equation of the arch

              C (crown hinge)
            .-'''-.            rise f = 5 m
        .-'         '-.
     .-'                 '-.
   A o                       o B
   |<---------- 40 m ------>|
   UDL 4 kN/m on A-C      60 kN at 8 m from B

With the origin at $A$ and $x$ measured along the springing line, the circle with centre $(20, -37.5)$ and radius $42.5$ gives

$$y = \sqrt{42.5^2 - (x-20)^2} - 37.5$$

Check: $y(0) = \sqrt{1806.25-400} - 37.5 = 0$ and $y(20) = 42.5 - 37.5 = 5$ m.

Loads: UDL 4 kN/m on the left half ($x = 0$ to 20 m); 60 kN at 8 m from $B$ ($x = 32$ m). Section at $x = 10$ m.

Step 1: Slope at the section

$$\tan\theta = \frac{dy}{dx} = \frac{20-x}{\sqrt{R^2-(x-20)^2}}$$

At the section $x = 10$ m: $y = 3.807$ m, $\tan\theta = 0.2421$, so $\theta = 13.61^\circ$, $\sin\theta = 0.2353$, $\cos\theta = 0.9719$.

Step 2: Vertical reactions

Taking moments about $A$ for the whole arch:

$$R_B \times 40 = (4\times 20)\times 10 + 60\times 32 = 2720 \Rightarrow R_B = 68\ \text{kN}$$ $$R_A = 140 - 68 = 72\ \text{kN}$$

Step 3: Horizontal thrust

The bending moment at the crown hinge $C$ is zero. Taking moments of the forces on the left of $C$ about $C$ (rise $f = 5$ m):

$$H \times 5 = 72\times 20 - (4\times 20)\times 10 = 640 \Rightarrow H = 128\ \text{kN}$$

Step 4: Forces at the section (x = 10 m)

Simple-beam values at the section: $M_b = 520$ kN m, $V_b = 32$ kN.

$$\begin{aligned} M &= M_b - H\,y = 520 - 128\times 3.807 = 32.73\ \text{kN m}\\ V &= V_b\cos\theta - H\sin\theta = 32\times 0.9719 - 128\times 0.2353 = 0.98\ \text{kN}\\ N &= V_b\sin\theta + H\cos\theta = 32\times 0.2353 + 128\times 0.9719 = 131.94\ \text{kN} \end{aligned}$$

Answer: at 10 m from the left support: bending moment $= 32.73$ kN m, radial shear $= 0.98$ kN, normal thrust $= 131.94$ kN.