With the origin at A and the arch passing through A, the parabola is written y = ax^2 + bx. Why is there no constant term?

Because y = 0 at x = 0 (the curve passes through the origin A)

Because the arch is symmetric about A

Why are two moment conditions (about B for the whole arch and about hinge C for the left part) solved together?

Moments about A and B alone cannot separate V and H when the supports are at different levels, so the hinge condition is needed as well

Because the load is at the section, so one equation is zero

Because the arch is symmetric, so V and H are equal

Because the hinge C carries a vertical reaction

What is the bending moment at the section 15 m from A, and why does the 10 kN load not enter?

M = V_A x 15 - H_A x y = 23.44 kN m; the load acts at the section so its moment arm is zero

M = V_A x 15 + H_A x y = 23.44 kN m; the load is balanced by V_B

M = V_A x 15 - 10 x 15; the horizontal thrust has no effect

M = H_A x y only; the vertical reaction acts through the section

2081 Chaitra · 8 marks

Parabolic arch at different levels: BM at 15 m

Short answer

From equilibrium of the whole arch and the left part about the crown, horizontal thrust is 8 kN, vertical reactions are 7.5 kN (left) and 2.5 kN (right). At the section 15 m from the left support, where the 10 kN load acts, the bending moment is 35.16 kN m (sagging).

A parabolic three hinged arch shown in figure below has a span of 40 m and is supported at different levels as shown. Find bending moment at a section 15 m from left support when a 10-kN vertical load at this section. [Figure: crown hinge, left support A at the lower level with the crown 11.25 m above A, right support B 5 m below the crown level as read from the scan; 10 kN at C, 15 m from A; horizontal span 40 m. Dimensions partly unclear.]

Assumptions (dimensions partly unclear in the figure): horizontal span 40 m; the left support $A$ is 11.25 m and the right support $B$ is 5 m below the crown hinge $C$ , which is the highest point (vertex) of the parabola. So $B$ is $11.25 - 5 = 6.25$ m above $A$ . A 10 kN vertical load acts at the section $D$ , 15 m from $A$ .

Position of the crown

For a parabola with its vertex at $C$ , the depth below the crown is proportional to the square of the horizontal distance from it. If $L_1$ and $L_2$ are the horizontal distances of $A$ and $B$ from $C$ :

\frac{L_1}{L_2} = \sqrt{\frac{11.25}{5}} = 1.5, \qquad L_1 + L_2 = 40 \Rightarrow L_1 = 24\ \text{m},\ L_2 = 16\ \text{m}

Equation of the arch and height of the section

With the origin at $C$ and depth $y'$ measured downwards: $y' = kx^2$ , $k = \dfrac{11.25}{24^2} = \dfrac{5}{16^2} = 0.019531$ .

The section is $24 - 15 = 9$ m left of $C$ : $y' = 0.019531\times 81 = 1.582$ m, so the section is $11.25 - 1.582 = 9.668$ m above $A$ .

Reactions

Let the reaction at $A$ be $V_A$ (up) and $H$ (to the right); at $B$ , $V_B$ (up) and $H$ (to the left). Because the supports are at different levels, the vertical and horizontal components cannot be separated by moments about $A$ and $B$ alone, so both conditions are used together (anticlockwise moments positive).

Moments about $B$ ( $B$ is 40 m right of and 6.25 m above $A$ ; the load is 25 m left of $B$ ):

-40V_A + 6.25H + 10\times 25 = 0 \qquad (1)

Moments about the hinge $C$ for the left part ( $C$ is 24 m right of and 11.25 m above $A$ ; the load is 9 m left of $C$ ):

-24V_A + 11.25H + 10\times 9 = 0 \qquad (2)

From (1): $V_A = \dfrac{6.25H + 250}{40}$ . Substituting in (2): $-3.75H - 150 + 11.25H + 90 = 0 \Rightarrow H = \dfrac{60}{7.5} = 8$ kN.

V_A = \frac{6.25\times 8 + 250}{40} = 7.5\ \text{kN}, \qquad V_B = 10 - 7.5 = 2.5\ \text{kN}, \qquad H = 8\ \text{kN}

Bending moment at the section (x = 15 m)

M = V_A\times 15 - H\times y = 7.5\times 15 - 8\times 9.668 = 112.5 - 77.34 = 35.16\ \text{kN m}

(The 10 kN load acts at the section, so it has no moment about it.)

Answer: bending moment at the section = 35.16 kN m (sagging), with $H = 8$ kN, $V_A = 7.5$ kN and $V_B = 2.5$ kN.

✓ Answer checked by Claude Opus 5.5 (high effort)

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All Theory of Structures I chapter 6 questions →

Parabolic arch at different levels: BM at 15 m

Position of the crown

Equation of the arch and height of the section

Reactions

Bending moment at the section (x = 15 m)

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