---
title: "Compatibility equation for two-hinged arch"
subject: "Theory of Structures II"
chapter: 2
key: "semester_5/theory_of_structures_ii:arch-compatibility-equation"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/arch-compatibility-equation
site: BCT Engineering
---

# Compatibility equation for two-hinged arch

Theory of Structures II · Chapter 2: Force method

- Asked in: 2076 Asoj, 6 marks
- Video: [Two Hinged Arch Numerical | Theory of structure 2 | 2069 TU | 2018 PU | Horizontal Reaction](https://www.youtube.com/watch?v=QwBVb5R7igE) — Sudip Dhakal
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/arch-compatibility-equation
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-2

## Question

Write down the compatibility equation for a two hinged parabolic arch due to external loads, variation in temperature, rib shortening and yielding of supports.

## Short answer

The compatibility equation for a two-hinged parabolic arch is H = (∫(M0 y)/EI ds + αtL – δ) / (∫ y²/EI ds + ∫ cos²θ/AE ds), where the numerator includes external load, temperature, and support yield effects, and the denominator includes bending and rib shortening contributions.

## Answer

A two-hinged arch is indeterminate to the first degree. Take the horizontal thrust $H$ as the redundant, release the hinge at B (roller), and write that the net horizontal movement of B is equal to the yield of the support.

```text
          C
       .-----.
     /         \
   A o           o B  <-- H (redundant)
   |<----- L ----->|
```

#### Displacement of B in the released structure (outward positive)
1. **External loads:** $\Delta_{10} = \displaystyle\int \frac{M_0\,y}{EI}\,ds$ ($M_0$ = BM of the equivalent simple beam, $y$ = ordinate of the arch axis). The axial-force term is small and is usually neglected.
2. **Temperature rise $t$:** $\Delta_{1t} = \alpha\,t\,L$ (a fall in temperature gives a negative value).
3. **Redundant $H$ (unit horizontal force at B gives $m = -y$ and $N = -\cos\theta$):**

$$
\Delta_{1H} = -H\,f_{11}, \qquad f_{11} = \int \frac{y^2}{EI}\,ds + \int \frac{\cos^2\theta}{AE}\,ds
$$

The first term of $f_{11}$ is bending, and the second is the **rib shortening** due to the axial thrust $H\cos\theta$.

4. **Yielding of support:** B moves outward by $\delta$ (positive outward).

#### Compatibility equation
The final horizontal movement of B must equal the support yield $\delta$:

$$
\Delta_{10} + \alpha\,t\,L - H\,f_{11} = \delta
$$

$$
\boxed{H = \frac{\displaystyle\int \frac{M_0\,y}{EI}\,ds + \alpha\,t\,L - \delta}{\displaystyle\int \frac{y^2}{EI}\,ds + \int \frac{\cos^2\theta}{AE}\,ds}}
$$

#### Remarks
- For $I = I_c\sec\theta$ the bending integrals become $\dfrac{1}{EI_c}\int M_0\,y\,dx$ and $\dfrac{1}{EI_c}\int y^2\,dx$, and with $A = A_c\sec\theta$ the rib shortening term is $\dfrac{1}{EA_c}\int\cos^2\theta\,dx$.
- A temperature rise or an inward movement of B increases $H$. An outward yield and rib shortening reduce $H$ (rib shortening acts like an elastic spring in the denominator).
- After $H$ is found, the moment at any section is $M = M_0 - Hy$, the normal thrust is $N = H\cos\theta + V_0\sin\theta$ and the radial shear is $V_r = V_0\cos\theta - H\sin\theta$.

## Check yourself

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

1. For a two-hinged arch, which quantity is taken as the redundant?
   A. Vertical reaction at A
   B. Horizontal thrust H
   C. Bending moment at the crown
   D. Radial shear at the support
   Answer: B. A two-hinged arch is indeterminate to the first degree; releasing one hinge to a roller and treating the horizontal thrust H as the redundant gives the primary structure. (Key point: Redundant choice)

2. What does the compatibility equation state?
   A. The net horizontal movement of B is zero in all cases
   B. Sum of vertical forces equals zero
   C. The net horizontal movement of B equals the support yield δ
   D. Thrust H equals the rib shortening
   Answer: C. Loads, temperature and H together move B by an amount that must equal the actual support yield, giving Δ10 + αtL - H f11 = δ. (Key point: Compatibility condition)

3. In f11, what does the term ∫cos²θ/(AE) ds represent?
   A. Rib shortening due to axial thrust
   B. Shear deformation of the rib
   C. Temperature expansion
   D. Support settlement
   Answer: A. The axial thrust H cosθ shortens the rib, which adds a spring-like term to the denominator. (Key point: Rib shortening term)

4. How do a temperature rise and an outward support yield affect H?
   A. Both increase H
   B. Both reduce H
   C. Rise reduces H, outward yield increases H
   D. Rise increases H, outward yield reduces H
   Answer: D. H = (Δ10 + αtL - δ)/f11, so the temperature term adds to H while the outward yield δ subtracts from it. (Key point: Effect of temperature and yield)

## Related in Theory of Structures II

- [Two-hinged arch: thrust with rib shortening, support yield](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aarch-60m-6m-rib-shortening) — Theory of Structures II, chapter 2
- [Two-hinged parabolic arch: horizontal reaction, symmetric loads](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aarch-secant-symmetric-loads-80m) — Theory of Structures II, chapter 2
- [Compatibility condition; portal frame temperature variation](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acompatibility-condition-portal-temperature) — Theory of Structures II, chapter 2
- [Two-hinged parabolic arch: BM at D](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aarch-bm-point-d-30-50) — Theory of Structures II, chapter 2
- [Two-hinged arch: BM, thrust, radial shear, support yield](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aarch-bm-thrust-shear-yield-support) — Theory of Structures II, chapter 2
- [Two-hinged arch: horizontal reaction, UDL on half](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aarch-12knm-horizontal-reaction-50m-8m) — Theory of Structures II, chapter 2

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