# A general condition for the existence of unconnected equilibria for symmetric arches

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Abstract: This paper presents a semi-analytical study of unconnected equilibrium states for symmetric curved beams. Using the Fourier series approximation, a general condition for the existence of unconnected equilibria for symmetric shallow arches is derived for the first time. With this derived condition, we can directly determine whether or not a shallow arch with specific initial configuration and external load has remote unconnected equilibria. These unconnected equilibria cannot be obtained in experiments or nonlinear finite element simulations without performing a proper perturbation first. The derived general condition is then applied to curved beams with different initial shapes and external loads. It is found that initially symmetric parabolic arches under a uniformly distributed vertical force can have multiple groups of unconnected equilibria, depending on the initial rise of the structure. However, small symmetric geometric deviations are required for parabolic arches under a central point load, and half-sine arches under a central point load or a uniformly distributed load to have unconnected equilibria. The validity of the analytical derivations of the nonlinear equilibrium solutions and the general condition for the existence of unconnected equilibria are verified by nonlinear finite element methods.

## Summary (2 min read)

### 1. Introduction

- Curved beams have been studied extensively due to their rich nonlinear structural behavior and broad applications in aerospace, civil and mechanical engineering.
- Therefore, only determining snap-through buckling loads is not sufficient and it is also necessary to gain insights into the post-snap responses.
- In the current paper, the authors find that initially symmetric arches may also have unconnected “hidden” equilibria, which is not as initially expected.

### 2.2. Existence of unconnected equilibria

- Here the superscript ′ denotes the partial derivative with respect to p. A(p), B(p) and C(p) are defined in Eqs. (7).
- Therefore, a general condition that unconnected equilibria starts occurring for symmetric shallow arches can be obtained by solving Eq. (12).

### 3.1.1. Perfect parabolic arches

- In Fig. 2, solid and dashed lines represent the primary and bifurcated equilibria obtained from the analytical formula respectively.
- In order to obtain remote unconnected equilibria in FEA, the structure is first perturbed to a remote unconnected equilibrium configuration using the information from the analytical solutions.
- When the rise of the arch h = 6.95 is slightly larger than the first critical rise h = 6.91, one group of unconnected equilibria appears, and on this path two critical points (zero tangent, singular stiffness) are present (gray curves in Fig. 2b), separating equilibria with different number of negative eigenvalues.
- For these higher dimension responses, the higher modes corresponding to the unstable equilibria of the system become quite relevant.

### 3.1.2. Half-sine arches with symmetric geometric deviations

- The derived formulas are applied to half-sine arches with symmetric geometric deviations and under a uniformly distributed load.
- Fig. 7a and 7b show the boundaries separating the cases that have and do not have unconnected equilibria for half-sine arches with symmetric geometric deviations in e3h sin(3ξ) and e5h sin(5ξ) respectively.
- The growing rate of the geometric deviations coefficients tends to become smaller as the rise h increases and the splitting boundaries eventually flatten out.
- When the deviation coefficients are slightly above the splitting boundaries (e3 = 0.0345, cross marker in Fig. 7a, or e5 = 0.0048, cross marker in Fig. 7b), remote unconnected equilibrium states appear for these arches (Fig. 8b and 9b respectively).
- The influence of symmetric geometric deviations on the appearance of unconnected equilibria is however similar to that of asymmetric geometric imperfections [42].

### 3.2. Concentrated load at the midspan

- From Eq. (7) and (12), the authors can tell that both initial shapes and external loading types can influence the appearance of unconnected equilibria.
- The authors focus on shallow arches subjected to another type of external load, vertical concentrated load applied at the midspan.

### 3.2.1. Parabolic arches with symmetric geometric deviations

- Unlike arches subjected to uniformly distributed load, perfect parabolic arches under a point load at the midspan do not have unconnected equilibria.
- Small symmetric geometric deviations can lead to the appearance of unconnected equilibrium states.
- Other initial mode coefficients can be found from Eq. (16).
- Fig. 11a and 11b illustrate the splitting boundaries for parabolic arches with symmetric geometric deviations in e3h sin(3ξ) and e5h sin(5ξ) respectively when they are subjected to a central point load.

### 3.2.2. Half-sine arches with symmetric geometric deviations

- Similar to perfect half-sine arches under uniformly distributed load, the perfect half-sine arches subjected to a point load at the midspan also do not have unconnected equilibria.
- Similar to the previous section, geometric deviations in e3h sin(3ξ) and e5h sin 5ξ are considered, where h is the non-dimensional initial rise of a perfect half-sine arch.
- Fig. 15 and 16 show the equilibria of several representative half-sine arches with geometric deviations that are within and beyond the splitting boundaries.
- Fig. 15a and 16a illustrate the equilibria of arches corresponding to the circle markers in Fig. 14a and 14b, while Fig. 15b and 16b describe the equilibria of arches corresponding to the cross markers in Fig. 14a and 14b.
- As expected, no unconnected equilibria exist in Fig. 15a and 16a and unconnected equilibria appear in Fig. 15b and 16b.

### 4. Conclusions

- A general condition for the existence of unconnected equilibria for initially symmetric arches is derived by utilizing the Fourier series.
- Under displacement or arc-length control, these unconnected equilibrium states can be obtained in experiments or nonlinear finite element simulations if the system is perturbed to one of the remote unconnected equilibrium configuration under the guidance of analytical solutions.
- The existence of unconnected equilibria is sensitive to the initial shapes and external load types that shallow arches have.
- For perfect parabolic arches under a point load and perfect half-sin arches under a point load or a uniformly distributed load, when certain symmetric geometric deviations for instance e3h sin(3ξ) or e5h sin(5ξ) are added to the structures, unconnected equilibrium states can appear.
- The absolute value of the geometric deviation coefficient e3 or e5 on the splitting boundaries tends to grow and eventually stop varying when the rise of the perfect shaped arch increases.

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