静的決定のガイド, 不確定性, と不安定性

Once the structure is completely modeled and the loads are applied, 分析段階から得られるすべての結果を見つけるために、システムを解決する時が来ました: 反応, 内力 (剪断力, 曲げモーメント, 軸力, とねじれ), 変位, とストレス. 物事が期待どおりに進まない場合があり、それを解決するとき, you may get an annoying message saying “Solve Failed! Check your structure set up, as the model is probably unstable”, and chances are that you do not know exactly what is wrong with the model. この記事では、非常に関連性の高いトピックである静的決定性と不確定性とともに、構造安定性の概念について説明します。, it also presents some tips for troubleshooting an unstable model.

一般に, a structure is considered internally stable if it holds its shape when supports are removed. 逆に, a structure is considered internally unstable if once it is detached from the supports, the structure cannot maintain its shape and may undergo large displacements or even collapse. とはいえ, 静的決定式を詳しく見ていきます, both for internally unstable and stable structures. The static determinacy formula helps to classify a structure as externally unstable, 静的に決定する, または静的に不確定.

Static determinacy of internally stable structures

An internally stable structure can be taken as statically determinate externally if all its support reactions can be solved by solving the equations of equilibrium. For planar structures under in-plane loads, there need to be three supporting reactions for the structure to be in equilibrium. さらに, since there are only three equilibrium equations available, three reactions can be found at the most, so there should be exactly three reactions supporting the plane structure. Some examples of externally statically determinate planar structures are shown in the following picture:

When more than three reactions are supporting the structure, it is called a statically indeterminate externally structure, since it is not possible to solve all the reactions using the three equations of equilibrium. Those reactions are called external redundant and the number of external redundant is known as the degree of external indeterminacy, written as an equation it looks like this:

\(i_e = r – 3\)

どこ \(r) is the number of reactions that support the structure.

3つ未満の支持反応で構造が支持されている場合, それらは、構造がその平面上で移動するのを防ぐのに十分ではありません, since that structure would not remain in equilibrium under a general system of loads it is called a statically unstable externally. Based on the number of reactions \(r), 静的決定性を決定することが可能です, 不確定性, and instability of internally stable structures using the criteria that follows:

\( rbegin{ASCEの使用} < 3, \; \テキスト{外部的に静的に不安定な構造} \\ = 3, \; \テキスト{外部的に静的に決定する構造} \\ > 3, \; \テキスト{外部的に静的に不確定な構造} \終わり{ASCEの使用}\)

最後の 2 つの条件が満たされると、, 構造が安定するという保証はない. It means that even when the structure is supported by more than three reactions if supports and not arranged properly the structure might still be unstable. This is called geometrical external instability and is caused by two main reasons:

• When all the reactions’ lines of action are parallel because the structure will not be prevented from displacing in the perpendicular direction of the reactions
• When all the reactions’ lines of action are concurrent at one point because the structure will be allowed to rotate about the concurrency point

Static determinacy of internally unstable structures

Internally unstable structures are those that when detached from the supports cannot hold their original shape. サポートなし, 構造のメンバーは、互いに対して大きな変位を受ける. 静的決定性を定義するには, 不確定性, and external stability of these structures, the equations change due to the additional equations of condition that come into play accounting for the special connections between members. 特別な接続は瞬間を送信しないものです, 剪断力, or axial force.

例として, consider the structure in the picture below (top section), it is clearly supported by three reactions, but those are not enough for guaranteeing equilibrium under the applied in-plane forces thanks to the presence of the hinge between the members, it needs a fourth non-parallel, non-concurrent reaction to ensure equilibrium (bottom section).

一般に, for internally unstable structures, based on the number of reactions \(r) and equations of condition \(e_c), 静的決定性を決定することが可能です, 不確定性, and instability of internally stable structures using the criteria that follows:

\( rbegin{ASCEの使用} < 3 + e_c, \; \テキスト{外部的に静的に不安定な構造} \\ = 3 + e_c, \; \テキスト{外部的に静的に決定する構造} \\ > 3 + e_c, \; \テキスト{外部的に静的に不確定な構造} \終わり{ASCEの使用}\)

For indeterminate externally internally unstable structures, the degree of external indeterminacy can be expressed as follows:

\(i_e = r – (3 + e_c)\)

Wrap-up

For planar structures that are internally stable to be externally stable, meaning that it remains in equilibrium under a general system of loads, the following must hold:

• The structure is supported by three or more reactions
• All the reactions cannot be parallel nor concurrent

For planar structures that are internally unstable to be externally stable, meaning that it remains in equilibrium under a general system of loads, the following must hold:

• The number of reactions supporting the structure is greater than or equal to the number of equations of condition plus three
• All the reactions cannot be parallel nor concurrent

Troubleshooting an unstable structure in S3D

There are a few things you can try when the model is not solving, and you are getting the “Unstable structure” エラーメッセージ:

• Select all the members and ensure they have their member end fixities as “FFFFFF”, and try to run the model. If some members need to have a different end fixity in the final model, start changing it one by one and solve the structure to see which is the exact member or set of members causing the instability.
• Select all the members and change their type from normal to continuous, which would properly connect members that do not have a node in the intersection but are meant to be connected.
• Suppress from the analysis all the cables, and try to solve the model. To learn how to suppress members, refer to この記事.

参考文献

Kassimali, Aslam. “3.4 Static Determinacy, 不確定性, and Instability.” 構造解析, センゲージ ラーニング, 2011.