Note on Stability of Structures

COLORADO SCHOOL OF MINES DIVISION OF ENGINEERING EGGN 441 / EGES 541 ADVANCED STRUCTURAL ANALYSIS A NOTE ON STABILITY OF STRUCTURES By Professor Panos D. Kiousis INTRODUCTION

The stability of a structural system should always be examined before the internal forces are calculated. To make the analysis of stability of a structure easier, we often examine internal and external stability separately. When we say that a structure is internally unstable we mean that the structure is not rigid by itself and needs additional external constraints (in the form of additional supports) to carry external loads. An internally unstable structure must be supported by an enhanced reactionary system (i.e. more than three unknown support forces). The structure of Figure 1a is an unstable stable structure in this sense. Additional supports applied to this structure to make it stable are shown in Figure 1b. On the other hand, an internally stable structure can be externally unstable, as shown in figure 1c. In this case the structure it self is formation but it is not supported adequately externally. In our example of Figure 1c the structure cannot carry horizontal loads due to lack of an external support to transfer this load to the foundation. A METHOD TO DETERMINE STABILITY.

We shall start our stability analysis by recognizing the obviously stable structures, such as a rigid beam or a rigid bar (no hinges to interrupt their continuity). The following two simple rules will be used to define stability: A) Two stable formations, connected with three bars, not all of which are parallel and not all passing through the same point, form a stable formation (Keep in mind mind here that a hinge is equivalent to two bars and a roller is equivalent to one bar).

B) Three stable formations connected to each other by two bars (total of six bars with four bars attached to each formation) form a stable structure if the following two restrictions are met: 1) Not all four bars attached to any of the three formations are parallel to each other. 2) Each pair of bars connecting two formations has an intersection point. These define a total of three intersection points that must not be on a straight line.

To illustrate these principles we shall start with one of the simplest stable formations, the triangle (Figure 2a). The triangle ABC is formed by two stable structures AB and BC, which are connected with three bars as follows: Hinge B (two bars) and bar AC. Based on rule A, the triangle is stable formation. Alternatively, the triangle ABC is formed by the three stable structures AB, BC, CA. We note that AB is connected to BC with the hinge B (i.e. two bars). Similarly AB is connected to AC with the hinge A, and finally, BC is connected to AC with the Stability of Structures Notes by Professor P. D. Kiousis

Last Update: Aug 26, 2005 Page 1 of 6

hinge C. Based on rule B, the triangle is a stable formation. Using this result we can prove that the structure ABCD of two triangles in Figure (2b) is also stable: The triangle ABC is a stable formation, and so is the bar BD. These two stable formations are connected with the pin B (two bars) and the bar CD. Based on rule A this is a stable structure. We can extend this logic to prove that any structure formed by consecutive triangles (Figure 2c) is also stable. The proof of stability of a structure is not always as straight forward as the one of the examples in Figure 2. A "feel" for the behavior of the structure must be developed before more difficult problems can be solved. Let us consider the examples of figure 3: Figure 3a: To prove stability of this structure we observe that it consists of the triangle ABC (stable formation), and the triangle DEF (which is also stable). These two stable structures are connected with the bars EA, CD, and BF which are not parallel and do not pass through the same point; Based on rule A, this structure is stable. Figure 3b: To prove the stability of this structure we observe that it consists of the following three stable formations: triangle ABC, bar DE and bar FG. These three stable "substructures" are connected with each other as follows: I) ABC is connected to DE with bars AD and EC. II) ABC is connect to FG with bars AF and BG. Ill) DE and FG are connected with bars FD and EG. Based of rule B this is a stable structure. From this point on, you are advised to try to solve the problems yourselves before you read the solution. Figure 3c: A similar approach to the one of the previous problem is taken here. The structure consists of the bar GH (stable), the square CDFE which is stable since it consists of two triangles, and the bar AB (stable). GH is connected to CDEF with bars GD and FH. Also, GH is connected to AB with bars AG and BH. Finally, CDEF is connected to AB with bars AC and BE. According to rule B, this is a stable structure. Figure 3d: This is a rather simpler problem. Our structure consists of the triangles ABF and CDE which are connected with bars AD, BC, and EF. According to rule A our structure is stable. Figure 3e: We have already seen this problem before (three hinged arch), and we accepted its stability based on our intuition. Here it is how we can prove it based on our rules. We have three stable structures, i.e. AC, BC, and the ground. AC is connected to BC with the hinge C (two bars) and it is also connected to the ground with the hinge A. Finally, BC is connected to the ground with the hinge B. According to rule B, this is a stable structure. Figure 3f. This is an integrated problem of the form of our previous example. We start by realizing that structure ABHCDEFGI is a three hinged arch and therefore it is a stable structure. This means that points H and I can be considered as rigid supports (since they belong to the stable structure ABHCDEFGI). We can now see that HJI is also a three hinged structure that is supported on a stable structure and therefore our structure is stable.

Stability of Structures Notes by Professor P. D. Kiousis


Figure 3g: Our final example is analyzed as follows: First (and maybe misleading) observation: We have three stable structures FG, CE, and AB which are connected with each other with two bars (FG and CE are connected with bars CF and GE; CE and AB are connected with AC and BE; finally, FG and AB are connected with BF and GA). This observation could lead us to conclude that this is a stable structure. Unfortunately, this would not be a correct conclusion, because one of the restrictions of rule B is not satisfied! Note that the points of intersection of AG and BF, CF and GE, and CD and BE all lie on one line. This violates the requirement of rule B that these points MUST NOT all lie at the same line. This structure is unstable.



Figure 1: Examples of stable and unstable structures



Figure 2: Building a stable structure from a stable nucleus.



Figure 3: Examples of complex stable and unstable structures.



Note on Stability of Structures

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