Determinacy and Stability on the FE Civil Exam
For those taking the FE Civil exam, one of the structural topics one should be familiar with is determinacy and stability. This blog post will provide a brief overview of these concepts, review how basic structures can be evaluated for determinacy, and briefly discuss some of the basic methods for solving the reactions of indeterminate structures.
1. Statically Determinate and Statically Indeterminate Structural Systems
In structural analysis, there are considered to be two different types of stable structural systems: statically determinate and statically indeterminate. A stable structure is one whose forces are in equilibrium (as opposed, for example, to one which is in motion). Those stable structures which are determinate can be solved by statics using the three familiar equations of equilibrium. Namely, these are: The sum of forces in the vertical (or y-axis) direction is equal to zero; the sum of forces in the horizontal (or x-axis) direction is equal to zero; and the sum of moments is equal to zero. Again, such structural problems which can be solved using only these equations are known as statically determinate structures. Statically indeterminate structures, by contrast, are those which cannot be solved by these equations alone. These are also sometimes referred to as redundant structures.
The simple method by which one can determine whether a structure is statically determinate or indeterminate (and if indeterminate, by what degree) is the following: Determine the number of support reactions to solve for, and then compare them to the number of static equilibrium equations, which is three. If the number of support reactions is less than or equal to three, then it is a statically determinate structure. If it contains more than three support reactions to solve for, then it is statically indeterminate. The degree to which a structure is indeterminate is equal to the support reactions minus three.
For example, if a simple beam is supported on one end by a pin support and on the other end by a roller support, the structure can be evaluated for determinacy as follows. The pin support, since it prevents translation in the horizontal and vertical directions (but allows for rotation) has two unknown reaction forces, that of the x-direction and the y-direction. The roller support, since it prevents translation in the vertical direction (but allows for translation in the horizontal direction and for rotation), has one unknown reaction force, that of the y-direction. With a total of three unknown support reactions, which is equal to the number of static equilibrium equations, the structure is statically determinate, and (whatever the loading conditions on the beam) the reactions can consequently be solved for utilizing those three equations alone.
If, however, one were to take the example of a beam which is supported with two pins and a roller, it would be evaluated as follows. Each of the two pin supports would have two unknowns (again, the vertical and horizontal reaction force components), and the roller would have one (again, the vertical reaction force). The total number of the support reaction forces, therefore, is five. As this is in excess of the number of static equilibrium equations by two, the system is statically indeterminate by a degree of two.
Taking one further example - a cantilevered beam with a fixed support connection at one end and a roller at the center, it can be evaluated for determinacy as follows. The fixed support, as it prevents translation in the vertical and horizontal directions and, additionally, as it prevents rotation at the support, is found to have three unknown reaction forces (x-direction, y-direction, and moment about the connection point). Adding these three unknowns to the one unknown of the roller support yields a total of four, and thus the structure is found to be indeterminate to the degree of one.
It should be noted that the above examples relate to what is known as external determinacy (or indeterminacy, as the case may be), but it is also possible to have structures which are internally indeterminate (even if they are externally determinate). One example of an internally indeterminate structure is that of a truss with an excess of members such that the forces within them cannot be calculated using the static equilibrium equations alone.
In evaluating a truss (within a two-dimensional plan) for internal determinacy, the following method can be used. Count the number of truss members and add this number to the number of support reactions. If this number is greater than the number of joints in the truss multiplied by two, then the structure is indeterminate. If this is the case, the degree to which it is indeterminate would be the difference then between the two numbers.
In terms of stability, while statically determinate structures are stable, a failure at any restraint of any of the supports results in instability of the system. By contrast, indeterminate structures, by virtue of the fact that there are more restraints than necessary for stability, are considered to be redundant systems. That is to say, the overall structure has greater potential to remain stable if a local failure were to occur within the system.
4. General Approaches to Problem-Solving
While exam-takers of the FE exam are unlikely to need to solve for reactions in a full analysis problem of an indeterminate structure, it is worth having knowledge of the general approaches to solving for these types of problems. These include the general force (or unit load) method, Castigliano's method, the moment distribution method, and the slope deflection method. For indeterminate building frames, there are still other methods, including the portal method and the cantilever method. As the general force method is one of the more common basic approaches, a brief overview of the method is provided below.
5. General Force Method
In the general force method, a loaded indeterminate structure is examined such that all support reaction components are identified. Then the structure is reimagined as one in which all redundant support components are removed so that a determinate stable structure remains (that is to say, one with only three support reaction components supporting the structure). This imagined version of the structure is sometimes referred to as the "primary structure" for the method. The primary structure is analyzed under the given loading conditions to determine the displacements which occur at the locations of the redundant support components of the original structure. The next step is to remove the original loading conditions and analyze a "secondary structure" which corresponds to the primary structure along with a unit force load applied at the location of the redundant support of the original structure. The displacement which occurs for this secondary structure is then solved for under this unit force load. This is done individually for each of the structure's redundant support restraints. Finally, through setting up equations in which the sums of the displacements are set to zero, the unknown reaction components of the structure's redundant supports can be solved for.
In summary, exam-takers of the FE Civil exam should be familiar with the concepts of determinacy and stability and be able to determine whether a particular two-dimensional structure is statically determinate or indeterminate and, if indeterminate, by what degree. The above examples describe the approach. While support reactions of determinate structures can be solved for using only the static equations of equilibrium, solving for indeterminate structures is a more involved process. There are a number of methods which have been developed for solving problems involving indeterminate structures. While exam-takers will not be calculating reactions for indeterminate structures on the exam, a general understanding of the most common methods, such as the general force method, aids in an understanding of the general approach to solving for indeterminate structures.
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