civil engineering-->>Determinacy, Indeterminacy and Stability of Frames

Determinacy, Indeterminacy and Stability of Frames :

Structural engineers must be able to apply judgment rather than stated rules. The most important aspect of structural design is not the ability to apply formulas or manipulate mathematics. The most important skill for the structural engineer is to be able to stand back, look at a drawing or sketch and determine whether a structure is stable, and if it is stable, to be able to determine how it will carry the applied loads. For a very complicated structure this might be more difficult and a computer can provide some help, but ultimately it is the skill and concern of a good structural engineer which produces good structural designs which have integrity. Another important closely related skill is the ability to determine whether, and to what degree, a structure is statically indeterminate.

There really aren’t  many rules and rules may be difficult to apply in any case. For this reason, we need experience. A skilled structural engineer can spot instability very quickly. Students studying structural analysis for the first time take longer.

In the discussion which follows that we will explore the application of Newton’s First Law and look more deeply into the concepts of determinacy, indeterminacy and stability.

                                                                      Fig. 2-2 Simple Beam

For the beam in Fig. 2-2 the equations,

∑Fx=0

∑Fy=0

∑Fn=0

are not independent, and cannot be solved from the 3 reactions. The last equation is automatically satisfied if the first two are. The equation for Fn contains a combination of the force components in the x and y directions. To solve the reactions, a moment equation about one of the support is used. We note further that there is nothing sacred about ‘x’ and ‘y’. The coordinate system is really an artifice created by the analyst. Nor do these axes strictly have to be at right angles, although there are very good practical reasons why we choose them this way. Newton’s Law demands that for equilibrium, the forces in any direction must balance and that there must be balanced moments about any point. In a plane we guarantee that this requirement will be satisfied when 3 independent conditions are satisfied.

                                                          Fig. 2-3 (a) Cantilever; (b) Propped Cantilever

The cantilever also has 3 reaction quantities, and these are all at one location. One reaction is a moment type. Three independent conditions can again be employed to solve for the reactions, and when this is done Newton’s Law is satisfied for the forces in any direction, and the moments about any point in the plane. Note how in the two support systems, simple support and cantilever, the structure has the ability to provide reaction to forces in two directions and a couple, and that the couple in the simply supported system is resisted by 2 forces separated by a distance. By adding another support to the cantilever, i.e. propped cantilever in Fig. 2-3(b), we have one more reaction which is not necessary for equilibrium. We call extra reactions “redundant”.

Redundant Force: In a statically indeterminate structure the forces (often external supports but possibly internal actions) which are not strictly needed for stability (or equilibrium) are called “redundant”. In statically indeterminate structural analysis the redundants are the primary unknowns in the solution. That is, we solve for these first and then use equilibrium to complete the solution.

The term “redundant” should not be taken to have a negative connotation, that is, these force quantities are not important. Redundancy may often be a desirable attribute in structure providing alternate load paths in the event of a member’s failure, or in providing additional stiffness. For all but the simplest of structures redundancy cannot be avoided, and in general all building frames and most concrete structures comprised of more than a single member, have redundant forces. From our discussion of ideal versus real pinned joints in a truss in the previous chapter we can see that even trusses which are designed to be statically indeterminate have some level of redundancy, though this is not considered in their design.

Degree of Indeterminacy: The number of redundant forces over and above that strictly required for stability is the “degree of indeterminacy”. In a sense this may be regarded as the “degree of difficulty” in solving a statically indeterminate problem (by calculator and pencil), since solving for these primary unknowns involves the generation and solution of simultaneous equations.

                                                Fig. 2-4: Beam 2-degrees statically indeterminate

The cantilever with 2 extra supports (Fig. 2-4) is 2 degrees indeterminate.

                                                                    Fig. 2-5: Frames: (a) Determinate; (b) Indeterminate.

The frame shown on the left in Fig. 2-5(a) is stable and statically determinate. Replacing the right hand roller with a hinge Fig. 2-5(b), the frame on the right is statically indeterminate to one degree.

                                                                               Fig. 2-6: Unstable Support System.

A structure supported as in Fig. 2-6 is considered to be unstable, because the 3 reactions all meet at one point (remember the pin-ended members only carry forces in line with the member axis). This structure would not necessarily collapse, since the geometry of the support system will change as the structure moves, but it would tend to shake excessively under moving loads because it does not have any stiffness against rotation at the given geometry. The highly simplified drawing in Fig. 2-6 makes this conclusion fairly obvious, but it might not be so obvious in some of the examples presented at the end of this chapter.
In the following examples, note that each added external support increases the level of indeterminacy by one, while each added internal articulation reduces the level by one.

                                                                        Fig. 2-7: Articulated Beam

The beam in Fig. 2-7 is determinate and stable. It appears to have an extra support but an internal hinge is also included.
The reader may find it convenient to perform this inspection as follows. Ignoring the internal pin in Fig. 2-7, we see a simple supported beam with an extra redundant reaction, giving a structure statically indeterminate to one degree. Subtracting one for the internal release renders the structure is statically determinate. The principle involved is very general and works when there are several additional supports or internal releases. It also works for frames, but one must be careful. For almost every rule we can find an example that breaks it.

                                                               Fig. 2-8: Determinate or Unstable?

In the following example in Fig. 2-8 we apparently have a beam which is statically determinate. Only by inspection, however, can we determine that the structure is unstable.
The student is advised to pause at this point before proceeding and carry out another inspection. Try to sketch the collapse mode for the beam shown in Fig. 2-8.

                                                             Fig. 2-9: Determinate or Indeterminate?

The beam shown in Fig. 2-9 is determinate and stable. It could be viewed as a cantilever with 2 extra supports (increase indeterminacy to level 2) and 2 internal hinges (reduce indeterminacy by 2 levels back to zero). This inspection technique can be applied to frames. But we have to take care that the structure is not unstable.
The reader might wonder at this point: “How can I be certain that a structure is stable when the number of articulations and reactions indicates stability?” Unfortunately the answer may not satisfy some. We must be able to see it collapse in our minds while looking at the sketch on paper. If we are not sure, then we must keep trying until we are sure. Much of our engineering studies in school may seem straightforward (or algorithmic), and assigned problems are generally benign. In the real world, one needs experience before one can have confidence in the work of structural design. There is no guarantee that all problems actually have a stable solution, and some of the problems in this text may test you on this. Every structural concept on paper that you are given, you should analyze critically with imagination together with guidelines.

                                                                                   Fig. 2-10: Unstable Frame

The frame structure in Fig. 2-10 is unstable (and should at this point be obviously so). If the right hand roller were made a pin, the instability is technically removed, but is this a good design concept? Probably not. The pin is not well located if the horizontal member is carrying loads and deflections are to be kept reasonably low. Perhaps, since the corner of the frame is a connection detail, we could locate the pin there!
These type of inspections are carried out during preliminary designs, where the overall concept is checked for integrity. Once the concept is accepted structural analysis is used to determine the forces carried by the structural members. The member sizes are selected from an inspection of the member stresses. More than one cycle may have to be done before the final structural design is achieved. We will not consider the full design cycle in this text, and some of the problems may not be very practical in order to keep them simple, interesting, and to illustrate important points relating to analysis.

                                                                                       Fig. 2-11: Unsymmetric hingeless arch.

In general, the hingeless arch is indeterminate to 3 degrees. The structure shown in Fig. 2-11 does not have symmetry about a centreline. We will find out later that we can make a simplification if the structure is both symmetrical in geometry and loading.
The quickest way to establish the level of indeterminacy is to imagine all 3 reactions at the right hand support to be removed, making the structure a stable cantilever. By reapplying the three support reactions we can see that we have added 3 redundant forces to support the system.

                                                                                             Fig. 2-12: Unsymmetric 2-hing arch

For the 2 hinge arch we have reduced the indeterminacy from the previous example by 2, by eliminating the 2 moment reactions which exist at the supports. This structure is indeterminate to 1 degree. Again there is no symmetry.

                                                                                          Fig. 2-13: Unsymmetric 3-hinge arch

The 3 hinge arch is statically determinate. Note that for this structure, the equilibrium equations for the structure as a free body contain the four unknown external reactions. Thus we will have to use the condition of zero moment at the pin just as we would have done in the beam example in Fig. 2-7.
This provides another useful inspection procedure for frames and arches which do not act the same as beams. If we add additional support to the arch examples given above we increase the level of indeterminacy. If we add more internal releases we reduce indeterminacy.

                                                                              Fig. 2-14: Stable or determinate?

Inspect the frame in Fig. 2-14. Is it stable or unstable? If it is unstable can you sketch the collapse mode? If it is stable, is it determinate? If it is indeterminate, can you state to what degree? You will have the opportunity to practice with such problems at the end of the chapter.

                                                                       Fig. 2-15: Building Frame

For the structure in Fig. 2-15, the determination of the degree of indeterminacy is an academic exercise. (It is 12 degrees indeterminate!) This is a two storey, two bay rectangular building frame. We do not analyze such structure using indeterminate analysis. As a beam-column structure it is analyzed by a method known as the stiffness method which we will study later in Part II. As a slab and column reinforced concrete system it is analyzed using procedures given in the Code for the Design of Reinforced Concrete Structures for Buildings XX.