Internal Stability
A structure is considered to be internally stable, or rigid, if it maintains its shape and remains a rigid body when detached from the supports.
Conversely, a structure is termed internally unstable (or nonrigid) if it cannot maintain its shape and may undergo large displacements under SECTION 3.4 Static Determinacy, Indeterminacy, and Instability 47
small disturbances when not supported externally. Some examples of internally stable structures are shown in Fig. 3.7. Note that each of the structures shown forms a rigid body, and each can maintain its shape under loads. Figure 3.8 shows some examples of internally unstable structures. A careful look at these structures indicates that each struc- ture is composed of two rigid parts,ABandBC, connected by a hinged joint B, which cannot prevent the rotation of one part with respect to the other.
Category Type of support Symbolic representation Reactions Number of unknowns
Roller
1 The reaction forceRacts perpendicular to the supporting surface and may be directed either into or away from the structure.
The magnitude ofRis the unknown.
I Rocker
Link
1
The reaction forceRacts in the direction of the link and may be directed either into or away from the structure. The magnitude ofRis the unknown.
II Hinge
2
The reaction forceRmay act in any direction. It is usually convenient to representRby its rectangular components,RxandRy. The magnitudes ofRxandRyare the two unknowns.
III Fixed
3
The reactions consist of two force componentsRxandRyand a couple of momentM. The magnitudes ofRx,Ry, andMare the three unknowns.
FIG.3.3 Types of Supports for Plane Structures
It should be realized that all physical bodies deform when subjected to loads; the deformations in most engineering structures under service conditions are so small that their e¤ect on the equilibrium state of the structure can be neglected. The termrigid structureas used here implies that the structure o¤ers significant resistance to its change of shape, SECTION 3.4 Static Determinacy, Indeterminacy, and Instability 49
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whereas a nonrigid structure o¤ers negligible resistance to its change of shape when detached from the supports and would often collapse under its own weight when not supported externally.
Static Determinacy of Internally Stable Structures
An internally stable structure is considered to be statically determinate externally if all its support reactions can be determined by solving the equations of equilibrium. Since a plane internally stable structure can be treated as a plane rigid body, in order for it to be in equilibrium under a general system of coplanar loads, it must be supported by at least three reactions that satisfy the three equations of equilibrium (Eqs. 3.2, 3.3, or 3.4). Also, since there are only three equilibrium equations, they cannot
FIG.3.7 Examples of Internally Stable Structures
FIG.3.8 Examples of Internally Unstable Structures
be used to determine more than three reactions. Thus, a plane structure that is statically determinate externally must be supported by exactly three reactions. Some examples of externally statically determinate plane structures are shown in Fig. 3.9. It should be noted that each of these structures is supported by three reactions that can be determined by solving the three equilibrium equations.
FIG.3.9 Examples of Externally Statically Determinate Plane Structures
SECTION 3.4 Static Determinacy, Indeterminacy, and Instability 51
FIG.3.10 Examples of Externally Statically Indeterminate Plane Structures
If a structure is supported by more than three reactions, then all the reactions cannot be determined from the three equations of equili- brium. Such structures are termedstatically indeterminate externally. The reactions in excess of those necessary for equilibrium are called external redundants, and the number of external redundants is referred to as the degree of external indeterminacy. Thus, if a structure has r reactions ðr>3ị, then the degree of external indeterminacy can be written as
ieẳr3 (3.7)
Figure 3.10 shows some examples of externally statically indeterminate plane structures.
If a structure is supported by fewer than three support reactions, the reactions are not su‰cient to prevent all possible movements of the structure in its plane. Such a structure cannot remain in equilibrium under a general system of loads and is, therefore, referred to asstatically unstable externally. An example of such a structure is shown in Fig.
3.11. The truss shown in this figure is supported on only two rollers. It should be obvious that although the two reactions can prevent the truss from rotating and translating in the vertical direction, they cannot pre- vent its translation in the horizontal direction. Thus, the truss is not fully constrained and is statically unstable.
The conditions of static instability, determinacy, and indeterminacy of plane internally stable structures can be summarized as follows:
r<3 the structure is statically unstable externally rẳ3 the structure is statically determinate externally r>3 the structure is statically indeterminate externally
(3.8)
where rẳnumber of reactions.
It should be realized that the first of three conditions stated in Eq.
(3.8) is both necessary and su‰cient in the sense that ifr<3, the struc- ture is definitely unstable. However, the remaining two conditions,rẳ3 and r>3, although necessary, are not su‰cient for static determinacy and indeterminacy, respectively. In other words, a structure may be sup- ported by a su‰cient number of reactions ðrb3ị but may still be un- stable due to improper arrangement of supports. Such structures are
FIG.3.11 An Example of Externally Statically Unstable Plane Structure
SECTION 3.4 Static Determinacy, Indeterminacy, and Instability 53
referred to as geometrically unstable externally. The two types of re- action arrangements that cause geometric instability in plane structures are shown in Fig. 3.12. The truss in Fig. 3.12(a) is supported by three parallel reactions. It can be seen from this figure that although there is a su‰cient number of reactions ðrẳ3ị, all of them are in the vertical direction, so they cannot prevent translation of the structure in the horizontal direction. The truss is, therefore, geometrically unstable. The other type of reaction arrangement that causes geometric instability is shown in Fig. 3.12(b). In this case, the beam is supported by three nonparallel reactions. However, since the lines of action of all three re- action forces are concurrent at the same point, A, they cannot prevent rotation of the beam about pointA. In other words, the moment equili- brium equation P
MAẳ0 cannot be satisfied for a general system of coplanar loads applied to the beam. The beam is, therefore, geometri- cally unstable.
Based on the preceding discussion, we can conclude that in order for a plane internally stable structure to be geometrically stable ex- ternally so that it can remain in equilibrium under the action of any arbi- trary coplanar loads, it must be supported by at least three reactions, all of which must be neither parallel nor concurrent.
Static Determinacy of Internally Unstable Structures—Equations of Condition
Consider an internally unstable structure composed of two rigid mem- bersABandBC connected by an internal hinge atB, as shown in Fig.
3.13(a). The structure is supported by a roller support atAand a hinged support at C, which provide three nonparallel nonconcurrent external reactions. As this figure indicates, these reactions, which would have
FIG.3.12 Reaction Arrangements Causing External Geometric Instability in Plane Structures
been su‰cient to fully constrain an internally stable or rigid structure, are not su‰cient for this structure. The structure can, however, be made externally stable by replacing the roller support at A by a hinged sup- port to prevent the horizontal movement of end A of the structure.
Thus, as shown in Fig. 3.13(b), the minimum number of external reac- tions required to fully constrain this structure is four.
Obviously, the three equilibrium equations are not su‰cient to de- termine the four unknown reactions at the supports for this structure.
However, the presence of the internal hinge at B yields an additional equation that can be used with the three equilibrium equations to
FIG.3.13
SECTION 3.4 Static Determinacy, Indeterminacy, and Instability 55
determine the four unknowns. The additional equation is based on the condition that an internal hinge cannot transmit moment; that is, the moments at the ends of the parts of the structure connected to a hinged joint are zero. Therefore, when an internal hinge is used to connect two portions of a structure, the algebraic sum of the moments about the hinge of the loads and reactions acting on each portion of the structure on either side of the hinge must be zero. Thus, for the structure of Fig.
3.13(b), the presence of the internal hinge at B requires that the alge- braic sum of moments aboutBof the loads and reactions acting on the individual members ABandBCmust be zero; that is,P
MBAB ẳ0 and PMBBC ẳ0. Such equations are commonly referred to as theequations of condition or construction. It is important to realize that these two equations are not independent. When one of the two equations—for example,P
MBAB ẳ0—is satisfied along with the moment equilibrium equation P
Mẳ0 for the entire structure, the remaining equation PMBBC ẳ0 is automatically satisfied. Thus, an internal hinge connect- ing two members or portions of a structure provides one independent equation of condition. (The structures that contain hinged joints con- necting more than two members are considered in subsequent chapters.) Because all four unknown reactions for the structure of Fig. 3.13(b) can be determined by solving the three equations of equilibrium plus one equation of condition (P
MBABẳ0 orP
MBBC ẳ0), the structure is considered to be statically determinate externally.
Occasionally, connections are used in structures that permit not only relative rotations of the member ends but also relative translations in certain directions of the ends of the connected members. Such con- nections are modeled as internal roller joints for the purposes of analy- sis. Figure 3.13(c) shows a structure consisting of two rigid membersAB andBCthat are connected by such an internal roller atB. The structure is internally unstable and requires a minimum of five external support reactions to be fully constrained against all possible movements under a general system of coplanar loads. Since an internal roller can transmit neither moment nor force in the direction parallel to the supporting sur- face, it provides two equations of condition;
PFxAB ẳ0 or P
FxBC ẳ0
and P
MBAB ẳ0 or P
MBBCẳ0
These two equations of condition can be used in conjunction with the three equilibrium equations to determine the five unknown external reactions. Thus, the structure of Fig. 3.13(c) is statically determinate externally.
From the foregoing discussion, we can conclude that if there areec equations of condition (one equation for each internal hinge and two equations for each internal roller) for an internally unstable structure, which is supported byrexternal reactions, then if
r<3þec the structure is statically unstable externally rẳ3ỵec the structure is statically
determinate externally r>3þec the structure is statically
indeterminate externally
(3.9)
For an externally indeterminate structure, the degree of external in- determinacy is expressed as
ieẳr ð3ỵecị (3.10)
Alternative Approach An alternative approach that can be used for de- termining the static instability, determinacy, and indeterminacy of inter- nally unstable structures is as follows:
1. Count the total number of support reactions,r.
2. Count the total number of internal forces, fi, that can be transmitted through the internal hinges and the internal rollers of the structure.
Recall that an internal hinge can transmit two force components, and an internal roller can transmit one force component.
3. Determine the total number of unknowns,rþfi.
4. Count the number of rigid members or portions, nr, contained in the structure.
5. Because each of the individual rigid portions or members of the structure must be in equilibrium under the action of applied loads, reactions, and/or internal forces, each member must satisfy the three equations of equilibrium (P
Fxẳ0,P
Fyẳ0, andP
Mẳ0).
Thus, the total number of equations available for the entire struc- ture is 3nr.
6. Determine whether the structure is statically unstable, determinate, or indeterminate by comparing the total number of unknowns, rþfi, to the total number of equations. If
rþ fi<3nr the structure is statically unstable externally rỵ fiẳ3nr the structure is statically
determinate externally rþ fi>3nr the structure is statically
indeterminate externally
(3.11)
For indeterminate structures, the degree of external indeterminacy is given by
ieẳ ðrỵ fiị 3nr (3.12) SECTION 3.4 Static Determinacy, Indeterminacy, and Instability 57
Applying this alternative procedure to the structure of Fig. 3.13(b), we can see that for this structure,rẳ4, fiẳ2, andnrẳ2. As the total number of unknownsðrỵ fiẳ6ịis equal to the total number of equa- tions ð3nrẳ6ị, the structure is statically determinate externally. Sim- ilarly, for the structure of Fig. 3.13(c),rẳ5, fiẳ1, andnrẳ2. Since rỵfiẳ3nr, this structure is also statically determinate externally.
The criteria for the static determinacy and indeterminacy as de- scribed in Eqs. (3.9) and (3.11), although necessary, are not su‰cient because they cannot account for the possibility of geometric instability.
To avoid geometric instability, the internally unstable structures, like the internally stable structures considered previously, must be supported by reactions, all of which are neither parallel nor concurrent. An addi- tional type of geometric instability that may arise in internally unstable structures is depicted in Fig. 3.14. For the beam shown, which contains three internal hinges at B;C, andD,rẳ6 and ecẳ3 (i.e., rẳ3ỵec);
therefore, according to Eq. (3.9), the beam is supported by a su‰cient number of reactions, and it should be statically determinate. However, it can be seen from the figure that portionBCD of the beam is unstable because it cannot support the vertical load P applied to it in its un- deformed position. Members BCandCDmust undergo finite rotations to develop any resistance to the applied load. Such a type of geometric instability can be avoided by externally supporting any portion of the structure that contains three or more internal hinges that are collinear.
Example 3.1
Classify each of the structures shown in Fig. 3.15 as externally unstable, statically determinate, or statically in- determinate. If the structure is statically indeterminate externally, then determine the degree of external indeterminacy.
Solution
(a) This beam is internally stable withrẳ5>3. Therefore, it is statically indeterminate externally with the degree of external indeterminacy of
ieẳr3ẳ53ẳ2 Ans.
(b) This beam is internally unstable. It is composed of two rigid membersABandBCconnected by an internal hinge atB. For this beam,rẳ6 andecẳ1. Sincer>3ỵec, the structure is statically indeterminate externally with the degree of external indeterminacy of
ieẳr ð3ỵecị ẳ6 ð3ỵ1ị ẳ2 Ans.
FIG.3.14
continued
FIG.3.15
Alternative Method fiẳ2, nrẳ2, rỵ fiẳ6ỵ2ẳ8, and 3nrẳ3ð2ị ẳ6. Asrỵ fi>3nr, the beam is statically indeterminate externally, with
ieẳ ðrỵ fiị 3nrẳ86ẳ2 Checks (c) This structure is internally unstable withrẳ4 andecẳ2. Sincer<3ỵec, the structure is statically unstable externally. This can be verified from the figure, which shows that the memberBCis not restrained against movement in
the horizontal direction. Ans.
Alternative Method fiẳ1, nrẳ2, rỵ fiẳ4ỵ1ẳ5, and 3nrẳ6. Sincerỵ fi<3nr, the structure is statically
unstable externally. Checks
(d) This beam is internally unstable withrẳ5 andecẳ2. Becauserẳ3ỵec, the beam is statically determinate
externally. Ans.
Alternative Method fiẳ4,nrẳ3,rỵfiẳ5ỵ4ẳ9, and 3nrẳ3ð3ị ẳ9. Becauserỵfiẳ3nr, the beam is stat-
icaly determinate externally. Checks
continued
SECTION 3.4 Static Determinacy, Indeterminacy, and Instability 59
(e) This is an internally unstable structure withrẳ6 andecẳ3. Sincerẳ3ỵec, the structure is statically deter-
minate externally. Ans.
Alternative Method fiẳ6,nrẳ4,rỵ fiẳ6ỵ6ẳ12, and 3nrẳ3ð4ị ẳ12. Becauserỵfiẳ3nr, the structure is
statically determinate externally. Checks
(f ) This frame is internally unstable withrẳ4 andecẳ1. Sincerẳ3ỵec, the frame is statically determinate ex-
ternally. Ans.
Alternative Method fiẳ2, nrẳ2, rỵ fiẳ4ỵ2ẳ6, and 3nrẳ3ð2ị ẳ6. Since rỵfiẳ3nr, the frame is stat-
ically determinate externally. Checks
(g) This frame is internally unstable withrẳ6 and ecẳ3. Sincerẳ 3ỵec, the frame is statically determinate
externally. Ans.
Alternative Method fiẳ6, nrẳ4, rỵ fiẳ6ỵ6ẳ12, and 3nrẳ3ð4ị ẳ 12. Becauserỵ fiẳ3nr, the frame is
statically determinate externally. Checks