We consider a truss to be statically determinate if the forces in all its members, as well as all the external reactions, can be determined by using the equations of equilibrium.
Since the two methods of analysis presented in the following sections can be used to analyze only statically determinate trusses, it is
important for the student to be able to recognize statically determinate trusses before proceeding with the analysis.
Consider a plane truss subjected to external loadsP1;P2, andP3, as shown in Fig. 4.14(a). The free-body diagrams of the five members and the four joints are shown in Fig. 4.14(b). Each member is subjected to two axial forces at its ends, which are collinear (with the member cen- troidal axis) and equal in magnitude but opposite in sense. Note that in Fig. 4.14(b), all members are assumed to be in tension; that is, the forces are pulling on the members. The free-body diagrams of the joints show the same member forces but in opposite directions, in accordance with Newton’s third law. The analysis of the truss involves the calculation of
FIG.4.13 Equations of Condition for Plane Trusses
SECTION 4.4 Static Determinacy, Indeterminacy, and Instability of Plane Trusses 101
the magnitudes of the five member forces, FAB;FAC;FBC;FBD, and FCD
(the lines of action of these forces are known), and the three reactions, Ax;Ay, andBy. Therefore, the total number of unknown quantities to be determined is eight.
Because the entire truss is in equilibrium, each of its joints must also be in equilibrium. As shown in Fig. 4.14(b), at each joint the internal and external forces form a coplanar and concurrent force system, which must satisfy the two equations of equilibrium,P
Fxẳ0 andP Fyẳ0.
Since the truss contains four joints, the total number of equations avail- able is 2ð4ị ẳ8. These eight joint equilibrium equations can be solved to calculate the eight unknowns. The plane truss of Fig. 4.14(a) is, therefore, statically determinate.
FIG.4.14
Three equations of equilibrium of the entire truss as a rigid body could be written and solved for the three unknown reactions (Ax;Ay, andBy). However, these equilibrium equations (as well as the equations of condition in the case of internally unstable trusses) are not in- dependentfrom the joint equilibrium equations and do not contain any additional information.
Based on the preceding discussion, we can develop the criteria for the static determinacy, indeterminacy, and instability of general plane trusses containingmmembers and jjoints and supported byr(number of ) external reactions. For the analysis, we need to determinemmember forces and r external reactions; that is, we need to calculate a total of mþrunknown quantities. Since there are j joints and we can write two equations of equilibrium (P
Fxẳ0 and P
Fyẳ0) for each joint, the total number of equilibrium equations available is 2j. If the number of unknowns ðmỵrị for a truss is equal to the number of equilibrium equations ð2jị—that is, mỵrẳ2j—all the unknowns can be de- termined by solving the equations of equilibrium, and the truss is stat- ically determinate.
If a truss has more unknowns ðmỵrị than the available equili- brium equations ð2jị—that is, mỵr>2j—all the unknowns cannot be determined by solving the available equations of equilibrium. Such a truss is called statically indeterminate. Statically indeterminate trusses have more members and/or external reactions than the mini- mum required for stability. The excess members and reactions are calledredundants, and the number of excess members and reactions is referred to as the degree of static indeterminacy, i, which can be ex- pressed as
iẳ ðmỵrị 2j (4.3)
If the number of unknowns ðmỵrị for a truss is less than the number of equations of joint equilibrium ð2jị—that is, mỵr<2j—
the truss is calledstatically unstable. The static instability may be due to the truss having fewer members than the minimum required for internal stability or due to an insu‰cient number of external reactions or both.
The conditions of static instability, determinacy, and indeterminacy of plane trusses can be summarized as follows:
mþr<2j statically unstable truss mỵrẳ2j statically determinate truss mþr>2j statically indeterminate truss
(4.4)
The first condition, for the static instability of trusses, is both necessary and su‰cient in the sense that ifm<2jr, the truss is definitely stat- ically unstable. However, the remaining two conditions, for static SECTION 4.4 Static Determinacy, Indeterminacy, and Instability of Plane Trusses 103
determinacyðmẳ2jrịand indeterminacyðm>2jrị, are necessary but not su‰cient conditions. In other words, these two equations simply tell us that thenumberof members and reactions is su‰cient for stabil- ity. They do not provide any information regarding theirarrangement.
A truss may have a su‰cient number of members and external reactions but may still be unstable due to improper arrangement of members and/
or external supports.
We emphasize that in order for the criteria for static determinacy and indeterminacy, as given by Eqs. (4.3) and (4.4), to be valid, the truss must be stable and act as a single rigid body under a general system of coplanar loads when attached to the supports. Internally stable trusses must be supported by at least three reactions, all of which must be nei- ther parallel nor concurrent. If a truss is internally unstable, then it must be supported by reactions equal in number to at least three plus the number of equations of conditionð3ỵecị, and all the reactions must be neither parallel nor concurrent. In addition, each joint, member, and portion of the truss must be constrained against all possible rigid body movements in the plane of the truss, either by the rest of the truss or by external supports. If a truss contains a su‰cient number of members, but they are not properly arranged, the truss is said to havecritical form.
For some trusses, it may not be obvious from the drawings whether or not their members are arranged properly. However, if the member ar- rangement is improper, it will become evident during the analysis of the truss. The analysis of such unstable trusses will always lead to incon- sistent, indeterminate, or infinite results.
Example 4.2
Classify each of the plane trusses shown in Fig. 4.15 as unstable, statically determinate, or statically indeterminate. If the truss is statically indeterminate, then determine the degree of static indeterminacy.
Solution
(a) The truss shown in Fig. 4.15(a) contains 17 members and 10 joints and is supported by 3 reactions. Thus, mỵrẳ2j. Since the three reactions are neither parallel nor concurrent and the members of the truss are properly ar-
ranged, it is statically determinate. Ans.
(b) For this truss,mẳ17, jẳ10, andrẳ2. Becausemỵr<2j, the truss is unstable. Ans.
(c) For this truss,mẳ21, jẳ10, andrẳ3. Becausemỵr>2j, the truss is statically indeterminate, with the de- gree of static indeterminacy iẳ ðmỵrị 2jẳ4. It should be obvious from Fig. 4.15(c) that the truss contains four
more members than required for stability. Ans.
(d) This truss hasmẳ16, jẳ10, andrẳ3. The truss is unstable, sincemỵr<2j. Ans.
continued
FIG.4.15
continued
SECTION 4.4 Static Determinacy, Indeterminacy, and Instability of Plane Trusses 105
(e) This truss is composed of two rigid portions,ABandBC, connected by an internal hinge atB. The truss has mẳ26, jẳ15, andrẳ4. Thus,mỵrẳ2j. The four reactions are neither parallel nor concurrent and the entire truss
is properly constrained, so the truss is statically determinate. Ans.
(f ) For this truss,mẳ10, jẳ7, andrẳ3. Becausemỵr<2j, the truss is unstable. Ans.
(g) In Fig. 4.15(g), a member BC has been added to the truss of Fig. 4.15(f ), which prevents the relative rotation of the two portions ABE and CDE. Since m has now been increased to 11, with j and r kept con- stant at 7 and 3, respectively, the equation mỵrẳ2j is satisfied. Thus, the truss of Fig. 4.15(g) is statically
determinate. Ans.
(h) The truss of Fig. 4.15(f ) is stabilized by replacing the roller support atDby a hinged support, as shown in Fig.
4.15(h). Thus, the number of reactions has been increased to 4, butmand jremain constant at 10 and 7, respectively.
Withmỵrẳ2j, the truss is now statically determinate. Ans.
(i) For the tower truss shown in Fig. 4.15(i),mẳ16, jẳ10, andrẳ4. Becausemỵrẳ2j, the truss is statically
determinate. Ans.
( j) This truss hasmẳ13, jẳ8, and rẳ3. Althoughmỵrẳ2j, the truss is unstable, because it contains two rigid portionsABCDand EFGH connected by three parallel members,BF;CE, and DH, which cannot prevent the relative displacement, in the vertical direction, of one rigid part of the truss with respect to the other. Ans.
(k) For the truss shown in Fig. 4.15(k),mẳ19, jẳ12, andrẳ5. Becausemỵrẳ2j, the truss is statically de-
terminate. Ans.