Consider a simply supported beam subjected to a vertical concentrated load P, as shown in Fig. 3.26. By applying the moment equilibrium equations,P
MBẳ0 andP
MAẳ0, we obtain the expressions for the vertical reactions at supportsAandB, respectively, as
AyẳP b
S and ByẳP a
S (3.13)
where, as shown in Fig. 3.26,aẳdistance of the loadPfrom supportA (measured positive to the right);bẳdistance ofPfrom supportB(meas- ured positive to the left); andSẳdistance between supportsAandB.
The first of the two expressions in Eq. (3.13) indicates that the mag- nitude of the vertical reaction atAis equal to the magnitude of the loadP times the ratio of the distance of P from support B to the distance be- tween the supportsAandB. Similarly, the second expression in Eq. (3.13) states that the magnitude of the vertical reaction at B is equal to the magnitude ofPtimes the ratio of the distance ofPfromAto the distance betweenAandB. These expressions involving proportions, when used in conjunction with the principle of superposition, make it very convenient to determine reactions of simply supported structures subjected to series of concentrated loads, as illustrated by the following example.
FIG.3.26
Example 3.12
Determine the reactions at the supports for the truss shown in Fig. 3.27(a).
Solution
Free-Body Diagram See Fig. 3.27(b).
Static Determinacy The truss is internally stable withrẳ3. Therefore, it is statically determinate.
Support Reactions
þ !P Fxẳ0
Axẳ0 Ans.
continued
SECTION 3.7 Reactions of Simply Supported Structures Using Proportions 79
Ayẳ66:75 6
4 þ133:5 5 4þ3
4
þ111:25 2
4 þ89 1 41
4
þ44:5 2 4
ẳ400:5 kN
Ayẳ400:5 kN" Ans.
Byẳ66:75 2
4 þ133:5 1 4 þ1
4
þ111:25 2
4 þ89 3 4þ5
4
þ44:5 6 4
ẳ267 kN
Byẳ267 kN" Ans.
Checking Computations þ "P
Fyẳ 66:752ð133:5ị 111:252ð89ị 44:5ỵ400:5ỵ267ẳ0 Checks
FIG.3.27
SUMMARY
In this chapter, we have learned that a structure is considered to be in equilibrium if, initially at rest, it remains at rest when subjected to a system of forces and couples. The equations of equilibrium of space structures can be expressed as
PFxẳ0 P
Fyẳ0 P Fzẳ0 PMxẳ0 P
Myẳ0 P Mzẳ0
(3.1)
For plane structures, the equations of equilibrium are expressed as PFxẳ0 P
Fyẳ0 P
Mzẳ0 (3.2)
Two alternative forms of the equilibrium equations for plane structures are given in Eqs. (3.3) and (3.4).
The common types of supports used for plane structures are sum- marized in Fig. 3.3. 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.
A structure is called statically determinate externally if all of its support reactions can be determined by solving the equations of equili- brium and condition. For a plane internally stable structure supported byrnumber of reactions, if
r<3 the structure is statically unstable externally
rẳ3 the structure is statically determinate externally (3.8) r>3 the structure is statically indeterminate externally The degree of external indeterminacy is given by
ieẳr3 (3.7)
For a plane internally unstable structure, which has r number of external reactions andecnumber of equations of condition, if
r<3þec the structure is statically unstable externally
rẳ3ỵec the structure is statically determinate externally (3.9) r>3þec the structure is statically indeterminate externally The degree of external indeterminacy for such a structure is given by
ieẳr ð3ỵecị (3.10)
In order for a plane structure to be geometrically stable, it must be supported by reactions, all of which are neither parallel nor concurrent.
A procedure for the determination of reactions at supports for plane structures is presented in Section 3.5.
The principle of superposition states that on a linear elastic struc- ture, the combined e¤ect of several loads acting simultaneously is equal to the algebraic sum of the e¤ects of each load acting individually. The determination of reactions of simply supported structures using propor- tions is discussed in Section 3.7.
Summary 81
PROBLEMS
Section 3.4
3.1 through 3.4 Classify each of the structures shown as externally unstable, statically determinate, or statically inde-
terminate. If the structure is statically indeterminate ex- ternally, then determine the degree of external indeterminacy.
FIG.P3.1
FIG.P3.2
FIG.P3.3
FIG.P3.4
Sections 3.5 and 3.7
3.5 through 3.13 Determine the reactions at the supports for the beam shown.
3 m 6 m 4.5 m
29.2 kN/m
A B
FIG.P3.5
3 m 3 m 6 m
100 kN 20 kN/m
A B
FIG.P3.6
12 m
25 kN/m B
A FIG.P3.7
3 m 9 m 3 m
21.9 kN/m
A B
FIG.P3.8
Problems 83
4 m 2 m 70 kN 30 kN/m
A
150 kN – m
FIG.P3.9
2 m 2 m 4 m 3 m
222.5 kN
22 kN/m
A B
4 3 30°
135.7 kN-m
FIG.P3.10
3 m 6 m 3 m
A B
29.2 kN/m 43.8 kN/m
133.5 kN
81.4 kN – m
FIG.P3.11
3 m 5 m 2 m
29.2 kN/m
43.8 kN/m
B A
FIG.P3.12
10 m 30 kN/m
A B
3 4
FIG.P3.13
3.14The weight of a car, moving at a constant speed on a beam bridge, is modeled as a single concentrated load, as shown in Fig. P3.14. Determine the expressions for the ver- tical reactions at the supports in terms of the position of the car as measured by the distancex, and plot the graphs showing the variations of these reactions as functions ofx.
5 m 8 m 3 m
A B
x
W = 20 kN
FIG.P3.14
3.15 The weight of a 5-m-long trolley, moving at a constant speed on a beam bridge, is modeled as a moving uniformly dis- tributed load, as shown in Fig. P3.15. Determine the expressions for the vertical reactions at the supports in terms of the position of the trolley as measured by the distancex, and plot the graphs showing the variations of these reactions as functions ofx.
25 m
A B
x 5 m
w = 10 kN/m
FIG.P3.15
3.16 through 3.41Determine the reactions at the supports for the structures shown.
70 kN
50 kN 50 kN
A B
4 at 6 m = 24 m
5 m
FIG.P3.16
106.8 kN 66.75 kN
106.8 kN 106.8 kN 106.8 kN 53.4 kN
6 at 6.1 m = 36.6 m
5 m A
B
FIG.P3.17
FIG.P3.18
A B
10 m 15 m
200 kN 35 kN/m
FIG.P3.19
A B
12.2 m 6.1 m
6.1 m
133.5 kN
66.75 kN 18.2 kN/m
36.4 kN/m
FIG.P3.20
100 kN
4 m 12 m 4 m
A B
5 m
5 m 40 kN/m
20 kN/m
FIG.P3.21
Problems 85
6 m 3 m A
B 36.5 kN/m
36.5 kN/m
4.6 m
FIG.P3.22
6 m 6 m 6 m
8 m 20 kN/m
40 kN/m
150 kN
A
B
FIG.P3.23
FIG.P3.24
10 m 6.1 m 6.1 m
133.5 kN 21.9 kN/m
Hinge A
B
FIG.P3.25
Hinge
4.6 m 4.6 m
1.2 m 1.2 m
111.25 kN
29.2 kN/m A
B
FIG.P3.26
FIG.P3.27
20 m 10 m 10 m 30 kN/m
A C
B Hinge
FIG.P3.28
5 m 5 m 6.1 m 5 m 5 m
43.8 kN/m
A D
Hinge B C Hinge
FIG.P3.29
3 m 3 m 3 m 10 m
12 kN/m
B A
130 kN 5 12
FIG.P3.30
FIG.P3.31
FIG.P3.32
100 kN
4 m 6 m 6 m 4 m
A B
5 m
5 m
40 kN/m 20 kN/m
Hinge
FIG.P3.33
A B
5 m 3 m
3 m
111.25 kN
29.2 kN/m
43.8 kN/m Hinge
FIG.P3.34
Problems 87
FIG.P3.35
5 m 5 m
5 m 5 m
117 kN/m
A C
B Hinge
Hinge
FIG.P3.36
8 m 8 m 8 m 8 m 8 m 5 m
20 kN/m A
B C D
Hinge Hinge
FIG.P3.37
FIG.P3.38
12.2 m 267 kN
36.5 kN/m
7.5 m
7.5 m 7.5 m
Hinge
A
B
FIG.P3.39
6.1 m 6.1 m
6.1 m
6.1 m
Hinge
Hinge Hinge
133.5 kN
A B
FIG.P3.40
FIG.P3.41
4
Plane and Space Trusses
4.1 Assumptions for Analysis of Trusses
4.2 Arrangement of Members of Plane Trusses—Internal Stability 4.3 Equations of Condition for Plane Trusses
4.4 Static Determinacy, Indeterminacy, and Instability of Plane Trusses 4.5 Analysis of Plane Trusses by the Method of Joints
4.6 Analysis of Plane Trusses by the Method of Sections 4.7 Analysis of Compound Trusses
4.8 Complex Trusses 4.9 Space Trusses
Summary Problems
89 Atrussis an assemblage of straight members connected at their ends by flexible connections to form a rigid configuration. Because of their light weight and high strength, trusses are widely used, and their applications range from supporting bridges and roofs of buildings (Fig. 4.1) to being support structures in space stations (Fig. 4.2). Modern trusses are con- structed by connecting members, which usually consist of structural steel or aluminum shapes or wood struts, to gusset plates by bolted or welded connections.
As discussed in Section 1.4, if all the members of a truss and the applied loads lie in a single plane, the truss is called aplane truss.
Plane trusses are commonly used for supporting decks of bridges and roofs of buildings. A typical framing system for truss bridges was described in Section 1.4 (see Fig. 1.13(a)). Figure 4.3 shows a typical framing system for a roof supported by plane trusses. In this case, two or more trusses are connected at their joints by beams, termedpurlins, to form a three-dimensional framework. The roof is attached to the pur- lins, which transmit the roof load (weight of the roof plus any other load due to snow, wind, etc.) as well as their own weight to the supporting trusses at the joints. Because this applied loading acts on each truss in its own plane, the trusses can be treated as plane trusses. Some of the Truss Bridges
Terry Poche/Shutterstock
common configurations of bridge and roof trusses, many of which have been named after their original designers, are shown in Figs. 4.4 and 4.5 (see pp. 92 and 93), respectively.
Although a great majority of trusses can be analyzed as plane trusses, there are some truss systems, such as transmission towers and latticed domes (Fig. 4.6), that cannot be treated as plane trusses because
FIG.4.1 Roof Trusses. Plum High School. Large Bow Truss and Supporting Truss for Gymnasium,
Camber Corporation. Web address: http://
www.cambergroup.com/g87.htm
FIG.4.2 A Segment of the Integrated Truss Structure which forms the Backbone of the International Space Station
Courtesy of National Aeronautics and Space Administration 98_05164
of their shape, arrangement of members, or applied loading. Such trusses, which are calledspace trusses, are analyzed as three-dimensional bodies subjected to three-dimensional force systems.
The objective of this chapter is to develop the analysis of member forces of statically determinate plane and space trusses. We begin by discussing the basic assumptions underlying the analysis presented in this chapter, and then we consider the number and arrangement of members needed to form internally stable or rigid plane trusses. As part of this discussion, we definesimpleandcompoundtrusses. We also pres- ent the equations of condition commonly encountered in plane trusses.
We next establish the classification of plane trusses as statically deter- minate, indeterminate, and unstable and present the procedures for the analysis of simple plane trusses by the methods of joints and sections.
We conclude with an analysis of compound plane trusses, a brief dis- cussion of complex trusses, and analysis of space trusses.