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Chapter 3

By Diane Gardner,2014-09-22 11:15
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Structural Mechanics:Truss

    Chapter 3 (Chapter 7 in book)

    Analysis of statically Determinate Truss

    ?1. Introduction

    A truss is defined as a structure formed by a group of members which are assumed to be connected by pins and uniform in cross-section.

    Trusses use material very efficiently and are consequent economical for spanning long distances. They are commonly used in bridges, towers and roof. Crane, etc.

     A classic truss is shown in Fig.3-1

    Fig.3-1

    The truss is comprised by chords and webs .Chords can be divided into top and bottom, and web into vertical and diagonal.

    The point at which web members connect to chord is called panel point (joint). The length between two joints on the same chord is called panel.

    The following assumptions are made in order to simplify the analysis of trusses: a) Truss members are connected together at their ends by frictionless pins.

    Fig.3-2

    b) The centroidal axes of members are all straight lines .those meeting at a joint all

    intersect at a common point.

    Fig.3-3

    c) External loads and reactions are applied on the truss only at its joints.

    Fig.3-4

    According to above assumptions members of a truss are subjected to axial force only. In other words, the members in a truss are all two-force members. It is customary to designate the tension as the positive axial force and compression as the negative.

    Fig.3-5

    ?2. Classification of Trusses

    1. Simple truss

    A plane truss is formed by beginning triangle, and then units of two components are added on the triangle. Trusses formed in this way are called simple trusses.

    Fig.3-6

2. Compound truss

    Two or more trusses are connected according to two rigid pieces rule or three rigid pieces rule, the trusses formed in this way are termed compound trussed.

    Fig.3-7

    3. Complex truss

    Trusses that cannot be classified as either simple trusses or compound are called.

    Fig.3-8

    ?3. Method of Joints

    In this method by a section passed completely around a joint, the joint is isolated from the rest of truss.

    The equilibrium equations and may be applied to the joint free Y0X0

    body to determine the unknown forces in members meeting there.

    This can always be done successively in the order reverse to that they were set up (built) in simple trusses.

    In general, the axial force N of an inclined member will be resolved into the horizontal component X and vertical component Y in the calculation (Fig.3-9).

    Fig.3-9

Length of AB=L, its projection in horizontal ,and in vertical. llxy

    Similarly: the axial force =N, projection of N in horizontal =x, in vertical =y. Two triangles are similar to each other .

    So:

    NXY (3-1) lllxy

    We can obtained components X,Y using this formula.

    In the calculation it is customary to assume the unknown axial forces to be tension. If the obtained solution is a negative one, it indicates that the axial force is compression.

    Fig.3-10

    Ex. 3-1

    Find (determine) forces of members using joints method.

    Fig.3-11

    Solution:

    (1) We determined the reactions of entire truss at first.

     M08

     12R409;606;803R80N11

    ; Y0

     R;R40;60;80R100N188

    We can find that the order set up of this truss is 1,2,……8,or 8,72,1 We isolate the joints reverse to the order.

    (2) Joint 1 is isolated as free body. We suppose the axial forces of members 12, 13 are positive forces as Fig.3-12.

    Fig.3-12

    According to equilibrium equation.

     Y0

     (compression) Y;800Y80(N)1313

     Equation (3-1)

    35 XY60NNY100N1313131344

    We also have,

    X0

     (tension) N;N0N60N121312

Joint 2:

    Fig.3-13

     X0NN60N2412

     Y0N40N23

     Joint 3:

    Fig.3-14

     Y0Y;N;Y0132334

     YYY804040N341323

     X30NN40N3434

     X0

     XXN0133435

     (compression) NXX0603090N351334

    Joint 4, 5,6,7,8 can be done like this.

    Axial forces of all members can be found.

    In the calculation, it is necessary to draw a free body sketch. The member in which the axial force is equal to zero is called inactive member.

    In the following cases, the inactive members may be determined easily by the

    method of joints.

    A. If no load is applied at a joint between only two truss members, both the two

    members are inactive members.

    Fig.3-15

    B. If a concentrated load P acting at a joint between two truss members is along one

    of the member axes, the other member is inactive member.

    Fig. 3-16

    C. If no load is applied at a joint between three members, of which two lie on the

    same line, the third one is inactive member and the axial forces of the other two

    members are the same.

    D. If no load is applied at a joint between four members, of which two lie on the

    same line and other two lie on the other same line, the force of two members on

    the same line are the same.

    Fig.3-16

    ?4. Method of Sections

    If the axial force in only one member is desired and for a compound truss, the method of joints is less convenient, the method of sections may be applied to determine the unknowns in these members.

    of sections involves ? isolating a portion of the truss by cutting The method

    certain members and ? solving for the axial forces in these members with the equilibrium equations for the isolated free body.

    In general, only there members are cut by a section, because three unknowns can be obtained by three statically equilibrium equations.

    Ex. 3-2 (Fig.3-17)

    Fig.3-17

    Determine the internal forces of a, b, c members.

    Solution:

    (1) Find reactions of the truss

     ; R10NR20N17

    (2) Pass section I-I through members a, b, c, take the left portion of the truss as

    free body (Fig.3-18).

    Fig.3-18

    (3) Find Na

    M0 4

     N3R60a1

     N20Na

     (4) Also, M03

    40 X2.5;R40X16NC1C2.5

    X41c YXccY14c

     Y4Nc

    2222 NX;Y4;16ccc

    (5) X0

     X;N;X0cab

     XXN16204Nbca

    In some special cases, although four or more members are cut by a section, we can still determine some unknown force among them provided that the forces comply with some rules.

    How to determine internal force of member 1 () (Fig.3-19) N1

    Fig.3-19

    Pass section ?-?though truss as shown (indicated) by the dash lines in Fig.3-19. Although 5 members are cut, force can be found because all members cut by N1

    section intersect at the same point C except,M0. Nc1

    For the truss shown in Fig.3-20, section ?-?cut 5 members. It is obvious that

    member 5 is inactive member; we set coordination system X, Y shown in Fig. We can find force of member 1 because of forces in members 2, 3, 4 are parallel and

    perpendicate to X.

    Fig.3-20

    ; can be obtained. X0N1

    The method of joints and the method of sections may be combined when a truss is

    analyzed.

    Ex. 3-3 P90

    Find force in member 1, 2.

    Fig.3-21

    Solution:

    (1) Reactions=60kN

    (2) Pass section ?-?though the truss take the left portion as free body .shown in

    3-22. are parallel each other and horizontal. N,N,N023

    Fig.3-22

    So

    Y0, N60kN1

    Then, take joint B as free body (3-23)

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