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ENGR-1100 Introduction to Engineering Analysis. Lecture 16. Today Lecture Outline. Structures: Trusses Frames Trusses analysis- method of joints Stability criteria. Trusses : Structures composed entirely of two force members.

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Lecture 16

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ENGR-1100 Introduction to Engineering Analysis

Lecture 16


Today Lecture Outline

  • Structures:

    • Trusses

    • Frames

  • Trusses analysis- method of joints

  • Stability criteria.


  • Trusses: Structures composed entirely of two force members.

    Frames: Structures containing at least one member acted on by forces at three or more points.

    Two important structures types


    Plane trusses: lie in a single plane.

    Space trusses: not contained in a single plane and/or loaded out of the structure plane.

    Plane Trusses


    Assumptions

    1) Truss members are connected together at their ends only.

    2) Truss members are connected together by frictionless pins.

    3) The truss structure is loaded only at the joints.

    4) The weight of the member may be neglected.


    The Four Assumptions

    Truss members are two-force members

    F

    F

    Truss member

    F

    F


    Compressive forces tend to shorten the member.

    Tensile forces tend to elongate the member.

    F

    F

    F

    F

    Straight Members

    Forces act along the axis of the member


    “Rigid” trusses

    “Rigid”- the truss will retain its shape when removed from its support

    Simple truss- constructed by attaching several triangles together.

    Allows a simple way to check rigidity.


    What are we looking for?

    The support reaction .

    The force in each member.

    How many equations are available? How many unknowns?

    Each joint- 2 equations

    Unknowns- number of members+ support reaction.


    Stability Criteria

    m=2j-3

    2j- number of equations to be solved.

    m- number of members.

    3- number of support reaction

    m<2j-3

    Truss unstable

    m>2j-3

    Statically indeterminate


    m=2j-3

    Example

    m (Number of members) =

    13

    j (Number of joints) =

    8

    Number of supports=

    3


    Method of Joints

    Separate free-body diagrams for:

    each member

    each pin

    Equilibrium equations for each pin:

    SF=0

    no moment equation


    Example 7-3

    Use the method of joints to determine the force in each member of the truss shown in Fig. P7-3. State whether each member is in tension or compression.


    Joint B

    Joint D

    ---------

    ---------

    y

    y

    From a free-body diagram on joint D:

    x

    TBD

    Fy = TBD - 3000 = 0

    x

    TAD

    TCD

    TBD = 3000 lb = 3000 lb (T)

    From a free-body diagram on joint B:

    TAB = - 1500.0 lb = 1500 lb (c)

    TBC = - 2598 lb  2600 lb (c)

    TAB

    TBC

    TBD

    Solution

    Fx = TBC sin 30º - TAB sin 60º = 0

    Fy = - TBC cos 30º - TAB cos 60º - TBD = 0


    Joint D

    Joint C

    ---------

    ---------

    y

    y

    x

    TBC

    TBD

    x

    From a free-body diagram on joint C:

    TAD

    TCD

    TCD

    Cy

    From a free-body diagram on joint D:

    known

    Fx = - TCD - TBCsin 30

    = - TCD -(2598) sin 30 = 0

    TCD = 1299 lb = 1299 lb (T)

    Fx =-TAD + TCD = - TAD + 1299 = 0

    TAD = 1299 lb = 1299 lb (T)


    Class Assignment: Exercise set 7-4

    please submit to TA at the end of the lecture

    Answer:

    TBC=14.1 kN (T)

    TAC=5.13 kN (T)

    TAB=28.2 kN (C)


    Class Assignment: Exercise set 7-6

    please submit to TA at the end of the lecture

    Answer:

    TBD=3.23 kN (C)

    TCD=4.62 kN (T)

    TAD=0.567 kN (T)

    TBC=4 kN (C)

    TAB=2.38kN (C)


    Free body diagram on joint C

    y

    FBC

    x

    FCD

    FCD=0

    FBC=0

    SFx=0

    Member BC and DC are zero force members

    Zero Force Members

    SFy=0


    Free body diagram on joint B

    y

    y

    FCB

    FAB

    x

    x

    FBD

    FBD=0

    Free body diagram on joint D

    FDC

    FAD

    FAD=0

    FED

    SFy=0


    Example 7-19

    The truss shown in figure P7-19 support one side of a bridge; anidentical truss supportsthe other side. Floor beams carry vehicle loads to the truss joints. Calculate the forces inmembersBC, BG, andCGwhen a truck weighing 7500 lb is stopped in the middle of the bridge as shown. The center of gravity of the truck is midway between the frontand rear wheels.


    Free body diagram on floor beams FH

    Fy = 2FG - 2(3750) + 4(1125) = 0

    FG= 1500 lb

    Solution

    From symmetry: FH = FF

    Free body diagram on floor beams

    GF

    MG = 2FF (10) – 3750 (6) = 0

    FF = 1125 lb


    f= tan-1 8/5 = 57.99

    = tan-1 5/10 =26.57

    = tan-1 8/10 =38.66

    Free body diagram on complete truss

    Fx = Ax = 0

    Ax = 0

    ME= Ay (30) -1125 (25) - 1500(15) - 1125(5) = 0

    Ay = 1875 lb = 1875 lb


    Free body diagram on joint A

    Fx = TAB cos 57.99 + TAH = 0

    Fy = TAB sin 57.99 + 1875= 0

    TAB = - 2211 lb  2210 lb (c)

    Free body diagram on joint H

    Fy = TBH - 1125 = 0

    TBH = 1125 lb = 1125 lb (T)


    Free body diagram on joint B

    Fx = TBC cos 26.57 + TBG cos 38.66

    + 2211cos 57.99 = 0

    Fy = TBC cos 26.57 - TBG sin 38.66

    + 2211 sin 57.99 - 1125 = 0

    TBC = -1451.2 lb  1451 lb (C)

    TBG = 161.31 lb  161.3 lb (T)

    Free body diagram on joint C

    Fx = TCD cos 26.57+ 1451.2 cos 26.57 = 0

    Fy = TCD sin 26.57 - TCG + 1451.2 sin 26.57 = 0

    TCD = -1451.2 lb  1451 lb (C)

    TCG = 1298.2 lb  1298 lb (T)


    Class Assignment: Exercise set 7-23

    please submit to TA at the end of the lecture


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