Chapter 4.b(Method of Section)

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    Engineering Mechanics :Engineering Mechanics :STATICSSTATICS

    Lecture #09By,

    Noraniah KassimUniversiti Tun Hussein Onn Malaysia

    (UTHM),

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    STRUCTURAL ANALYSIS – THE METHOD OF SECTIONS

    Learning Topic:

    • Applications

    • The Method of Sections

    Today’s Objectives:

    Students will be able to determine:

    1. Forces in truss members using the

    method of sections.

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    READING QUIZ

    1. In the method of sections, generall a !cut" passes through nomore than ##### members in which the forces are un$nown.

      A% 1 &% '

      (% ) *% +

     '. If a simple truss member carries a tensile force of T along its

    length, then the internal force in the member is ###### .

      A% tensile with magnitude of T'&% compressi-e with magnitude of T'

      (% compressi-e with magnitude of T

    *% tensile with magnitude of T

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    APPLICATIONS

    ong trusses are often used to construct bridges.

    The method of /oints re0uires that man /oints be analed before we can determine the forces in the middle part of the

    truss.

    Is there another method to determine these forces directl2

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    THE METHOD OF SECTIONS

    In the method of sections, a truss is di-ided into two parts b

    ta$ing an imaginar !cut" 3shown here as a4a% through the truss.

    Since truss members are sub/ected to onl tensile or compressi-eforces along their length, the internal forces at the cut member will

    also be either tensile or compressi-e with the same magnitude.

    This result is based on the e0uilibrium principle and 5ewton6s

    third law.

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    STEPS OF ANALYSIS

    1. *ecide how ou need to !cut" the truss. This is based on:

    a% where ou need to determine forces, and, b% where the total

    number of un$nowns does not e7ceed three 3in general%.

    '. *ecide which side of the cut truss will be easier to wor$with 3minimie the number of reactions ou ha-e to find%.

    ). If re0uired, determine the necessar support reactions b

    drawing the F&* of the entire truss and appling the 84of48.

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    STEPS OF ANALYSIS (Continued)

    +. *raw the F&* of the selected part of the cut truss. 9e need to

    indicate the un$nown forces at the cut members. Initiall we

    ma assume all the members are in tension, as we did when

    using the method of /oints. pon sol-ing, if the answer is positi-e, the member is in tension as per our assumption. If the

    answer is negati-e, the member must be in compression. 3;lease

    note that ou can also assume forces to be either tension or

    compression b inspection as was done in the figures abo-e.%

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    STEPS OF ANALYSIS (Continued)

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    EAMPLE

    Given: oads as shown on the

    roof truss.

    Find: The force in members

    *8, *, and M.

    Plan:

    a% Ta$e a cut through the members *8, *, and M.

     b% 9or$ with the left part of the cut section. 9h2

    c% *etermine the support reaction at A. 9hat are the2

    d% Appl the 8of8 to find the forces in *8, *, and M.

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      EAMPLE  (continued)

    Analing the entire truss, we get Σ F= > A= > ?. &

    smmetr, the -ertical support reactions are

    A@ > I@ > ) $5

    B M*  > C ) 3D% B 3D% B 1' 3+% B FM  3 ?

      FM  > )D.+ $5 3 T %

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      EAMPLE  (continued)

      BΣ M > C) 31'% B 31'% B 1' 3D% B 1' 3+% C F*8 3 +√1E%3% > ?

      F*8 > C)E.11 $5 or )E.1 $5 3(%

    → B Σ F= > )D.+ B 3+√1E% 3C)E.11% B 3+√+1% F* > ?

    F* > C).D+ $5 or ).D+ $5 3(%

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    CONCEPT QUIZ

    1. (an ou determine the forcein member 8* b ma$ing the

    cut at section a4a2 87plain

    our answer.

      A% 5o, there are + un$nowns.  &% @es, using Σ M* > ? .

      (% @es, using Σ M8  > ? .

      *% @es, using Σ M& > ? .

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    CONCEPT QUIZ  (continued)

    '. If ou $now F8*, how will ou determine F8& 2

      A% & ta$ing section b4b and using Σ M8 > ?

      &% & ta$ing section b4b, and using Σ F= > ? and Σ F@ > ?

      (% & ta$ing section a4a and using Σ M& > ?

      *% & ta$ing section a4a and using Σ M* > ?

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    IN CLASS TUTORIAL (GROUP PRO!LEM SOL"ING)

    Given: oading on the

    truss as shown.

    Find: The force in

    members &(, &8,

    and 8F.

    Plan:

    a% Ta$e a cut through the members &(, &8, and 8F.

     b% Anale the top section 3no support reactions%.

    c% *raw the F&* of the top section.

    d% Appl the e0uations of e0uilibrium such that e-er

    e0uation ields answer to one un$nown.

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    PRO!LEM  (continued)

     B → ΣF=  > < B 1? C F&8 cos + ?

    F&8 > '1.' $5 3T%

      B Σ M8 > C ?

    F(& > < $5 3T%

      B Σ M&  > C < 3D% C 1? 3+% C < 3+% C F8F 3+% > ?

    F8F

      > C '< $5 or '< $5 3(%

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    ATTENTION QUIZ

    1. As shown, a cut is madethrough members H, &H

    and &( to determine the

    forces in them. 9hich section

    will ou choose for analsis

    and wh2

      A% Jight, fewer calculations.

      &% eft, fewer calculations.

      (% 8ither right or left, same

    amount of wor$.

      *% 5one of the abo-e, too

    man un$nowns.

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    ATTENTION QUIZ

    '. 9hen determining the force in

    member H in the pre-ious

    0uestion, which one e0uation of

    e0uilibrium is best to use2

      A% Σ M > ?

      &% Σ MH  > ?

      (% Σ M&

      > ?

      *% Σ M(  > ?

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    HOME#OR$ TUTORIAL

    K1 34)?% :

    The Home bridge truss is sub/ected to the loading shown. *etermine theforce in members *8, 8, and H, and state if the members are in

    tension or compression.

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    HOME#OR$ TUTORIAL (continued)

    K' 34))% :

    The roof truss supports the -ertical loading shown. *etermine the forcein members &(, (L, and L and state if these members are in tension

    or compression

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    HOME#OR$ TUTORIAL (continued)

    K) 3

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