Engineering Mechanics: STATICS

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Engineering MechanicsSTATICS

• Anthony Bedford and Wallace Fowler
• SI Edition

Teaching Slides Chapter 6 Structures in
Equilibrium
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Chapter Outline

• Trusses
• The Method of Joints
• The Method of Sections
• Space Trusses
• Frames Machines
• Computational Mechanics

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6.1 Trusses

• Truss structures such as the beams supporting a
  house

• Starting with very simple examples
• 3 bars pinned together at their ends to form a
  triangle with supports as shown ? structure that
  will support a load F

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6.1 Trusses

• Construct more elaborate structures by adding
  more triangles
• The bars are the members of these structures
  the places where the bars are pinned together are
  called joints
• Even though these examples are quite simple, they
  begin to resemble the structures used to support
  bridges roofs of houses

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6.1 Trusses

• If these structures are supported loaded at
  their joints we neglect the weight of the bars,
  each bar is a 2-force member
• We call such a structure a truss

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6.1 Trusses

• Drawing the free-body diagram of a truss
• Force T axial force in the member
• When the forces are directed away from each
  other, the member is in tension
• When the forces are directed toward each other,
  the member is in compression

• Because it is a 2-force member, the forces at the
  ends, which are the sums of the forces exerted on
  the member at its joints, must be equal in
  magnitude, opposite in direction directed along
  the line between the joints

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6.1 Trusses

• If we cut the member by a plane draw the
  free-body diagram of the part of the member on 1
  side of the plane
• We represent the system of internal forces
  moments exerted by the part not included in the
  free-body diagram by a force F acting at the
  point P where the plane intersects the axis of
  the member a couple M

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6.1 Trusses

• The sum of the moments about P must equal zero,
  so M 0
• Therefore, we have a 2-force member, which means
  that F must equal in magnitude opposite in
  direction to the force T acting at the joint
• The internal force is a tension or compression
  equal to the tension or compression exerted at
  the joint
• Notice the similarity to a rope or cable, in
  which the internal force is a tension equal to
  the tension applied at the ends

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6.1 Trusses

• Although many actual structures, including roof
  trusses bridge trusses, consist of bars
  connected at the ends, very few have pinned
  joints
• E.g. a joint of a bridge truss

• The ends of the members are welded at the joint
  are not free to rotate
• Such a joint can exert couples on the members

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6.1 Trusses

• However, these structures are designed to
  function as trusses
• They support loads primarily by subjecting their
  members to axial forces
• They can usually be modeled as trusses, treating
  the joints as pinned connections under the
  assumption that couples they exert on the members
  are small in comparison to axial forces

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6.2 The Method of Joints

• The method of joints involves
• Drawing free-body diagrams of the joints of a
  truss 1 by 1
• Using equilibrium equations to determine the
  axial forces in the members

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6.2 The Method of Joints

• Before beginning, it is usually necessary to draw
  a free-body diagram of the entire truss (i.e.
  treat the truss as a single object) determine
  the reactions at its supports
• E.g. consider the Warren truss which has members
  2 m in length support loads at B D

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6.2 The Method of Joints

• From the equilibrium equations
• S Fx Ax 0
• S Fy Ay E ? 400 N ? 800 N 0
• S Mpoint A ?(1 m)(400 N) ? (3 m)(800 N) (4
  m)E 0
• We obtain the reactions
• Ax 0, Ay 500 N E 700 N
• The next step is to choose a joint draw its
  free-body diagram
• Isolate joint A by cutting members AB AC

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6.2 The Method of Joints
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6.2 The Method of Joints

• The terms TAB TAC are the axial forces in
  members AB AC, respectively
• Although the directions of the arrows
  representing the unknown axial forces can be
  chosen arbitrarily, notice that we have chosen
  them so that a member is in tension if we obtain
  a positive value for the axial force
• Consistently choosing the directions in this way
  helps avoid errors

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6.2 The Method of Joints

• The equilibrium equations for joint A are
• S Fx TAC TAB cos 60 0
• S Fy TAB sin 60 500 N 0
• Solving these equations, we obtain the axial
  force TAB ?577 N TAC 289 N
• Member AB is in compression member AC is in
  tension

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6.2 The Method of Joints

• Next, obtain a free-body diagram of joint B by
  cutting members AB, BC BD
• From the equilibrium equations for joint B
• S Fx TBD TBC cos 60 577 cos 60 0
• S Fy ?400 N 577 sin 60 ? TBC sin 60 0
• We obtain TBC 115 N TBD ?346 N
• Member BC is in tension member BD is in
  compression

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6.2 The Method of Joints

• By continuing to draw free-body diagrams of the
  joints, we can determine the axial forces of all
  the members
• In 2 dimensions, you can obtain only 2
  independent equilibrium equations from the
  free-body diagram of a joint
• Summing the moments about a point does not result
  in an additional independent equation because the
  forces are concurrent

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6.2 The Method of Joints

• Therefore when applying the method of joints, you
  should choose joints to analyze that are
  subjected to no more than 2 unknown forces
• In our example, we analyzed joint A first because
  it was subjected to the known reaction exerted by
  the pin support 2 unknown forces, the axial
  forces TAB TAC
• We could then analyze joint B because it was
  subjected to 2 known forces 2 unknown forces,
  TBC TBD
• If we had attempted to analyze joint B first,
  there would have been 3 unknown forces

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6.2 The Method of Joints

• When determining the axial forces in the members
  of a truss, it will be simpler if you are
  familiar with 3 particular types of joints
• 1.Truss joints with 2 collinear members no
  load the sum of the forces must equal zero, T1
  T2. The axial forces are equal.

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6.2 The Method of Joints

• 2.Truss joints with 2 noncollinear members no
  load because the sum of the forces in the x
  direction must equal zero, T2 0. therefore T1
  must also equal zero. The axial forces are zero.

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6.2 The Method of Joints

• 3.Truss joints with 3 members, 2 of which are
  collinear no load because the sum of the
  forces in the x direction must equal zero, T3
  0. The sum of the forces in the y direction must
  equal zero, so T1 T2. The axial forces in the
  collinear members are equal the axial force in
  the 3rd member is zero.

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Example 6.1 Applying the Method of Joints

• Determine the axial forces in the members of
  the truss in Fig. 6.12.
• Strategy
• 1st, draw a free-body diagram of the entire
  truss, treating it as a single object determine
  the reactions at the supports. Then apply the
  method of joints, simplifying the task by
  identifying any special joints

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Example 6.1 Applying the Method of Joints

• Solution
• Determine the Reactions at the Supports
• Draw the free-body diagram of the entire truss

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Example 6.1 Applying the Method of Joints

• Solution
• From the equilibrium equations
• S Fx Ax B 0
• S Fy Ay ? 2 kN 0
• S Mpoint B ?(6 m) Ax ? (10 m)(2 kN) 0
• We obtain the reactions Ax ?3.33 kN, Ay 2 kN
  
• B 3.33 kN.

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Example 6.1 Applying the Method of Joints

• Solution
• Identify Special Joints
• Because joint C has 3 members, 2 of which are
• collinear no load, the axial force in member BC
  is
• zero, TBC 0 the axial forces in the collinear
  
• members AC CD are equal, TAC TCD.
• Draw Free-Body Diagrams of the Joints
• We know the reaction exerted on joint A by the
• support joint A is subjected to only 2 unknown
• forces, the axial forces in members AB AC.

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Example 6.1 Applying the Method of Joints

• Solution

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Example 6.1 Applying the Method of Joints

• Solution
• The angle ? arctan (5/3) 59.0
• The equilibrium equations for joint A are
• S Fx TAC sin ? ? 3.33 kN 0
• S Fy 2 kN ? TAB ? TAC cos ? 0
• Solving these equations, we obtain TAB 0
• TAC 3.89 kN.

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Example 6.1 Applying the Method of Joints

• Solution
• Now draw the free-body diagram of joint B

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Example 6.1 Applying the Method of Joints

• Solution
• From the equilibrium equation
• S Fx TBD 3.33 kN 0
• We obtain TBD ?3.33 kN. The negative sign
• indicates that member BD is in compression.
• The axial forces in the members are
• AB 0
• AC 3.89 kN in tension (T)
• BC 0
• BD 3.33 kN in compression (C)
• CD 3.89 kN in tension (T)

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Example 6.1 Applying the Method of Joints

• Critical Thinking
• Observe how our solution was simplified by
  recognizing that joint C is the type of special
  joint with 3 members, 2 of which are collinear
  no load
• This allowed us to determine the axial forces in
  all members of the truss by analyzing only 2
  joints

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Example 6.2 Determining the Largest Force a Truss
Will Support

• Each member of the truss in Fig. 6.13 will
  safely support a tensile force of 10 kN a
  compressive force of 2 kN. What is the largest
  downward load F that the truss will safely
  support?

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Example 6.2 Determining the Largest Force a Truss
Will Support

• Strategy
• This truss is identical to the one we analyzed
  in Example 6.1. By applying the method of joints
  in the same way, the axial forces in the members
  can be determined in terms of the load F. The
  smallest value of F that will cause a tensile
  force of 10 kN or a compressive force of 2 kN in
  any of the members is the largest value of F that
  the truss will support.

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Example 6.2 Determining the Largest Force a Truss
Will Support

• Solution
• By using the method of joints in the same way as
• in Example 6.1, we obtain the axial forces
• AB 0
• AC 1.94F (T)
• BC 0
• BD 1.67F (C)
• CD 1.94F (T)

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Example 6.2 Determining the Largest Force a Truss
Will Support

• Solution
• For a given load F, the largest tensile force
  is 1.94F (in members AC CD) the largest
  compressive force is 1.67F (in member BD).
• The largest safe tensile force would occur
  when 1.94F 10 kN or when F 5.14 kN.
• The largest safe compressive force would occur
  when 1.67F 2 kN or when F 1.20 kN.
• Therefore, the largest load F that the truss
  will safely support is 1.20 kN.

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Example 6.2 Determining the Largest Force a Truss
Will Support

• Critical Thinking
• This example demonstrates why engineers analyze
  structures
• By doing so, they can determine the loads that an
  existing structure will support or design a
  structure to support given loads
• In this example, the tensile compressive loads
  the members of the truss will support are given

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Example 6.2 Determining the Largest Force a Truss
Will Support

• Critical Thinking
• Information of that kind must be obtained by
  applying the methods of mechanics of materials to
  the individual members
• Then statics can be used, as we have done in this
  examples, to determine the axial loads in the
  members in terms of the external loads on the
  structure