Engineering Mechanics: STATICS

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Engineering MechanicsSTATICS

• Anthony Bedford and Wallace Fowler
• SI Edition

Teaching Slides Chapter 11 Virtual Work
Potential Energy
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Chapter Outline

• Virtual Work
• Potential Energy
• Computational Mechanics

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1.1 Virtual Work

• The principle of virtual work is a statement
  about work done by force couples when an object
  or structure is subjected to various hypothetical
  motions

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1.1 Virtual Work

• Work
• Consider a force acting on an object at a point
  P
• Suppose that the object undergoes an
  infinitesimal motion, so that P has a
  differential displacement dr
• The work dU done by F as a result of the
  displacement dr is defined to be
• dU F dr
  (11.1)

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1.1 Virtual Work

• From the definition of the dot product
• dU (F cos ?) dr
• where ? is the angle between F dr
• The work is equal to the product of the component
  of F in the direction of dr the magnitude of dr
• Notice that
• If the component of F parallel to dr points in
  the direction opposite to dr, the work is
  negative
• If F is perpendicular to dr, the work is zero
• Dimensions (force) (length)

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1.1 Virtual Work

• Consider a couple acting on an object
• The moment due to the couple is M Fh in the
  counterclockwise direction
• If the object rotates through an infinitesimal
  clockwise angle d?, the points of application of
  the forces are displaced through differential
  distances ½h d?

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1.1 Virtual Work

• Consequently, the total work done is
• Therefore, when an object acted on by a couple M
  is rotated through an angle d? in the same
  direction as the couple, the resulting work is
• dU M d? (11.2)
• If the direction of the couple is opposite to the
  direction of d?, the work is negative

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1.1 Virtual Work

• Principle of Virtual Work
• The homogenous bar is supported by the wall by
  the pin support at A
  is loaded by a couple M
• The free-body diagram

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1.1 Virtual Work

• The equilibrium equations are
• (11.3)
• (11.4)
• (11.5)

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1.1 Virtual Work

• Consider if the bar is acted on by the forces
  couples we subject it to a hypothetical
  translation in the x direction

• The hypothetical displacement dx virtual
  displacement
• Resulting work dU virtual work
• The pin support the wall prevent the bar from
  actually moving in the x direction the virtual
  displacement is a theoretical artifice

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1.1 Virtual Work

• The objective is to calculate the resulting
  virtual work
• dU Axdx (?N)dx (Ax ?
  N)dx (11.6)
• The forces Ay W do no work because they are
  perpendicular to the displacements of their
  points of application
• The couple M also does no work because the bar
  does not rotate
• Comparing this equation with Eq. (11.3), the
  virtual work equals zero

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1.1 Virtual Work

• Give the bar a virtual translation in the y
  direction
• The resulting virtual work is
• dU Aydy (?W)dy (Au ? W)dy
  (11.7)
• From Eq. (11.4), the virtual work again equals
  zero

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1.1 Virtual Work

• Give the bar a virtual rotation while holding
  point A fixed
• The forces Ax Ay do no work because their point
  of application does not move
• The work done by the couple M is ?M d? because
  its direction is opposite to that of the rotation

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1.1 Virtual Work

• Displacements of the points of application of the
  forces N W
• Components of the forces in the direction of the
  displacements

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1.1 Virtual Work

• The work done by N (N sin ?) (L d?)
• The work done by W (?W cos ?) (½ L d?)
• The total work is
• (11.8)
• From Eq. (11.5), the virtual work resulting from
  the virtual rotation is also zero

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1.1 Virtual Work

• Therefore, if an object is in equilibrium, the
  virtual work done by the external forces
  couples acting on it is zero for any virtual
  translation or rotation
• dU 0
  (11.9)
• This principle can be used to derive the
  equilibrium equations for an object
• Subjecting the bar to virtual translations dx
  dy a virtual rotation d? results in Eqs.
  (11.6)(11.8)
• Because the virtual work is zero in each case, we
  obtain Eqs. (11.3)(11.5)

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1.1 Virtual Work

• But there is no advantage to this approach
  compared to simply drawing the free-body diagram
  of the object writing the equations of
  equilibrium in the usual way
• The advantages of the principle of virtual work
  become evident when we consider structures

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1.1 Virtual Work

• Application to Structures
• The principle of virtual work applies to each
  member of a structure
• By subjecting certain types of structures in
  equilibrium to virtual motions calculating the
  total virtual work, we can determine unknown
  reactions at their supports as well as internal
  forces in their members
• The procedure involves finding virtual motions
  that result in virtual work being done both by
  known loads unknown forces couples

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1.1 Virtual Work

• Suppose that we want to determine the axial load
  in member BD of the truss
• The other members of the truss are subjected to
  the 4-kN load the forces exerted on them by
  member BD

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1.1 Virtual Work

• If we give the structure a virtual
  rotation d?, virtual work is done
  by the force TBD acting at B by
  the 4-kN load at C
• The virtual work done by these 2 forces is the
  total virtual work done on the members of the
  structure because the virtual work done by the
  internal forces they exert on each other cancels
  out

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1.1 Virtual Work

• E.g. consider joint C
• The force TBC is the axial load in member BC
• The virtual work done at C on member BC is
  TBC(1.4 m) d? the work done at C on member CD
  is (4 kN ? TBC)(1.4 m) d?
• When we add up the virtual work done on the
  members to obtain the total virtual work on the
  structure, the virtual work due to the internal
  force TBC cancels out

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1.1 Virtual Work

• If the members exerted an internal couple on each
  other at C for example, as a result of friction
  in the pin support the virtual work would not
  cancel out
• Therefore, we can ignore internal forces in
  calculating the total virtual work on the
  structure
• dU (TBD cos ? )(1.4 m) d? (4 kN)(1.4 m) d?
  0
• The angle ? arctan (1.4/1) 54.5
• Solving this equation, we obtain TBD ?6.88 kN

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1.1 Virtual Work

• Using virtual work to determine reactions on
  members of structures involves 2 steps
• 1.Choose a virtual motion identify a virtual
  motion that results in virtual work being done by
  known loads by an unknown force or couple you
  want to determine
• 2.Determine the virtual work calculate the
  total virtual work resulting from the virtual
  motion to obtain an equation for the unknown
  force or couple

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Example 11.1 Applying Virtual Work to a Structure

• For the structure in Fig. 11.8, use the
  principle of virtual work to determine the
  horizontal reaction at C.

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Example 11.1 Applying Virtual Work to a Structure

• Strategy
• Notice that even though the structure is fixed
  at A C, it can be subjected to hypothetical
  virtual motions. We can choose a virtual motion
  for which the horizontal reaction at C the
  external force couple acting on the structure
  do work. By calculating the resulting virtual
  work we can determine the horizontal reaction at
  C.

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Example 11.1 Applying Virtual Work to a Structure

• Solution
• Choose a virtual Motion
• Draw the free-body diagram of the structure

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Example 11.1 Applying Virtual Work to a Structure

• Solution
• If we hold point A fixed subject bar AB to a
  virtual rotation d? while requiring point C to
  move horizontally, the virtual work is only done
  by the external loads on the structure by Cx.
  The reactions Ax Ay do no work because A does
  not move the reaction Cy does no work because
  it is perpendicular to the virtual displacement
  of point C.

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Example 11.1 Applying Virtual Work to a Structure

• Solution
• Determine the virtual work
• The virtual work done by the 400-N force is
• (400 sin 40 N) (1 m) d?.
• The bar BC undergoes a virtual rotation d? in the
  
• counterclockwise direction, so the work done by
  the
• couple is (500 N-m) d?.
• In terms of the virtual displacement dx of point
  C, the
• work done by the reaction Cx is Cxdx.

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Example 11.1 Applying Virtual Work to a Structure

• Solution
• The total virtual work is
• dU (400 sin 40 N)(1 m) d? (500 N-m) d?
  Cxdx 0
• To obtain Cx from this equation, we must
  determine the relationship between d? dx.
• From the geometry of the structure, the
  relationship between the angle ? the distance
  x from A to C, in m, is
• x 2(2 cos ?)

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Example 11.1 Applying Virtual Work to a Structure

• Solution
• The derivative of this equation with respect
  to ? is
• Therefore, an infinitesimal change in x is
  related to an infinitesimal change in ? by
• dx ?4 sin ? d?
• Because the virtual rotation d? is a decease
  in ?, we conclude that dx is related to d? by
• dx 4 sin 40 d?

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Example 11.1 Applying Virtual Work to a Structure

• Solution
• Substituting this expression into our equation
  for the virtual work gives
• dU (400 sin 40 N-m) (500 N-m) Cx (4 sin
  40 m) d? 0
• Solving, we obtain Cx ?294 N.

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Example 11.1 Applying Virtual Work to a Structure

• Critical Thinking
• Notice that we ignored the internal forces the
  members exert on each other at B
• The virtual work done by these internal forces
  cancels out
• To obtain the solution, we needed to determine
  the relationship between the virtual
  displacements dx d?
• Determining the geometrical relationships between
  virtual displacements is often the most
  challenging aspect of applying the principle of
  virtual work

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Example 11.2 Applying Virtual Work to a Machine

• The extensible platform in Fig. 11.9 is raised
  lowered by the hydraulic cylinder BC. The total
  weight of the platform the men is W. The
  weights of the beams supporting the platform can
  be neglected. What axial force must the hydraulic
  cylinder exert to hold the platform in
  equilibrium in the position shown?

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Example 11.2 Applying Virtual Work to a Machine

• Strategy
• We can use a virtual motion that coincides
  with the actual motion of the platform beams
  when the length of the hydraulic cylinder
  changes. By calculating the virtual work done by
  the hydraulic cylinder by the weight of the men
  platform, we can determine the force exerted by
  the hydraulic cylinder.

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Example 11.2 Applying Virtual Work to a Machine

• Solution
• Choose a Virtual Motion
• Draw the free-body diagram of the platform
  the beams
• If we hold point A fixed subject point C to
  a horizontal virtual displacement dx, the only
  external forces that do virtual work are F the
  weight W.
• (The reaction due to the roller support at C
  is perpendicular to the virtual displacement)

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Example 11.2 Applying Virtual Work to a Machine

• Solution
• Determine the Virtual Work
• The virtual work done by the force F as point
  C undergoes a virtual displacement dx to the
  right is ?F dx.

To determine the virtual work done by the
weight W, we must determine the vertical
displacement of point D when point C moves to the
right a distance dx.
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Example 11.2 Applying Virtual Work to a Machine

• Solution
• The dimensions b h are related by
• b2 h2 L2
• where L is the of the beam AD.
• Taking the derivative of this equation with
  respect to b, we obtain
• which we can solve for dh in terms of db

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Example 11.2 Applying Virtual Work to a Machine

• Solution
• Thus, when b increases an amount dx, the
  dimension h decreases an amount (b/h) dx.
• Because there are 3 pairs of beams, the
  platform moves downward a distance (3b/h) dx
  the virtual work done by the weight is (3b/h) W
  dx.
• The total virtual work is
• we obtain F (3b/h) W.

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Example 11.2 Applying Virtual Work to a Machine

• Critical Thinking
• This example was designed to demonstrate how
  advantageous the method of virtual work can be
  for certain types of problems
• It would be very tedious to draw the free-body
  diagrams of the individual members of the frame
  supporting the platform solve the equilibrium
  equations to determine the force exerted by the
  hydraulic cylinder
• In contrast, it was relatively simple to
  determine the virtual work done by the external
  forces acting on the frame