Engineering Mechanics: Statics

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Engineering Mechanics Statics

• Douglas Gabauer
• Virginia Tech Campus
• gabauer_at_vt.edu

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Context

• Mechanics Study of bodies subject to forces

Statics ? Bodies at Rest
Rigid Bodies
Dynamics ? Bodies in Motion
Deformable Bodies
Solid Mechanics
Fluids
Fluid Mechanics
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Background

• Newtons Laws
• 1st Law Body at rest tends to stay at rest
• 2nd Law Unbalanced force causes acceleration
  (Fma)
• 3rd Law For every action there is an equal and
  opposite reaction

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More Background

• Units
• SI System
• US Customary
• Dimensional Homogeneity
• Vector and Scalar Quantities

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Vectors

• Vector Magnitude and Direction
• Ex. Forces, Velocities, Accelerations
• Representation
• Manipulation
• Addition/Subtraction
• Scalar Product
• Components

B
rAB
A
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Vector Manipulation

• Components

A
A
y
A Ax i Ayj
Ay
x
Ax

• Addition

AB (AxBx) i (AyBy)j

• Scalar Multiplication

cA cAx i cAyj
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Component Example

• A person exerts a 60-lb force F to push a crate
  onto a truck. Express F in terms of components.
• Fx 60cos(20) Fy 60sin(20)

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Fundamental Equations

• Statics implies equilibrium
• No Acceleration ? SF ma 0
• Sum of Forces in all directions is ZERO!
• SFx 0
• SFy 0
• SFz 0
• No Rotation ? SM 0
• Sum of Moments in all directions is ZERO!

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Force Terminology

• Terms to be familiar with
• Line of Action (c)
• Concurrent Forces (a)
• Parallel Forces (b)
• Coplanar Forces

(c)
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More Force Terminology
Center of Mass

• Gravity
• W mg
• G 9.81 m/s2 32.2 ft/s2
• Body Force
• Ex Gravity
• Surface Force
• Normal Force (N)
• Frictional Force (f)

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Free Body Diagrams

• Most important analysis tool
• Aids in identification of external forces
• Procedure
• Identify the object to be isolated
• Draw the object isolated (with relevant
  dimensions)
• Draw vectors to represent all external forces

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Free Body Diagrams

• Internal/External Force
• Depends on choice of object

Person Chair
Person Only
WT
RC
WP
RF
RF
RC
RC
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Example Problem

• 1. The sum of the forces FA FB FC 0. If the
  magnitude of FA 100 N and the angle
  . Determine FB and FC.

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Example Problem

• 2. A zoologist estimates that the jaw of a
  predator is subjected to a force P as large as
  800 N. What forces T and M must be exerted by
  the temporalis and masseter muscles to support
  this value of P?

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Example Problem

• 3. The inclined surface is smooth. The two
  springs are identical, with unstretched lengths
  of 250 mm and spring constants k 1200 N/mm.
  What are the masses of blocks A and B?

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Example Problem

• 4. The force F 5i (kN) acts on point A where
  the cables AB, AC, and AD are joined. Determine
  the tensions in the three cables.

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Moment (Torque) Concept

• A rotation-causing force
• Vector with direction perpendicular to plane
  (right-hand rule)
• MP D F
• Units N-m or lb-F

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Vector Representation

• Mp r x F
• r is vector from point of interest to any point
  along line of action of F
• This method is more convenient for 3D problems

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Computing the Cross Product

• NOT Commutative (Order MATTERS!)
• U x V (Uxi Uyj) x (Vxi Vyj)
• UxVx(i x i) UxVy(i x j) UyVx(j x
  i) UyVy(j x j)

()
(-) (-) (-) () () ()
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Example Problem

• 5. Two forces of equal magnitude are applied to
  the wrench as shown. If a 50 N-m moment is
  required to loosen the nut, what is the necessary
  value of F?

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Example Problem

• 6. The moment exerted about point E by the weight
  is 299 lb-in. What moment does the weight exert
  about point S?

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Example Problem

• 7. The force F points in the direction of the
  unit vector e 2/3i - 2/3j 1/3k. The support
  at O will safely support a moment of 560 N-m
  magnitude. Based on this criterion, what is the
  largest safe magnitude of F?

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Objects in Equilibrium

• SF 0
• 2D ? x,y ? 2 Equations, 2 Unknowns
• 3D ? x,y,z ? 3 Equations, 3 Unknowns
• SM 0 (about any point)
• Rigid body assumption

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Supports

• When drawing free body diagram
• If you remove a support, you must replace it with
  appropriate reaction forces
• Think What movements does the support restrict?

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Example Problem

• 8. The ironing board has supports at A and B that
  can be modeled as roller supports. Draw a free
  body diagram of the ironing board and determine
  the reactions at A and B.

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Example Problem

• 9. The person doing push-ups pauses in the
  position shown. His mass is 80 kg. Assume that
  his weight, W, acts at the point shown. The
  dimensions shown are a 250 mm, b 740 mm, and
  c 300 mm. Find the normal force exerted by the
  floor on each hand and each foot.

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More Free Body Diagrams

• Frames and Machines