28 Statics, Vectors, Forces, Moments - Baylor ECSweb.ecs.baylor.edu/faculty/grady/_28_Statics,...
Transcript of 28 Statics, Vectors, Forces, Moments - Baylor ECSweb.ecs.baylor.edu/faculty/grady/_28_Statics,...
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Slide 1
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Slide 2
EGR 1301Introduction to Static Analysis
Presentation adapted from Distance Learning / Online Instructional Presentation
Originally created by Mr. Dick Campbell
Presented by:Departments of Engineering Baylor University
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Slide 3
• Understand the definition of Mechanics• Learn the difference between static and dynamic analysis• Understand the concept of force as a vector• Separate vector into x and y components.• Apply this concept to analyzing sums of forces (ΣF=0), as well as sum of moments (ΣM=0).
• Understand the importance of engineering analysis design.• Apply these concepts to a foot bridge, wall hanging, and even biomechanics
Outline
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Slide 4
• Mechanics - the study of objects at rest or in motion, the effects of forces on a body, and the prediction motion.
• The fundamentals of Mechanics were formulated by Isaac Newton, using his three Laws:1. A body at rest or in constant
motion remains in that state until acted upon by an external unopposed force.
2. An unopposed force causes a mass to accelerate.
3. Every force action has an equal and opposite reaction.
Mechanics
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Slide 5
• Mechanics is divided into the study of Statics and Dynamics.
• Newton’s 2nd Law is expressed as:
Mechanics
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Slide 6
Static analysis
• Therefore...a = 0 • In Statics, nothing is accelerating!
•Newton’s 1st Law• Statics is the study of forces acting on a non-accelerating body,
and the reaction of that body
Dynamic Analysis• If an unopposed force acts:
• Acceleration is proportional to the mass of the body• Dynamics is the study of the motion of a body, both in translation and
rotation.
0F
0F
Statics vs. Dynamics
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Slide 7
• Baylor Engineers in Africa, May 2005– 3 professors, 6 students– Kenya, Africa– Engineering services to Kenya’s
poor population• Foot Bridge Project
– 40-m wide river– Analyze for safety and possible
design improvements
The Example of Engineering Analysis
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Slide 8
• Village was divided, far side had trouble:– Taking their farm produce
to market– Attending school– Getting medical care
• Situation– 5 miles to nearest bridge
(20 mile round trip)– Several drownings per year
Need For Bridge
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Slide 9
• Estimates of approximately 400 crossings per day– Saving 1,460,000 miles of
walking per year• Approximate cost: $5000
– 1/3 ¢ per mile per year– Great impact at minimal
cost
Need For Bridge
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Slide 10
• Cable used is rated to withstand a maximum load in tension of 16,000 lbs.– How much can the bridge support?– How is cable failure considered in the design?– If six people (est. 1000 lbs.) stand in the center, what is the cable
tension?– What is the “safety factor” (SF)?
How Does Engineering Analysis Help?
LoadRatingFS ..
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Slide 11
• Tension – pull apart• Compression – push together• Moment – rotation• Shear – distort shape
Types of Loading
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Slide 12
• Scalar– Has magnitude only (i.e. it’s “just a number”)
• Vector – Has magnitude and direction– In an x-y coordinate system, force may be broken down into
“components”• X-component parallel to x-axis• Y-component parallel to y-axis
Θ
Force as a Vector
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Slide 13
• Use Trigonometry to:– Calculate the magnitude and angle of the vector from the
magnitude of its components (Pythagorean theorem & inverse tangent)
– Calculate the magnitude of the components from the magnitude and angle of the vector (sine and cosine)
)cos( FFx
)sin( FFy
22yx FFF
x
y
FF1tan
Force as a Vector
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Slide 14
• Addition can be interpreted:– Graphically: move the “start” of b to the “tail” of a– Mathematically: add the components that lie in the same
direction
xxx bac
Adding Vectors
xc
a
xa
ya
b
xbyb
cyc
yyy bac
bac
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15
Figure 2.6
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16
Figure 2.18
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Slide 17
• Multiplying by a scalar means:– Changing the magnitude of the vector– But not changing direction– Except for negative numbers
)cos( acbx
Multiplying a Vector by a Scalar
b
acb
a
xa
yayb
xb
where c is a scalar (i.e. “just a number”)
acb
)sin( acby
b
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Slide 18
• î and ĵ indicate direction of vector components– î has magnitude of 1 unit in the x-direction– ĵ has magnitude of 1 unit in the y-direction
• When a vector component is multiplied by î or ĵ– Magnitude of the vector component remains unchanged– The direction of the vector component is defined as parallel to
the x- or y-axis
Unit Vectors
jFiFF yxˆˆ
Can now write a vector as:
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Slide 19
• Simplifying assumptions– Loaded with six persons (approx. 1000 lbs) at the center – Cables are straight– Neglect the weight of the bridge
i
j
Force Vectors & Static Analysis ofFoot Bridge
lbs F 10003
?2 F ?1 F
ft 130
ft 5
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Slide 20
• Draw a sketch of the forces as vectors• Write each force vector in terms of i and j components
– Components perpendicular to each other can be treated separately• Invoke Newton’s 1st Law
– Sum of i components=zero (x direction)– Sum of j components = zero (y direction)
• Solve the two equations with two unknowns
Setting Up the Analysis
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Slide 21
• Resolve the three forces into î, ĵ components• Identify our unknowns (i.e., F1 and F2)• Set up summation equations in î, ĵ directions
– Solve these two equations for the unknowns
i
j
Static Analysis of the Foot Bridge’s Cable
4.4)(tan 6551
lbs F 10003
?1 F ?2 F
ft 130
ft 5
?
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Slide 22
i
j
?2 F
lbs F 10003
?1 F
ft 130
ft 54.4˚4.4˚
jFiFF ˆsinˆcos 111 jFiFF ˆsinˆcos 222
0ˆ0ˆcosˆcosˆ21 ilbsiFiFiF
0ˆ1000ˆsinˆsinˆ21 jlbsjFjFjF
jlbsilbsF ˆ1000ˆ03
Static Analysis of the Foot Bridge’s Cable
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Slide 23
Rearranging and substituting in angles:
2121 ),4.4cos()4.4cos( FFFF lbsFF 1000)4.4sin()4.4sin( 21
lbs 652721 FF
Safety Factor (Cable Strength = 16,000 lbs):
45.2lbs 6527lbs 16000..
FS
Static Analysis of the Foot Bridge’s Cable
0ˆ0ˆcosˆcosˆ21 ilbsiFiFiF
0ˆ1000ˆsinˆsinˆ21 jlbsjFjFjF
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Slide 24
• If we allow the bridge to have larger sag, what will happen to the tension in the cables?
• What will happen to the required anchors for the cables at the two ends of the bridge?
• What is the disadvantage of having larger sag in the bridge?
Engineering is an Exercisein Trade-Offs!
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Slide 25
Then: lbsF 1000)10sin(2 1 lbsFF 287921
56.5lbs 8792lbs 16000.. FS
What happens to the Safety Factor if we increase thesag? Let : o10
Static Analysis of the Foot Bridge’s Cable
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Slide 26
• What did our model (sketch) not take into account?• How will that affect our calculations, in particular, the safety factor?• Does our safety factor (2.45) now seem too low, too high, or about right?• How do you determine what constitutes a “good” safety factor?
Engineering is an Estimate
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Slide 27
• The wood decking was built using 3-1”x 6” planks laid side by side.– The bridge span is 130 feet. – The wood’s density is 30 lbs/ft3.– The weight of the cable and
hangers is approx. 112 lbs. – Can you calculate the weight of
the decking? • What does this do to our safety
factor?
What About the Weight of the Bridge?
130 ft
32
25.1613012161)3( ftft
inftininV
lbsftftlbsVm
48825.1630 3
3
lbslbslbs 600112488
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Slide 28
What happens to the Safety Factor if we include the weight of the bridge? Let: lbs 6001000 and 10 Wo
Then: lbsF 1600)10sin(2 1
lbs 460721 FF
47.3lbs 4607lbs 16000..
FS
Static Analysis of the Foot Bridge’s Cable
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Slide 29
• How does a suspension bridge like the Golden Gate provide a way to achieve the goals of lower cost and convenience of use?
• What are the disadvantages in the suspension bridge design?
Suspension Bridges
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Slide 30
Engineering Disasters:Tacoma Narrows Bridge
http://archive.org/details/SF121
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Slide 31
What is the best way to hang a 40 lb mirror on a wall? How much tension is in the hanger wire in each case?
Picture Frame Sample Problem
Option 1: One Hanger in Wall Option 2: Two Hangers in Wall
48 in
12 in 12 in
12 in
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Slide 32
Picture Frame Sample Problem
Option 1: One Hanger in Wall
i
j
jlbsilbsF ˆ40ˆ01
6.262412tan 1
jFiFF ˆsinˆcos 222
2F
1F
3F
jFiFF ˆsinˆcos 333
0coscos0 32ˆ FFFi
32 FF
0sinsin40 32ˆ FFlbsFj
sin240 2 Flbs
48 in
12 in
3F
lbslbsFF
7.44
)6.26sin(240
32
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Slide 33
Picture Frame Sample Problem
i
j
jlbsilbsF ˆ20ˆ01
451212tan 1
jFiFF ˆsinˆcos 222
2F
1F
3F
jiFF ˆ0ˆ33
0cos0 32ˆ FFFi
32 cos FF
0sin20 2ˆ FlbsFj
lbslbsF
3.28sin
202
lbslbsF 0.20cos3.283
Option 2: Two Hangers in Wall
12 in
12 in
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Slide 34
• Moments occur at a given point and are caused by a force that causes rotation about a point or an axis.
• A moment is equal to the Force times the perpendicular distance from the force to the point that is being evaluated.
• Force F causes a clockwise rotation if unopposed about point A in the picture below.
• 4in is the perpendicular distance and 100lb is the force in the picture, so the Moment M caused by this force is: M=-(100lb x 4in) = -400 lb-in
• The moment is negative because it is clockwise.• Be careful with units, Moments can be measured in lb-in, lb- ft, N-m, or other units.
Moments
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Slide 35
Calculate the moment at point “A”.
Moments Example 1
125 lbs
500 lbs8 ft 2 ft
Remember the sign convention for moments:
+
M1 = -(8 ft)(125 lbs) = -1000 lb-ftM2 = +(10 ft)(500 lbs) = 5000 lb-ft
MA= M1 + M2 = 4000 lb-ft
A
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Slide 36
Calculate the moment at point “A”.
Moments Example 2
200 lbs
10 ft
Remember the sign convention for moments:
+
60o
lbslbsFx 100)60cos(200lbslbsFy 2.173)60sin(200
ftlbsftlbsM Fy 1732)10)(2.173(ftlbsftlbsM Fx 0)0)(100(
A
ftlbsftlbsftlbsM A 1732)0)(100()10)(2.173(
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37
Figure 3.13
Moment of a Force About a PointProduct of a Force and the
Perpendicular Distance to the Point
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Example Problem
• The moment exerted about point E by the weight is 300 lb-in. What moment does the weight exert about point S?
38
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Equilibrium of A Particle