dvc calculus field trip welcome! deepti chatti ([email protected]) carollo engineers april 23,...
TRANSCRIPT
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Carollo Engineers
We provide water and wastewater engineering services.
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Carollo offices
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CALCULUS
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When do we need calculus?
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When do we need calculus?
When using algebra becomes too difficult…!!
Example: To calculate the distance traveled by a moving object
a) Constant velocityb) Constant acceleration c) Varying acceleration
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Constant velocity – Easy!
Distance = Velocity x Time
TIME
VELOCITY
0 1.0 1.5
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Constant Acceleration – Easy!
TIME
VELOCITY
01.0
Distance = ½ x Velocity x Time
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Varying acceleration – Umm..
Distance = ???
TIME
VELOCITY
F(t)
0 T
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Varying acceleration – Easy!
Distance =
TIME
VELOCITY
F(t)
0 T
∫T
dttF0
)(
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How do we use calculus at Carollo Engineers?
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Who knows what a centroid is?(You get a prize!)
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Centroids
∫ A
xdxcx =
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Structural elements
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Shear and Moment
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Shear at a point
….equals the integral of the load function w(x) or the area under the load diagram up to that point.
∫=−2
1
12 )(x
x
xwVV
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Example..
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Shear diagram
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Moment at a point
…equal to the integral of the shear function, or the area under the shear diagram up to the point.
∫=−2
1
)(12
x
x
dxxVMM
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Shear and Moment
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Deflection of a beam..
∫=AE
Pdlδ
So you will know how much longer the beam will get when acted upon by a force.
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Pressure in a fluid
∫∫ =2
1
2
1
y
y
P
P
dydP γ
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Radiant Time Series: Outside Wall
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CTS for Light to Heavy Walls
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CTS for Walls with Similar Mass and Increasing Insulation
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Special Thanks To
Dave KraskaLyn Gomes
Scott Emberlin
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"As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality." -Albert Einstein