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DVC Calculus Field Trip

Welcome!

Deepti Chatti(dchatti@carollo.com)

Carollo EngineersApril 23, 2008

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Carollo Engineers

We provide water and wastewaterengineering 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 amoving object

Constant velocity Constant acceleration 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

0 1.0

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

dttF

0

)(

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How do we use calculus at CarolloEngineers?

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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) orthe area under the load diagram up to thatpoint.

!="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 upto the point.

!="2

1

)(12

x

x

dxxVMM

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Shear and Moment

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Deflection of a beam..

!=AE

Pdl"

So you will know how much longerthe beam will get when acted uponby a force.

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Pressure in a fluid

!! =

2

1

2

1

y

y

P

P

dydP "

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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 Massand 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 referto reality, they are not certain, and as faras they are certain, they do not refer toreality." -Albert Einstein