DVC Calculus Field Trip

Welcome!

Deepti Chatti(dchatti@carollo.com)

Carollo EngineersApril 23, 2008

Filename.ppt

Carollo Engineers

We provide water and wastewater engineering services.

Filename.ppt

Carollo offices

Filename.ppt

CALCULUS

Filename.ppt

When do we need calculus?

Filename.ppt

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

Filename.ppt

Constant velocity – Easy!

Distance = Velocity x Time

TIME

VELOCITY

0 1.0 1.5

Filename.ppt

Constant Acceleration – Easy!

TIME

VELOCITY

01.0

Distance = ½ x Velocity x Time

Filename.ppt

Varying acceleration – Umm..

Distance = ???

TIME

VELOCITY

F(t)

0 T

Filename.ppt

Varying acceleration – Easy!

Distance =

TIME

VELOCITY

F(t)

0 T

∫T

dttF0

)(

Filename.ppt

How do we use calculus at Carollo Engineers?

Filename.ppt

Who knows what a centroid is?(You get a prize!)

Filename.ppt

Centroids

∫ A

xdxcx =

Filename.ppt

Structural elements

Filename.ppt

Shear and Moment

Filename.ppt

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

Filename.ppt

Example..

Filename.ppt

Filename.ppt

Shear diagram

Filename.ppt

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

Filename.ppt

Shear and Moment

Filename.ppt

Deflection of a beam..

∫=AE

Pdlδ

So you will know how much longer the beam will get when acted upon by a force.

Filename.ppt

Pressure in a fluid

∫∫ =2

1

2

1

y

y

P

P

dydP γ

Filename.ppt

Filename.ppt

Radiant Time Series: Outside Wall

Filename.ppt

CTS for Light to Heavy Walls

Filename.ppt

CTS for Walls with Similar Mass and Increasing Insulation

Filename.ppt

Special Thanks To

Dave KraskaLyn Gomes

Scott Emberlin

Filename.ppt

"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