Mathematics Mathematics in in

EngineeringEngineering Philip B. Bedient

Civil & Environmental Engineering

Mathematics & EngineeringMathematics & Engineering

Engineering problems are Engineering problems are often solved with often solved with mathematical models of mathematical models of physical situationsphysical situations

Purpose is to focus on Purpose is to focus on important mathematical models important mathematical models & concepts to see why & concepts to see why mathematics is important in mathematics is important in engineeringengineering

Greek SymbolsGreek Symbols

Simple Linear ModelsSimple Linear Models Simplest form of Simplest form of equationsequations

Linear springLinear spring• F = kxF = kx where where

FF = spring force = spring forcekk = spring constant = spring constantxx = deformation of = deformation of

springspring FF is dependent variable is dependent variable FF depends on how much spring depends on how much spring is stretched or compressedis stretched or compressed x is independent variablex is independent variable

Linear ModelsLinear Models Temperature distribution across wallTemperature distribution across wall

• T(x) = (TT(x) = (T22 – T – T11)x/L + T)x/L + T11

• T is dependent variable, x T is dependent variable, x independent variableindependent variable

Equation of a straight lineEquation of a straight line• Y = ax + bY = ax + b

Relationship b/w Fahrenheit & Relationship b/w Fahrenheit & Celsius scalesCelsius scales• T(°F) = 9/5 T(°C) + 32T(°F) = 9/5 T(°C) + 32

Linear Linear ModelsModels

HaveHave

ConstantConstant

SlopesSlopes

Nonlinear ModelsNonlinear Models Used to describe relationships b/w Used to describe relationships b/w dependent & independent variablesdependent & independent variables

Predict actual relationships more Predict actual relationships more accurately than linear modelsaccurately than linear models

Polynomial FunctionsPolynomial Functions• Laminar fluid velocity inside a Laminar fluid velocity inside a pipe- how fluid velocity changes pipe- how fluid velocity changes at given cross-section inside pipeat given cross-section inside pipe

Parabolic FunctionParabolic Function Velocity distribution inside pipe Velocity distribution inside pipe given by given by

U = VU = Vcc[1-(r/R)[1-(r/R)22] ] wherewhere

UU = = UU((rr)= fluid velocity at the )= fluid velocity at the radial radial distance r distance r

VVcc = center line velocity = center line velocity rr = radial distance measured = radial distance measured

      from center of       from center of pipepipe

RR = radius of pipe where = radius of pipe where UU = 0 = 0

Parabolic FunctionsParabolic Functions Velocity distribution inside pipe given by Velocity distribution inside pipe given by

U U = = VVcc[1-(r/R)[1-(r/R)22] ]

Explore the rate of change of Explore the rate of change of UU with with rr

dU/dr = 0 - 2VdU/dr = 0 - 2Vcc (r/R) (r/R)-3 -3 == 2V2Vcc /(r/R) /(r/R)3 3

dU/dr = k/rdU/dr = k/r33

Thus, the rate of change is largest at Thus, the rate of change is largest at small rsmall r

And is smallest near the pipe wall And is smallest near the pipe wall

Non-linear fluid velocity Non-linear fluid velocity modelmodel

Velocity Velocity distribution distribution where Vwhere Vcc = 0.5 = 0.5 m/s & R = 0.1 m/s & R = 0.1 mm

Nonlinear Nonlinear second-order second-order polynomialpolynomial

Greatest slope Greatest slope changes with changes with r r near zeronear zero

Manning’s EquationManning’s Equation Calculates flows for uncovered Calculates flows for uncovered channels that carry a steady uniform channels that carry a steady uniform flowflow

V = (1.49/n)RV = (1.49/n)R2/32/3√(S)√(S)WhereWhere

V = channel velocityV = channel velocity

n = Manning’s roughness n = Manning’s roughness coefficient coefficient

R = hydraulic radius = A/PR = hydraulic radius = A/P

S = slope of channel bottom (ft/ft)S = slope of channel bottom (ft/ft)

A = cross sectional area of channelA = cross sectional area of channel

P = wetted perimeter of channelP = wetted perimeter of channel

Wetted Perimeter

Area

Manning’s EquationManning’s Equation V = (1.49/n) RV = (1.49/n) R2/3 2/3 √(S)√(S) A non-linear equation used in A non-linear equation used in civil engineering - V fcn of n, R, civil engineering - V fcn of n, R, and Sand S

V is inverse with n, goes as V is inverse with n, goes as SS1/21/2

Dependent variable V changes by Dependent variable V changes by different amountsdifferent amounts•depends on values of independent depends on values of independent variables R & Svariables R & S

Manning’s EquationManning’s Equation V = (1.49/n)RV = (1.49/n)R2/32/3√(S)√(S) Solve a simple problem here.Solve a simple problem here. Show some picsShow some pics

Other Non-linear examplesOther Non-linear examples Many other engineering situations Many other engineering situations w/ 2w/ 2ndnd & higher order polynomials & higher order polynomials•Trajectory of projectile Trajectory of projectile •Power consumption for resistive Power consumption for resistive elementelement

•Drag forceDrag force•Air resistance to motion of Air resistance to motion of vehiclevehicle

•Deflection of cantilevered beamDeflection of cantilevered beam

Polynomial Model Polynomial Model CharacteristicsCharacteristics

General formGeneral form

y = f(x) = ay = f(x) = a00 + a + a11x + ax + a22xx22 + … + a + … + annxxnn

Unlike linear models, 2Unlike linear models, 2ndnd & higher- & higher-order polynomials have variable order polynomials have variable slopesslopes

Dependent variable Dependent variable yy has zero value has zero value at points where intersects at points where intersects xx axis axis

Some have real roots &/or imaginary Some have real roots &/or imaginary rootsroots

Exponential ModelsExponential Models Value of dependent Value of dependent variable levels variable levels off as independent off as independent variable value variable value gets largergets larger

Simplest form Simplest form given by given by ff((xx) = ) = ee--

xx

Exponential FunctionsExponential Functions ff((xx) = ) = ee-x-x22

Symmetric Symmetric fcn used in fcn used in expressing expressing probability probability distributiodistributionsns

Note bell Note bell shaped shaped curvecurve

Logarithmic FunctionsLogarithmic Functions The symbol “log” reads logarithms The symbol “log” reads logarithms to base-10 or common logarithmto base-10 or common logarithm

If 10If 10xx = = yy, then define log , then define log yy = x = x Natural logarithm Natural logarithm lnln reads to base- reads to base-ee• If If eexx = = yy, then define , then define ln yln y = = xx

Relationship b/w natural & common Relationship b/w natural & common loglog• ln xln x = ( = (lnln 10)(log 10)(log xx) = 2.3 log x) = 2.3 log x

Matrix AlgebraMatrix Algebra Formulation of many engineering Formulation of many engineering problems lead to set of linear problems lead to set of linear algebraic equations solved togetheralgebraic equations solved together

Matrix algebra essential in Matrix algebra essential in formulation & solution of these formulation & solution of these modelsmodels

A A matrixmatrix is array of numbers, is array of numbers, variables, or mathematical termsvariables, or mathematical terms

Size defined by number of Size defined by number of mm rows & rows & nn columnscolumns

MatricesMatrices Describe situations that Describe situations that require many values (vector require many values (vector variables- posses both variables- posses both magnitude & direction)magnitude & direction)

[N] = 6 5 9

1 26 14 3X3 MATRIX

-5 8 0

X = {x1 x2 x3 x4} Row Matrix

Differential CalculusDifferential Calculus

Important in determining Important in determining rate of rate of changechange in engineering problems in engineering problems

Engineers calculate rate of Engineers calculate rate of change of variables to design change of variables to design systems & servicessystems & services

If If differentiatedifferentiate a function a function describing speed, obtain describing speed, obtain acceleration acceleration

a = dv/dta = dv/dt

Integral CalculusIntegral Calculus Second moment of Second moment of inertia- property of inertia- property of area (chap. 7)area (chap. 7)

Property that provides Property that provides info on how hard to info on how hard to bend something (used bend something (used in structure design)in structure design)

For small area element For small area element AA at distance at distance xx from from axis axis y-yy-y, area moment , area moment of inertia isof inertia is

IIy-yy-y = = xx22AA

Integral CalculusIntegral Calculus For more small area elements, area For more small area elements, area moment of inertia for system of moment of inertia for system of discrete areas about y-y axis isdiscrete areas about y-y axis is

IIy-yy-y = = xx1122AA11 + x + x22

22AA22 + x + x3322AA33

Integral CalculusIntegral Calculus 22ndnd moment of inertia for cross- moment of inertia for cross-sectional area: sum area moment of sectional area: sum area moment of inertia for all little area elementsinertia for all little area elements

For continuous cross-sectional area, For continuous cross-sectional area, use integrals instead of summing use integrals instead of summing xx22AA termsterms

IIy-yy-y = = ∫ x∫ x22dAdA Formula for cross-Formula for cross-section about section about y-yy-y axes axes

IIy-yy-y = = 1/12 h w1/12 h w33

Integral CalculusIntegral Calculus

IIy-yy-y = = ∫ x∫ x22dA = ∫ xdA = ∫ x22hdx = h ∫ xhdx = h ∫ x22dx dx

Integrating Integrating IIy-yy-y = = ww33/24 + w/24 + w33/24/24 = =

-w/2 -w/2

w/2 w/2

IIy-yy-y = = 1/12 h w1/12 h w33

Integral CalculusIntegral Calculus Used to determine force exerted by water Used to determine force exerted by water stored behind damstored behind dam

Pressure increases w/ depth of fluid Pressure increases w/ depth of fluid according toaccording toP = P = gy gy wherewhere

P = fluid pressure distance P = fluid pressure distance       y below water surface       y below water surface

= density of fluid= density of fluidg = acceleration due g = acceleration due to gravityto gravityy = distance of point y = distance of point below fluid surfacebelow fluid surface

Variation of Pressure with yVariation of Pressure with y Must add pressure exerted on Must add pressure exerted on areas at various depths to areas at various depths to obtain net forceobtain net force

Consider force acting at depth Consider force acting at depth yy over small area over small area dA, or dF = dA, or dF = PdAPdA

Then use P = gyAssume const and gSubst dA = w dyIntegrate y = 0 to H

Net F = 1/2 gwH2

Integral CalculusIntegral CalculusExample 1Example 1Evaluate ∫ (3xEvaluate ∫ (3x22 – 20x)dx – 20x)dxUse rules 2 & 6 from tableUse rules 2 & 6 from table

∫∫(3x(3x22 – 20x)dx – 20x)dx= ∫3x= ∫3x22dx + ∫-20xdx = 3∫xdx + ∫-20xdx = 3∫x22dx - dx - 20∫xdx 20∫xdx = 3[1/(3)]x= 3[1/(3)]x33 – 20[1/(2)]x – 20[1/(2)]x22 + C + C

Answer = xAnswer = x33 – 10x – 10x22 + C + C

Differential EquationsDifferential Equations Contain derivatives of functions or Contain derivatives of functions or differential terms - MATH 211differential terms - MATH 211

Represent balance of mass, force, Represent balance of mass, force, energy, etc.energy, etc.

Boundary conditions tell what is Boundary conditions tell what is happening physically at boundarieshappening physically at boundaries

Know initial conditions of system at Know initial conditions of system at tt = 0 = 0

Exact solutions give detailed Exact solutions give detailed behavior of systembehavior of system