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Mechanics of Materials

by: Lora Metz

Mechanics of Materials EGN 3331

Lora Metz
University of Central Florida
GPA 3.64


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This 72 page Class Notes was uploaded by Lora Metz on Thursday October 22, 2015. The Class Notes belongs to EGN 3331 at University of Central Florida taught by Staff in Fall. Since its upload, it has received 52 views. For similar materials see /class/227561/egn-3331-university-of-central-florida in General Engineering at University of Central Florida.

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Date Created: 10/22/15
Applications of Calculus I Application of Maximum and Minimum Values and Optimization to Engineering Problems by Dr Manoj Chopra PE gy UCF EXCEL Outline 0 Review of Maximum and Minimum Values in Calculus 0 Review of Optimization 0 Applications to Engineering quot5 iquot 2 v lNSFk SQ UCF EXCEL Maximum and Minimum Values 0 You have seen these in Chapter 4 0 Some important applications of differential calculus need the determination of these values 0 Typically this involves nding the maximum and or minimum values of a Function 0 Two Types Global or Absolute or Local or Relative gy UCF EXCEL quotf 3 Local Maxima 0r Minima Fermat s Theorem If a function fx has a local maximum or minimum at c and if f C GXiStS then f C O Critical Number 0 of a function f x is number such that either f c O or it does not eXist 4 n7 YA axNSFa gy UCF EXCEL Closed Interval Method Used to nd the Absolute Global Maxima or Minima in a Closed Interval ab Find f at the critical numbers of f in ab Find f at the endpoints Largest value is absolute maximum and smallest is the absolute minimum gquot C EXCEL Engineering Demo 0 Highlights the importance of the following Understanding of Math Understanding of Physics In uence of Several Independent Variables Fun 4 n7 YA axNSFa gy UCF EXCEL Calculus Application Graphing and Finding Maxima or Minima Section 41 66 On May 7 1992 the space shuttle Endeavor was launched on mission STS49 the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite The table gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters up n YA axNSFa 39 quotv g9 UCFEXCEL Calculus Application Graphing and Finding Maxima or Minima Event Time s Velocity fts Launch 0 0 Begin roll maneuver 10 185 End roll maneuver 15 3 19 Throttle to 89 20 447 Throttle to 67 32 742 Throttle to 104 59 1325 Maximum dynamic pressure 62 1445 Solid rocket booster separation 125 4151 9 y39LNSFrq D g UCF EXCEL Shuttle Video g UCF EXCEL Calculus Application Graphing and Finding Maxima or Minima Use a graphing calculator or computer to nd the cubic polynomial that best models the velocity of the shuttle for the time interval 0 S t S 125 Then graph this polynomial Find a model for the acceleration of the shuttle and use it to estimate the maximum and minimum values of acceleration during the rst 125 seconds Q UCFEXCEL Strategy Let us use a computer program MSEXCEL to graph the variation of velocity With time for the first 125 seconds of ight after liftoff The graph is first created as a scatter plot and then a trendline is added The trendline menu allows for the selection of a polynomial fit and a cubic polynomial is picked as required in the problem description above Q UCFEXCEL Velocity fts 4500 4000 3500 3000 2500 2000 1500 1000 500 500 Shuttle Velocity Profile y 00015x3 01155x2 24982x 21269 Time s g UCFEXCEL Solution 0 From the graph the function yx or vt can be expressed as W 000153 01155t2 24982t 21269 0 Acceleration is the derivative of velocity with time dvt dz az 000452 2 0231 24982 39 1 n7 YA axNSFa gy UCF EXCEL Solution Continued During the rst 125 seconds of ight that is in the interval 0 S t S 125 apply the Closed Interval Method to the continuous function at on this interval The derivative is dat a39t 0009t 0231 The critical number occurs When a39t O whichgiVeSUS t1 zwzm 0009 seconds 99 a u Q U C F EXCEL A Solution Continued Evaluating the acceleration at the Critical Number and at the Endpoints we get a2567 220 frs2 a0 24982 s2 a125 6642 s2 Thus the maximum acceleration is 6642 fts2 and the minimum is 220 fts2 gy UCF EXCEL Calculus Application Optimization Section 47 34 0 A fence is 8 feet tall and runs parallel to a tall building at a distance of 4 feet from the building 0 What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building agNSFrq 39239 V g UCFEXCEL Calculus Application Optimization LADDER BUILDING l FENCE I 6 a l l 39 g UCF EXCEL Calculus Application Strategy From the gure using trigonometry the length of the ladder can be expressed as H D LABBC s1n 6 cos 6 Next nd the critical number for 6 for Which the length L of the ladder is minimum Differentiating L With respect to 6 and setting it equal to zero 39 1 n7 YA vi Iv gy UCF EXCEL Engineering Courses With Math Some future Engineering Courses at UCF that you may take are EGN3310 Engineering Mechanics Statics EGN3321 Engineering Mechanics Dynamics EGN 3331 Mechanics of Materials EML 3601 Solid Mechanics and several of your engineering major courses Q UCFEXCEL Use of Calculus in Engineering Realworld Engineering Applications that use Calculus Concepts such as Derivatives and Integrals Global and Local Extreme Values are often needed in optimization problems such as Structural or Component Shape Optimal Transportation Systems Industrial Applications Optimal Biomedical Applications SQ UCFEXCEL Calculus Topics Covered Global and local extreme values Critical Number Closed Interval Method Optimization Problems using Application to Engineering Problems gy UCF EXCEL Applications to Engineering 0 Maximum Range of a projectile Mechanical and Aerospace engineering 0 Optimization of Dam location on a River Civil engineering 0 Potential Energy and Stability of Equilibrium Mechanical Civil Aerospace Electrical Engineering SQ UCFEXCEL Applications to Engineering Optimal Shape of an Irrigation Channel Civil engineering Overcoming Friction and other Forces to move an Object Mechanical Aerospace Civil engineering g UCFEXCEL Application to Projectile Dynamics 0 Maximum Range for a Projectile 0 May also be applied to Forward Pass in Football 0 Goal 1 To nd the Maximum Range R of a projectile With Muzzle Discharge Velocity of 12 meters sec 0 Goal 2 Find Initial Angle of Elevation to achieve this range SQ UCFEXCEL Engineering Problem Solution 0 Gather All Given Information 0 Establish a Strategy for the Solution Collect the Tools Concepts Equations 0 Draw any FiguresDiagrams Solve the Equations 0 Report the Answer 0 Consider Is the answer Realistic g UCF EXCEL quotf Given Information 0 The Range R is a function of the muzzle velocity and initial angle of elevation R v2 sin26 g R 0 9 is the angle of elevation in radians and g is the acceleration due to gravity equal to 98 ms2 SQ UCFEXCEL Strategy We need to find the maximum value of the range R With respect to different angles of elevation Differentiate R With respect to 6 and set it to zero to find the global maXima Note that in this case v and g are constants The end points for the interval for forward motion are 0 S 6 S E 2 g UCFEXCEL Solution RV2i1129 dRv22cos2l9O g d9 g As v and g are both nonzero 2 00526 Zoos 9 1 0 cos 26 O 1 cos 9 J5 Using trigonometric double 0 49 2 angle formula g UCFEXCEL Solution Continued Evaluating the range at the Critical Value gives 2 IKE V And At the End Points 4 RO 0 R7z 2 0 Maximum range for the projectile is reached When zz or45O 4 SQ UCFEXCEL Optimizing the Shape of Structures 0 Relates to Fluid Mechanics and Hydraulics in Civil Engineering 0 Civil Engineers have to design Hydraulic Systems at Optimal Locations along Rivers 0 They also have to Optimize the Size of the Dam for Cost Constraints SQ UCFEXCEL Optimal Location of Dam Depth of Water Doc 2 20x 10 St Johns River City of Rock Sp ngs Width OfRiver Wx 10x2 8x 22 Q UCFEXCEL Example of a Dam on a River Given Constraints and Questions If the dam cannot be more than 310 feet Wide and 130 feet above the riverbed and the top of the dam must be 20 feet above the present river water surface What is a range of locations that the dam can be placed A What are the dimensions of the Widest and narrowest dam B that can be constructed in accordance With the above constraints If the cost is proportional to the product of the Width and the height of the dam Where should the most economical dam be located C SQ UCFEXCEL Strategy Use the Closed Interval Method to nd the Widest and narrowest dam in the range of acceptable locations of the dam De ne the Cost Function as proportional to the product of Width and height Minimize Cost Function With respect to the location x measured from Rock Springs g UCFEXCEL Solution A Based on the Speci ed Constraints Width must be less than 310 Wx10x2 8x22s310 gt leS9 Depth must be less than 110 Dx 20xlO s 110 gt x s 5 Range of locations for the Dam 03x35 SQ UCFEXCEL Solution B To obtain the Widest maximum W and narrowest minimum W for the dam apply the Closed Interval Method for the function WX in the interval 0 S x S 5 Wm 10x2 8x 22 Critical Value Differentiating dWOC 20x 80 0 dx x 4 g UCFEXCEL Solution B Continued Corresponding Width WM 60 feet is the Minimum Width Next checking the endpoints of the interval we obtain the following values W5 70 feet and WO 220 feet Maximum Width of the dam is 220 feet at Rock Spring x 0 SQ UCFEXCEL Solution C Cost Minimization Height of Darn must be 20 feet HIGHER than Depth of Water there Hx Dx 20 20x 30 102x 3 Cost Function is Proportional to Product of H and W Cx Fx2 8x 222x 3 Where F is a positive Constant Simplifying Cx F2x3 13x2 20x 66 SQ UCFEXCEL Solution C Continued To Find the Critical Number dCx O or dCx dx 2F3x lOx 1 0 Solving for two values of X Cheaper Dam is at x 10 3 Cost of Dam at this location 623OF Checking Endpoints at xO Cost 66F and at X 5 Cost 91F MINIMUM COST 623OF at x 103 or xlO3 Q UCFEXCEL My Current Research Areas Permeable Concrete Pavements Soil Erosion and Sediment Control Slope Stability of Soil Structures and Land lls Modeling of Structures Pile Foundations g UCFEXCEL Permeable Concrete Pavements Optimization of Water Transport Channel 0 Applies to Land Development and Surface Hydrological Engineering 0 Such applications are common in Water and Geotechnical areas of Civil Engineering 0 Part of the Overall Design of the Irrigation Channel other areas Structural design Fluid Flow Calculations and Location g UCFEXCEL Irrigation Water Transport Channel g UCF EXCEL Objective A trapezoidal channel of uniform depth d is shown below To maintain a certain volume of ow in the channel its crosssectional areaA is xed at say 100 square feet Minimize the amount of concrete that must be used to construct the lining of the channel Irrigation Channel 9 is the angle of inclination of each side The other relevant e1 b e2 dimensions are labeled on the gure g UCFEXCEL a Strategy Make Simplifying Assumptions at this level 6126226 and 6126226 Minimize the Length L of the Channel Perimeter excluding the Top surface Length SQ UCFEXCEL Solution Based on Geometry L hl b 12 d h h 2 sin 6 Since the Crosssectional Area of the Channel must 100 sq ft A 100 bd2ed 100 100 d or b e d d tanH SQ UCFEXCEL Solution Continued Wetted length Length in contact With water When full 100 d 2d L d tan6 sin6 Minimizing L as a function of 9 and d requires advanced multivariable calculus To simplify let us make a DESIGN AS SUMPTION Assume one of the two variables 91 3 Q UCFEXCEL Solution Continued Expression for L now is Lfd x d d L f d 400012 0 To get Global Minimum dd for L for 100 00 or d2 O lt d lt 5 or d 75984 i5 g UCFEXCEL Solution Continued Since f d 200 d3 gt 0 in the interval 09 oo Length of the Channel With d 75984 f75984 a 26322 is the Global Minimum SQ UCFEXCEL Soil Erosion Test Laboratory Minimizing Energy to Build Stable Systems 0 Applies to both Mechanical and Civil Engineers 0 Potential Energy is encountered in Mechanics and in Machine Design and Structural Analysis 0 Minimizing Potential Energy maintains Equilibrium State and helps in Stability SQ UCFEXCEL Example Pinned Machine Part g9 UCF EXCEL Given Information Pinned Bars form the parts of a Machine Held in place by a Spring Each Bar weighs W and has a length of L Spring is UNSTRETCHED When 06 O and in equilibrium When a 600 Q UCFEXCEL Objective 0 Find the value of the Spring Constant such that the system is in Equilibrium 0 Determine if this Equilibrium Position is Stable or Unstable Q UCFEXCEL Strategy 0 Note that the forces that do the work to generate potential energy are Weight of the Bars Force in the Spring pulling to the right 0 Express Potential Energy U as a function of the angle 06 and solve for k using dU 0 da g UCFEXCEL Solution Setting the Reference State or Datum at A Potential Energy Sum of Weight of each Bar times the Translations or movement of each Bar 39r i1 I 2Lcosa g i 39 39i l I Q UCFEXCEL Solution continued Due to the two Bars potential energy is l U1 W 5Ls1n0 W L s1n 05 WL s1n05 Change in spring length I or stretch of spring 39 v 39 522L 2Loosal Potential Energy due to Spring U 1 k2L 2 L cos 002 2 E Q UCFEXCEL Solution continued Total Potential Energy becomes Umm m U WL sin 05 2ch2 l cos 002 When in Equilibrium state Total U is in a MINIMUM state With respect to the rotation from REST STATE dU da Q UCFEXCEL 0 Solution continued Differentiating and setting equal to O dU WL cos a 4kL2 sin 01 cos 0 0 da Given that the angle is Solving for k Wcosa Wcos 6O 0289L 4L sin al cos a 4L sin 601 cos 60 W gy UCF EXCEL Stability Cheek Second Derivative of Potential Energy is an INDICATOR of Stability of the System If the Second Derivative of U is a POSITIVE Number the System is STABLE dZU 2 2 2 d 2 WLs1na4kL eosa eos as1n 05 05 WL sin 60 4ch2 cos 60 eos2 60 sin2 60 As W L and k are positive quantities the Equilibrium Position at is ltSTABLEgt gy UCFEXCEL Application to Beams Design of Beams requires knowledge of forces inside the beam Two types Shear and Bending Moment Design engineers PLOT the distribution along the beam aXis Use Derivatives to determine Maximum and Minimum values and other parameters n A 611351 1 g UCF EXCEL ff Application of Calculus to Friction and Static Equilibrium Problem 0 Friction is Important for Different Areas of Engineering ME AE CE and IE 0 This example deals With a Concept you will see shortly in Engineering Mechanics Class 0 Concepts Include Freebody Diagrams Friction Newton s Laws 3 and Equations of Equilibrium Q UCFEXCEL Forces Needed to Move a Stuck Car Man exerts a force P on the Car at an angle of a Car is Front Wheel Drive With Mass 1727 kN Driver in Car is able to Spin the Front Wheels uk 2 002 Snow behind the back tires F L 255m has built up and exerts a Force S kN g UCF EXCEL Objective Getting the Car UNSTUCK and moving requires Overcoming a Resisting Force of S 420 N 0 What angle 06 minimizes the force P needed to overcome the resistance due to the snow SQ UCFEXCEL Strategy 0 Draw Pictorial Representation of ALL forces on the Car FreeBody Diagram FBD 0 Apply Equations of Equilibrium to this FBD will learn in PHY and use in EGN Classes 0 Express P as a function of angle of push a 0 Find the Global Minimum for P in the range Oltalt9m g UCF EXCEL quotf Freebody Diagram Fgf r om DJAHEP u39h Q UCFEXCEL Solution 0 Equations of Equilibrium are applied to the FBD 0 This implies the BALANCE of all the FORCES and MOMENTS Rotations on the System SQ UCFEXCEL Equations of Equilibrium S kNF Pcosa0 NR NF W Psina0 W162NF255Pcosa09O Psina340 O NF iS Pcosa k NR NF WPsina SQ UCFEXCEL Expression for Force P angle l62W 255iS Fees 05 090Peosa 340Psina 0 k Differentiating using the Chain Rule to nd 07 da and setting it equal to 0 gives us the minimum value critical of 06 SQ UCFEXCEL Computations 23955 d Pcosa Psin a 090d Pcosa 090P sina 340d Psin 05 340P cosa 0 M da da da P 255 090sin05 34000505 amp k da 0 090 cos 0 340 sin 05 23955 cos a k SQ UCFEXCEL Minimum Value of Angle of Push 340 255 090 k 0r 0 21540 tana Q UCFEXCEL The End 0 You now know more about how Differential Calculus is used in Engineering Good Luck 0 ohopramailuefedu SQ UCF EXCEL


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