Mathematics in Structural Engineering - colincaprani.com

1 Dr Colin CapraniPhD, BSc(Eng), DipEng, CEng, MIEI, MIABSE, MIStructECol iste Cois Life 5th and 6th YearStructural AnalysisMaterialsStress ResultantsLoadsShapesMathematicsin Structural EngineeringMathematics in Structural EngineeringDr Colin CapraniAbout Me Degree in Structural Engineering 1999 Full time consultancy until 2001 PhD in UCD from 2001 to 2006 Lecturing in DIT and UCD Consultant in buildings & bridgesGuess my Leaving result!C1 in Honours MathsYou don t have to be a in Structural EngineeringDr Colin CapraniDefinition of Structural EngineeringInstitution of Structural Engineers: ..the science and art of designingand making with economyand elegance buildings, bridges, frameworks and other similar structuresso that they can safely resistthe forcesto which they may be subjected Prof.

2 Tom Collins, University of Toronto: ..the art of moulding materialswe do not really understand into shapeswe cannot really analyzeso as to withstand forceswe cannot really assess in such a way that the public does not really suspect Some examples of Structural in Structural EngineeringDr Colin CapraniImportant Maths TopicsEssential maths topics are:1. Algebra2. Calculus differentiation and integration3. Matrices4. Complex numbers5. Statistics and probabilityFor each of these, I ll give an example of its in Structural EngineeringDr Colin CapraniAlgebraHow stiff should a beam be?For a point load on the centre of a beam we will work it in Structural EngineeringDr Colin CapraniCalculus IBeam deflection:Given the bending in a beam, can we find the deflection?

3 In Structural EngineeringDr Colin CapraniCalculus IIVibration of structuresappliedstiffnessdampinginertia FFF F=++stiffnessdampinginertiaFFFkucumu===& &&() () () ()mu tcu tku tF t++=&& &Fundamental Equation of Motion: Mathematics in Structural EngineeringDr Colin CapraniMatrices IIn Structural frames displacement is related to forces:= FK Force VectorDisplacement VectorStiffness MatrixTo solve, we pre-multiply each side by the inverse of the stiffness matrix:111 = = = KFKK I KFMathematics in Structural EngineeringDr Colin CapraniMatrices IIEach member in a frame has its own stiffness matrix:These are assembled to solve for the whole structure displacementsMathematics in Structural EngineeringDr Colin CapraniMatrices IIILinPro Software: Displays the stiffness matrix for a memberMathematics in Structural EngineeringDr Colin CapraniMatrices IVAssembling the simple matrices for each member lets us calculate complex structures: Mathematics in Structural EngineeringDr Colin CapraniComplex Numbers IFree vibration.

4 2()() 0utut +=&&220 +=1,2i = ()()()12cossincossincossinutC titC titAtB t =++ =+()00cossinuut utt =+ &()12ititut CeCe + =+()1212ttut CeCe =+Sincecos siniei = e (s)Displacement (mm)(a)(b)(c)m= 10 kgk= 100 N/m Mathematics in Structural EngineeringDr Colin CapraniComplex Numbers II()00cossinuut utt =+ &00u=050mm/su=&020mmu=050mm/su=&00u=&020 mmu= Mathematics in Structural EngineeringDr Colin CapraniComplex Numbers IIIAre used to model complex PartA Function of Complex NumbersImaginary PartFunction valueMathematics in Structural EngineeringDr Colin CapraniComplex Numbers IVAerofoil lift-505-5-4-3-2-1012345 Flow Around a Circle. [N/m]-505-5-4-3-2-1012345 Flow Around the Corresponding Airfoil.

5 [N/m] Mathematics in Structural EngineeringDr Colin CapraniComplex Numbers VWhy does the ball curl?Speed in Structural EngineeringDr Colin CapraniStatistics and Probability IHow strong is a structure?How much load is on a structure? Mathematics in Structural EngineeringDr Colin CapraniStatistics and Probability IIHow strong is a beam?What is the effect of the load on the beam? Mathematics in Structural EngineeringDr Colin CapraniStatistics and Probability IIIWhat about bridges?Loading event dataStructure AssessmentTruck traffic on bridgeStatistical analysisRepresents my area of interestMathematics in Structural EngineeringDr Colin CapraniStatistics and Probability IVSimulated bridge loading in Structural EngineeringDr Colin CapraniMaths for the sake of voted the most beautiful relation in maths:10ie +=It links the five most important numbers in === Of this, a professor once said.

6 It is surely true, it is paradoxical, we can t understand it, and we don t know what it means, but we have proved it, and therefore we know it is the truth Mathematics in Structural EngineeringDr Colin CapraniConclusion All designed objects require Mathematics to describe them I ve just shown you my area of Structural Engineering Maths is essential for any profession involved in technical design It can also be enjoyable for its own sakeThanks for one last questionMathematics in Structural EngineeringDr Colin CapraniQuestionIf there are 23 people in a room, what are the chances two of them share a birthday?a) Over 80%b) Over 50%c) Over 20%d) Almost zilch!