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Dr Colin CapraniPhD, BSc(Eng), DipEng, CEng, MIEI, MIABSE, áiste Cois Life – 5th and 6th Year
Mathematics 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 genius
Mathematics in Structural EngineeringDr Colin CapraniDefinition of Structural EngineeringInstitution of Structural Engineers:“ the science and art of designing and making with economy andelegance buildings, bridges, frameworks and other similar structures sothat they can safely resist the forces to which they may be subjected”Prof. Tom Collins, University of Toronto:“ the art of moulding materials we do not really understand into shapeswe cannot really analyze so as to withstand forces we cannot really assessin such a way that the public does not really suspect”Some examples of structural engineering
Mathematics 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 application
Mathematics in Structural EngineeringDr Colin CapraniAlgebraHow stiff should a beam be?For a point load on the centre of a beam we will work it out -100.0000-0.0421741
Mathematics in Structural EngineeringDr Colin CapraniCalculus IBeam deflection:Given the bending in a beam, can we find the deflection?-100.00000.0010.00150.00
Mathematics in Structural EngineeringDr Colin CapraniCalculus IIVibration of structuresFapplied Fstiffness Fdamping FinertiaFstiffness kuFdamping cu&Finertia mu&&Fundamental Equation of Motion:mu&&(t ) cu& (t ) ku (t ) F (t )
Mathematics in Structural EngineeringDr Colin CapraniMatrices IIn structural frames displacement is related to forces:F K δForceVectorStiffnessMatrixDisplacementVectorTo solve, we pre-multiply each side by the inverse of the stiffness matrix:K 1 F K 1K δ I δ δ K 1 F
Mathematics in Structural EngineeringDr Colin CapraniMatrices IIEach member in a frame has its own stiffness matrix:These are assembled to solve for the whole structure displacements
Mathematics in Structural EngineeringDr Colin CapraniMatrices IIILinPro Software:Displays the stiffnessmatrix for a member
Mathematics in Structural EngineeringDr Colin CapraniMatrices IVAssembling the simple matrices for each member lets us calculate complexstructures:
Mathematics in Structural EngineeringDr Colin CapraniComplex Numbers IFree vibration:u&&(t ) ω 2u (t ) 0λ2 ω2 0λ1,2 iωu ( t ) C1eλ1t C2eλ2tu ( t ) C1e iωt C2 e iωtSincee iθ cos θ i sin θu ( t ) C1 ( cos ωt i sin ωt ) C2 ( cos ωt i sin ωt ) A cos ωt B sin ωt u&0 u ( t ) u0 cos ω t sin ω t ω
Mathematics in Structural EngineeringDr Colin CapraniComplex Numbers II u&0 u ( t ) u0 cos ω t sin ω t ω 3020Displacement (mm)k 100 N/m10000.511.522.533.54m 10 kg-10-20(a)-30(b)Tim e (s)(c)u0 20mm u&0 0u0 0u&0 50mm/su0 20mm u&0 50mm/s
Mathematics in Structural EngineeringDr Colin CapraniComplex Numbers IIIAre used to model complex geometries:A Function of Complex Numbers10.8Function value0.60.40.20-0.2-0.42010201000-10Imaginary Part-10-20-20Real Part
Mathematics in Structural EngineeringDr Colin CapraniComplex Numbers IVAerofoil liftFlow Around a Circle. Lift:1.0195 [N/m]Flow Around the Corresponding Airfoil. Lift:1.0195 [N/m]554433221100-1-1-2-2-3-3-4-4-5-505-5-505
Mathematics in Structural EngineeringDr Colin CapraniComplex Numbers VWhy does the ball curl?Speed Vectors21.510.50-0.5-1-1.5-2-2-1.5-1-0.500.511.52
Mathematics 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 abeam?What is theeffect of the loadon the beam?
Mathematics in Structural EngineeringDr Colin CapraniStatistics and Probability IIIWhat about bridges?Represents my area of interestTruck traffic on bridgeLoading event dataStructure AssessmentStatistical analysis
Mathematics in Structural EngineeringDr Colin CapraniStatistics and Probability IVSimulated bridge loading events
Mathematics in Structural EngineeringDr Colin CapraniMaths for the sake of it Once voted the most beautiful relation in maths:iπe 1 0It links the five most important numbers in maths:e 2.718281.π 3.141592.i 110Of this, a professor once said:“it is surely true, it is paradoxical,we can’t understand it, and wedon’t know what it means, but wehave proved it, and therefore weknow 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 listening but one last question
Mathematics 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!
Mathematics in Structural Engineering Dr Colin Caprani About 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 & bridges Guess my Leaving result! C1 in Honours Maths You don't have to be a genius
