Mathematics in structural engineering

Dr Colin Caprani PhD, BSc(Eng), DipEng, CEng, MIEI, MIABSE, MIStructE Mathematics inShapes Structural Structural Engineering Loads Materials Analysis Stress Resultants Coláiste Cois Life – 5th and 6th Year 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… Mathematics in Structural Engineering Dr Colin Caprani Definition of Structural Engineering Institution of Structural Engineers: “…the science and art of designing and making with economy and elegance buildings, bridges, frameworks and other similar structures so that they can safely resist the forces to which they may be subjected” Prof Tom Collins, University of Toronto: “…the art of moulding materials we not really understand into shapes we cannot really analyze so as to withstand forces we cannot really assess in such a way that the public does not really suspect” Some examples of structural engineering… Mathematics in Structural Engineering Dr Colin Caprani Matrices II Each member in a frame has its own stiffness matrix: These are assembled to solve for the whole structure displacements Mathematics in Structural Engineering Dr Colin Caprani Matrices III LinPro Software: Displays the stiffness matrix for a member Mathematics in Structural Engineering Dr Colin Caprani Matrices IV Assembling the simple matrices for each member lets us calculate complex structures: Mathematics in Structural Engineering Dr Colin Caprani Complex Numbers I Free vibration: u&&(t ) + ω 2u (t ) = λ2 +ω2 = λ1,2 = ±iω u ( t ) = C1eλ1t + C2eλ2t u ( t ) = C1e+ iωt + C2 e−iωt Since e± iθ = cos θ ± i sin θ u ( 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 Engineering Dr Colin Caprani Complex Numbers II ⎛ u&0 ⎞ u ( t ) = u0 cos ω t + ⎜ ⎟ sin ω t ⎝ω ⎠ 30 20 Displacement (mm) k = 100 N/m 10 0 0.5 1.5 2.5 3.5 m = 10 kg -10 -20 (a) -30 (b) Tim e (s) (c) u0 = 20mm u&0 = u0 = u&0 = 50mm/s u0 = 20mm u&0 = 50mm/s Mathematics in Structural Engineering Dr Colin Caprani Complex Numbers III Are used to model complex geometries: A Function of Complex Numbers 0.8 Function value 0.6 0.4 0.2 -0.2 -0.4 20 10 20 10 0 -10 Imaginary Part -10 -20 -20 Real Part Mathematics in Structural Engineering Dr Colin Caprani Complex Numbers IV Aerofoil lift Flow Around a Circle Lift:1.0195 [N/m] Flow Around the Corresponding Airfoil Lift:1.0195 [N/m] 5 4 3 2 1 0 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -5 -5 Mathematics in Structural Engineering Dr Colin Caprani Complex Numbers V Why does the ball curl? Speed Vectors 1.5 0.5 -0.5 -1 -1.5 -2 -2 -1.5 -1 -0.5 0.5 1.5 Mathematics in Structural Engineering Dr Colin Caprani Statistics and Probability I How strong is a structure? How much load is on a structure? Mathematics in Structural Engineering Dr Colin Caprani Statistics and Probability II How strong is a beam? What is the effect of the load on the beam? Mathematics in Structural Engineering Dr Colin Caprani Statistics and Probability III What about bridges? Represents my area of interest Truck traffic on bridge Loading event data Structure Assessment Statistical analysis Mathematics in Structural Engineering Dr Colin Caprani Statistics and Probability IV Simulated bridge loading events… Mathematics in Structural Engineering Dr Colin Caprani Maths for the sake of it… Once voted the most beautiful relation in maths: iπ e +1= It links the five most important numbers in maths: e = 2.718281 π = 3.141592 i = −1 Of this, a professor once said: “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 Engineering Dr Colin Caprani Conclusion • 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 sake Thanks for listening…but one last question Mathematics in Structural Engineering Dr Colin Caprani Question If 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! ... Consultant in buildings & bridges Guess my Leaving result! C1 in Honours Maths You don’t have to be a genius… Mathematics in Structural Engineering Dr Colin Caprani Definition of Structural Engineering. . .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... in Structural Engineering Dr Colin Caprani Matrices III LinPro Software: Displays the stiffness matrix for a member Mathematics in Structural Engineering Dr Colin Caprani Matrices IV Assembling