Mathematics in structural engineering
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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 do 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
Important Maths Topics
Essential maths topics are:
1. Algebra
2. Calculus – differentiation and integration
3. Matrices
4. Complex numbers
5. Statistics and probability
For each of these, I’ll give an example of its application…
Mathematics in Structural Engineering
Dr Colin Caprani
Algebra
How stiff should a beam be?
For a point load on the centre of a beam we will work it out…
-100.00
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0
-0.042174
1
Mathematics in Structural Engineering
Dr Colin Caprani
Calculus I
Beam deflection:
Given the bending in a beam, can we find the deflection?
-100.00
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1
0.00
150.00
Mathematics in Structural Engineering
Dr Colin Caprani
Calculus II
Vibration of structures
Fapplied = Fstiffness + Fdamping + Finertia
Fstiffness = ku
Fdamping = cu&
Finertia = mu&&
Fundamental Equation of Motion:
mu&&(t ) + cu& (t ) + ku (t ) = F (t )
Mathematics in Structural Engineering
Dr Colin Caprani
Matrices I
In structural frames displacement is related to forces:
F = K ⋅δ
Force
Vector
Stiffness
Matrix
Displacement
Vector
To 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 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 ) = 0
λ2 +ω2 = 0
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
0.5
1
1.5
2
2.5
3
3.5
4
m = 10 kg
-10
-20
(a)
-30
(b)
Tim e (s)
(c)
u0 = 20mm u&0 = 0
u0 = 0
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
1
0.8
Function value
0.6
0.4
0.2
0
-0.2
-0.4
20
10
20
10
0
0
-10
Imaginary Part
-10
-20
-20
Real Part
