P.K. Koliopoulos, Department of Structural Engineering,. Technological ... Chopra
'Dynamics of Structures: Theory and Applications to Earthquake. Engineering' ...
FUNDAMENTALS OF STRUCTURAL DYNAMICS Original draft by Prof. G.D. Manolis, Department of Civil Engineering Aristotle University, Thessaloniki, Greece
Final draft - Presentation Prof. P.K. Koliopoulos, Department of Structural Engineering, Technological Educational Institute of Serres, Greece
• Topics : • Revision of single degree-of freedom vibration theory • Response to sinusoidal excitation • Response to impulse loading • Response spectrum • Multi-degree of freedom structures References : R.W. Clough and J. Penzien ‘Dynamics of Structures’ 1975 A.K.. Chopra ‘Dynamics of Structures: Theory and Applications to Earthquake Engineering’ 20011 G.D. Manolis, Analysis for Dynamic Loading, Chapter 2 in Dynamic Loading and Design of Structures, Edited by A.J. Kappos, Spon Press, London, pp. 31-65, 2001.
Why dynamic analysis? Æ Loads change with time Unit impulse f(t)
Harmonic load
f(t) 1/ε t
τ Ground acceleration
ε
Single degree of freedom (sdof) system u(t)
c m k
mass-spring-damper system f(t)
Μass m (kgr, tn), spring parameter k (kN/m), viscous damper parameter c (kN*sec/m), displacement u(t) (m), excitation f(t) (kN).
Definitions of restoring force parameter k
Dynamic equilibrium – D’Alembert’s principle f(t) = fI(t) + fD(t) + fS(t) Inertia force fI(t), Damping force fD(t) Restoring (elastic) force fS(t)
fI(t) fD(t)
f(t)
fS(t)
Setting response parameters as: displacement u(t) (in m), velocity u’(t) (in m/s) and acceleration u’’(t) (in m/s2), then: fI(t) = m u’’(t) , fD(t) = c u’(t) , fS(t) = k u(t) .
Shear plane frame - dynamic parameters q u C
B h
k A
D l
Rigid beam, mass less columns. Total weight (mass) accumulated in the middle of the beam. AB – Fixed end CD – Hinged end
m = w/g = (ql)/g k = fst(u=1) = VBA + VΓ∆ = 12EI/h3 + 3EI/h3 = 15EI/h3
Free vibration with no damping u
q B
C
h
k A
l
D
FI B VΒΑ
C VCD
No external force f(t). Oscillations due to initial conditions at t = 0. Initial displacement u0 or/and initial velocity u’0 m u’’(t) + k u(t) = 0 u(t) = R1 sin ωt + R2 cos ωt = R sin(ωt+θ) where R2 = R12 + R22 and tan θ = R2/R1 Natural frequency ω = [k/m]1/2 (rad/s), Nat. period T = 2π/ω (sec)
u’0 2 R
u0
1
3
t(s)
5 R
4 To = 2π/ωο 1
2
3
Unrealistic – no decay
4
5
m
Free vibration with damping
B c
u(t) Ι ∞
Γ
∆
A
Equation of motion Æ Homogeneous 2nd order-ODE: m u’’(t) + c u’(t) + k u(t) = 0 Characteristic equation
(mr2 + cr + k) = 0
and roots: r1,2 = ±
c2 k 2 m ( 2m)
2
k c 2 m ( 2m)
{
>0 =0 5.16 Modal combination rule SRSS 2 2 2 = 4.912 + 1.562 + 0.102 = 5.15 ≈ 5.16 u 31 + u 32 + u 33
THE END
