Dynamics Flashcards
What happens to tanks during an earthquake?
The soil may liquefy if it does, the tanks can FLOAT OUT OF THE GROUND.
How is the equation G = (3-e)^2/(1+e) * (p’)^0.5 applied?
p’ is taken below the structure (and so includes the structure’s bearing pressure).
p’ = 1 + 2Ko/3 sigma’v
e is the voids ratio
A bridge has mass moment of inertia I and natural frequency omega. What is its stiffness?
Meq = \int{I * sin(x)^2}
Keq = omega^2 MeqName 3 elements of blast resistant design
Stand Off
Redundancy
Appropriate Glazing
What is admittance?
H(omega)
Where is ground displacement on a tripartite graph?
k->0 T->inf
Where is ground acc’n on a tripartite graph?
k->inf T->0
What is the column shear?
=12EI/h^3 * (u1-u2)
If there are multiple modes, need (u1-u2)*=sqrt(sum( {u1-u2}^2 ) )
What happens in partial liquefaction?
p’ lowers, so G falls, resonance may get worse.
In what kinds of soil can liquefaction occur?
Loose, saturated sands.
What is the displacemene of mode u_i in an earthquake?
= \Gamma_i * S_d * u_i
\Gamma_i is modal participation factor
S_d(\omega_i) is the displacement on the tripartite of T_i
Don’t forget u_i (the mode shape itself!) e.g. if u_i = 2 on the top floor, the top floor displacement is 2 * ug* gamma
How is the soil natural frequency calculated?
Imagine a sinusoid to bedrock. The wavelength is 4h. The speed is vs.
f = vs / 4h
How is the ductility factor defined?
\mu = u_f / u_y
u_f is failure
u_y is yield
What is eccentric bracing?
In a K-brace, the two diagonals don’t quite meet. This eccentricity allows a plastic hinge to form.
What does eccentric bracing allow?
A plastic hinge can form, dissipating energy without buckling.
Why is concentric bracing bad?
Members will buckle as the building sways one way then the other. As a result, there is no ductility.
What do helical strakes achieve?
They prevent vortices synchronising up a structure.
What is galloping?
On eg a hexagonal shape, the seperation points move when the structure sways because the apparent wind direction changes. This causes self-reinforcing swaying
What are the scaling laws for blast?
z* = z/(w^1/3)
t* = t/(w^1/3)
i* = i/(w^1/3)
i is the impulse = \int{P}dt
What is the damping ratio of a welded steel building?
1%
What is the damping ratio of a masonry building?
10%
What is SRSS?
Square Root Sum of Squares.
Name 5 solutions to liquefaction.
Air Sparging Bio-Cementation in-situ Densification improved drainage lower water table
How is concrete designed in seismic zones?
Require ductility, so ensure under-reinforced, lots of shear reinforcement.
Lots of reinforcement lapping and anchorage
Why is timber suited to seismic zones?
Nails provide ductility.
What is R_aa(\tau) for \tau = 0?
Raa(0) = 1/T * \int{a(t)*a(t) dt} = \sigma_a ^ 2
Why is spectral analysis not valid for soils?
Soils are non-linear, so we can’t just superpose spectral results.
What are flutter derivatives?
Flutter is controlled by two coupled differential equations:
m \ddot{h} + c \dot{h} + k h = c1 h + c2 \dot{h} + c3 alpha + c4 \dot{alpha}
I \ddot{alpha} + c \dot{alpha} + k alpha = c5 h + c6 \dot{h} + c7 alpha + c8 \dot{alpha}
c1 - c8 are flutter derivatives.
What is flutter?
Coupled vertical and torsional motions of flat plate-like structures.
How are flutter derivatives calculated?
By the scanlan method:
c1 - c8 are measured in a wind tunnel by applying a particular wind speed, and harmonic motions are applied (ie f is also chosen). c1 - c8 are found by measuring pressures.
Note that c1 - c8 are for a particular wind speed and frequency.
When do the flutter equations hold?
They hold true at the point of instability. We are therefore interested in the wind speed at which they hold true.
m \ddot{h} + c \dot{h} + k h = c1 h + c2 \dot{h} + c3 alpha + c4 \dot{alpha}
I \ddot{alpha} + c \dot{alpha} + k alpha = c5 h + c6 \dot{h} + c7 alpha + c8 \dot{alpha}
How can the flutter equations be simplified?
Sometimes, can set h = 0 and look at a purely torsional flutter:
I \ddot{alpha} + c \dot{alpha} + k alpha = c7 alpha + c8 \dot{alpha}
Flutter occurs when c8 > c and so there’s negative damping
Glazing options include:
Annealed, toughened, polycarbonate, laminated, and…
anti-shatter film ; curtains
Glazing options include:
Annealed, toughened, anti-shatter film, curtains, and…
polycarbonate, laminated
Discuss benefits of polycarbonate glass.
V good for blast resistance, but scratches easily and degrades in UV.
Discuss benefits of laminated glass.
2 + sheets of glass with interlayer. The interlayer can undergo large plastic strains.
Why is toughened glass better than annealed glass?
It breaks into smaller fractions.
In wind engineering, unsteady velocities lead to a static force and a fluctuating force. What is this fluctuating force?
Say U(t) = u + d(t)
u is constant.
F(t) = 1/2 rho Cd A ( U(t) )^2
= f + rho Cd A u * d(t)
How are spatial decorrelations accounted for in wind engineering?
Consider the area average of the forcing term. Then have two terms:
Sxx(omega) = mechanical admittance * aerodynamic admittance * Suu(omega)
Aerodynamic admittance <1 -> Sxx(omega) reduced due to area effects. Happens when length scale of fluctuations similar to length scale of building ( sqrt{A} )
A cantilever of length L has a lumped mass at the top. Modelling it as a third order polynomial, what is a sensible mode shape?
The mode shape needs to be compatible with B.C.s, which are u(0) = u’(0) = 0
and u’‘(L) = 0 due to no B.M.
u’’‘(L) =/= 0 because shear force is needed to move the lumped mass.
A cantilever of length L has uniform mass per unit length. Could it be modelled as a third order polynomial?
The x^3 term would have to be zero because the shear force at the top of the tower is zero, so u’’‘(L) = 0.
A cantilever is modelled with u(z) = z/L + z^2/L^2.
This finds that Keq = 5 ; Feq = 1.
What is the static displacement at the top of the tower?
d_st = u(L) * Feq/Keq
= 2 * 1 / 5 = 0.4
Is u = z a suitable Rayleigh guess for a cantilever?
No. The B.C.s are u(0)=u’(0)=u’‘(L)=u’’‘(L)=0
This doesn’t satisfy the B.C.s
What are the B.C.s for the Rayleigh estimate of a cantilever with uniform mass?
u(0) = u'(0) = 0 u''(L) = u'''(L) = 0
Is u = z^2 a suitable Rayleigh estimate for a cantilever with uniform mass?
Strictly, no. The B.C.s are u’‘(L) = 0
But u(z) = z^2 gives u’‘(z) = 2, which is non-zero.
How does wind engineering differ from earthquake engineering?
It takes account of SPATIAL DECORRELATION of forces.
What is the stiffness of a double encastre beam under shear?
12 EI / h^3
Found from the end moments = 6 EI / h^2
Take moments about one end,
F * h = 2 * 6 EI / h^2
What is the stiffness of an encastre-pinned beam under shear?
3EI/h^3 from data book.
What is wrong with the equation
delta_ static = Feq / Keq
?
delta_static = u(z) * Feq / Keq
Need to account for magnitude of the mode shape!
A structure of mass M_1
moves with soil of mass M_s. The sructure has stiffness K_1, and the soil K_s. What is the equivalent 2DOF oscillator?
A spring K_s connects to M_s + M_f, where M_f is foundation mass.
A second spring 12EI/h^3 connects to M_1
Why is partial liquefaction bad?
It reduces G, making resonance worse.
Why does galloping occur in cables that are wet?
The water forms rivulets, two lumps towards the leeward side of the cable. If unable to visualise, Google ‘ rivulets cable’.
What is static divergence?
The wind-induced moment (prop to alpha) exceeds the restoring moment (also prop to alpha), and so the structure is unstable in a torsional mode and fails abruptly.
What is aerodynamic admittance?
X_aero relates the wind speed to a force.
X_mech then relates the force to e.g. a bending moment.
Hence:
M(w) = |Xaero(w)|^2 * |X_mech(w)|^2 * U(w)
M is the spectrum of the bending moment.
I have found the response spectrum
X(w) = ｜H(w)｜^2 * ｜A(w)｜^2 * U(w)
What comes next?
It is typical to design for the mean response + 3 or 3.5 * s.d.
The s.d. is the square root of the area under the curve X(w)
Wind engineering differs from earthquake engineering because the wind speeds vary with space as well as time. What impact does this have?
1) The structure has an aerodynamic admittance as well as a structural admittance.
2) Everything needs to be mode-generalised because the load varies with position and so affects different modes differently.
3) Design for X_avg + 3 * sigma_x
In wind engineering, there are TWO ways of thinking about unsteadiness. Expalin them both.
- The wind speed = U + u(t)
so F = A* 1/2 rho Cp V^2 = |F| + A * U * rho * Cp * u(t) - The force = X + sigma_x
Design for Xd = X + g * simga_x
When does SELBERG predict that flutter is a major issue?
Selberg says that Ucrit \propto sqrt( w_tor^2 - w_vert^2 )
So if the torsional vibration and vertical vibration frequencies are the same, we have a problem.
What is the Discrete Vortex Method?
A computer throws thousands of vortices at the bridge, and keeps track of them. This allows any fluid behaviour to be modelled (including von Karman vortex streets etc.)
What is the name given to placing a model in a wind tunnel and obtaining flutter coefficients by varying wind speed and vibration frequency?
The Scanlan method.
The scanlan method derives flutter coefficients in a wind tunnel. On what two parameters do the flutter coefficients depend?
They depend on wind speed AND vibration frequency.
One way to calculate base shear is to say that:
V = k1 * u1 = k1 * \Gamma * Sd * u1
What is another way?
V = \Gamma ^2 * Meq * Sa
Is sin( \pi x /(2L) ) a good model for a cantilever beam?
No! it has d/dx (x = 0) = 1. That does not satisfy the clamped condition!
What is a good model for a cantilever beam?
1 - cos ( \pi x / (2L) )
OR
2/3 mass at 2/3 height (and model as D.B. cantilever). The remaining mass is at the bottom.
Helical strakes are good at reducing the severity of vortex-induced vibration. What is a disadvantage?
They increase the static drag.
What is the name of the computational method in which thousands of vortices are simulated interacting with a structure?
The Discrete Vortex Method.
What is the scanlan method?
Place structure in wind tunnel, simulate flutter to obtain flutter derivatives. Flutter derivatives are functions of wind speed and vibration frequency.
Note that we need to consider dimensional analysis in order to scale the model appropriately.
What needs to be done for the Scanlan method to predict flutter derivatives for a full-scale structure?
Need to non-dimensionalise, so that scale model reflects true structure.
What two spectrums does X_aero relate?
The oscillatory wind speed spectrum to the force spectrum.
For large areas, it is LESS than rho * Cd * U * A
What is the value of X_aero for small areas?
It relates buffetting wind velocity to force, so has the form rho * Cd * |U| * A
|U| is the average wind velocity.
A lumped mass model places 2/3 of the mass at a height 2/3 of the original length. What else needs to happen?
The remaining mass is at the base, and so will increase the mass of whatever the structure is sitting on.
How is a lumped mass model used?
Consider a cantilever of height h= 2L/3, with k = 3EI/h^3
The equivalent mass is 2M/3 at the tip, and 1M/3 at the base (which rarely matters)
What is the S.D.O.F. flutter equation?
I a’’ + c a’ + k a = c1 a’ + c2 a
If c1 > c, then we have negative damping and flutter occurs.
A structure has yield limit u_y but ductility m. What is the equivalent elastic limit?
u_e = u_y * sqrt( 2*m - 1 )
I have used design spectra to check a ductile building in an earthquake zone. This has issues, including
- Don’t account for structure-soil interaction
- Higher order modes generally ignored
- Brittle connections / Localised damage
And what else?
- Design spectra are averages
A ductile design spectra is not accurate because it is an average, doesn’t account for structure-soil interaction, and ignores higher modes. What else?
Localised damage might lead it to be unconservative.
In S.D.O.F. flutter, we have an equation like:
I a’’ + c a’ + k a = c1 a’ + c2 a
At what point does flutter occur?
When c1 > c and so we get negative damping.
A 2 d.o.f. system has k1, m1, k2, m2. What is the equivalent mass matrix?
Meq = m1 u1^2 + m2 u2^2
A 2 d.o.f. system has k1, m1, k2, m2. What is the equivalent stiffness?
Keq = k1 * u1^2 + k2 * (u2-u1)^2
Using a matrix expansion gives the same answer.
A 3 d.o.f. system has k1, m1, k2, m2, etc. What is the equivalent stiffness matrix?
Keq = k1 * u1 ^2 + k2 * (u2-u1)^2 + k3 * (u3-u2)^2
A second order equation has the form
Ax’’ + Bx’ + Cx = 0
When is this stable? Unstable?
B > 0 is stable (can verify in D.B.
B < 0 is unstable.
What is the displacement of the first floor under earthquake loading if the mode shape is:
[first floor, second floor] = [0.5 , 1]
0.5 * \Gamma * S_d
The displacement of a floor under earthquake loading is:
\Gamma * S_d * u_i
How is this independent of the size of u_i?
\Gamma effectively has units of 1/mode shape size, and so this expression is independent of mode shape size.
What might the discrete vortex method be able to pick up on?
von Karman vortices ; buffetting (by releasing vortices randomly) ; Galloping.
The Messina bridge has two decks. Which wind engineering method is suitable to predict its behaviour?
The discrete vortex method.
The assumption of a tri-partite graph is that
w^2 Sd = w Sv = Sa
This is true for elastic, lightly damped structures under what condition?
It doesn’t work for low frequencies.
The assumption of a tri-partite graph is that
w^2 Sd = w Sv = Sa
which breaks down at low frequencies. When else does it break down?
When there is damping, or the structure is inelastic.
A two-storey structure has a stiffness 2k to the ground, and k between the first and second stories. What is the stiffness matrix?
[3k -k
-k k]
A two-storey structure has a stiffness k to the ground, and k between the first and second stories. What is the stiffness matrix?
[2k -k
-k k]
A 2dof structure has k1, k2. What is the stiffness matrix?
[k1 + k2 ; -k2
-k2 ; k2]
What is the key step to showing that u_i and u_j are orthgonal w.r.t. to the stiffness matrix?
When taking the transpose of
u2T M u1
Note that M is symmetric so MT = M and we get
u1T M u2
What shape is a hurricane wind?
Air flows inwards towards the eye and rises just outside the eye. This is driven by the latent heat released when it condenses.
What shape is a thunderstorm wind?
Air sinks in the centre and flows outwards. The ‘first gust’ is a long and continuous wind associated with this flow.
What is ‘first gust’?
A long and continuous wind associated with an incoming thunderstorm. It is due to air sinking and flowing outwards from the centre of the storm.
Why does air sink in the centre of a thunderstorm?
Viscous drag of raindrops.
Why does air just outside the eye of a hurricane?
Water evaporates from sea and rises.
What is a Foehn/Bora wind?
Wind goes over a hill. It expands and cools over the top, often raining. On the far side, it compresses and warms and a strong wind is experienced on the far side.
What is D’Alambert’s paradox?
In inviscid theory, there is no flow seperation and no viscosity. Therefore, form and friction drag are zero. There is still drag on such an object, because of flow seperation (more important) and viscosity (less important).
How does FE apply to a suspension bridge?
Need to account for non-linear tension-stiffening of self-weight in cables. Therefore, non-linear static model of deadweight, then linear dynamic model of stiffened bridge for eigen modes.
What shape is the pressure wave of a high explosive blast?
Very high positive pressures decaying rapidly (right triangle), then small negative pressures as air is sucked back in. These negative pressures are small, but td is much greater and so they can be damaging.
What shape is the pressure wave of a gas or dust explosion?
A low ‘poisson-like’ curve that increases to a peak and then gradually decays.
