Time History Analysis: MDF Frame

import ema as em
import matplotlib.pyplot as plt
import numpy as np
%config InlineBackend.figure_format = 'svg' # used to make plots look nicer
from ema_examples.dynamics import P09_07
from ema.utilities.ipyutils import disp_sbs

ft = 12
h = 12*ft
m = 80/386.4
E = 29000
I = 1000
EI = E*I
RHA = False

N = I = 3
mdl = P09_07(h = h, m = m, EI = EI)
# fig, ax = plt.subplots()
# em.plot_structure(mdl, ax)

m, k = em.Mass_matrix(mdl), em.K_matrix(mdl)
k, m = em.analysis.StaticCondensation(k, m)
disp_sbs(m.df, k.df)

The function ema.analysis.ModalAnalysis wraps a scipy routine for solving the eigenvalue problem.

freq2, Phi = em.analysis.ModalAnalysis(mdl, norm='last')
Phi

array([[ 5.00000000e-01, -1.00000000e+00,  5.00000000e-01],
       [ 8.66025404e-01,  2.28035323e-16, -8.66025404e-01],
       [ 1.00000000e+00,  1.00000000e+00,  1.00000000e+00]])

omega = np.array([np.sqrt(np.real(freq)) for freq in freq2])
omega

array([17.36843442, 47.45144529, 64.81987972])

M = Phi.T@m@Phi
K = Phi.T@k@Phi

Modal expansion of earthquake forces

\[\mathrm{p}_{\mathrm{eff}}(t)=-\mathrm{m} \iota \ddot{u}_{g}(t)\]

\[\mathbf{m} \iota=\sum_{n=1}^{N} \mathbf{s}_{n}=\sum_{n=1}^{N} \Gamma_{n} \mathbf{m} \phi_{n}\]

I = N = 3
iota = np.ones(I)
L = np.array([sum(Phi.T[n,i]*sum(m[i,j]*iota[j] for j in range(I)) for i in range(I)) for n in range(N)])
L = Phi.T@m@iota
L

Structural_Vector([ 0.38634066, -0.10351967,  0.02773801])

gamma = np.array([L[n]/M[n,n]  for n in range(N)])
gamma

array([ 1.24401694, -0.33333333,  0.0893164 ])

s = np.array([gamma[n]*(m@Phi.T[n]) for n in range(N)]).T
s

array([[ 1.28780221e-01,  6.90131125e-02,  9.24600388e-03],
       [ 2.23053886e-01, -1.57374274e-17, -1.60145485e-02],
       [ 1.28780221e-01, -3.45065562e-02,  9.24600388e-03]])

a) Determine \(A_n\) and \(D_n\)

# Values read from response spectrum:
D = np.array([0.877, 0.10, 0.04]) # inches
D

array([0.877, 0.1  , 0.04 ])

# if RHA:
D = []
u = []
for i, w in enumerate(omega):
    zeta = 0.05
    t, d = em.analysis.ElcentroRHA(zeta, w)
    D.append(max(d))
    u.append([t,d])
print(D)

[0.923352660303864, 0.09304519274915218, 0.03763691127115581]

Plot modes:

fig2, ax2 = plt.subplots()
em.plot_structure(mdl, ax2)
for i in range(3):
    plt.plot(10*u[i][0],200+300*u[i][1], linewidth=0.5)
plt.show()

A = np.array([D[n]*omega[n]**2 for n in range(N)])
A

array([278.54088513, 209.50424621, 158.13587895])

b) Modal response quantities

Floor displacements

Un = np.array([[gamma[n]*Phi[i,n]*D[n] for n in range(N)]for i in range(I)])
Un

array([[ 5.74333174e-01,  3.10150642e-02,  1.68079666e-03],
       [ 9.94774237e-01, -7.07253019e-18, -2.91122522e-03],
       [ 1.14866635e+00, -3.10150642e-02,  3.36159333e-03]])

Story shears

Vin = np.array([[sum(s[j,n]*A[n] for j in range(i, I)) for n in range(N)] for i in range(I)])
Vin

array([[133.87074037,   7.22927006,   0.3917752 ],
       [ 98.0001836 ,  -7.22927006,  -1.07034975],
       [ 35.87055677,  -7.22927006,   1.46212495]])

Floor and base moments

M_base = np.array([sum(s[i,n]*h*(i+1)*A[n]  for i in range(I)) for n in range(N)])
M_base # kip-inch

array([38554.77322731, -1041.01488801,   112.8312575 ])

H = [h*(i+1) for i in range(I)]
H

[144, 288, 432]

M_floor = np.array([[sum((H[j]-h*(i+1))*s[j,n]*A[n] for j in range(i,N)) for n in range(N)] for i in range(I)])
M_floor # kip-inch

array([[19277.38661365, -2082.02977602,    56.41562875],
       [ 5165.36017531, -1041.01488801,   210.54599284],
       [    0.        ,     0.        ,     0.        ]])

c) Peak modal response combination

For well-seperated modal frequencies, the SRSS method is employed.

def ro(rno):
    return np.sqrt(sum(rn**2 for rn in rno))

Floor displacements

text

ro(Un.T)

array([0.57517246, 0.9947785 , 1.14908991])

Story shears

ro(Vin.T)

array([134.06636775,  98.27229508,  36.62099122])

Floor and base overturning moments

ro(M_base)

38568.98989471707

ro(M_floor)

array([19957.41918167,  2327.78005518,   217.97325126])