Truss Domes
6 min read • 1,272 wordsSeveral periodic truss domes are generated using OpenSeesRT.
A variety of periodic truss structures are investigated. This class of
models was investigated by .
All figures have been produced with
veux
.
This model is investigated by .
model = create_dome("120")
This model is investigated by
This model is investigated by
This model is investigated by
The following code block contains the source code used to generate these
models. In particular, the function revolve() takes a representative segment
and generates a full model by revolving the nodes and elements.
# ===----------------------------------------------------------------------===//
#
# OpenSees - Open System for Earthquake Engineering Simulation
# Structural Artificial Intelligence Laboratory
# gallery.stairlab.io
#
# ===----------------------------------------------------------------------===//
"""
References
==========
Koohestani, K., and A. Kaveh.
“Efficient Buckling and Free Vibration Analysis of Cyclically Repeated Space Truss Structures.”
Finite Elements in Analysis and Design 46, no. 10 (October 2010): 943–48. https://doi.org/10.1016/j.finel.2010.06.009.
"""
from math import pi, sin, cos, sqrt
import opensees.openseespy as ops
def create_dome(design: str, *args, **kwds):
return _DOMES[design](*args, **kwds)
def dome52():
"""
References
==========
Degertekin, S.O., G. Yalcin Bayar, and L. Lamberti.
“Parameter Free Jaya Algorithm for Truss Sizing-Layout Optimization under Natural Frequency Constraints.”
Computers & Structures 245 (March 2021): 106461. https://doi.org/10.1016/j.compstruc.2020.106461.
"""
pass
def dome120():
"""
Create a segment of the 120-bar dome. Use with revolve() to generate
a full structural model as follows:
nodes, elems = revolve(*dome120())
References
==========
Lieu, Qui X., Dieu T. T. Do, and Jaehong Lee.
“An Adaptive Hybrid Evolutionary Firefly Algorithm for Shape and Size Optimization of Truss Structures with Frequency Constraints.”
Computers & Structures 195 (January 15, 2018): 99–112. https://doi.org/10.1016/j.compstruc.2017.06.016.
Kaveh, A., and M. Ilchi Ghazaan.
“Optimal Design of Dome Truss Structures with Dynamic Frequency Constraints.”
Structural and Multidisciplinary Optimization 53, no. 3 (March 1, 2016): 605–21. https://doi.org/10.1007/s00158-015-1357-2.
"""
# Number of repeated segments
s = 12
# Number of nodes per segment
n = 4
scale = 1/100
r1 = 273.26*scale
r2 = 492.12*scale
r = 625.59*scale
h = 275.59*scale
h1 = 196.85*scale
h2 = 118.11*scale
c2 = cos(pi/s)
s2 = sin(pi/s)
nodes = {
0: ( 0 , 0.0 , h ),
1: ( r1, 0.0 , h1),
2: ( r2, 0.0 , h2),
3: ( r , 0.0 , 0 ),
4: (r2*c2, r2*s2, h2),
}
elems = [
(0, 1),
(1, 2),
(2, 3),
(3, 4),
(2, 4),
(1, 4),
(1+n, 4),
(1+n, 1),
(3+n, 4),
(2+n, 4),
(1+n, 4)
]
return nodes, elems, {3}, s, {0}
def dome600():
"""
Create a segment of the 600-bar dome. Use with revolve() to generate
a full structural model as follows:
nodes, elems = revolve(*dome600())
References
==========
Kaveh, Ali, Kiarash Biabani Hamedani, and Bamdad Biabani Hamedani.
“Optimal Design of Large-Scale Dome Truss Structures with Multiple Frequency Constraints Using Success-History Based Adaptive Differential Evolution Algorithm.”
Periodica Polytechnica Civil Engineering, September 28, 2022. https://doi.org/10.3311/PPci.21147.
"""
# Number of repeated segments
s = 24
# Number of nodes per segment
n = 9
nodes = {
1: ( 1.0, 0.0, 7.0) ,
2: ( 1.0, 0.0, 7.5) ,
3: ( 3.0, 0.0, 7.25) ,
4: ( 5.0, 0.0, 6.75) ,
5: ( 7.0, 0.0, 6.0) ,
6: ( 9.0, 0.0, 5.0) ,
7: (11.0, 0.0, 3.5) ,
8: (13.0, 0.0, 1.5) ,
9: (14.0, 0.0, 0.0)
}
elems = [
(1, 2),
(1, 3),
(1, 1+n),
(1, 2+n),
(2, 3),
(2, 2+n),
(3, 2+n),
(3, 3+n),
(3, 4),
(4, 3+n),
(4, 4+n),
(4, 5),
(5, 4+n),
(5, 5+n),
(5, 6),
(6, 5+n),
(6, 6+n),
(6, 7),
(7, 6+n),
(7, 7+n),
(7, 8),
(8, 7+n),
(8, 8+n),
(8, 9),
(9, 8+n)
]
return nodes, elems, {9}, s, {}
def dome1180():
"""
Create a segment of the 1180-bar dome. Use with revolve() to generate
a full structural model as follows:
nodes, elems = revolve(*dome1180())
Kaveh, Ali, Kiarash Biabani Hamedani, and Bamdad Biabani Hamedani. “Optimal Design of Large-Scale Dome Truss Structures with Multiple Frequency Constraints Using Success-History Based Adaptive Differential Evolution Algorithm.” Periodica Polytechnica Civil Engineering, September 28, 2022. https://doi.org/10.3311/PPci.21147.
Kaveh, Ali, and M. Ilchi Ghazaan. “Optimal Design of Dome Truss Structures with Dynamic Frequency Constraints.” Structural and Multidisciplinary Optimization 53, no. 3 (March 1, 2016): 605–21. https://doi.org/10.1007/s00158-015-1357-2.
"""
s = 20
n = 20
nodes = {
1: ( 3.1181, 0.0 , 14.6723),
2: ( 6.1013, 0.0 , 13.7031),
3: ( 8.8166, 0.0 , 12.1354),
4: (11.1476, 0.0 , 10.0365),
5: (12.9904, 0.0 , 7.5000),
6: (14.2657, 0.0 , 4.6358),
7: (14.9179, 0.0 , 1.5676),
8: (14.9179, 0.0 , -1.5677),
9: (14.2656, 0.0 , -4.6359),
10: (12.9903, 0.0 , -7.5001),
11: ( 4.5788, 0.7252, 14.2657),
12: ( 7.4077, 1.1733, 12.9904),
13: ( 9.9130, 1.5701, 11.1476),
14: (11.9860, 1.8984, 8.8165),
15: (13.5344, 2.1436, 6.1013),
16: (14.4917, 2.2953, 3.1180),
17: (14.8153, 2.3465, 0.0),
18: (14.4917, 2.2953, -3.1181),
19: (13.5343, 2.1436, -6.1014),
20: ( 3.1181, 0.0 , 13.7031)
}
elems = [
(i, i+1) for i in range(1, 10)
] + [
(i, i+n) for i in range(1, 10)
] + [
(i, i+9) for i in range(2, 10)
] + [
(i, i+10) for i in range(2, 10)
] + [
(i+n, i+9) for i in range(2, 10)
] + [
(i+n, i+10) for i in range(2, 10)
]
return nodes, elems, {None}, s, {}
def dome1410():
"""
Create a segment of the 1410-bar dome. Use with revolve() to generate
a full structural model as follows:
nodes, elems = revolve(*dome1410())
Koohestani, K., and A. Kaveh.
“Efficient Buckling and Free Vibration Analysis of Cyclically Repeated Space Truss Structures.” Finite Elements in Analysis and Design 46, no. 10 (October 2010): 943–48. https://doi.org/10.1016/j.finel.2010.06.009.
"""
s = 30
n = 13
nodes = {
1: ( 1.0,0.0,4.0) ,
2: ( 3.0,0.0,3.75) ,
3: ( 5.0,0.0,3.25) ,
4: ( 7.0,0.0,2.75) ,
5: ( 9.0,0.0,2.0) ,
6: (11.0,0.0,1.25) ,
7: (13.0,0.0,0.0),
8: ( 1.989,0.209, 3.0),
9: ( 3.978,0.418,2.75),
10: ( 5.967,0.627,2.25),
11: ( 7.956,0.836,1.75),
12: ( 9.945,1.0453,1.0),
13: (11.934,1.2543,-0.5)
}
elems = [
( 1, 2),
( 2, 3),
( 3, 4),
( 4, 5),
( 5, 6),
( 6, 7),
( 8, 9),
( 9, 10),
(10, 11),
(11, 12),
(12, 13),
( 8+n, 8),
( 9+n, 9),
(10+n, 10),
(11+n, 11),
(12+n, 12),
(13+n, 13),
( 7, 13),
( 7+n, 13)
]
for i in range(2, 7):
elems.extend([(i , i+6),
(i , i+7),
(i+n, i+6),
(i+n, i+7)])
return nodes, elems, {None}, s, {}
_DOMES = {
"600": dome600,
"120": dome120,
"1410": dome1410,
"1180": dome1180
}
def revolve(ref_nodes, ref_elems, fixed, count, key_nodes=None, scale=1, shift=None):
"""
Create nodes and connectivity of a structure that is generated by revolving a
reference assembly idenfied by ref_nodes and ref_elems.
wr
"""
nodes = {}
elems = []
if key_nodes is None:
key_nodes = set()
nn = len(ref_nodes) - len(key_nodes)
for i in range(count):
angle = 2*pi*i/count
cs = cos(angle)
sn = sin(angle)
if 0 in ref_nodes:
node = ref_nodes[0]
nodes[0] = (cs*node[0] - sn*node[1],
sn*node[0] + cs*node[1],
node[2])
# Create nodes
for j,node in ref_nodes.items():
if j : # not in key_nodes:
nodes[j+i*nn] = (cs*node[0] - sn*node[1],
sn*node[0] + cs*node[1],
node[2])
for elem in ref_elems:
elems.append(tuple(
node if node in key_nodes else (node-1+i*nn)%(count*nn)+1 for node in elem
))
return nodes, elems
def create_truss(nodes, elems, areas=None):
"""
Create a truss model in OpenSees
"""
# Create a model in 3 dimensions with 3 degrees of freedom
model = ops.Model(ndm=3, ndf=3)
# Define a linear-elastic material
model.uniaxialMaterial('Elastic', 1, 3000)
area = 1.0
# Add nodes to the model
for tag, node in nodes.items():
model.node(tag, node)
# Add elements to the model
for tag, nodes in enumerate(elems):
model.element("Truss", tag, nodes, area, 1)
return model
if __name__ == "__main__":
import veux
nodes, elems = revolve(*dome1180())
model = create_truss(nodes, elems)
artist = veux.render(model, vertical=3)
# Show the rendering
veux.serve(artist)