Lecture 11 s.s.iii Design of Steel Structures - Faculty of Civil Engineering Iaşi

STRUCTURESWITHSPATIALGRIDS
(RETICULARSTRUCTURES)
C.Teleman_S.S.III_Lecture11
1

GENERALASPECTS
 Structures with one, two or three layers-spatial lattice systems obtained from steel
members interconnected;
 Used in plan dimensions of the building close to square,; economy of 10...12% in
comparison with lattice structures;
 May have a flat shape (rigid plane rectangular structure) or in the shape of a cupola
for buildings that have circular, polygonal or ovoid plane. In particular, big
structures that sustain radio telescopes;
 Modern structures; numerous constructive systems in the last 50 years;
 The spatial behavior determines a light weight and consequently, a reduction in
steel consumption and small heights of the roofs;
 Wide spans of the roofs;
 Great stiffness in the plane of the roof, small general deformations;
 Short time of mounting due to prefabrication in great extend;
 Low costs due to fast execution but also due to the fact that transportation and
depositing of the prefabricated units are not expensive.
2

 Plane (flat) grids;
 Curved grids (with single or double curvature): cupola, cylindrical, or rotational surfaces
obtained from hyperbolical parables;
 Towers with grids;
 Other combined structural shapes.
 The maximum spans for the grids with one layer do not exceed 10 m.
 When the necessities exceed these limits a two layers system is used as the solution or
three layers system placed at the edges and two layers placed in the middle of the plane
surface.
 The spatial planar grids combine the effect of a lattice girder with the effect of shell. The
planar grids with have limited spans of around 60…65 m imposed by the stiffness of the
whole system (the maximum deflection).
 The mesh of the grid may be triangular, square or hexagonal their stiffness decreasing
from the first to the last.
3

VARIOUS DESTINATIONS OF GRIDS
4

VARIOUS
DESTINATIONS OF
GRIDS
Structure of the
roof
Structure of
the
envelope
5

VARIOUS
DESTINATIONS OF
GRIDS
Domes
6

WIDE SPANNED STRUCTURES
 Spatial frames are the
result of optimization of
wide spanned structures
with special destinations;
 In order to improve the
behaviour of planar
trusses we have to insure a
spatial collaboration with
other structural systems;
the result is a spatial grid
7

BASIC CONCEPT
The chords of the truss must
change the shape in order to cope
with increasing spans
The new spatial system is made of
two planar systems that take
together the loads and the
deformations
yx
yx
ii
iii
ff
PPP


ASSUMPTIONS
I. The connections are perfect spherical articulations, only axial efforts may result at the end of the
convergent bars (no bending and no torsion);
II. The bars converge axially (perfect) in the connection;
III. Actions are forces acting only in joints.
8

Spatial grinds are obtained from nb members interconnected in nc joints
 TRIANGULAR MODULES are efficient in transferring stresses. With little to no bending moments, they are
more stable and stronger than 90 degree frames.
 3-D LATTICE STRUCTURES can cover larger areas at a lower weight. The many lightweight members in a
lattice structure distribute loads evenly and efficiently through the structure in three dimensions, making it
more efficient and lighter than a conventional two-dimensional frame.
 DOUBLY CURVED GEOMETRIES have the ability to span long distances. Their curvature transfers stresses
more efficiently with little to no bending moments, making them stiffer than conventional flat surfaces. Doubly
curved geometries now offer infinite possibilities of free-style designs
CLASSIFICATION OF THE STRUCTURES WITH GRIDS
9

TRIANGULARGRIDS
• Domes-double curvature in one
direction on circular plan
• Parabolic-compound or elliptical
inverted surfaces
10

HYPERBOLOIDPARABLESANDCYLINDRICALSURFACES
o double curvature in
opposite directions
Shells made of one layer grids: a)- cupola; b)- cylindrical; c)- hyperbolical parable
11

CLASSIFICATIONBYTHENUMBEROFLAYERS
 Single layer membrane
maximum spans <10 m
 Double layer with diagonals
 Double layer with posts
(Vierendeel)
 Three layer systems
12

Assemblingthetriangularsystems
13

TRIANGULARANDSQUAREPLANARGRIDSWITHTWOLAYERS
 Limited spans of 60…65 m imposed by the stiffness of the whole system (the maximum
deflection is 1/300…1/400 of the span);
 The grids may be: triangular, square or hexagonal, their stiffness decreasing from the
first to the last;
 Triangular planar grids: two layers translated relative one to the other; 3 diagonals
emerge from every joint and link the two surfaces;
 Square planar grids: simple, oblique diagonal etc. In the case of the simple and oblique
grid, 8 members are interconnected in a joint, 4 from the face and another 4 being the
diagonals placed at 450; in the case of the diagonal grid a number of 6 members meet in the
joints placed in the top face from these 2 being diagonals.
Planar grids with two layers and different arrangements of the internal members
14

Spatial planar
square simple
Spatial planar
square
diagonal
structure
Planar square systems with
internal members eliminated
15

PLANARGRIDS
• The in-deformability of the system must be maintained (stiffness)
Hexagonal systems of spatial planar structures: a)- simple; b) double
16

CONSTRUCTIVESOLUTIONSFORTHECONNECTIONS
OFTHEMEMBERSOFTHEGRID
•Some of the constructive systems adopted are:
•TRIODETIC (Canada): the members are CHS (circular hollow sections) flattened at the ends. They
are fixed in the joint with two washers and a bolt and may be easily dismounted;
•SPACE-DECK (U.K.): a square base pyramid made of hot rolled sections (angles) is place at the top of
the grid upside down; the bars are filleted in the joint at the top part of the pyramid;
•MERO (Germany and other European countries) a sphere in metal with up to 18 holes with filets
inside which CHS or RHS (rectangular hollow sections) are fixed with HSFG Bolts;
•UNISTRUT (SUA): the connection is made of a gusset spatially shaped with holes in which up to 8
bars may be fixed with bolts. The bars are channels (C) and can be hot rolled or cold formed. Sections;
•Other systems like: PYRAMITEC, TRIDIMATEC, TUBACCORD, SDC (France), UNIBAT and NODUS
(UK), OKTAPLATTE (Germany) are also used.
Constructive solutions for the connections between the internal members of a grid:
a)- TRIODETIC; b)- SPACE-DECK; c)- MERO; d)- UNISTRUT; e)- TRIDIMATEC
17

TYPESOFCONNECTIONSADOPTEDINROMANIA,ACCORDINGTOSTO13-1997
Welded spherical connections CHS (bottom face) welded on a disc
18

DESIGNOFTHESTRUCTURALELEMENTSANDCONNECTIONS
ACTIONS
 a. Permanent actions
 b. Variable actions - in particular:
 uneven sink at the foundations, variation of temperature due tot technological causes;
settlements at the supports;
 important snow deposits in the case of skylights, gables, attic placed on perimeter or higher
buildings placed in the close neighborhoods;
 wind;
 effect of temperature variations;
 all kind of loads or forces due to mounting stage that modify the static scheme designed for the
service life.
 c. Combinations of actions -exploitation state and the mounting stage.
GEOMETRIC INVARIANCE AND STATIC EQUILIBRIUM
Basic assumptions:
 The connections are perfect spherical articulations;
 The joints maintain their position relative to each other as long as we consider that the length of
the bars is constant.
 The condition of geometric invariance - in two alternatives:
A - the internal constraints in the connections and the external restraints at the supports act as a
single rigid system;
B - geometric invariance and static equilibrium of the grid insured only by the constraints in the
structural system
Computation of the grids may be done with the following methods:
1. Slope-deflection method – we develop the matrix analysis by the direct stiffness method;
2. A finite element method may be applied with computer aids;
3. Assimilation of the structure with an equivalent shell.
19

A. The condition of geometrical invariance is expressed with: 03  nrb nnn
where:
nb, nn - the total number of bars and internal joints, respectively;
nr - number of bars that connect the grid to the supports.
B. A minimum number of bars (nb=6) is necessary in order to insure the connection between the
rigid plane (considered as a free body in space) and the ground. A common type of grid is the two
layer grid and it contains a total number of bars:
bdbibsb nnnn 
nbs, nbi and nbd are the number of bars in the top layer, in the bottom layer and in the diagonals.
  42
12
8



nmn
nmmnn
mnn
r
n
b
EXEMPLE
    111  mnnmi
The redundancy is determined with the following relationship:
20

21

22

23

24

25

26

GEOMETRICELEMENTSOFTHEGRID
• Spacing between two running joints: 1.5… 3.0 m;
• Height (h): 1/15…1/20 of the minimum span;
•  = 450…600;
• Square spatial planar grids:


sincos2
;
2 hl
l
l
h
tg d 




sincos2
;
2 hl
l
l
h
tg d 


Optimum steel consumption the cross section of the internal members differentiated according to
distinct areas (maximum three) on the surface of the mesh
Recommended surfaces for different sections of the steel elements
Member
Area
Central Intermediary Marginal
Inside the top face As 2/3As 1/3As
Inside the bottom face Ai 2/3Ai 1/3Ai
In diagonals 0.4As or 0.4Ai
Simple grid:
Diagonal grid:
27

CONNECTIONS
• Verified for limit situations:
a- sections of failure;
b- crushing under compression efforts;
c- shear of the walls of the elements, gussets or spheres;
d- local buckling of the walls in compression.
• The dimensions of spherical connections: diagram,
depending on the values of the critical efforts Pl based on
the maximum effort in the members converging in a specific
joint multiplied with a safety factor of 2.5;
• Diameter of the sphere: de aprox. 1.8…2.0 dCHS.
 sphere
diamExt
wallThick
spherediamExt
CHSdiamExt
..
.
;
..
..
 
AkC The specific steel consumption:
k= 1.1 - span < 24m;
k= 1.5…1.68 - span > 24m.
Minimum thickness of the wall is 4 mm;
Bolted connections: bolts in 6.6 category and slip resistance bolts
28

COMPUTATIONOFINTERNALFORCESINTHEMEMBERSOFTHEGRID
 The connections are perfect spherical articulations, only axial efforts may result at the end of the
convergent bars (no bending and no torsion);
 The bars converge axially (perfect) in the connection;
 Actions are forces acting only in joints.
 General methods:
- slope-deflection method – we develop the matrix analysis by the direct stiffness method;
- a finite element method may be applied with computer aids;
- assimilation of the structure with an equivalent shell.
DIRECT STIFFNESS METHOD
we write the joint equilibrium equations in terms of unknown joint displacements and stiffness
coefficients, respectively. The stiffness coefficients are in fact the forces due to unit displacements).
PkkF
kkF
yxy
yxx




2221
1211
0
;00














xx
i
ii
x
i
ii
xx
i
ii
x
i
ii
L
EA
k
L
EA
k
L
EA
k
L
EA
k




cossin
sin
sincos
cos
12
2
22
21
2
11
Fk 
1
22212
12111
FFF
FFF
yx
yx


x
xx
L
AE
FFF
L
AE
F
L
AE
F


sin
sin;cos
2221
1211


2221
1211
kk
kk
k 
y
x



P
F
0

29

DIRECTSTIFFNESSMETHOD
Stiffness coefficients for an axially loaded bar: a)-
forces created by a unit horizontal displacement;
b)- forces created by a unit vertical displacement
System of two bars (truss system) subjected to a force
acting in the joint 2: a)- actual forces acting on the
original structure; b)- case I-displacements under
horizontal component of force; c)- case II-
displacements under vertical component of force.
30

SLOPEDEFLECTIONMETHOD-GENERALEQUATIONSOFTHESYSTEM
• An important “degree” of redundancy implies a great number of equations of equilibrium so in
fact the slope-deflection method will also be using the computer aids, basically starting with
• Then:
     FK 
     FK 
1
Knowing the translations of the joints “i” and “j” in the global system of coordinates ix, iy,
iz, and jx, jy, jz, the elongation of the member “ij” will be determined (translations and
rotations of the joints “i”, “j” in the loaded structure, in the figure)
      zizjzyiyjyxixjxijl  coscoscos 
specific elongation:
ij
ij
ij
l
l

ij
ij
ij
ijijijij A
l
l
AEN 

 
Forces in the internal members vary with the 1/h and in particular the efforts in the diagonals
vary with 1/sin. The deflection varies with 1/h2
forces in the member “ij”:
h
aP
h
ap
N
23












31

a/b  a/b  a/b 
0.5 0.1935 1.05 0.0419 1.6 0.00680
0.55 0.1702 1.1 0.0357 1.65 0.00576
0.6 0.1500 1.15 0.0303 1.7 0.00487
0.65 0.1322 1.2 0.0257 1.75 0.00412
0.7 0.1162 1.25 0.0218 1.8 0.00349
0.75 0.1018 1.3 0.0185 1.85 0.00295
0.8 0.0888 1.35 0.0156 1.9 0.00249
0.85 0.0771 1.4 0.0132 1.95 0.00210
0.9 0.0666 1.45 0.0112 2.0 0.00176
0.95 0.0573 1.5 0.00948 - -
1.0 0.0491 1.55 0.00803 - -
n  n  n 
5 3.333 14 9.286 23 15.333
6 3.889 15 10.000 24 15.972
7 4.667 16 10.625 25 16.667
8 5.250 17 11.333 26 17.308
9 6.000 18 11.963 27 18.000
10 6.600 19 12.667 28 18.643
11 7.333 20 13.300 29 19.333
12 7.944 21 14.000 30 19.978
13 8.667 22 14.636 - -
Values of the coefficient 
Values of the coefficient 
OLT 35 OL 44 and OLT 45 OL 52
 rectified  rectified  rectified
2080 -12 2075 -11 2070 -10
80100 -7 7590 -7 7080 -6
>100  >90  >80 
i
lf

i
lf

Slenderness ratios rectified for CHS
i
lf

32

ANALYSISOFRETICULATEDSTRUCTURESASSHELLS
• The first type of analysis consists in modeling a discrete structure and study
the stresses and strains in the internal members by using mathematical
discrete variables.
• For reticular structures much more intricate and non symmetric the explicit
solutions are not acceptable and numerical methods along with approximate
analysis techniques are adopted. In 1927 F. Bleich and E Melan developed the
discrete structural computation methods but only after 1960 these methods
were applied for reticulated structures.
• The second type of analysis is adopted for structures with a very big number
of element; the basic concept replaces the reticular space with a continuous
equivalent space, the methods of equivalence being either with
interdependent solutions between the two spaces, or by conversion of the
finite difference equations into approximate differentials.
• Wright developed the method of interdependent equations for “unistrat”
systems based on the shell theory.
33

Static equilibrium: a)- in the triangular spatial grid; b)-
in the equivalent continuous space
 
 
 3NN
3
L
P
;3NN
3
L
P
;NN3
32
L
P
xyy3
xyy2
yx1



'
;
'
;
'
'
;
'
'
'
;
'
'
'
t
N
t
N
t
N
GEEEE
xy
xy
y
y
y
x
x
x
xy
xy
x
x
y
y
y
y
y
y
x
x
x
x













3
1
'''
'4
3
'
3'
2
'''





 yx
yx
Lt
AE
G
Lt
AE
EEE
''' ttt yx 
 
 
xyxy
xyy
yxx
Lt
AE
Lt
AE
Lt
AE














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'4
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 2
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


tE
D
it  32'
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d
t 
34

35

36

37

38

Lecture 11 s.s.iii Design of Steel Structures - Faculty of Civil Engineering Iaşi

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Lecture 11 s.s.iii Design of Steel Structures - Faculty of Civil Engineering Iaşi