STRUCTURES WITH SPATIAL GRIDS (RETICULAR STRUCTURES)

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GENERAL ASPECTS

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.

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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.

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VARIOUS DESTINATIONS OF GRIDS

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VARIOUS DESTINATIONS OF

GRIDS

Structure of the roof

Structure of the

envelope

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VARIOUS DESTINATIONS OF

GRIDS

Domes

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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

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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.

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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

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TRIANGULAR GRIDS

• Domes-double curvature in one direction on circular plan

• Parabolic-compound or elliptical inverted surfaces

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HYPERBOLOID PARABLES AND CYLINDRICAL SURFACES

o double curvature in opposite directions

Shells made of one layer grids: a)- cupola; b)- cylindrical; c)- hyperbolical parable

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CLASSIFICATION BY THE NUMBER OF LAYERS

Single layer membrane maximum spans <10 m

Double layer with diagonals

Double layer with posts (Vierendeel)

Three layer systems

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Assembling the triangular systems

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TRIANGULAR AND SQUARE PLANAR GRIDS WITH TWO LAYERS

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

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Spatial planar square simple

Spatial planar square

diagonal structure

Planar square systems with internal members eliminated

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PLANAR GRIDS

• The in-deformability of the system must be maintained (stiffness)

Hexagonal systems of spatial planar structures: a)- simple; b) double

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CONSTRUCTIVE SOLUTIONS FOR THE CONNECTIONS

OF THE MEMBERS OF THE GRID

•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

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TYPES OF CONNECTIONS ADOPTED IN ROMANIA, ACCORDING TO STO 13-1997

Welded spherical connections CHS (bottom face) welded on a disc

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DESIGN OF THE STRUCTURAL ELEMENTS AND CONNECTIONS

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.

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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:

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GEOMETRIC ELEMENTS OF THE GRID

• 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 hll

l

htg d

sincos2;

2 hll

l

htg 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:

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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.

spherediamExt

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

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COMPUTATION OF INTERNAL FORCES IN THE MEMBERS OF THE GRID

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

EAk

L

EAk

L

EAk

L

EAk

cossin

sin

sincos

cos

12

2

22

21

2

11

Fk 1

22212

12111

FFF

FFF

yx

yx

x

xx

L

AEFFF

L

AEF

L

AEF

sin

sin;cos

2221

1211

2221

1211

kk

kkk

y

x

PF

0

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DIRECT STIFFNESS METHOD

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.

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SLOPE DEFLECTION METHOD-GENERAL EQUATIONS OF THE SYSTEM

• 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

ijl

l

ij

ij

ij

ijijijij Al

lAEN

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

apN

23

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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

2080 -12 2075 -11 2070 -10

80100 -7 7590 -7 7080 -6

>100 >90 >80

i

l f

i

l f

Slenderness ratios rectified for CHS

i

l f

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ANALYSIS OF RETICULATED STRUCTURES AS SHELLS

• 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.

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Static equilibrium: a)- in the triangular spatial grid; b)- in the equivalent continuous space

3NN3

LP

;3NN3

LP

;NN332

LP

xyy3

xyy2

yx1

';

';

'

';

''

';

''

'

t

N

t

N

t

N

GEEEE

xy

xy

y

y

y

x

xx

xy

xy

x

xy

y

y

y

y

y

x

x

xx

3

1'''

'4

3'

3'

2'''

yx

yx

Lt

AEG

Lt

AEEEE

''' ttt yx

xyxy

xyy

yxx

Lt

AE

Lt

AE

Lt

AE

'4

3

;3'4

3

;3'4

3

2

3

'112

''

tED

it 32'3

2'

dt

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