Advances in Steel Structures - part 25 pptx

Quasi- Tensegric Systems and Its Applications 221 The external force space {f} can be split in two subspaces: one is r-dimensioned and is a fitted external force subspace forces can be

220 L Yuxin and L Zhitao

{X}, - - { X } ,-1 q- [D],_, {a},_l (14)

in which {o~} ,-1 is unknown According to the criteria of energy, we can select

Here, it is clearly that ~b is a size parameter that controls the size of displacement increments in the gradient direction Substituting Eqn 15 to Eqn 13 yields

Repeating the procedure from Eqn 13 to Eqn 16, we shall find out the nodal coordinates in GSES step

by step Finally, the final space coordinates of nodes can be expressed as follows

oo

i=l

The convergence of Eqn 17 depends on the extemal wind force {f} This is the same behavior for any unstable structure (Liu and Motro,1995)

STRUCTURAL ANALYSIS AND VECTORIAL SUBSPACES

Eqn.6 which describe the kinematic and static relations in a reticulated system connect two vectorial spaces, namely the node space R, and the member space Rb External forces and displacements are related to the first one, internal forces and elongation to the second one If we call r the rank of the equilibrium matrix [A], a Gaussian elimination procedure on the augmented matrix [A:I], [/] being a diagonal unit matrix, leads to the transformation below (Pellegrino and Calladine,1986)

Gaussian transformation ~- ~-

which can be put in the form

Applying the transformation to static relationship Eqn.6, we get

Quasi- Tensegric Systems and Its Applications 221 The external force space {f} can be split in two subspaces: one is r-dimensioned and is a fitted external force subspace (forces can be equilibrated), the second one, of m-dimension with m n-r corresponds

to the forces which activate the mechanisms Similar derivation (with [J] being a diagonal unit matrix) can be achieved on the compatibility equation 2 and lead to

The deformation space is composed of two subspaces, a "fitted" and a "non-fitted" subspace Only the former is compatible with the displacements If we consider Eqns.19, 20 under an energy view, each row of [It] and [lm], can be assimilated to displacements and each row of [Jr], [Js] to internal forces In Eqn.20, when components of {d} belong to the mechanism subspace, corresponding values of {e} are equal to zero and corresponding rows of [Br], which are related to the external forces are orthogonal to the displacements The m mechanism vectors are included in [lm] from Eqn.19 and the corresponding displacements can be computed by

with {a} being composed of arbitrary constants combining elementary mechanisms.[D]=[lm] r is known as the displacement mode matrix Similar analysis leads to the computation of any self-stress vector by

with {/3} being composed of arbitrary constants combining elementary self-stress states.[D]=[J,] r is known as the self-stress mode matrix We note that for a structure that verifies compatibility condition, the self-stress subspace is orthogonal to the elastic deformation one If the structure is in equilibrium, the mechanism displacement space is orthogonal to the external force space

Governing Equations for GSES

Considering the compatibility equation, the matrix [B] is a bxn matrix, and generally this is not a square and even in this case its rank is not equal to n (=b); we can't use traditional procedures to solve

it We must introduce the Moore Penrose inverse and more precisely the { 1 }-inverse [B]- With this, the general solution of the compatibility equation can be put in the form

Where {7"} is a n-dimensional vector The second term in Eqn.23 belongs to the mechanism displacement space and Eqn.23 is equivalent to

222 L Yuxin and L Zhitao

When {e}={0}, i.e for a rigid displacement of the structure, the space coordinates can be found either

by general inverse method or by an iterative altemate method (Liu and Motro,1995), [D] is called as a displacement model matrix Similar derivations give access to the intemal forces as

Matrix [S] is called as a self-stress model matrix, {fl} is an unknown vector

Elastic Stable Equilibrium State

In order to reach the ESES with a known GSES, it is necessary to determinate the values of vectors { a} and {fl} in Eqns.24, 25 Assuming that {t}0 is the initial value of internal forces in GSES, we can obtain the elastic deformation vector {e} by combining Eqn.25 and 7 which describes the constitutive law

{e} = [F]([A]- {f} - {t}0 ) + [F] [S] {/3} (26)

From the previous discussion, for an incompatible structure, the deformation space is orthogonal to the self-stress space This condition gives the needed equation for computing {fl}

[S] v {e} = [S] r [F]([A]- {f} - {t}0 ) + [S] r [F][S]{fl} = {0} (27)

Since the product [S] T [F][S] is not singular and the first term of Eqn.27 is known, we can find out {fl} ,and the relevant internal force {t} (Eqn.25) and {e} with the constitutive law (Eqn.7) The second term can be equated by considering the orthogonality between the mechanism displacement and the non- fitted external forces, when structure is in equilibrium Matrix [D] is a base of non-fitted external force space, so we get with Eqn.23 and 24

[D] ~ {d} = [D] r [B]- {e} + [D] T [D] {a} = {0} (28)

where [D] r [D] is not singular, so we can find out { a} and simultaneously {d} Then the position vector {x} is calculated with the GSES coordinates as first value A new matrix [B] is derived and the process is repeated until the increments become close to zero Hence the ESES is found The convergence depends on the properties of the structure If the rank is equal to the number of members there is no self-stress state and the elastic deformation {e} is compatible with {d}: the process is

always convergent When r<b, {e} doesn't fit with {d} In this case, the external forces must be

divided into small increments, so that the fitting between {e } and {d} can be nearly satisfied

NUMERICAL EXAMPLES

Unstable Cable

Quasi-Tensegric Systems and Its Applications 223 The first example is an unstable cable that has been researched by F Baron and M.S Vendatesan (1971) The sectional area of the cable is 1.465 cm 2 and the stiffness is EA=12119.51kN The initial ESES is reached under two symmetric loads 17.8kN and an added 13.3kN local load applied to node 2 Comparison with Baron's results is done for the GSES and the ESES Coordinates of nodes are listed

in Table 1 and internal forces in Table 2 Baron's results are derived from a Newton-Raphson method

TABLE 1 SPACE COORDINATES CALCULATED

xl (m)

F Baron

GSES

ESES

TABLE 2 STRESES IN BARS (MPa)

Our results are close to those of Baron It can be noticed that the difference between GSES and ESES

is very small and could be neglected: a non-extension hypothesis could be acceptable

Tensegrity System

A tensegric system (see Figure 1) is analyzed, in which an initial state is shown in Figure l a The calculating results by the method proposed in this paper are Shown in Figure lb and c The initial nodal

X2

0.5m 0.5m

(a) Initial state

6

J

~ 1 x2

(b) Final State: vertical view (c) Final State: elevation

X 2

Figure 1" Tensegric element

224 L Yuxin and L Zhitao

coordinates are listed in Table 3 The nodal 5,6,7,8 are fixed, the nodal 1,2,3,4 are free The final free nodal coordinates are listed in Table 4 In dynamic relaxation method (Liu, 1998), the coefficient damping takes 500kg-m/s, the time incremental is 0.2s, nodal mass is 5000kg at each node After

73695 times iteration get the final form For incremental iterative method, the coefficient ~b =0.001, iterate 14166 times get the final results Finally we reach the results which show in Table 4 and Table 5, respectively The two methods are effective for calculating tensegric system

TABLE 3 THE INITIAL COORDINATES OF THE TENSEGRIC SYSTEM

xl(m) -0.35355 -0.35355 -0.35355 0.35355

x2(m) 0.35355 0.35355 -0.35355 -0.35355

x3 (m) 0.97832 0.97832 0.97832 0.97832

5 0.5 0.5

0

6 -0.5 0.5

0

-0.5 0.5 -0.5 -0.5

TABLE 4 THE FINAL FREE COORDINATES OF THE TENSEGRIC SYSTEM

Methods

Nodal No

x,(m)

x2(m)

x3(m)

Dynamic Relaxation (Liu, 1998)

0.0000 -0.50000 0.00000 0.50000

0.5000 0.00000-0.50000 0.00000

0.86603 0.86603 0.86603 0.86603

Incremental iterative

-0.00001 -0.50023 -0.00001 0.50021 0.50022 0.00000 -0.50022 -0.00000 0.86701 0.86701 0.86701 0.86701

TABLE 5 THE INITIAL INTERNAL FORCES OF THE TENSEGRIC SYSTEM

Members 12,23,34,41 15,26,37,48 18,25,36,47 12,23,34,41 15,26,37,48 18,25,36,47 Stresses(kN) 0.50000 0.70711 -1.00000 0.49952 0.70673 -1.00000

THE CONSTRUCTION OF QUASI-TENSEGRIC SYSTEM

Cable net is a good structural system that has ability to span large space, but the stiffness is a big problem Tensegric system is a maximum economic form of structure, but its construction is very complicate This paper tries to combine the advantages of the two systems to form quasi-tensegric systems

According to the area to be covered, we at first design a cable-net, for example as shown in Figure 2a

Of cause, in practical engineering the plan of structure may not be a rectangular Then using the tensegric element of Figure lb,c, we connect the tensegrity to cable-net one element by one element,

as shown in Figure 2b Finally we can form a double-layer quasi-tensegric spatial frame In construction, the key problems are the application of prestressed force technology and the connecting

Quasi-Tensegric Systems and Its Applications 225 form at nodes These will be researched further in the future And the whole structural analysis after integrating quasi-tensegric system is also important

Tensegric element Tensegric element

i

I

Tensegric element

Tensegric element

Cable

Tensegric element

Cable

(a) Cable-net

(b) Tensegric elements build on cable-net Figure 2: Quasi-tensegric system

CONCLUSION

The results of our numerical examples assess the validity of the iterative alternate method we developed As far as unstable system are concerned this method could be useful and specifically in the field of tensegric systems Quasi-tensegric system is a new system that has the advantage of cable-net and tensegrity It will be applied widely large span structure in practice

ACKNOWLEDGMENT

This work was supported by National Natural Science Foundation of China (Project number

59508010 ) We express heartfelt thanks

REFERENCES

Liu Y (1998) Analysis of unstable systems and of tensegrity by dynamic relaxation Chinese Journal

of Spatial structures, 4:3, 26-30

Liu Y and Motro R.(1995) Shape analysis and internal force in unstable structures Journal of Southeast University, No.IA, 262-267

Liu Y., Lu Z., Han X., Jing J (1995) Analysis for unstable cable-nets under static wind loads Journal

of Southeast University, No.IA, 531-535

Pellegrino S and Calladine C.R.(1986) Matrix analysis of statically and kinematically indeterminate frameworks, lnt J Solids Structures, 22:4, 409-428

Baron F and Vendtesan M.S (1971) Nonlinear analysis of cable and truss structures Journal of the structure Division, ASCE, 97:2, 679-710

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Connections

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THE DESIGN OF PINS IN STEEL STRUCTURES

R.Q Bridge School of Civic Engineering and Environment, University of Western Sydney, Nepean,

Kingswood, NSW 2747, Australia

ABSTRACT

There has been a recent upsurge in the use of pins, particularly in architectural steel structures with visible tension and compression members However, the rules for the design of pins vary quite considerably from code to code and this has caused some confusion amongst consulting structural engineers operating internationally A comprehensive testing program has examined the influence of such parameters such as pin diameter, material properties of the pin, thickness of the loading plates, material properties of the loading plates and the distance of the pin to the edge of the loading plates Modifications to current design procedures are proposed that take into account the different possible modes of failure

KEYWORDS

Bearing, design, failure, pins, shear, steel structures, strength, tests

INTRODUCTION

As they have no head and are not threaded, pins cannot carry any axial forces and can only carry shear forces transverse across the pin Despite this limitation, they are often used in structural applications

by designers and architects for steel structures with visible tension and compression members, particularly in applications such as canopies, sporting stadiums, convention centres and bridges In these cases, the pins are essentially subjected to static conditions and rotations are generally small The design procedures for pins can be found in most structural steel codes, standards and specifications However, there is some disparity in the design values for three major of the major design conditions: shear of the pin; bearing on the pin; and bearing on the plies (plates) that load the pin For instance, the Australian Standard AS4100-1998 has an apparently high design value for the strength of a ply (plate) in bearing and yet a low value for the strength of a pin in shear whereas Eurocode 3-1992 has a low value for plate bearing strength but a high value for pin shear strength To explore this disparity, the behaviour of pins under load has been examined experimentally to determine the effects of the material and geometrical properties of both the pin and the loading plate on the

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