Advances in Steel Structures - part 24 pptx

210
L. Xiliang et al.
many tiny cross corrugations, they become curved, forming arched trough plates. Because arch
structure can translate the applied loads mainly into forces in the plane of its surface, so such
arched trough plate can be employed in larger span buildings (more than 30m) not only as an
accessory material to be used for simple coating, but also as load bearing skeleton. With the plate-
skeleton-combined structural style and the highly mechanized construction procedure, arched
corrugated metal roof possesses such advantages as strong spanning ability, light self-weight, fast
and easy construction, good waterproofing quality and attractive appearance etc. The combination
of these advantages certainly can result in cost saving. It is very suitable to be used in single layer
buildings, and if hoisting condition permits, it can also be used in multistory buildings. According
to the different sectional configurations of arched trough plates, this kind of structure can be
classified into several types. In China there are mainly three kinds of sectional configurations, so
there are three types of this kind of structure, which are respectively named MMR-118, MMR-178
and MMR-238. Figure l shows the outlines of their cross sections.
Figure 1
Design specifications and recommendations for cold-formed steel structural members are now
available in many countries, but none of the rules for the design and construction of arched
corrugated metal roof have been published all over the world till now. Even though it is a typical
kind of thin-walled steel structure, because of its peculiar characteristics, its performance under
load differs in several significant respects from that of ordinary cold-formed structural members.
As a result, design specifications for cold-formed steel structural members cannot possibly cover
the design features of this kind of structure completely, so it needs an appropriate design
specification. With no provision of certain design code, engineering accidents will be inevitable. In
the winter of 1996, a heavy fall of snow in the northeast of China caused more than 30000 m 2 of
this kind of roof to collapse.
According to former research work, there are mainly two kinds of mechanic models for this kind of
structure, namely arch and shell. However, for some reasons, none purely theoretical analysis on
this kind of structure can make satisfactory results [1 ], so experimental studies are essential here.
Nevertheless, just because of its special construction characteristics, it is almost impossible to
carry out scale model test, full-sized model tests are indispensable to the research of this kind of

structure.
After the engineering accidents mentioned above, the authors had carried out nine groups of large-
span model experiments on the spots of these accidents. Through these model tests the cause of
Study on Full-Sized Models of Arched Corrugated Metal Roof
211
these accidents and the load bearing performance of this kind of structure could be understood. By
comparing the theoretical results with the testing results, the great divergences between them could
be seen clearly. Aiming at reducing these divergences, some recommendations for further studies
are proposed.
2 OUTLINE OF EXPERIMENT
2.1 Model specimens
All of these tests were on-the-spot tests. The models studied here were the very structures that
survived from that heavy fall of snow. The steel plate used in these models had the yield strength
of 280Mpa and Young's modulus of 2.00 • 105 MPa. The sectional configurations of these trough
plates of these models were the same as that shown in fig.lc, namely trapezoid section. Five of
these models spaned 33m and the others spaned 22m. For the convenience of the application of
load, only one model was made up of six pieces of arched trough plates, the others were all made
up of four pieces. The cross section is shown in fig.2. In order to search for an effective measure to
raise the load bearing capacity of this kind of structure, three models were reinforced with tension
chords. The reinforcing pattern is shown in fig.3. The geometrical size and load patterns of these
models are described in tab.1. Because the width-to-span ratios of these models were very small,
their lateral rigidity was quite low. To avoid lateral buckling and something unwanted scaffolds
were placed under and by both sides of these models. The outlook of a model after being put in
order is shown in fig.4
Figure2: The cross section of models
Figure 3: The reinforcing pattern Figure 4: Testing ground
212
No.
1
2

3
4
5
6
7
8
9
L. Xiliang et al.
TABLE 1
Arch span
33(m)
33(m)
33(m)
33(m)
33(m)
22(m)
22(m)
22(m)
22(m)
Arch rise
6.6(m)
6.6(m)
6.6(m)
6.6(m)
6.6(m)
4.4(m)
4.4(m)
Plate thick,
1.25(mm)
1.25(mm)

1.25(mm)
1.25(mm)
1.25(mm)
1.00(mm)
1.00(mm)
Lateral
width
2440(mm)
2440(mm)
3660(mm)
2440(mm)
2440(mm)
2440(mm)
2440(mm)
Load pattern
Full span
Half span
Half span
Full span
Half span
Remarks
Local distributed load
Reinforced
Reinforced
Full span
Half span
Half span Triangular load distribution
Half span Reinforced
4.4(m)
4.4(m)

1.00(mm)
1.00(mm)
2440(mm)
2440(mm)
2.2 Loading method
As a kind of thin-walled structure, arched corrugated metal roof is very sensitive to concentrated
load which may cause local buckling of the structure at a relatively low load lever. In actual
engineering, large concentrated load should be avoided. To simulate the actual load-bearing pattern,
distributed loads were applied by using sandbags. From tab.1 we can see that No.3 model bore
local half-span distributed load, which means that only four out of the six trough plates bore half-
span distributed load, while the two edge trough plates were free from any external direct loads.
Tab.1 tells us that No.8 model bore triangularly distributed load. This loading pattern is to imitate
non-uniformly distributed snow load.
2.3 Observation method
Because this is a kind of flexible structure, its deformations are so large that any displacement
measuring instruments with conventional precision can not cover its deformation scope, therefore
levelling instruments were used to survey the vertical displacements, and theodolites were used to
measure the rotary angles of those surveying points. Through the values of these rotary angles, we
can figure out the horizontal displacements. 7V08 static electrical resistance strainometer was
employed to observe the distribution of strains in the models. The surveying points of
displacements and strains were arranged at such locations as two springs, L/8 section, L/4 section,
L/2 section, 3L/4 section and 7L/8 section.
Study on Full-Sized Models of Arched Corrugated Metal Roof
3 EXPERIMENTAL RESULTS
213
The ultimate load, maximum horizontal displacement (U) and its location, maximum vertical
displacement (V) and its location of each model are listed in tab.2
No.
1
2

3
4
5
6
7
8
9
TABLE 2
Ultimate load
U
Location
V Location
0.87kN/m 2 38cm L/8 43cm L/2
0.56kN/m 2 52cm 3 L/4 57 cm 3 L/4
0.27kN/m 2 53cm L/4 54cm L/4
0.92kN/m 2 36cm L/8 42cm L/2
1.02kN/m 2 19cm L/4 23cm L/4
1.06kN/m 2 18cm L/8 27cm L/2
0.54kN/m 2 32cm 3 L/4 41 cm 3 L/4
1.02kN/m 2 31 cm 3L/4 39cm 3 L/4
1 lcm
1.28kN/m 2
L/4
16cm
L/4
Studying the data got from electric resistance strainometer, it's hard to find the laws of the stress
distribution in these models' sections. Although the cross sections of the models and patterns of
external load were symmetric, the stresses in one section didn't show symmetry. The direction of
principal stress of a certain point changed form time to time with the load added. The tiny ripples in the
trough plates and the out door wind load may account for this to a certain extend. Certainly the stresses

measured couldn't reflect the laws of the distribution of the actual stresses, but as few of them exceeded
the yield point stress of the material, so they could qualitatively tell us it isn't strength that determines
this kind of structure's load bearing capacity. Though the width-to-thickness ratios of the trough plates
in these models are very large, local buckling models which is common for thin-walled members didn't
appear during these tests. This demonstrates clearly that the tiny ripples can strengthen the local
stability of the plates.
Both No.1 and No.6 models bore full-span uniformly distributed load, so their performances were
similar. When the load level wasn't high, their deformations were symmetric, as shown in fig.5. But
when the load was close to the ultimate load (shown in tab.2), a sudden change from symmetric
deformation to non-symmetric deformation happened, which caused the internal forces around L/8 in
this side to increase steeply. With a little more loads, the model lost its stability and buckled.
The failure model of this kind of structure under half-span distributed load was shown in fig. 6. It's
easy to understand that the stability bearing capacity of this kind of structure under half-span load is
214 L. Xiliang et al.
much lower than that under full-span load, while the stresses and displacements were much bigger. The
reason accounted for this was that the deviation between arch axis and pressure line in the half-span
load model was much larger than that in the full-span load model, so bending moments were prominent
here, which was very disadvantageous to any structures. According to the data provided by the local
meteorological department, after that fall of snow the basic snow load of the zone where the
accidents happened was about 0.521kN/m 2, and the gale also blow snow from windward side to
leeward side during the snow-fall. So the uneven half-span snow load was close to ultimate half
span load listed in tab.2. It's quite sure that half-span load pattern is the most dangerous load
pattern for this kind of structure.
Original shape
~ L / 8 4 L / 8-~L L / 8 ~L / 8 4 L / 8-~L L / 8 4 L / 8 4 L / 8 *J
Original
I, L. I B -'-4 L I B J L I B I L I B J' L I B-'-~ L I B I L I B J' L I B "-~
Figure5: Deformation Shape of Full Span Load Model Figure6: Deformation Shape of Half Span
Though there were two pieces of
trough plate free from direct

external load, the ultimate load of
No.3 model is not bigger than that
of No.2 model. This model test
indicates that as a kind of thin-
walled member with open cross
Load Model
I~' L / 8 "#- L / 8 Jr L / 8"-"IL- L / 8 i L / 8~L L / 8 J,"-'L / 8 l L / 8 4
Figure7: Deformation Shape of Reinforced Half-Span Load
Model
section, the trough plate's torsional rigidity was very small and its capacity of resisting torsional load
was poor. From this test we also can see that the coordination between two pieces of plates was bad,
and the lateral widths of other models had little effects on their load bearing capacity.
Fig.7 shows the deformation shape of the models reinforced with tension chords subjected to half
span load. Tab.2 tells that the reinforcing pattern shown in fig.3 has little effect on the load bearing
capacity of the structure under full span load, while under half span load the load bearing capacity
of the same structUre can be doubled. From fig.7 we can see that two chords restrain the 3L/4
section, where the largest deformation will take place without these chords. The tension chords can
make the distribution of the internal forces even more uniform.
4 COMPARISON OF EXPERIMENTAL AND THEORETICAL RESULTS
Because of the symmetry of the configuration and the load distribution along the longitudinal
direction of this kind of structure, it can be looked as a kind of arch structure and modeled with
thin-walled beam elements. The material constants, such as bending rigidity, axial rigidity, etc, are
Study on Full-Sized Models of Arched Corrugated Metal Roof
calculated according to the geometric size of unit width of its cross section.
215
To reflect such structure characteristics as thin wall, tiny ripples, doubly curved space
configuration, shell element is the most ideal one. The shell element used here is a kind of
generalized conforming quadrilateral flat shell element [2]. A piece of arched trough plate is
chosen as calculating model. Because the length-to-width ratio of the trough plate is very big, in
order to avoid deformed grid dividing, the size of shell element should be very small. So the

number of the shell elements in a piece of trough plate is great. This of course increases the amount
of calculation, while on the other hand this also can raise the calculating precision. In general, the
steel material used in this kind of structure is isotropic. But because of the ripples on the webs and
flanges of the trough plate, the webs and flanges will respond to load orthotropically. To analyses
the effect of the ripples an equivalent orthotropic fiat sheet is defined for the shell FEA model. The
material constants of the equivalent flat sheet can be acquired according to the equivalent condition [3].
The above experiments had indicated that it is global stability, not material strength, that control
the load bearing capacity of this kind of structure, so only geometric nonlinearity is considered in
this paper. For the same reason, local buckling isn't considered. To avoid the problem of material
nolinearity in theoretical analysis, yield criterion is adopted as the failure criterion. By the
programs based on above mentioned theory, specimen 1, 2, 6 and 7 had been calculated. The
ultimate loads of theoretical analysis and experiments and the errors of theoretical results
compared with experimental results are listed in tab.3.
TABLE 3
Experiment
Arch model Error
Model No. Shell model Error
1 0.87kN/m 2 2.17kN/m 2 149.4% 1.26kN/m 2 44.83%
2 0.56kN/m 2 1.06kN/m 2 89.29% 0.67kN/m 2 19.64%
6 1.02kN/m 2 5.76kN/m 2 464.7% 3.23kN/m 2 216.7%

1.89kN/m 2
0.54kN/m 2
250.0%
1.14kN/m 2
111.1%
As a kind of thin-walled steel structure, it is very sensitive to defects. Because the models used in
these experiments were the survivors of accidents, initial deformation and initial stress were
inevitable. In addition, all the tests were carried out outdoor, wind load will bring harmful effect on
the tests too. So from tab.3 we can see all the theoretical results are much higher than the

corresponding experimental results. But compared with half-span loading models, the errors of
full-span loading models are even larger, which indicates that this kind of structure under full-span
load is more sensitive to defects than that under half-span load.
It's obviously that the results calculated with shell FEA model are much closer to the experimental
216
L. Xiliang et al.
results than that calculated with arch FEA model. This indicates that even though it's symmetric
along longitudinal direction, the arched trough plate, the structure's components have the property
of space load carrying because of its characteristics of thin wall and local ripple shape. The
construction of ripples on the plates certainly can strengthen the stiffness along longitudinal
direction, which makes the structure free from wavelike local buckling, but they weaken the
stiffness along span direction which is very disadvantageous for this kind of bearing structure.
Shell FEA model can reflect these factors to a certain extend.
From the analysis above, it's not difficult to find out that purely theoretical analysis on this kind of
structure has a distance from real application. Model test is indispensable here. But experimental
study requires testing of full-sized models, which are very expensive and the result is only
applicable for some special situations. So studying the relation between theoretical analysis and the
experimental results and finding out the appropriate calculating constants from experiments so as
to revise the calculation programs have great significance for the research of this kind of structure.
The authors of this paper are now preparing several groups of member tests in order to observe the
material constants of the arched corrugated trough plate. The material constants got from
experiments will be used in FEA.
5 CONCLUSION
Through the description of these full-sized model tests, the load bearing performance and the
failure model of arched corrugated metal roof are clear now. After pointing out that local buckling
and material strength are not the control factors to its load carrying capacity, two kinds of FEA
models were established for the its theoretical analysis. Though the theoretical results didn't agree
well with the test results, these deviations indicate that such structural characteristics of this kind
of structure as thin wall and local ripple shape have great effect on its load bearing performance.
To reduce the difference between theory analysis and experiment study, recommendations for

further research are proposed.
References
1. Zhang Yong, Liu Xiliang and Zhang Fuhai (1997) Experimental Study on Static Stability
Bearing Capacity of Milky Way Arched Corrugated Metal Roof. J. of Building Structures, 18:6,
46-54
2. Zhang Fuhai, Zhang Yong and Liu Xiliang (1997) A Generalized Conforming Quadrilateral
Flat Shell Element for Geometric Nonlinear Finite Element Analysis. J. of Building Structures
18:2, 66-71
3. Erdal Atrek, Arthur H.Nilson (1980) Nonlinear Analysis of Cold-Formed Steel Shear
Diaphragms, J. of the Structural Division 3,693-710
QUASI-TENSEGRIC SYSTEMS AND ITS
APPLICATIONS
Liu Yuxin 1 and Lti Zhitao 2
1Nanjing Architectural & Civil Engineering Institute; Nanjing 210009, China
2Southeast University, Nanjing 210018, China
ABSTRACT
Tensegric system is an optimum structural form in which the behavior of high strength in steel cable
can be utilized, but the reliability of this system is not very good because of the quasi-variable
characteristics. Cable-nets are also an effective structure that could span large space. This paper
proposes a new concept of spatial structure in which we combine tensegrity with cable-nets to form a
quasi-tensegric system. So we can make use of the advantages of these two systems. A construction
manner is developed. A quasi-tensegric system could be formed by the tensegric elements. This paper
divides the equilibrium state of quasi-tensegric system into two states: one is geometrical stable
equilibrium state, the other is elastic state equilibrium state. A method is developed to calculate the
form and internal forces in the geometrical stable equilibrium state and the convergence is provided.
The results of calculating show that the method proposed has a good convergence and a high precision.
Comparing incremental iterative method with dynamic relaxation method, the two methods are
effective and reliable in engineering design.
KEYWORDS
Quasi-tensegric system, tensegrity, cable-nets, geometrical stability, equilibrium state, prestressed

force, incremental iterative method, dynamic relaxation method
INTRODUCTION
Among reticulated structures composed of struts and cables, which require formfinding processes a
specific class can be defined as funicular system's class (Liu and Motro,1995). Their stable shape is
directly related to a set of external actions. Two equilibrium states are defined. The first one which
doesn't take into account the member deformations corresponds to geometrical stable equilibrium state
(GSES), the second one is related to a computation of the equilibrium in the deformed shape under
217
218
L. Yuxin and L. Zhitao
extemal actions and is named the elastic state equilibrium state (ESES). A method for computing the
coordinates for the GSES was obtained by using the theory of generalized inverse matrix (Liu and Lu
et a1,1995). In order to determine the ESES leading to the value of node coordinates and internal forces,
an alternate method was put forward. Computed results are compared with those obtained with a
Newton Raphson method. We shall introduce briefly these main results in this paper. After giving the
method of unstable systems, we discuss calculating procedure of cable-nets and simple tensegric
system. And finally give the construction rule of quasi-tensegric systems.
INCREMENTAL ITERATIVE METHOD FOR UNSTABLE SYSTEM
Kinematic Relationship
Static and kinematic equations are established assuming classical hypothesis for reticulated structures
with struts and cables. Assuming that free nodal displacements there are b members and n degreeS of
freedom, the kinematic relationship can be expressed in matrix as follows
{e} = ([B] + [AB]){d} (1)
{e } is elastic deformation vector, [B] is the compatibility matrix and [AB] an increment of [B], {d} is
the displacement vector in which boundary condition being included by deleting the corresponding
values. When
II {d} II
is very smaller, the second term can be neglected, then
[B] {d} = {e} (2)
For an unstable structure, there is no elastic deformation until the geometric stable equilibrium state

and Eqn.2 become
[B] {d} = {0} (3)
Static Equilibrium Relationship
Static equilibrium equation can be derived from principle of virtual work. For a set of extemal actions
{f} and a virtual displacement { rid}, corresponding elongation { de } and internal forces {t} must satisfy
{f} r {d} = {t} r {e} (4)
Substituting Eqn.2 into Eqn.4 yields
({f}r _{t}r [B]){fd} = {0}
(5)
It holds for arbitrary { 6at}, so that
Quasi-Tensegric Systems and Its Applications
[B] r {t}= {/}
or [A]{t}={/}
The constitutive law can be expressed in matrix form as follows
{e} = [F]{t}
219
(6)
(7)
Where [F] is b-order diagonal matrix with
F, = (L / EA) ,
i=l,b
(8)
Stability Criterion
and Convergence
When analyzing the form in geometric stable equilibrium state, we use the compatibility Eqn.3 instead
of static equilibrium equation Eqn.7. At equilibrium the total potential YI of the structure takes a local
minimum value. The necessary and sufficient conditions for equilibrium are
oTI=0 (9)
521 I 0 (10)
Where 5 is a variational symbol related to the displacement space. The equilibrium is arbitrary or

stable according to the value of 6 2H. Condition expressed by Eqn.9 will be used in next section in
order to choose a parameter leading to the stable equilibrium state. For the problem of GSES, we use
linear incremental method to solve the system of linear homogenous equation 3. That is to say an
iterative procedure will be used. As the incremental {d} is small, in each iterative step (say i-l), take
the first order approximation, then
{d},_ 1
={x'}i_lt
(11)
Where t is a small parameter, {x'},_~ is the first order derivation of the displacement vector with regard
to t. If {d},_~ have been found out,
[B]i. 1 can
be calculated[2]. So we have
IN]i_ 1
{dIi
{0) (12)
Based on the generalized inverse theory of matrix algebra, thesolution of Eqn.12) is
{d},
=([I]-[Bl+[B]{y},_l
= [D],_I {a},_~
(13)
where { y}
t-1
as well as {a} ,_~ are n and
(n-r)
dimension algebra vector, respectively. In this paper we
chose { 1 }-inverse, so { c~} i-1 is a
(n-r)
dimension vector. [D] ~-1 will be called as a displacement model
matrix. Hence from i-1 step to i step, nodal position coordinate vector is