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Robustness of Highway Overpasses
H. Stempfle
Swiss Federal Institute of Technology ETH, Zurich, Switzerland
T. Vogel
Swiss Federal Institute of Technology ETH, Zurich, Switzerland
ABSTRACT: The demands on bridges increase during their service life. Road traffic is growing
and additional actions like failure of a column, terrorism, settlements etc. arise that are normally
not considered when the structure was designed. A new approach concerning the robustness of
bridges is necessary. Bridges are individual and complex systems. To evaluate the robustness of
bridges we have to describe them as a system which consists of a complex combination of different subsystems. The whole system is divided in subsystems to investigate the interactions between the different subsystems. The robustness of a structure depends on the interactions between the subsystems. It has to be evaluated how they depend on each other and what are the
different influences of each subsystem concerning the whole system. Highway overpasses with
V-columns are treated as an example because they are widely common in Switzerland. Based on
the methods of systems theory and theory of plasticity a concept for evaluation of the robustness
of bridges will be developed.
1 INTRODUCTION
Bridges are planned for a long service life with changing demands that can hardly be fully anticipated. The traffic is still growing, the live loads increase and de-icing salt is used more and
more since the seventies of the last century. Due to the amount of traffic it is not possible anymore to reduce the number of traffic lanes or even to close the whole bridge during repairs. In
fact, during repair periods the full traffic is diverted on side-strips that are not designed for that
purpose. Although all these facts lead to higher risks, the society claims to limit risks more than
ever. A concept to overcome this situation is to design new bridges for robustness and to judge
the robustness of existing ones.
If different authors discuss about robustness of structures, they use a lot of terms (robustness,
redundancy, vulnerability, system, etc.) in different contexts (serviceability, structural safety
etc.). In this paper the robustness of structures is considered in the context of structural safety.
The robustness of the structure itself is investigated and the correlation between the environment and the structure is neglected.
An investigation of the robustness of a structure always concerns the whole system and not
only parts of the structure. To evaluate the robustness of a structure it is necessary to define both
the term robustness and the system itself. In the first part of the paper a proposal for a definition
of the term robustness is shown. Based on Systems Theory of Engineering by G. Ropohl [4] a
definition of a bridge system is described in the second part.
2 TERMS
2.1 Redundancy
The original meaning of the term redundancy is abundance and profusion. It is often used in different sciences (genetics, theory of communication, engineering science, etc.). Today in every
science the linguistic use of the term redundancy is similar. Often redundancy is used in the context that there exist an oversupply of information which means that a part of information can be
eliminated without loss of essential information.
In engineering science redundancy means that an additional expense is made to guarantee the
reliability of a system, although this additional expense is not necessary for the function of the
system. Redundancy is differed in active and passive redundancy. Passive redundancy means
that different apparatuses are connected in parallel but only one of them is activated. They will
be only activated if the running one fails to undertake the task.
Active redundancy means that different apparatuses are connected in parallel but they are
running at the same time, being independent of each other. In contrast to the passive redundancy
active redundancy means, if one system fails the other one will undertake the task without any
time-lag.
Normally, systems of structural engineering have neither a passive redundancy nor an active
one. All parallel systems or subsystems of a structure are working. It is not possible to activate
them as soon as one subsystem fails. To say it otherwise there is no passive redundancy in structures. Structures have many activated subsystems but they depend from each other. They fulfil
only a part of the definition of active redundancy.
Due to this we propose that the term redundancy should not be used describing structures.
2.2 Robustness
The term robustness is often used in different contexts (serviceability, structural safety, etc.).
Some authors use robustness in the context of both, serviceability and structural safety [1], [2].
These definitions are too general and they would not help to find a good definition of the term
robustness.
In the following the meaning of robustness refers to the structural safety only but covers both
structural elements and whole structure. It should be possible that components have a brittle behaviour but the damage of the structure should be limited compared to the whole structure. A
collapse of the whole structure should be excluded. Due to this we use the definition of the
Swiss code SIA 260 [3] as a possible definition of the term robustness: “Ability of a structure
and its members to keep the amount of deterioration or failure within reasonable limits in relation to the cause”.
3 SYSTEM-THEORETICAL DESCRIPTION OF A BRIDGE
3.1 Systems Theory of Engineering by G. Ropohl
Every structure is a complex system of different components. To evaluate the robustness of a
structure it is necessary to define a system and its characters, describing an artificial object as a
system. To describe a system we use Systems Theory of Engineering by G. Ropohl [4]. Most of
the systems theories are based on either a functional or a structural or a hierarchical concept of a
system. In contrast the theory of Ropohl integrates the three concepts in one systems theory.
The systems of structural engineering have functional, structural, and hierarchical characteristics. To make a robustness analysis, it is necessary to describe a bridge as a system with a comprehensive theory, which can define all these characteristics of the system as fulfilled by the
systems theory of Ropohl. Additionally, this theory is valid for deterministic, stochastic, dynamic and static systems.
The following chapter shows a system-theoretical description of a Standard Swiss Overpass
with V-columns based on Systems Theory of Engineering by G. Ropohl.
3.2 Definition of the system
A system (for example a bridge) is defined as a quadruple of the set αB, which consists of the set
of attributes Ai, the set ϕB, which consists of the set of functions Fj, the set σB, which consists of
the set of subsystems S´Bk and the set πB, which consists of the set of relations Pm.
B
B
B
B
S B = {α B , ϕ B , σ B , π B }
(1)
{ }
′ }, π B = {Pm }
with α B = {Ai }, ϕ B = F j , σ B = {S Bk
and i ∈ I , j ∈ J , k ∈ K , m ∈ M
(I, J, K, M ~
(2)
)
3.3 Functional concept of the system
The functional concept of the system is implemented with the sets αB and ϕB. Ai are the attributes of the bridge, which represent the characteristics of the bridge. Each attribute has a parameter value with Ai ={ain} ≠ Ø. The set of the parameter values have to be a non-null set. If it is a
null set the system has no attributes, which does not make sense. Possible attributes of a bridge
are e.g. load, settlement, material, cross section, etc. The attributes are classified in a set of inputs αx, in a set of outputs αy and in a set of states αz. This classification is possible, if αx, αy
and αz are pairwise disjoint subsets of the set of attributes α.
B
B
α x ⊂ α ,α y ⊂ α ,α z ⊂ α
(3)
αx ∪α y ∪αz = α
(4)
The attributes of these three sets include the material attributes Mi, the energy attributes Ei,
the coordinates of space R and the time T. The set of attributes of a bridge αB is a subset of these
sets.
B
{
}
α B ⊂ {M xi , E xi , R x , Tx } ∪ M yi , E yi , R y , T y ∪ {M zi , E zi , R z , Tz }
Input
Output
(5)
State
Material attributes are e.g. geometric variables, construction materials, ground, etc. concerning a bridge. Energy attributes are mechanical, other physical and chemical actions. A lorry as a
mechanical action for example, has a potential energy. If this lorry drives on the bridge, the
bridge will displace. Due to the displacement of the bridge, the lorry works on the bridge. The
level of potential energy of the lorry changes and the energetic exchange between the lorry and
the bridge happens. Based on these considerations and on the energy theorem bearing reactions,
internal forces or moments, displacements, etc. can be calculated. Due to this, these variables
can be considered as energetic attributes. The coordinates of space or of the bridge are normally
constant and do not change. Plastic displacements are an exception. The coordinates of the
bridge change, if dynamic actions affect the bridge, e.g. earthquakes, stimulation of the bridge
due to wind or pedestrians, etc. It can be chosen, if the coordinate system is fix or not, depending on the character of the investigation of the system. Time becomes important for dynamics
investigations but is not considered for static investigations. It becomes important, too, when
creep, shrinkage, impulsive actions, etc. is considered.
To describe the second part of the functional concept of the system a set of functions has to
be defined. There are a lot of types of functions between the attributes of a system, which can be
used to describe generally systems or different situations of systems. The set ψ as defined in
equation (6) is a range of functions. These two types of functions describe a changing state or
conserving state of the system.
ψ = {FE , FP }
and
ϕB ∩ψ ≠ φ
The set of functions of the bridge ϕB includes at least one of the functions of set ψ.
B
(6)
System S = (α , ϕB , σ , π )
B
B
B
B
Input
Output
A1
A2
Function F E , F P
Figure 1. Functional concept of a system of an overpass
The set of functions ψ include the set of functions FE and FP. FE describes the elastic behaviour i.e. that the state of the system or bridge does not change. FP describes the plastic behaviour
of the bridge i.e. that the state of the system or bridge does change. If Ai is a material, energetic
attribute or coordinate of space and the time T, the functions describe the elastic behaviour of
the bridge. FE is defined for any axin:
FE : Azi × Axi × Tz → Azi × Tz
with
a zin
= const.
(7)
If FE ∈ ϕB the system is called static. If the bridge is regularly loaded with a lorry, elastic deformations take place. For example, the input attribute Axi is a lorry or load and the attribute of
state Azi with the parameter value azin is the deformation. The Cartesian product between the input attribute, the attribute of state and the time is mapped to a new attribute of state. It depends
on the observation period, if the parameter value azin of the deformation is still constant or not.
At the moment the bridge is loaded, the parameter value azin is not constant. If the observation
period Tz is longer than just the moment the bridge is loaded, the parameter value azin=const.
Firstly, FP is defined in a similar way as FE with Ai as a material, energetic attribute or coordinate of space and the time T and secondly, it describes the change of the state of a system like
the plastic behaviour:
B
FP : Azi × Axi × Tz → Azi × Tz
with
a zin
≠ const.
(8)
If FP ∈ ϕB the system is called transient. If the bridge is overloaded it gives plastic deformations. The input is a lorry or a load and the attribute of state is the deformation with a parameter
value zero. The Cartesian product between the input attribute, the attribute of state and the time
is mapped to a new attribute of state. The new deformation is an enduring deformation. The parameter value azin ≠ const.
B
3.4 Structural concept of the system
A structural concept of a system consists of subsystems and relations between the subsystems.
The set of subsystems σB are defined in the same way as the system.
B
′
S Bk
′ , ϕ Bk
′ , σ Bk
′ , π ′Bk )
= (α Bk
(9)
To describe a bridge using the structural concept the bridge is divided into subsystems. These
subsystems are members and joints. Couplings interconnect the subsystems. The pattern of the
system consists of material and energetic couplings and consists of sterical and temporal relations.
π B = {PKMm } ∪ {PKEm } ∪ {PRm } ∪ {PTm }
(10)
Due to the couplings the attributes end-forces situated at the left of the subsystem S´B1 are allocated with the same parameter values to the end-forces situated at the right of the subsystem
S´B2. These couplings consist in the same way for material attributes.
System SB = (αB , ϕB , σB , πB )
Coupling
Set of the subsystems σB
Environmentγ =β /σ
B
B
B
Figure 2. Structural concept of a system of an overpass
The environment of a system is described in figure 2 as a difference quantity of the set βB and
σB. The set βB is a non-null set of artificial and concrete facts and σB is a subset of βB: σB ⊆ βB.
B
B
B
B
B
B
Adjacent roads and the road network are examples for the environment of a bridge. Other elements of the environment influencing the bridge are nature (floods, avalanches, etc.), traffic, etc.
3.5 Hierarchical concept of the system
The third concept is the hierarchical concept. It is established by the definition of the subsystems. The subsystems of the bridge are the members and joints. They are defined in the same
way as the systems and have also input, output, state attributes of the categories matter, energy,
space and time. These subsystems have also sets of functions, relations and subsystems.
Subsystem
S ′Bk = (α′Bk , ϕ′Bk , σ′Bk , π′Bk )
Input
Output
A′k1
A′k2
Function
F′Ek , F′Pk
Figure 3. Structural concept of a system of an overpass
Figure 3 shows an example of subsystems of a bridge. The member of the bridge could be a
system with inputs and outputs. These inputs and outputs are connected with the environment of
the system. In this example the environment is the joining members. The relations are directed
from or to the environment. The member with the cross section of a box girder is also a system
of different subsystems. In this case the webs and slabs are the subsystems of the system box
girder. The set of couplings π´Bk interconnect webs and the slabs. The degree of fragmentation
depends on the kind of investigation of the system. If a part of a system is investigated, a high
degree of fragmentation should be achieved. If the behaviour of the whole system is considered,
it is better not to divide the system too much, otherwise the investigation would become too
complex. If an investigation concerning the robustness of a bridge is to be done, we propose that
the lowest degree of the hierarchy should be the slabs, walls, steel girders, cables, etc. These
parts are not genuine systems and therefore called elements. The definition of an element is:
(1)
SB
(
= α B(1) , β B(1)
)
with
(1)
σB = φ
(10)
This definition does not include the structural concept. The element is described only by the
sets of attributes and functions. As the elements have no subsystems, the set of subsystems σΒ(1)
has to be a null set. If the elements are interconnected, they create a system of the next higher
degree. In this example it means that the slabs and plates are added to the system box girder.
The same procedure is also valid concerning the different degrees of hierarchy between the
box girders and the bridge. The bridges and tunnels of a road could be also seen as a system of
the super-system road. The road is a system which is one degree higher of the hierarchy like the
bridges and the road network is one degree higher of the hierarchy like the road etc.
The sequence of systems is called a hierarchy of systems and is defined:
HB
(
= S B(1) , S B(2 ) , S B(3) ,K, S B(b )
)
(11)
4 CONCEPT OF EVALUATION
1. To evaluate the robustness of a system, it is necessary to define the scope of the investigation and the system boundaries. These determinations influence the complexity of the
significance of the investigation. At the end it also influences how many degrees of the
hierarchy of the system are needed.
2. Having determined the scope of investigation, the system should be defined as shown in
chapter 3. This method makes it possible, that complex system can be described in a
concise way and that weak spots can be identified easier.
3. The definition of the system is valid for deterministic, stochastic, dynamic and static
systems. It can be chosen, which method should be used for further investigations of the
system. It is possible to use the methods of system reliability or the methods of structural analysis.
4. The specific investigations should be done using one of the mentioned methods. Detailed information about the combination of the system-theoretical methods and the
methods of the system reliability or the methods of structural analysis can not be shown
at the moment. The implementation of the methods of structural analysis in systemtheoretical methods is investigated in further research.
5 CONCLUSIONS AND OUTLOOK
The term redundancy is often used in different science. The linguistic use of the term in the sciences is similar but it is not transferable for the requirements of structural engineering. Due to
this comments, we state that the term redundancy should not be used describing structures.
The definition of the term robustness of the Swiss code SIA 260 [3] is a good base including the
requirements of the difficulty of the discussion about robustness.
The definition of the system is valid for deterministic, stochastic, dynamic and static systems. It
is a basic language for all specialists and all different methods investigating the robustness of
systems and structures, respectively.
The methods of systems theory allow describing complex systems in a concise way. The coherences of the different subsystems or elements concerning the structure can be illustrated in an
easier way.
Weak spots of a structure can be identified easier and it is a useful method to find out the consequences for the structure and the nearest subsystems in case of failure of one element or subsystem.
Further research is the implementation of the methods of structural analysis in the systemtheoretical methods. The upper bound theorem is described with system-theoretical methods to
investigate the plastic behaviour of a bridge in case of failure of one or more elements and subsystems, respectively.
6 REFERENCES
[1]
[2]
[3]
[4]
Michael Pötzl, Robuste Brücken: Vorschläge zur Erhöhung der ganzheitlichen Qualität,
Braunschweig [etc.]: Vieweg, cop. 1996
Kersken-Bradley M., Unempfindliche Tragwerke - Entwurf und Konstruktion, Bauingenieur 67, 1992, Heft 1, S. 1-5
SIA (Swiss Society of Engineers and Architects); SIA 260 Grundlage der Projektierung
von Tragwerken Zürich 2003
Ropohl Günter, Eine Systemtheorie der Technik - Zur Grundlegung der allgemeinen
Technologie, Carl Hanser Verlag, München Wien 1979, 336 pp.