Optimization of PT in Cantilever Construction of Prestressed Concrete Bridges

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37th IABSE Symposium Madrid 2014

Optimization of PT in Cantilever Construction of Prestressed Concrete Bridges Optimization of PT in cantilever construction of prestressed concrete bridges Santiago HERNANDEZ Civil Engineer Professor University of Coruna A Coruna, Spain

Pablo OURO Civil Engineer University of Coruna A Coruna, Spain [email protected] [email protected]

[email protected]

Luis E. ROMERA Mechanical Engineer Associate Professor University of Coruna A Coruna, Spain [email protected]

Santiago Hernandez, born in 1951, received his Ph. D in civil engineering from the University of Cantabria, Spain. He has more than thirty years of experience in structural optimization in mechanical, mechanical, civil and aircraft engineering.

Pablo Ouro, born in 1989, received his civil engineering degree from the University of Coruna, Spain. He has been a Research Assistant at the Mechanics of Structures Group of the University of Coruna.

Luis E. Romera, born in 1967, received his Ph. D in mechanical engineering from the University of Coruna, Spain. He has twenty years experience in analysis of mechanical, mechanical, civil and aircraft structures.

Summary In this paper an example of application of structural optimization methodologies is presented describing the minimization of the amount of prestressing steel required in a prestressed concrete bridge built by cantilever construction procedure. The numerical results obtained show that this approach reduces the quantity of material needed and does not introduce any additional complexity to bridge construction. Keywords: Prestressed concrete bridges, structural optimization, cantilever construction, multimodel analysis.

1.

Introduction

Cantilever construction is a very common procedure of prestressed concrete bridges [1,2]. The procedure is independent of the length of the span and thus longitudes of even 301 m. for the main span has been reached in the case of the Stolma bridge in Norway [3]. Figure 1.a shows a bridge during the cantilever construction and figure 1.b presents a picture of the aforementioned Stolma bridge.

a) Balanced cantilever construction

b) Stolma bridge, Norway

Figure 1. Examples of cantilever construction in concrete bridges .

IABSE Madrid Symposium Report, Vol. Vol. 102

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Cantilever construction is carried out by building in situ segments of the bridge or building prefabricated segments of the deck which are then put together after being lifted up. In the latter case some technical entities have published a set of predefined shapes for box girder decks [4,5]. During construction, after in situ or prefabricated segments of the deck are put in place, steel cables are introduced and prestressed. The amount of prestressing force needs to take in account two different requirements: a) The theoretical geometry of the bridge has to be maintained and therefore the vertical coordinates of the deck along the cantilever need to be inside the accepted tolerance values. b) Undesirable tensile stresses of excessive compressive stress in the concrete must be avoided. To achieve both objectives the number of steel strands varies along the span, diminishing in number from the supports to the span centre. Also the value of the prestressing force changes during the erection procedure because, as the weight of the deck increases, more prestressing is needed to accomplish the two conditions aforementioned. Usually the distribution of strands and the value of the prestressing force is decided by the practitioners using engineering judgement based upon previous realizations and experiences. Nevertheless nowadays more efficient techniques, for instance optimization methodologies, are at designers disposal facilitating to identify the best solution for the prestressing force of a given bridge.

2.

Definition of structural optimization

Modern structural optimization started in 1960 [6] and since then has matured from being an academic subject to a design method used in several engineering fields as mechanical and aircraft engineering. During these years several text books have been published [7-10] and some books describing the possibilities of this technique in real engineering problems exist. The main idea behind this technique is the substitution of heuristic rules, than are subjective and can be inefficient, by rigorous mathematical algorithms than proceed in a iterative way until reaching the best solution for the problem. Figure 2 shows two flowcharts indicating the conventional design and the structural optimization procedures.

a) Conventional Conventi onal design procedure

b) Structural Structura l optimization optimizat ion procedure

Figure 2. Flowcharts of design procedures

Applications of this design methodology in civil engineering are scarce but nevertheless some contributions to real problems have been carried out so far [13,14]. Proper formulation of a structural optimization problem requires the definition of the following - Set of design variables: Properties of the problem that can vary along the process. - Set of constraints: Groups of requirements than need to be accomplished in the process. - Objective function: Property of the design problem that is to be optimized.

3. Formulation of the optimization of the amount of prestressing cables in a prestressed concrete bridge The problem tacked out corresponds to the minimization of the amount of steel required for the prestressing cables of the portal bridge presented in figure 3 having a main span of 210 m length.


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Figure 3. Bridge elevation

Figure 4 shows the steps of the balanced cantilever construction with consisted in 17 phases and finished by building the central segment. Prestressing cables were designed as indicated in figure 5. It can be observed that some cables are anchored at the support, while others are anchored at two different locations inside the lateral span.

Figure 4. Phases of the cantilever construction procedure

Figure 5. Definition of the prestressing cables

Structural analysis of the bridge was carried out using a finite element model using beam elements as defined in the commercial code SAP2000 [15]. Beams elements were defined at the gravity centre of cross section bridge deck as indicated in figure 6. A different structural model was needed for each construction step. Figure 7 shows the full set of structural models considered in the analysis.


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Figure 6. Nodes location in the structural model of bridge main span.

Figure 7. Set of structural models considered in the study describing the construction phases.

The aim of this piece of research was to compare the results obtained by an engineering company using heuristic rules with those provided by optimization methodologies. In this case bridge designers were not inclined to change the location of the anchoring points of the tendons, therefore such characteristic was adopted as invariable in the optimization problem. Of course, anchoring locations could also be considered variables, but the main idea behind this work was to demonstrate that any formulation of bridge design can be worked out as an optimization problem. Therefore, in this case the objective of the problem was to minimize the amount of material required for the prestressing cables that will be defined as N

min V ( Pl )

=

min  l 1 =

Pl Ll ⋅

(1)

σ s

where Pl and Ll are the prestressing force and the length of the l-esime cable,


is the allowable

σ s

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stress of the steel and N = = 17. The set of design variables is composed by the prestressing forces Pl (l = 1,... N ). N ). Two types of the constraints were considered: a) Tolerances of vertical displacement of bridge deck : Each construction step leads to a geometry of the bridge deck that ideally should be a straight line connecting both ends of the central span. Nevertheless in practice some tolerances for positive and negative values of the vertical displacements, namely, k l and k u, need to be accepted. In this cases the following values were provided by the consultancy firm. k l = 0.2165 m for each construction phase

construction steps 1 to 5 k u = 0.0 m for the construction k u = 0.021 m for the remaining construction steps

Therefore the expression for the constraints related to the displacements of the nodes of the main span could be written as N

kl

≤

wi , ppj

+

 wi ,Pj ≤ ku

(2)

j =1

where

wi,ppj: vertical displacement at the i-esime node by bridge weight when construction has progressed up to the j-esime node. wi,Pj: vertical displacement at the i-esime node produce by the prestressing tendon at j-esime node. k l,l,k u: lower and upper limits of vertical displacements of nodes of cantilever structural models. wi,p1j: vertical displacement at the i-esime node produced by a unit prestresing load at the j-esime node.

Therefore expression (2) could be rewritten as N

kl

≤

wi , ppj

+

 Pj wi , p1 j ≤ ku

(3)

j =1

b) Stress constraints for bridge deck concrete: The limit values of tensile and compressive stresses for the concrete were defined by σ t = 0.0 MPa; σ c = 42 MPa. No condition was defined on the compressive stresses of the concrete because it was assumed that the prestressing forces will produce stress values much lower than the allowable limit. The stress values at the optimum design corroborated such assumption. The expression for the constraints related to the stress at the top fibre of the bridge deck, as described in figure 8, can be written as follows σ ppi , j

− σ Pi , j ≤ σ t

(4)

Figure 8. Cross section of bridge deck


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where : Stress produced by the prestressing forces at the top fibre of the i-esime node when bridge construction has reached the j-esime node.

σ Pi,j

: Stress produced by bridge weight at the top fibre of the i-esime node when bridge construction has reached the j-esime node. σ ppi,j

: limit value of tensile stress on concrete.

σ t

Stress σ Pi,j can be written as j

σ Pi , j =

Pl

 A l =i

+

i

M il ⋅ hi Ii

j

=

Pl

A l =i

i

+

Pl ⋅ ei ⋅ hi Ii

1

j

=

P  A l

l =i



+

i

ei ⋅ hi 

 (5)

I i 

where Pl: prestressing force of l-esime tendon. Ai: Area of bridge cross section at e-esime node I i: moment of inertia at i-esime node ei: Eccentricity of prestressing force at i-esime node. hi: distance between the gravity center of bridge deck cross-section and top fibre. M ilil: bending moment produced at the i-esime node by the j-esime prestressing force.

Similarly, Similarly, stress σ ppi,j , produced by bridge weight, can be expressed as σ ppi , j =

M ij ⋅ hi I i

(6)

where M ijij: bending moment produced by bridge weight at i-esime node when bridge construction has reached the j-esime node.

It can be observed that both, objective function and set of constraints, are linear expression in terms of the design variables Pl (l=1,...,N) which are the prestressing forces. Therefore from the mathematical point of view the optimization problem is a linear one. 4.

Numerical results

The optimization problem was solved using the SIMPLEX method. Distribution of axial forces of the initial design, given by an engineering firm and the optimal solution appears in figures 9 and 10. The numerical results of the prestressing forces at the initial design and the optimum design are presented in table 1.

Figure 9. Distribution of axial forces of the initial design


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Figure 10. Distribution of axial forces of the optimum design Table 1: Values of the prestressing prestress ing forces in the initial and optimal design PRESTRESSING FORCES

P1 (kN) P2 (kN) P3 (kN) P4 (kN) P5 (kN) P6 (kN) P7 (kN) P8 (kN) P9 (kN) P10 (kN) P11 (kN) P12 (kN) P13 (kN) P14 (kN) P15 (kN) P16 (kN) P17 (kN) Total value P (kN)

INITIAL DESIGN 9790 10261 10264 10558 10136 10242 10558 10347 10135 10649 10650 10649 10649 10649 10649 10649 10649 177485

OPTIMUM DESIGN 5486 5974 6494 7028 7615 8232 8687 9567 9976 10560 11097 11570 11974 12011 12248 12394 12443 163356

An admissible stress value of σ s = 1095 MPa was assumed for the steel. Considering the expression 3 of the volume of the prestressing cables given by (1) and the steel density of ρ = 7850 kg/m the numerical values of material weight for the initial and optimum design are presented in table 2 Table 2. Weight of prestressing steel s teel in the initial and optimal design OBJECTIVE FUNCTION

Amount of prestressing steel

INITIAL DESIGN 119340 kg

OPTIMUM DESIGN 112350 kg

The optimization procedure reduced the prestressing steel in an amount of 6990 kg. This quantity is for one of the cantilever spans therefore the saving for the bridge is twice that figure. In fact this project consisted in two twin bridges, one for each traffic direction of the highway, so the final weight reduction of the prestressing steel is almost 28000 kg.


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5.

Conclusions

The following conclusions can be extracted from this study: 1) A detailed analysis of a cantilever constructed prestressed concrete bridge has been carried out considering the full set of construction phases. 2) A formulation of the minimization of the prestressing forces has been set up taking in account the construction procedure and the limits imposed to the vertical displacement of bridge deck and the tensile stress of concrete. 3) The numerical results provided by the optimization approach showed that the quantity of prestressing steel is lower that the amount obtained by conventional design procedures.

6.

Acknowledgements

Information on the bridge used for the study s tudy was kindly given by the t he Spanish construction company Puentes y Calzadas. The authors duty recognize such contribution.

6.

References

[1]

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[2]

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[3]

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[4]

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[5]

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[6]

SCHMIT L,. "Structural Design by Systematic Synthesis", Second conference on Electronic Computation, ASCE , Pittsburg, Pa, pp 105-132, 1960.

[7]

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[11]

ARORA J.A., (ed.), Optimization of Structural and Mechanical Systems, Word Scientific, 2007

[12]

HERNANDEZ S. and FONTAN A.N., Practical applications of design optimization, WIT Press, 2002.

[13]

FONTAN A.N., DIAZ J., BALDOMIR A., HERNANDEZ S., "Improved optimization formulations for launching nose of incrementally launched prestressed concrete bridges", ASCE Journal of o f Bridge Engineering, 16 (3), pp 461-470.

[14]

aeroelasticity. JURADO J.A., HERNANDEZ S., NIETO F., and MOSQUERA A., Bridge aeroelasticity. Sensitivity analysis and optimal design, WIT Press, 2011.

[15]

SAP2000 V15, User´s Manual, Computers and structures, 2012.


Optimization of PT in Cantilever Construction of Prestressed Concrete Bridges

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