1

F

FC

T=C

Fig. 1 - Simple STM for a corbel

fib Symposium “Keep Concrete Attractive”, Budapest 2005 STRESS FIELD MODELS FOR STRUCTURAL CONCRETE João F. Almeida Civil Engineering, Associate Professor, IST. ICIST, Lisbon email:jalmeida@jsj.pt Miguel S. Lourenço Civil Engineering, JSJ Consult, Lisbon email:mlourenco@jsj.pt SUMMARY Strut and tie models (STM) are currently recognized as a valuable tool for the design and detailing of discontinuity regions in structural concrete. These regions are still weakly covered in technical documents, although its deficient design and detailing is frequently related to damages and even failures. Uniqueness and models validity is frequently mentioned as an obstacle for its practical implementation. A new approach, based on energetic criteria, applied to stress field models, is proposed to perform the assessment of the models. The method was tested by comparison with experimental results and known solutions for some typical cases. Its application to practical design situations is illustrated as well. The development and the practical application of such specific design tools will improve consistency and clearness of design methods, contributing to keep concrete attractive.

1. INTRODUCTION The application of strut and tie models (STM) will improve consistency in design concepts, allowing the treatment of discontinuity and current regions with comparable accuracy and emphasizing the essential role of detailing in the design process. Based on equilibrium and plasticity, such models intend to reproduce the stress trajectories, showing the flow of forces, throughout a concrete region. In a simplified way, service behaviour can be indirectly checked by ensuring that the model follows the elastic stress fields. In spite of such simple concepts, for different reasons the practical application of the method can not be considered widely disseminated. Some of the main aspects frequently mentioned refer to the model assessment validity and to its uniqueness. 2. MODEL UNIQUENESS The establishment of an equilibrate strut and tie model requires some engineering judgement. Modelling principles were printed out, e.g., by Schlaich et al (1987, 1991) and later by Schafer (1996). The designer would easily achieve the corbel design model shown at fig. 1, although, some doubts could come forward if the same corbel suspends the load. In that case, it is possible to transmit the load to the supports by vertical or inclined reinforcement, thus two different STM are possible (see fig. 2). Based on the lower bound of the plasticity theory, both models are correct, if the structure has enough ductility to adapt itself to the chosen

2

k.F

k.FC1

T1=C1

(1-k).FC2

T2=C2(1-k).F

Fig. 2 – STM for a corbel with indirect load.

model. A question remains: Which part of the load F should go through inclined stirrups, in order to avoid deficient service behaviour and assure enough ultimate resistance? In most cases the answer is based on the elastic theory or specific tests. Schlaich et al (1987) mentioned that the main struts and ties should follow the elastic stress trajectories in order to indirectly check serviceability. This criterion often leads to hyperstatic models obtained by superposition of two different models (see fig. 3).

3. STRESS FIELD MODELS The STM refinement can be the basis for the stress field model (SFM) which becomes very useful for many design issues, namely: -nodes geometry definition; -identification of distributed or concentrated ties; -definition of mechanical properties for the struts; and many others. For the illustration of SFM application a practical example of a viaduct fixed abutment shear wall is chosen, subject to several loads: -soil pressure; -deck’s vertical loads; -horizontal loads. Fig. 4 represents the shear wall loads and corresponding STM.

Load Case 1 Load Case 2

Fig. 4 – Abutment shear wall, loads and strut and tie model.

Fig. 3 – Hyperstatic models: a) concentrated load near support; b) internal anchorage load.

3

σ – stress; ε – deformation; t – boundary loads; uΓ– imposed displacements; Γu – cinematic boundary

The STM gives a good approximation of the D’region structural behaviour, in general better than other current design techniques, and provides good guidance for detailing. As known, the struts are defined by a single line that represents compression stress fields on concrete. The check of stresses along struts can be disregarded, because it spreads inside the region, and only concentrated nodes should be checked (see fig.5). A step forward could be the development of the SFM illustrated at fig. 6, giving an excellent overview of the stress flow throughout the region.

Load Case 1 Load Case 2

Fig. 6 – Abutment shear wall stress field model 4. MODELS ASSESSMENT Following the stress fields definition associated to a STM, struts and ties mechanical properties evaluation are possible. Energetic techniques can be applied in order to consider compatibility in the general model allowing the development of assessment techniques. For every possible static condition, the “exact” solution minimizes the functional (Koiter, 1960).

∫∫Γ

Γ Γ−εσ=Π uT

V

T dutdV21

Applying energetic theorems associated to SFM, it is possible to perform a nonlinear analysis, assuming nonlinear behaviour for concrete and tension stiffening effects (Sundermann, 1994). The proposed methodology can as well give some guidance for practical design and is based on deformation energy minimization, dU/dF=0, where F represents STM element forces.

5Ø205Ø20 2Ø20

8.93 MPa

0.56

x(0

.40)

3.4MPa

0.69x

(1.00)

2000 kN

2355 kN

T1=

1244

kN

1800

kN

Ø 0.37Ø 0.47

T2=1525 kN

4Ø20

4Ø20

4Ø20

1801kN

0.52x

(1.00)

0.42 x (1.00)

1050kN

10.38 MPa

3.5MPa

2.5M

Pa

N1

N2

1.33

0.64

Fig. 5 – Check of nodes

4

Considering elastic behaviour for concrete, the energy of each strut and each tie can be calculated as indicate at fig. 7:

cc

2

AELC

21U =

⋅

−=

ci

cj

cicjc

2

AA

ln)AA(E

LC21U

ss

2

AELT

21U = or generally smTL

21U ε=

Fig. 7 – Struts and ties deformation energy As an illustration, results were compared with deep beam tests performed by Leonhardt and Walther (1966) (see fig. 8). Large redistribution of internal stresses due to cracking was observed. In fact, WT2 test ultimate load was approximately the double of the obtained by elastic based STM. The inner lever arm z tends to use the deep beam full height, thus reducing considerably the bottom tie force. In case of the deep beam WT3, the bottom reinforcement area was doubled, stiffening the bottom tie. The corresponding inner level arm z at failure is smaller than that of deep beam WT2.

Test specimen WT2 geometry and loads

Test specimen WT2 at failure (Fu,exp=1195 kN)

Test specimen WT3 geometry and loads

Test specimen WT3 at failure (Fu,exp=1290 kN)

1.28

1.60

0.16 1.28 0.16

1.44

Ø8 (As=2.14 cm )2

0.02

5

Ø5

Ø5

0.03

Ø5

0.100

1.28

1.60

0.16 1.28 0.16

1.44

Ø8 (As=4.28 cm )2

0.02

5

Ø5

Ø5

3x0.

06

Ø5

0.100

Elastic based STM Ultimante load based STM Elastic based STM Ultimante load based STM (Fu,calc=580kN) (z/L=0.70)

(reinforcement failure) Fu,calc/2

1.44

0.04

1.00

68°

0.64

0.56

117 kN

Fu,calc/2

(Fu,calc=1195kN) (z/L=1.05) (reinforcement failure)

117

Fu,calc/2

1.44 0.04

1.51

0.64Fu,calc/2

78.0°

(Fu,calc=1082kN) (z/L=0.70) (node compression failure)

Fu,calc/2

1.44

0.04

1.00

68°

0.64

0.56

234 kN

Fu,calc/2

(Fu,calc=1290kN) z/L=0.90) (node compression failure)

136

Fu,calc/2

1.44 0.04

0.64

1.30

Fu,calc/2

77.0°

Fig. 8 – Deep beam tests (Leonhardt and Walther, Stuttgart, 1966)

C, T – compression and tension forces, respectively Ac – prismatic strut area. Aci, Acj –initial and end areasEc, Es – Young modulus of concrete and steel L – strut or tie length εsm – reinforcement mean strain

5

z

h

L

z h

L

z

h

L

1.0

6.0

11.0

16.0

21.0

26.0

31.0

36.0

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

z/L

U/U

mín

linear non linear As=2.14cm2 non linear As=4.28cm2

z

h

L

z h

L

z

h

L

1.0

6.0

11.0

16.0

21.0

26.0

31.0

36.0

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

z/L

U/U

mín

linear non linear As=2.14cm2 non linear As=4.28cm2

Fig. 9 – Def. energy vs deep beam z/L

Fig. 9 shows th numerical results obtained by minimizing the SFM total deformation energy, pointing out the different z/L values for each analysis: -elastic based SFM and ultimate load SFM for deep beam’s WT2 and WT3. This technique provides the designer a simple tool for the prediction of structural behaviour of D’regions and for the STM assessment validity.

5. HYPERSTATIC MODELS The STM for interior tendon anchorage is a hyperstatic model, since part of load must be tied back behind the anchorage plate. The SFM for this case is shown at fig.10 as well as the variation of deformation energy with the percentage of load tied back. The minimum energy is obtained for F2≈0.20F, which is in accordance with fib recommendations 1999.

F2/2

F1/2

F/2

F/2

F2/2

F1/2

Interior tendon anchorage

0

1

2

3

4

5

6

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

F2/F

U/U

mín

Fig. 10 – SFM for an interior tendon anchorage. STM for a concentrated load near support should predict that part of the load that must be carried by the stirrups, as Schlaich (1991) has shown with corbel tests. MC90 and fib recommendations 1999, provides a simple equation for the suspended load percentage as a function of a/z. Fig. 11 compares the elastic solution (plate elements, finite element analysis), the MC90/fib recommendations 1999 and the proposed method numerical results.

F

T

M

C

F1 F2

F

C1

C2

z

a Load near support

0.00.10.20.3

0.40.50.60.7

0.80.9

1.0

0.5 1.0 1.5 2.0

a/z

F2/F

Udef MEF MC90

Fig. 11 –Load near support

The presented results illustrate the application of the proposed technique to the analysis of hyperstatic models.

−= 1

za2

31

FF2

6. PRACTICAL APPLICATIONS The technique proposed has been applied to some practical design cases. Fig. 12 shows a cross section of a 750m long viaduct in Lisbon, with current spans of 52.3m and two major spans of 90m. It is an urban viaduct, near the ground, so aesthetics aspects where carefully considered. Fig. 13 shows the column STM and SFM for vertical loads. The STM “outer part” is quite similar to a reverse deep beam with a distributed load at the top. Due to the column top geometry and the load eccentricity, it is necessary to conspart”. The inner level arm for the “deep bapproximately 0.7 of the span. Nevertheless sz, which is most relevant for the horizontanchorage. Minimizing deformation energy used for the columns design and detailing.

Fig. 13 – Vertical loads STM and SFM and re 7. FINAL REMARKS The paper points out the importance of the dmodels to the design and detailing of structuresults and its comparison with experimentalstandard cases, show the adequacy of the preassessment/validity “problems”, frequently reFIP Recommendations (1999): Practical Design of S

Sept. 1996. Publ.: SETO, London, Sept. 1999. CEB-FIP MC 90 (1993): Design of concrete structureLeonhardt F. and Walther R. (1966): “Wandartiger T

Heft 178, Wilhelm Ernst & Sohn, Berlin Schlaich, J.; Schäfer, K; Jennewein, M. (1987): Tow

design for structural concrete. PCI-Journ. V.32 (1Schlaich, J.; Schäfer, K. (1991): Design and detaili

concrete using strut-and-tie models. The StructuraShafer, K. (1996) - “Structural Concrete”, Textbook o

a 203, Dec. 1999. Sundermann, W. (1994): Tragfähigkeit und Tragver

für Tragwerksentwurf und -konstruktion, Univ. St

Fig. 12 – Cross section

ider an U’loop represented by the model “inner eam part” is well known and was considered ome doubts occur for the U’loop inner level arm al tie force and for the vertical reinforcement (fig. 13), a value of z=3.2m was obtained and

Deformation energy

1.00

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.10

1.7 2.2 2.7 3.2 3.7 4.2 4.7

z [m]

U/U

mín

spective deformation energy function of z.

evelopment and application of stress field based ral concrete discontinuity regions. The obtained results and the proposed solutions to hyperstatic sented methodology to deal with uniqueness and ferred in STM design field. tructural Concrete. FIP-Commission 3 "Practical Design",

s. CEB-FIP-Model-Code 1990. Thomas Telford, 1993. räger”, DAfStb

ard a consistent 987), No.3 ng of structural l Engineer, Vol 69n behaviour, Desig

halten von Stahlbeuttgart, 1994.

6

, No. 6, 1991 n and Performance, Volume 3, pp. 141

ton-Scheibentragwerken. Diss., Institut