Insumos

This method was originally designed to adapt and expand the analysis method using the truss model to the treatment of areas called type D (D as in disturbance) that may consist of geometric discontinuities or disturbances due to the application of concentrated loads. The method was developed by the German school, particularly by Jörg Schlaich, and has been used and presented by the Comité Européen de Béton (CEB) in its Model Code. Eurocode 2 presents only a summary of it through its clauses “Analysis with strut and tie model”

[EC2-1-1 5.6.4] and “Dimensioning using strut/tie models” [EC2-1-1 6.5]. For a correct usage of the method the designer is urged to consult, for example, the excellent report published in English by the PCI JOURNAL of May-June 1987 on the subject, entitled "Toward a Consistent Design of Structural Concrete" by Schlaich et al.

This guide mentions only a few essential rules.

II.1.Principle

The principle is as follows:

Special prescriptions and justifications Chapter 10 - Special prescriptions and justifications

• Choice of a diagram of transmission of the forces that allow building of the strut-tie model ensuring routing of the forces by straight elements (bars) and their deviation by nodes.

• determination of statically balanced forces in the struts and ties

• and finally, dimensioning and verification of struts, ties and nodes.

Fig./Tab.II.(1): Schematic diagram of a case of application of the strut method

II.2.Dimensioning of struts, ties and nodes II.2.1.Resistance of struts

The corresponding clauses are in [EC2-1-1 6.5.2].

Two cases are highlighted: the one where the strut is transversely compressed or not and the one where it is subjected to transverse tension, which leads to a reduction in its resistance.

Fig./Tab.II.(2): Resistance of struts subjected and not subjected to a transverse tension In the figure on the right the vertical arrows represent a transverse compressive stress possibly nil.

The resistances to compression of the struts are thus:

σRd,max = fcd for transverse compression. [EC2-1-1 Expr.(6.55)]

σRd,max = 0.6 ν’fcdfor transverse tension. [EC2-1-1 Expr.(6.56)]

with ν’ = 1-f /250 [EC2-1-1 Expr.(6.57N)]

This value of 0.6 ν’ may be brought closer to that of ν = 0.6(1-fck/250) where ν is the reduction factor of the resistance of the cracked concrete to the shear stress.

II.2.2.Resistance of ties

The corresponding clauses are in [EC2-1-1 6.5.3].

The design resistances of the transverse ties and the tendons meet the same limits as given by the general rules [EC2-1-1 3.2 et 3.3].

In particular, for reinforcement of reinforced concrete it is advisable on the one hand to anchor them according to the principles of section [EC2-1-1 Sect.8] and on the other hand to limit their stress to:

fyd = fyk/γs

Eurocode 2 part 1-1 recommends not to concentrate the ties in their theoretical model position but to distribute them on the zone where stresses increase [EC2-1-1 6.5.3(3)].

Eurocode 2 part 1-1 then mentions the expressions of the tensile forces for two simple cases:

a) for the case of regions of partial discontinuity ( b ≤ H/2) , [EC2-1-1 Fig.6.25 a) ] , b F

a b 4

T= 1 − [EC2-1-1 Expr.(6.58)]

b) for the case of regions of total discontinuity ( b > H/2), [EC2-1-1 Fig.6.25 b) ] , h F

7a , 0 4 1

T 1 



 

 −

= [EC2-1-1 Expr.(6.59)]

Special prescriptions and justifications Chapter 10 - Special prescriptions and justifications

B Region without discontinuity D Region of discontinuity a) partial discontinuity b) total discontinuity

Fig./Tab.II.(3): Transverse tensile forces in a field of compressive stresses with distributed reinforcement [EC2 -1-1 Fig.6.25]

II.2.3.Resistance of nodes

The corresponding clauses are in [EC2-1-1 6.5.4].

2.3.0.a)Compression limit in typical nodes

The compressive stress limits to meet for the three types of joint most commonly found are mentioned here:

(1) nodes subjected to compression with no tie anchored in them

Maximum acceptable compressive stress:

σRd,max = k1 ν’fcd

with k1= 1.0 value recommended by Eurocode 2 part 1-1 and the national annex

ν’ = 1-fck/250 (remember)

The national annex authorizes, on special justification, taking k1

to k1 = 1/ ν’

Fig./Tab.II.(4): Node subjected to compression with no tie [EC2-1 - 1 Fig 6.26]

(2) nodes subjected to tension and compression with tie s in one direction

Maximum acceptable compressive stress:

σRd,max = k2 ν’fcd

with k2= 0.85 value recommended by Eurocode 2 part 1-1 and the national annex

The national annex authorizes, on special justification, taking k2 to k2 = 1.0

Fig./Tab.II.(5): Nodes subjected to compression and tension with ties in one direction [EC2-1-1 Fig.6.27]

(3) Nodes subjected to tension and compression with ties in two directions

Maximum acceptable compressive stress:

σRd,max = k3 ν’fcd

with k3= 0.75 value recommended by EC2-1-1 and the national annex

The national annex authorizes, on special justification, taking k3 to k3 = 0.9

Fig./Tab.II.(6): Node subjected to compression and tension with tie s in two directions [EC2-1-1 Fig.6.28]

2.3.0.b)Increase of compression limit for special conditions

For certain nodes the acceptable compressive stress values may be increased by 10% if at least one of the following conditions is verified:

• a tri-axial compression is assured,

• all the angles between struts and ties are ≥ 55°,

• the stresses at the level of the supports or the points loads are uniform, and the node is bordered by transverse reinforcement ,

• the reinforcement are arranged on several courses,

• the node is securely confined by a special support arrangement or by friction.

Special prescriptions and justifications Chapter 10 - Special prescriptions and justifications

2.3.0.c)Case of nodes subjected to tri-axial compression

The special case of nodes subjected to tri-axial compression may be dealt with as that of confined concrete [EC2 1-1 3.1.9] whose characteristic resistance may be defined by the expressions:

fck,c = fck (1.000 + 5.0 σ2/fck) pour σ2 ≤ 0.05fck [EC2-1-1 Expr.(3.24)]

fck,c = fck (1.125 + 2.50 σ2/fck) pour σ2 > 0.05fck [EC2-1-1 Expr.(3.25)]

where σ2 is the lateral compressive stress due to confinement.

The compression limit is however maximized at the value σRd,max = k4 ν’fcd,c

with fcd,c=fck,c/γc. et k4= 3.0 value recommended and accepted by the national annex which, moreover, authorizes on special justification an increase in k4 to the value of k4 = 3/ν’.