Zonas de Descontinuidade de Betão Estrutural Campos de Tensões Tridimensionais

Inês Figueira Sousa Rodrigues

Dissertação para a obtenção do grau de Mestre em

Engenharia Civil

Extended Abstract

Orientador: Professor Doutor João Almeida

Outubro de 2008

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DISCONTINUITY REGIONS OF STRUCTURAL CONCRETE

THREE-DIMENSIONAL STRESS FIELDS

Inês Figueira Sousa Rodrigues Instituto Superior Técnico, Universidade Técnica de Lisboa

1. INTRODUCTION The discontinuity regions of an element are the regions where no linear distribution of the deformation field is verifiable. In these regions the Bernoulli hypothesis is not verifiable which precludes the enforcement of the conventional methods of prevailing usage of analysis and section design into structural concrete. The methods based on the plastic analysis supported by stress field models lead to safe and functional results and their enforcement is currently recognized by the most recent technical and normative documents for concrete structures design, such as Eurocode 2, ACI Comittee 318, FIP/fib Recommendations, CEB-FIP MC 90, American Concrete Institute and Canadian Standards Association. The aspects regarding the modeling and safety verification methodology concerning in-plan stress states can today be considered to be solved. This matter has been given great attention and merited the due interest of the international technical and scientific community, MacGregor, J.G.; Wight, J. K. (2004), Schlaich, J., Schäfer, K (1991), Schlaich, J.; Schäfer, K.; Jennewein, M. (1987). The practical situations concerning three-dimensional discontinuity regions are very common and, generally, resolution proposals based on the overlayering of in-plan models are the ones given. As a matter of fact, the enforcement of this type of models to three-dimensional cases, in which singularities are verifiable in what mainly concerns the aspects of the building of the model and the safety verification of its elements, is far from having been suficiently studied and divulged. The case of the pile cap, whose resolution is presented as an illustration of the enforcement of the assignment, can be shown as a startling example because of its frequence in practical situations.

2. INTRODUCTION TO MODELING WITH

STRESS FIELDS 2.1. DISCONTINUITY REGIONS – REGIONS

– D The discontinuity regions are elements or part of elements where no linear distribution of the deformation field is verifiable. Examples of these regions are holes and other openings, deep-beams, corbels and frame corners. Regions that can also be identified as discontinuity regions are the element regions where concentrated charges appear such as, for instance, prestress anchorage regions. These regions are confined to the area that goes from the source of the disturbance until approximately an equal distance to the width of the element. If the width of the sections is different for each side, then the length of the discontinuity region, for each side, matches each width. The analysis and the structural design of the discontinuity regions should be made resorting to model solutions that bear in mind the transverse stress deformations. The methods based on the plastic analysis supported by the strut-and-tie model lead to safe and functional results and its enforcement does not pose great difficulties. (based on fib Bulletin 3, 1999). Region-D was used as a simplification of the discontinuity region or disturbed region. For regions where the Bernoulli hypothesis is applicable, bending theory and the prevailing design methods are applied, i.e., continuity regions, called Regions-B.

2.2. DISCONTINUITY REGIONS DESIGN

THROUGH STRUT-AND-TIE MODELS The tensions or the internal forces of a structure can be diagrammatized through trajectories. In order to build a simple model of the structure, the tensions trajectories can be condensed and confined in order to obtain struts. Therefore, the forces that are on the

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strips where the main reinforcements will be put can be idealized as the model ties. After or during the development of the strut-and-tie model, the forces on the struts and ties are defined. Thus, the tension on the concrete and on the reinforcements is checked or designed after these forces. This design is completed by the use of reinforcements to control the crack occurence or distributed reinforcement, so that the ductile behaviour can be assured. There are several rules to avoid these errors that are articulated in the chapter 2.2.3. in Rodrigues (2008). An important conclusion has been reached from the experience and the practice of developing strut-and-tie models: some typical nodes will appear under different ways and arrangements, even in rather distinct elements. These typical nodes are due to the existence of very few discontinuities with a fairly different tension pattern. Thereby, it is possible to relate and combine several models, adjusting them to the geometry of the element. This fact is of great importance and outmost relevance to the development of the model.

3. IDENTIFICATION AND TYPIFICATION OF THE NODES As it is referred in fib Bulletin 2 (1999), the safety verification for a concrete element is not complete if the nodal zones are not taken into account, since they are zones where a high concentration of tensions in the concrete exists. To the design of these regions one should keep in mind that it occurs in them a shift in the direction of the forces that concur there. In given situations, the force transfer is done in a relatively large region and in a gradual manner, preventing thus the occurence of large tension concentrations. These regions are called smeared nodes. On the other hand, it is common situations where the forces result of concentrated stress fields, a fact which forces the transfer region to be reduced in size. This situation causes tension concentrations in small regions that are called singular or concentrated nodes. Figure 1 shows the types of nodes mentioned. Generally speaking, smeared nodes do not present resistance problems, as long as the reinforcement anchorages are given the due detail. As for the concentrated nodes, it will be necessary a solid

analysis of its resistance and of the reinforcement detailing that concur there. This being so, to design the concentrated nodes one has to obtain its geometry, characterize the installed tension state, define the concrete resistance and check the reinforcement anchorage.

Figure 1 – Singular nodes (B) and smeared nodes (A). From Schlaich, J.; Schäfer, K.; Jennewein, M. (1987).

3.1. IN-PLAN CASE In an in-plan structure, three or more forces have to meet in one node so that the balance is verifiable. In order to have balance, the sum of the forces in both directions has to be zero, as well as the sum of the bending moments. Nodal regions are classified as C-C-C if three compression forces meet in one node; as C-C-T if one of those forces is a tensile force; as C-T-T if only one compression force and two tensile forces exist. The T-T-T nodes can also occur and are constituted solely by tensile forces.

C-C-C NODES (FIB BULLETIN 2 , 1999) The necessary verification measures given the geometry of a C-C-C node are the ones enounced in the following paragraphs. If all sides of the node are perpendicular to the stress fields, the tensions on the three sides of the node and on the inside of the node zone will be constant and equal on all directions – in-plan hydrostatic pressure. In this case, only

one tension needs to be checked – 1cσ , being

1cσ the tension on the support.

If only one tension is orthogonal, such as 1cσ

in Figure 2 a), and the remaining stress fields are not perpendicular to the sides of the node,

1cσ is one of the two main tensions.

Therefore, 0cσ will be smaller than 1cσ , if 0a

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is larger than 0a matching a hydrostatic node

with the same 1a basis.

This type of modeling allows for a more realistic analysis of the structure’s behavior.

In case 0a is smaller than 0a of the

hydrostatic node, the main tension 0cσ will

become critical and will have to be checked. Sometimes, it might be useful to proceed to the division of the triangular node into two right triangles such as in the case of Figure 2 a). In

this case, the forces 1cF and the vertical

component of 2cF are equal, such as rcF 1 and

the vertical component 3cF , the existence of

the 0cF force being thus explained.

In Figure 2 c) the other C-C-C node of triangular geometry is represented that appears, for instance, in extremities supports of prestress beams and in corner frames. In

this case, the main tensions are 1cσ and 2cσ

and will have to be checked, the value of 3cσ

will be between the value of 1cσ and 2cσ .

Figure 2 - Standard C-C-C nodes. From fib Bulletin 2

(1999).

C-C-T NODES(FIB BULLETIN 2, 1999) In order to do a structural analysis in the force plane, it is necessary to assume that the reinforcements are parallelly disposed throughout the depth of the element, b, and that the node has a effective depth, u, where the stress fields face a change in direction, as it is shown in Figure 3 a). The representation of the nodal zone appears in Figure 3 b). The precise height of the node, u, can be

calculated according to the formulas professed by Baumann (1998).

Figure 3 – Standard end support node. From fib Bulletin 2

(1999). a) Idealized node region for one-layer reinforcement with excess length; b) Flow of forces for

multi-layer reinforcement with excess length and corresponding idealized node region.

In the C-C-T nodes, θ angles less than 55º should be avoided, for they can lead to

extremely high 2cσ tensions, unless there is

prestress. If it is not possible to avoid θ less than 55º, the problem can be dealt with through the usage of several reinforcement layers or additional reinforcement, loops, underneath the main reinforcement. These processes increase the precise height of the node, u, and consequently the a2 width will

increase, reducing thus the 2cσ tension.

These nodes can appear sometimes in supports that are located precisely in the extremity of an element, forcing the anchorage to be made at a given distance since the end of the nodal zone, this distance is necessary to the reinforcement cover. In these cases, the cover must resist a considerable part of the tensions generated by the compression field deviation. So that the splitting of the cover can be avoided and the reinforcement anchorage forces reduced, its reinforcement disposal should be such as to lead to a precise height, u, that is sufficient. One can also opt for stirrups orthogonally placed to the compression field and this process must always be the first option.

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C-T-T NODES (FIB BULLETIN 2, 1999) The C-T-T nodes can occur when there are reinforcements with bent bars and when there are tensions in orthogonal directions. When there are bent bars, these nodes are formed because the force in the struts is balanced by the radial forces of the bent bars. The nodal zone can be pictured such as the one presented in Figure 4, with a strut width, a

. As for the standard C-T-T nodes, with ties in orthogonal directions, the anchorage is essentially made through diagonal compressions on the concrete along with transversal reinforcements. The bars that make up this transversal reinforcement must have a fairly reduced diameter, since their anchorage must be obtained within the stress field of the tie. The concrete compression in these nodes will only be critical when there is a large quantity of reinforcement with small spacings.

Figure 4 – Node with bent bars. Scheme of forces, standard node region and cross-section. From fib Bulletin

2 (1999).

NODES WITH REDUCED SUPPORT WIDTH – THREE-DIMENSIONAL LOCATED EFFECTS

IN-PLAN ELEMENTS (FIB BULLETIN 2, 1999) The promptest procedure for these nodes takes into account firstly the major load plan and then considers the effects of a third direction loading. Despite this procedure, one must always be aware that the effects in the three directions are interconnected.

SAFETY VERIFICATION CRITERIA In order to proceed to the safety verification of the tensions in several elements, there have been used the following technical and normative documents for the concrete

structures design, the Eurocode 2, the ACI Committee 318 and the FIP/fib Recommendations.

ACI CODE (ACI 318, 2002) Using ACI code, the design strength for a concrete strut, can be calculated using the

equation (1), where cf ' is the compression

strength of concrete.

cscu ff '85,0 β= (1)

(A.3.2.1) For struts in which the area of the midsection cross section is the same as the area at the nodes, such as the compression

zone of a beam: 0,1=sβ .

(A.3.2.2) For struts located such that the width of the midsection of the strut is larger than the width at the nodes (bottle-shaped struts):

With reinforcement: 75,0=sβ

Without reinforcement: λβ 60,0=s .

(A.3.2.3) For struts in tension members or the

tension flanges: 4,0=sβ .

(A.3.2.4) For all other cases: 6,0=sβ .

The design strength for a concrete in the node, can be calculated using the equation (2).

cncu ff '85,0 β= (2)

(A.5.2.1) C-C-C Nodes: 0,1=nβ .

(A.5.2.2) C-C-T Nodes: 8,0=nβ .

(A.5.2.3) C-T-T Nodes: 6,0=nβ

MC90 – FIP (CEB-FIP MC 90, 1993) The FIB recommendations for the concrete strength in nodes are:

- C-C-C Nodes - [ ] cdckcd fff 250185,01 −=

(3),

- C-T-T Nodes - 12 7,0 cdcd ff = (4),

- C-C-T Nodes - [ ] cdckcd fff 25017,03 −=

(5). The design strength for a concrete strut:

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-Without cracks:

[ ] cdckcd fff 250185,01 −= (6),

-With cracks:

[ ] cdckcd fff 250160,02 −= (7).

The definition of ckf and cdf is at the equation

(10).

EUROCÓDIGO 2 - ENV 1992-1-1 (NORMA

PORTUGUESA) The design strength for a concrete strut: - Without cracks:

cdRd f1max, =σ (8)

and

γαα ck

cdcd

fff

⋅=⋅=1

(9),

With 5,1=cγ e 85,0=α , and:

c

ckcd

ff γ=

(10)

Where:

ckf - Characteristic compressive cylinder

strength of concrete at 28 days

cγ - Partial factor for concrete

- With cracks:

cdRd fνσ 6,0max, = (11)

2501 ckf−=ν (12)

The design values for the compressive stresses within nodes may be determined by: - In compression nodes where no ties are anchored at the node:

cdrd fk νσ 1max, = (13)

The recommended value for 1k is 1,0. The

definition of ν is at the equation (12) and the

definition of cdf is at the equation (10).

- In compression - tension nodes with anchored ties provided in one direction:

cdrd fk νσ 2max, = (14)

The recommended value for 2k is 0,85. The

definition of ν is at the equation (12) and the

definition of cdf is at the equation (10).

- In compression - tension nodes with anchored ties provided in more than one direction:

cdrd fk νσ 3max, = (15)

The recommended value for 3k is 0,75, The

definition of ν is at the equation (12) and the

definition of cdf is at the equation (10).

3.2. GENERALIZATION TO THE THREE-DIMENSIONAL CASE – “3D” NODES As in the in-plan case, the nodes on the three-dimensional space appear as critical regions and as connection regions of struts and/or ties. The following types of nodes are to be differentiated:

- Compression nodes, that are solely constituted by struts, equal to the C-C-C nodes in the in-plan case;

- Compression/tensile nodes, that are nodal regions where struts and ties are found. There are two distinct types of compression/tensile nodes, concentrated nodes, where the existent reinforcement in the tie region is anchoraged, and smeared nodes, in which the placed reinforcement allows the struts deviation. These nodes are similar to the C-C-T and C-T-T nodes in the in-plan case.

- Tensile nodes, that are solely constituted by ties. As it happens with the T-T-T nodes in the in-plan case. The most frequent nodes are the compression nodes and the compression/tension nodes. The pure tension nodes are rarely ever formed and should always be avoided. Thereby, it is necessary to analyse the compression nodes and the compression/tension nodes that will be named C-C-C 3D and C-C-T 3D nodes, respectively. Based on Nguyen (2002).

TYPES OF “3D” NODES C-C-C 3D NODES For compression nodes, the tension behaviour and the characteristical values of resistance to concrete compression depend on the struts in the in-plan case and in the three-dimensional case. Hence, it differentiates the existent types of nodes. On type 1 of the compression nodes, two struts are united in one axis alone.

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The geometry type 1 nodes calculation can be looked down upon for this calculation is identical to the one of the geometry of the two struts that, when united, form the node. It is possible to conclude, then, that it is not necessary a modeling for C-C-C 3D type 1 nodes. The type 2 of these C-C-C 3D nodes is described as being formed by three or more struts on the same plane that meet in this node, forming then a stress plane state on the strut plane, albeit constituted by more than one direction, this one being perpendicular to the strut plane. An identical situation to the one of the type 1 is verifiable, the node is formed due to the union of three or more struts on the same plane. Thus, a modeling of the node as a three-dimensional element is not necessary because the design criteria for reduced width support nodes, described in the chapter 3.1.1 in Rodrigues (2008), are valid. At last, type 3 nodes are described as the junction of at least four struts not on the same plane. In the nodal region there is therefore a three-dimensional tension state, hence it is necessary a node modeling to determine the node geometry one must take into account: the element sections that have influence in the geometry of the node, for example, the area of the columns or piles, number and direction of the struts. In view of the modeling of this type of C-C-C 3D nodes, simplifications can be admitted, so that the safety conditions in the nodal region can be verifiable. This procedure is used, for instance, for pile cap so that the number of piles is equal to the number of node sections and the passage sections between the node and the struts are flat. Sometimes it is necessary then to modify the shape of the sections and some struts that have small deviation angles can be replaced in certain conditions for a resultant. The examples and the respective simplifications of the passage sections between the column and the cap are shown in Figure 5.

Figure 5 – Examples and simplifications of the section between the pier and the pile cap. From Nguyen (2002).

NÓS C-C-T 3D These nodes can be defined as the union of struts and ties with different directions. The number and direction of the struts and ties have a strong influence on the tension state and on the design strength for a concrete in a node, where the struts have a central role. Type 1 nodes are only constituted by one tie besides the struts. Where all struts and ties can be on the same plane or not on the same plane. Type 2 nodes are constituted by two ties, besides the struts. When the ties are not on the same plane they lead to an unfavourable bending. The C-C-T nodes with three ties or more are classified as type 3. Their behaviour is very unfavourable due to the bending state that they present. So, whenever possible, they should be avoided in the design of an element for even the building process is quite complicated.

SAFETY CRITERIA VERIFICATIONS STRUTS In a three-dimensional strut, the stress increases, in conformity with the relation between the compression field measures,

ab / (Figure 6 a)). The limit of the effective

tensions in a three-dimensional stress field,

cdmáxeffcd ff 1, 88,3)( = , is superior to the limit

of the effective tensions for a in-plan stress

field, cdmáxeffcd ff 1, 6,1)( = .

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a) b) Figure 6 – Geometry of a strut. From (2002).

The stress limit for a compression field:

0c

cA

F=σ

(16)

effcdc f ,≤σ

(17)

Effective compressive strength of concrete,

effcdf , :

)1(6,0

:50

, βα ⋅+⋅⋅=

≤

cdeffcd

ck

ff

MPaf

(18)

)(2,36,0

:50

, βα ⋅⋅+⋅=

>

cdcdeffcd

ck

fff

MPaf

(19)

effcdf , in MPa. The definition of cdf is at the

equation (10).

)1(6,0

:50

, βα ⋅+⋅⋅=

≤

cdeffcd

ck

ff

MPaf

(20)

The influence of the narrowing of a compression field, parameter, α :

Direction x: x

xxa

b

1

=γ (21.a)

Direction y: y

y

ya

b

1

=γ (21.b)

−+−

−⋅−+⋅=

22 )1()1(

)1()1(1

3

1

yx

yx

γγ

γγγ

2

1

3

1≤≤ γ , 3/1=γ se

1== yx γγ

(21.c)

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1 −

=

γ

αc

c

A

A

(22)

The influence of the length of a compression

field, parameter β :

laall xxequx ⋅≤+−= 221, (23)

laall yyequy ⋅≤+−= 221, (24)

13 1

1, ≤−⋅

−=

xx

xequx

xab

alβ

se

xequx al 1, >

(25)

0=xβ se xequx al 1, ≤

(26)

13 1

1, ≤−⋅

−=

yy

yequy

yab

alβ

se

yequy al 1, >

(27)

0=yβ se yequy al 1, ≤

(28)

( ) 12

10 ≤+⋅=≤ yx βββ

(29)

NODES C-C-C 3D NODE For nodes of the type 2:

- Calculate the values of the main stresses of

the node, Iσ and IIσ , compression values

are negative and III σσ ≥ .

- Calculate 1α using the equation (30), using

Iσ e IIσ .

2,14,00,11 ≤+=II

I

σσ

α (30)

- Condition: cdcdeffcdII fff 111, 92,1≤=≤ ασ

(31) - Where the load is applied, a: width of the node; t: thickness of the element:

cdcdeffcdII ffatf 111, 88,3)/( ≤=≤ ασ (32)

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For nodes of the type 3:

- Calculate the values of the main stresses of

the node Iσ , IIσ e IIIσ , with IIIIII σσσ ≥≥, and compression values are negative - Condition:

cdeffcdIII ff 1, 88,3≤≤σ (33)

With effcdeffcd ff ,'

, 8,0 ×= (34), where

effcdf ,' is the value from the abacus of the

Figure 7, using the values of Iσ , IIσ e IIIσ .

According to Sundermann (1994), from Nguyen (2002).

Figure 7 – Abacus to obtain effcdf ,

' for C-C-C 3D nodes,

of type 3, Schmidtgönner (1984), from Nguyen (2002).

These safety verifications when compared, to the similar case of in-plan models, shows that because there is one more direction, on the three-dimensional models, the value of effective compressive strength of concrete is

calculated with the value of effcdf ,'

from the

figure 7 and then multiplied by 0,8, reducing it. The value of effective compressive strength of concrete, by Sundermann (1994), to in-plan

models has to be lower than cdf192,1 , to three-

dimensional models has to be lower than

cdf188,3 .

The fact of the concrete in nodal zone is subject to triaxial compression, make the

compressive strength of concrete higher than when is subject to biaxial compression

NÓ C-C-T 3D The limit of the strength for the concrete in the

node is effcdci f ,≤σ .

For nodes of the type 1:

- In-plan struts:

cdeffcd ff 1, 8,0 ⋅=

(35.a)

Where the load is applied, a: width of the node; t: thickness of the element:

cdcdeffcd ffatf 11, 88,38,0)/( ≤⋅⋅= (35.b)

- When there is a least one strut from other plane:

cdeffcd ff 1, 8,0 ⋅= se º45º30 min <≤θ (36.a)

cdcdeffcd fff 11min, 88,3tan ≤⋅= θ se

º45min ≥θ (36.b)

(minθ is the smaller angle of iθ )

For nodes of the type 2:

- Singular nodes:

cdeffcd ff 1, 8,0 ⋅= (37)

- Smeared nodes:

For struts where º45min ≥θ on the plane

of the tie:

cdeffcd ff 1, 7,0 ⋅= (38)

For the other case: cdeffcd ff 1, 6,0 ⋅= (39)

Where the load is applied, a: width of the node; t: thickness of the element, like for

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the nodes of the type 2, to obtain effcdf , the

)/( at must be multiplied.

For nodes of the type 3:

- cdeffcd ff 1, 5,0 ⋅=

(40)

The calculation of the anchorage length was made by the FIP-Recommendations 1996. The analogy between these nodes and the in-plan models can be made comparing the C-C-T 3D nodes of the type 1 with C-C-T nodes of the in-plan models and comparing type 2 and type 3 with C-T-T nodes of the in-plan models. For C-C-T nodes of in-plan models and C-C-T 3D nodes of the type 1, the safety criteria verification is the same, except when all the angles between the struts and the tie are bigger than 45º, for this situation the limit of the strength for the concrete in the node is higher. The limit of the strength of C-T-T nodes of in-plan models is the same as C-C-T 3D smeared nodes of the type 2 with the smaller angle between a strut and a tie minor than 45º. For all other cases of the type 2, the factor is higher, because the concrete strength in the node is higher too.

EXAMPLE OF A THREE-DIMENSIONAL

CASE – PILE CAP The pile cap is constituted by four piles, its dimensions in plant are 2x2 m2 and is 1,3 metres thick. The piles have a section of 0,3x0,3 m2 and the column 0,4x0,4 m2. The concrete chosen for this cap was the C30/37 and the steel the A400NR. To design the strut-and-tie model, firstly one has to know the shape and the load path. In the case in study the load path must transmit the load of the loading region to the reaction region, hence, the model was constituted by four struts, four ties and five

nodal regions: a type 3 C-C-C 3D nodal region and four type 2 C-C-T 3D nodal regions. In the C-C-C 3D node construction, the first step was to calculate the height of the node, guaranteeing the tensions limit on the nodal region concrete. Furthermore, the geometric shape was developed based on the balance

verifications, vd. Figure 8. The representation of the tension state existing on this nodal region was made through the Mohr circle. Another construction was performed on this node, a prompter construction that only allows to obtain the main tension values, for in this simplification one can see an inconsistency with reality. The comparison of the values obtained through the two models is shown on Table 1.

Table 1 – Comparison of the values obtained through the

two models.

Primeiro modelo Modelo simplificado

MPa674,7

,3

=

=== IIIIII σσσ

MPa237,6

,3

=

=== IIIIII σσσ

MPa

IIIIII

750,18

,1

=

==σσ

MPa

IIIIII

750,18

,1

=

==σσ

a) b) Figure 8 – a) Quarter of the C-C-C 3D node; b) View of the

C-C-C 3D node.

To model the C-C-T 3D node (Figure 9) over a square or circular pile it was necessary to understand how the compression force deviation took place in the reinforcement regions to the pile region, thereby acknowledging the values of the existent tensions in the concrete of this nodal region, the tension stress in the reinforcements and the needed anchorage lengths. The necessary calculations to the geometric construction, the calculations of the existent strain in the nodal regions of the node and the calculations of the existent tensions in the concrete and in the reinforcements of the nodal region were made resorting to a calculation sheet, using the computer programme Excel,

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this calculation sheet was programmed by Lourenço (2009). As for the case of the four circular piles cap, some changes in the calculation sheet were made so that the circular shape of the pile were taken into consideration. This model was built so that the safety could be checked according to what was said in the safety verification criteria from chapter 3.2.

Figure 9 – View of the C-C-T 3D node, over a square pile.

THE DETAILING OF THE REINFORCEMENTS In order to proceed to the detailing of the reinforcements, the first step is to calculate the needed reinforcement quantities, a

reinforcement quantity of 2778,7 cm being

necessary. The chosen bars to arm the pile cap in the tie regions were three bars with a 20mm diameter, placed on one layer alone, distanced from the cap’s lower section in 0,100m, with a spacing between bars of 0,110m and a cover of 0,030m. These bars

total an area of 2420,9 cm .

At last, the anchorage width necessary to the bars is to be defined. The value of the

necessary anchorage width was m471,0 , and

the value of the design stress is MPa045,3 .

After these calculations, the chosen disposal of the reinforcements was the one presented in Figure 10.

Figure 10 – Reinforcement of the pile cap.

4. CONCLUSION

This assignment had as its main goal the start of the development and of the implementation of the three-dimensional stress fields study, as well as to analyse the procedures to generalize the nodes safety verification criteria established for in-plan cases. Particularly, this study was applied to the practical case of the pile cap. The assignment began with a synthesis of the aspects concerning in-plan models of stress fields, representative of a bidimensional stress state in discontinuity regions. With the knowledge acquired from the in-plan nodes, a typification of the three-dimensional nodes was made, such as the generalization of the safety verification criteria. The work of Nguyen (2002) was used as a basis, for it is virtually the only author that describes and analyses in detail the three-dimensional stress fields resorting to strut-and-tie models. In order to put in practice the concepts withdrawn from the three-dimensional stress fields theory, the pile cap with four square piles was chosen. This element was chosen because it is an element that is usually designed through strut-and-tie models, resorting to the overlaying of in-plan models.

ACKNOWLEDGEMENTS The author acknowledges particularly Eng. Miguel Lourenço for all the help and precious knowledge.

REFERENCES CEB-FIP MC 90 - Design of concrete

structures. CEB-FIP-Model-Code 1990. Thomas Telford, 1993. EUROCÓDIGO 2 - ENV 1992-1-1 (Norma Portuguesa) – Projecto de Estruturas de Betão - Laboratório Nacional de Engenharia Civil, Termo de Homologação nº68/98, Lisboa, Abril de 1998. FIP/FIB RECOMMENDATIONS - Practical

Design of Structural Concrete, SETO, London, September 1999. FIB BULLETIN 1 - Structural Concrete – Textbook on Behaviour, Design and

Performance – Vol.1, fib, Lausanne, July 1999.

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FIB BULLETIN 2 - Structural Concrete – Textbook on Behaviour, Design and

Performance – Vol.2, fib, Lausanne, July 1999. FIB BULLETIN 3 - Structural Concrete – Textbook on Behaviour, Design and

Performance – Vol.3, fib, Lausanne, September 1999. FIB BULLETIN 16 - Design Examples for the FIP Recommendations "Practical design of

structural concrete". fib, Lausanne, January 2002. LOURENÇO, M. -“Stress Field Models for Structural Concrete”. Tese de Doutoramento em preparação, Lisboa, 2009 MUTTONI, A.; SCHWARTZ, J.; THÜRLIMANN, B. - Bemessen und

Konstruieren von Stahlbetontragwerken mit

Spannungsfeldern, Vorlesungsunterlagen Stahlbeton AK, ETH-Zürich, Zurich, Switzerland, 1988. NGUYEN, D. T. - Räumliche Stabwerkmodelle zur Bemessung von Betontragwerken. Diss. Institut fur Leichtbau Entwerfen und Konstruieren, Stuttgart, Germany, 2002. Rodrigues, I. – Zonas de descontinuidade de betão estrutural. Campos de tensões

tridimensionais. Dissertação do Mestrado Integrado de Engenharia Civil, Universidade Técnica de Lisboa, Instituto Superior Técnico, Lisboa, 2008 SCHLAICH, J., SCHÄFER, K. - Design and detailing of structural concrete using strut-and-

tie models. The Structural Engineer, V.69, No.6, 1991. SCHLAICH, J.; SCHÄFER, K.; JENNEWEIN, M. - Toward a Consistent Design for Structural Concrete. PCI-Journal, V.32, No.3,1987. SUNDERMANN, W. - Tragfähigkeit und

Tragverhalten von Stahlbeton-

Scheibentragwerken, Untersuchung mit

Stabwerkmodellen. Dissertation, Institut für Tragwerksentwurf und – konstruktion, Universität Stuttgart, 1994.