A new technique for ultimate limit state design of arbitrary shape RC sections under biaxial bending

Engineering Structures 104 (2015) 1–17

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A new technique for ultimate limit state design of arbitrary shape RC sections under biaxial bending R. Vaz Rodrigues Departamento de Engenharia Civil Arquitectura e Georrecursos, Instituto Superior Técnico, Lisbon, Portugal

a r t i c l e

i n f o

Article history: Received 28 August 2014 Revised 1 September 2015 Accepted 9 September 2015

Keywords: Cross sections Reinforced concrete Biaxial bending Interaction diagram Ultimate bending strength Computer graphics

a b s t r a c t The design of reinforced concrete sections of arbitrary shape, namely with variable geometry, holes as well as with arbitrary distribution of reinforcing steel bars, is a very common task in civil engineering, reinforced concrete structures. The design of these sections requires the integration of non-linear stress fields on complex shapes, because of the non-linear behavior of concrete in compression. In this paper, a novel algorithm is proposed to compute the ultimate strength of reinforced concrete sections under biaxial bending. The algorithm includes section subdivision into trapezoidal elements using the techniques of polygon clipping algorithm proposed by Weiler–Atherton. Exact numerical integration for normal strength concrete (fck 6 50 MPa) is achieved, for each trapezoid, using the change of variables theorem followed by Gauss–Legendre integration. The proposed technique is hereafter referred to as WAGL (Weiler– Atherton, Gauss–Legendre). The verification of the proposed algorithm is performed by comparing analytical results between the WAGL technique and methods proposed by other authors (five examples). Additionally, the results obtained are also compared with experimental results available in the literature. The application of the WAGL technique is illustrated with two RC cross-section design examples. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The design of reinforced concrete cross sections is a common task in structural engineering and usually involves reinforced concrete buildings, bridges and underground structures. The shape of cross sections is found to be of various types, which together with the bending moments acting simultaneously in both axes and the axial force turns the design a strongly non-linear problem. Simplified methods for approaching the strength of cross sections are proposed by Bresler [1], who provides two techniques for evaluating the ultimate strength of rectangular columns, the load contour method and reciprocal load equation. An application of the load contour method, an approximate formulation for rectangular, circular or elliptical cross sections, can be found in Eurocode 2 (clause 5.8.9) [2]. The CEB/FIP manual on bending and compression [3] provides design charts for rectangular and circular reinforced concrete cross-sections with various distributions of reinforcement, which are commonly used in current practice. This analysis shows design charts for some particular distributions of reinforcement along the perimeter of cross sections. Walther and Houriet [4] provide design charts for hollow reinforced concrete cross sections under biaxial eccentric loads. Hsu [5] proposes an

E-mail address: [email protected] http://dx.doi.org/10.1016/j.engstruct.2015.09.016 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.

approximate equation of the failure surface for square and rectangular columns, obtaining good agreement with available test results. The approximate equation of the failure surface is obtained using the ratio of ‘‘nominal bending moments” about x and y axis and the ‘‘nominal bending moments at balanced strain condition”. The use of computer programs to design reinforced concrete cross sections is nowadays common. Bentz [6] presents the treatment of reinforced concrete sections of arbitrary shape, including arbitrary location of reinforcement bars, using a computer application. Papanikolaou [7] shows that the type of algorithm used to calculate the ultimate strength of arbitrary cross sections, along with the material constitutive laws, line subdivision and stress integration scheme are important aspects of the algorithms. Concerning the material constitutive laws for concrete in compression, Eurocode 2 defines concrete in compression as parabolic linear, for design of reinforced concrete cross sections (clause 3.1.7), therefore this type of relation will be followed in this study. The use of parabolic linear behavior for concrete was also adopted by Rosati et al. [8]. The consideration of fully arbitrary material law for concrete in compression is evaluated in the studies presented by Papanikolaou [7]. The most critical aspects in terms of efficiency of algorithms are section subdivision and the stress integration scheme. In order to calculate the contributions of the compression stress fields, Char-

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R. Vaz Rodrigues / Engineering Structures 104 (2015) 1–17

Notation M N

rc fcd fck fck,cube Ec

ec ec2 ecu2

n

rs

Es

es

fsd fy ft k

eud

bending moment axial force compressive stress in the concrete design value of concrete compressive strength characteristic compressive cylinder strength of concrete characteristic compressive cube strength of concrete modulus of elasticity of concrete compressive strain in the concrete compressive strain in the concrete at peak stress ultimate compressive strain in the concrete exponent value, integer stress in reinforcing steel modulus of elasticity of reinforcing steel strain of reinforcing steel design yield strength of reinforcement yield strength of reinforcement tensile strength of reinforcement coefficient for considering steel hardening design strain of reinforcing steel at maximum load (=0.9 euk)

alampakis and Koumousis [9] divides any section with curvilinear trapezoids and used closed-form solutions to compute the internal forces, obtaining exact results. Sfakianakis [10] proposes computation of internal forces without section subdivision, using fiber integration. However, this approach can give approximate results depending on the mesh division. Dias da Silva et al. [11] uses a closed-form algorithm applicable to multi-rectangular sections followed by Gauss–Legendre integration. This analysis shows that closed-form solutions for computation of the stress resultants are preferable in terms of computational efficiency. Rodriguez and Aristizabal-Ochoa [12] present an algorithm based on section subdivision into polygons and performed closed-form integration per polygon. Pallarès et al. [13] perform closed-form integration, but without section subdivision. Another interesting integration scheme using boundary integrals is proposed by De Vivo and Rosati [14]. Table 1 presents stress integration techniques available in selected previous studies, compared with the suggested methodology. It can be seen that most previous studies use analytical solutions (closed-form functions) that have the main advantage of providing exact results, however they have the inconvenient of being restricted to specific stress–strain material law or section shapes. A second option can be considered using stress integration schemes based on fiber integration, but they are usually associated with the inconvenient of providing approximate solutions, depending on fiber mesh density. A third option, the use of a suitable numerical integration (Green/Gauss) scheme has been successfully used as integration scheme, consisting of an excellent approach both in terms of accuracy and execution times. This paper presents

euk strain of reinforcing steel at maximum load x, y coordinates (domain X) n, g coordinates (domain T) g, f functions Ci,x, Ci,y, Ca/b/c constants As cross sectional area of reinforcement d effective depth h overall depth d0 distance from top fiber, to center of reinforcing bar b cross section width Mpred calculated bending moment Mtest measured bending moment, at failure Ptest measured axial force, at failure 1/R curvature a angle; ratio h angle of inclination of neutral axis esup strain at topmost fiber of cross section einf strain at bottommost fiber of cross section

a novel approach, which combines the advantages of numerical integration schemes based on Gauss–Legendre integration, combined with Weiler–Atherton [15] algorithm considered for section subdivision. This algorithm was developed for application in computer graphics, but has proved to be efficient in the division of any section in polygons for the computation of internal forces of arbitrary reinforced concrete cross sections. As presented by Lam et al. [16], the use of spreadsheets can constitute an advantage in terms of its familiarity to the computer user. The use of extended interpreted programming languages that are embedded into the main spreadsheet program constitute an additional advantage. An example of such programming languages is Visual Basic for Applications (VBA), which will be used in this study. It should be noted that classic languages (e.g. C) can also be embedded in spreadsheets using a dynamic link library implementation (DLL), providing substantially better computational performance. 2. Material properties 2.1. Stress strain relations of concrete for the design of cross sections Eurocode 2 [2], clause 3.1.7, gives the following stress–strain relations for the design of cross sections and which are used in the current study:

8

n  ec > > for 0 6 ec 6 ec2 < rc ¼ f cd 1  1  ec2

rc ¼ f cd for > > : rc ¼ 0 for

ec2 6 ec 6 ecu2

ec 6 0

Table 1 Stress integration schemes of selected previous studies. Stress integration schemes

Authors

Closed-form per trapezoid/polygon

Rosati et al. [8], Charalampakis et al. [9], Rodriguez and Aristizabal-Ochoa [12], Pallarès et al. [13] Dias da Silva et al. [11] Sfakianakis [10] Papanikolaou [7] Suggested methodology (WAGL)

Closed-form, multi-rectangular Fiber integration Numerical Green/Gauss with adaptive strain-mapping (Section subdivision using Weiler–Atherton algorithm) numerical Gauss– Legendre integration

ð1Þ

3

R. Vaz Rodrigues / Engineering Structures 104 (2015) 1–17

σc

σs

fcd

k fsd fsd

εc2

εc

εcu2

fsd

εc

ε ud

Es

(b)

(a)

Fig. 1. Unixial constitutive stress–strain relationships considered in present study for: (a) concrete under compression and (b) steel reinforcement under compression/ tension.

Table 2 Strength and strain values of concrete under compression according to Eurocode 2 (2004). fck (MPa)

12

ec2 (‰) ecu2 (‰)

2.0 3.5 2.0

n

16

20

25

30

35

40

The graphical representation of these equations is shown in Fig. 1(a). The values for compressive strain at reaching maximum strength ec2, ultimate strain in the concrete ecu2 and exponent value n are given as a function of the characteristic compressive strength fck, as shown in Table 2. Note that fcd is considered as fcd = a fck/cc, according to Eurocode 2 clause 3.1.6. Besides the tabulated values for the strains ec2, ecu2 and exponent n, Eurocode 2 provides analytical expressions for these values as a function of the characteristic compressive strength fck, only applicable for high strength concrete (fck P 50 MPa):

ec2 ð‰Þ ¼ 2:0 þ 0:085ðf ck  50Þ0:53

ð2Þ

ecu2 ð‰Þ ¼ 2:6 þ 35½ð90  f ck Þ=1004

ð3Þ

n ¼ 1:4 þ 23:4½ð90  f ck Þ=100

4

45

50

55

60

70

80

90

2.2 3.1 1.75

2.3 2.9 1.60

2.4 2.7 1.45

2.5 2.6 1.40

2.6 2.6 1.40

8 rs ¼ Es es for 0 6 es 6 f sd =Es > > > > < rs ¼ f sd þ ðk1Þ f sd ðes  f sd =Es Þ for f sd =Es 6 es 6 eud e f =Es ud

sd

> rs ¼ Es es for  f sd =Es 6 es 6 0 > > > : Þ f sd rs ¼ f sd þ eðk1 ðes þ f sd =Es Þ for  eud 6 es 6 f sd =Es f =Es ud

ð5Þ

sd

Accordingly to Eurocode 2, the characteristic value of the yielding strength ranges between 400 MPa and 600 MPa. In addition, Eurocode 2 states that the constant k and the strain at maximum stress euk depend on the ductility class, as shown in Table 3. However, one can consider k = 1, corresponding to horizontal yield plateau, in any case. Note that the design strain at maximum stress corresponds to eud = 0.9euk and that the design yield strength corresponds to fsd = fy/1.15. The graphical representation of these equations is shown in Fig. 1(b).

ð4Þ

As mentioned, the concrete model adopted in the paper is the model considered in Eurocode 2, clause 3.1.7 (stress–strain relations for the design of cross-sections), and is the model for unconfined concrete. The effect of confinement results in higher compressive strengths and higher strains which can be considered using values of fck,c, ec2,c and ecu2,c (clause 3.1.9 of Eurocode 2).

3. Computation of internal forces in a trapezoidal element 3.1. Axial force due to concrete in compression The calculation of resultant compression force N is expressed by the integral in integration domain W, as indicated in Fig. 2 and expressed by the following equation:

2.2. Stress strain relations of reinforcement steel

Z

The behavior of steel in compression or tension is assumed to be defined by a bilinear relation, with k = ft/fy, the ratio between the tensile strength and the yielding strength, as shown in Eq. (5).

Table 3 Values of k and strain at maximum stress euk. Class

k

euk (‰)

A B C

P1.05 P1.08 P1.15 <1.35

P2.5 P5.0 P7.5

N¼

rc ðx; yÞ dx dy

ð6Þ

W

In a local coordinate system (xlocal, ylocal), assuming that neutral axis is coincident with xlocal axis (Fig. 2(a)), the strains are expressed as:

1 R

ec ðx; yÞ ¼ y

ð7Þ

Replacing this result for the strain distribution, the concrete compressive stresses can be obtained as a function of the local coordinate y and of the curvature 1/R. Hence, two ranges can be identified:

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R. Vaz Rodrigues / Engineering Structures 104 (2015) 1–17

Strain profile

εsup

Neutral axis My (>0)

ylocal

(a)

x local (x0, y0 )

εinf

y x

Mx (>0)

N (>0) Compression N (<0) Tension

(b) (x4; y4)

(-1;1)

(x3; y3)

(1;1) T

W y

(x1; y1)

(x2; y2)

(1;-1)

(-1;-1)

(Local) x

Fig. 2. (a) Global and Local Coordinate systems, neutral axis inclination h and sign conventions. (b) Coordinate transformation g(n, g) from T domain (n, g), to W domain (x, y).

rc ðyÞ ¼

8 
  1y n > ec2 R < rc ¼ f for 0 6 y 6 1=R cd 1  1  ec2 > :r ¼ f c cd

for

ec2

cu2 6 y 6 e1=R 1=R

2 ð8Þ

In order to perform integration of the compression stresses along the domain W, defined by an arbitrary trapezoid, it is useful to use the change of variables theorem. Therefore, the function g, used for variable transformation (mapping), is defined for the corresponding points between domain T and domain W, as indicated in Fig. 2(b). Given a point of coordinates n and g, defined in the coordinate system of domain T, the corresponding point (x, y) in the domain W will be defined by the following function g (R2 ? R2):

ðx; yÞ ¼ gðn; gÞ ¼

8 4 X > > > Ni ðn; gÞ xi > g 1 ðf; gÞ ¼ < i¼1

4 > X > > > Ni ðn; gÞ yi : g 2 ðf; gÞ ¼

ð9Þ

¼ ð1  g  n þ gnÞ=4 ¼ ð1 þ g þ n þ gnÞ=4

ð10Þ

¼ ð1 þ g  n  gnÞ=4

As already mentioned, in order to perform integration of the compressive stresses in the trapezoid concrete element, the function g has been used for variable change, resulting in the following expression for integration on domain T (1 6 n 6 1 and 1 6 g 6 1):

Z N¼ X

rc ðyÞdxdy ¼

Z T

rc

! 4 X Ni ðn; gÞyi jdet½Dgðn; gÞjdndg

2 ðn;gÞ

@g 2 ðn;gÞ @g

D gðn; gÞ ¼ 4 @g

@n

3 5

ð12Þ

The derivatives shown in the expression above can be explicitly written as:

8



 @g 1 1þg g > x1 þ 14 g x2 þ 1þ4 g x3 þ 1 x4 > > @n ¼ 4 4 > > 1þn 1n 1þn 1n > > @g 1 < ¼ 4 x1 þ 4 x2 þ 4 x 3 þ 4 x 4 @g



 @g 1þg g > > y1 þ 14 g y2 þ 1þ4 g y3 þ 1 y4 > @n2 ¼ 4 4 > > >         > : @g2 ¼ 1þn y þ 1n y þ 1þn y þ 1n y @g

4

1

4

2

4

3

4

ð13Þ

4

After simplifying terms and grouping the terms of variables n and g, the following expression is obtained for the determinant of the Jacobian of function g:

j det½D gðn; gÞj ¼ jðC 2x C 1y  C 1x C 2y Þn þ ðC 1x C 3y  C 3x C 1y Þg ð14Þ

Next, the constants C1,x, C1,y, C2,x, C2,y, C3,x, C3,y and C4,y are defined as a combination of trapezoidal element coordinates, in a local coordinate system (x, y):

The functions Ni, with i = 1–4, are defined as follows:

¼ ð1  g þ n  gnÞ=4

@g 1 ðn;gÞ @g

þ ðC 2x C 3y  C 3x C 2y Þj

i¼1

8 N1 > > > N 3 > > : N4

@g 1 ðn;gÞ @n

ð11Þ

i¼1

where D g(n, g) is the Jacobian of the function g, as shown in the following equation:

8 3 x4 Þ > C 1;x ¼ ðx1 x2 þx > 4 > > > ðx1 þx2 þx3 x4 Þ > > > C 2;x ¼ 4 > > < C 3;x ¼ ðx1 x24þx3 þx4 Þ > > C 4;x ¼ ðx1 þx2 þx3 þx4 Þ > 4 > > > > C a ¼ ðC 2x C 1y  C 1x C 2y Þ > > > : C c ¼ ðC 2x C 3y  C 3x C 2y Þ

3 y4 Þ C 1;y ¼ ðy1 y2 þy 4

C 2;y ¼ ðy1 þy24þy3 y4 Þ C 3;y ¼ ðy1 y24þy3 þy4 Þ 3 þy4 Þ C 4;y ¼ ðy1 þy2 þy 4

ð15Þ

C b ¼ ðC 1x C 3y  C 3x C 1y Þ

Using the (above defined) constants Ca, Cb and Cc, the determinant of the Jacobian of function g can thus be written in the following form:

j det½D gðn; gÞj ¼ jC a n þ C b g þ C c j

ð16Þ

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R. Vaz Rodrigues / Engineering Structures 104 (2015) 1–17

Z

And Eq. (11) can now be written as:

M¼

4 X Ni ðn; gÞyi ¼ C 3;y g þ C 2;y n þ C 1;y gn þ C 4;y

þ1

Z

1

þ1

1

f cd ½C 3;y g þ C 2;y n þ C 1;y gn

 n # 1 ½C 3;y g þ C 2;y n þ C 1;y gn þ C 4;y  R þ C 4;y  1  1 

ð17Þ

"

ec2

i¼1

Therefore, the resultant axial force in a trapezoidal element, corresponding to the first range of concrete’s constitutive law Eq. (8), can be written as:

Z N¼

þ1

Z

1

"

þ1

1

f cd

 n # 1 ½C 3;y g þ C 2;y n þ C 1;y gn þ C 4;y  R 1 1

ec2

j½C A n þ C B g þ C C jdndg

ð18Þ

In order to numerically evaluate the double integral, Gauss–Legendre quadrature is used with 2  2 points, which gives an exact result for polynomial expressions up to degree of three. It should be noted that exact results can only be obtained for concrete strength up to fck = 50 MPa. For high-strength concrete (fck > 50 MPa), the first branch of the concrete stress–strain relationship ceases to be polynomial (non-integer exponent value) and hence the employed Gauss–Legendre integration yields approximate results. Therefore, the final expression of the axial resultant force in a trapezoidal element is:

        1 1 1 1 1 1 1 1 N ¼ f pffiffiffi ; pffiffiffi þ f  pffiffiffi ; pffiffiffi þ f pffiffiffi ;  pffiffiffi þ f  pffiffiffi ;  pffiffiffi 3 3 3 3 3 3 3 3 ð19Þ " With f ðn; gÞ ¼ f cd

 n # 1 ½C 3;y g þ C 2;y n þ C 1;y gn þ C 4;y  R 1 1 ð20Þ

Similarly, the explicit expression for the resultant axial force in a trapezoidal element, corresponding to the second range of concrete’s constitutive law of Eq. (8), results in:

Z

rc ðyÞ dxdy ¼

N¼ X

Z ¼

1

1

Z

Z

f cd j det½D gðn; gÞ jdndg

1

f cd j½C A n þ C B g þ C C j dgdn

ð21Þ

f ðn; gÞ ¼ f cd j½C A n þ C B g þ C C j

ð22Þ

The calculation of the resultant bending moment M, with respect to local axis x, is expressed by the integral on domain W, as shown in Fig. 2(a) and expressed in the following equation:

yrc ðx; yÞ dxdy

ð23Þ

W

In order to perform integration of the compressive stresses in the concrete trapezoidal element, the function g for variable transformation is used, resulting in the following expression for integration on domain T (1 6 n 6 1 and 1 6 g 6 1):

Z M¼ X

where h(n, g) is given by:

hðn; gÞ ¼ f cd ½C 3;y g þ C 2;y n þ C 1;y gn "  n # 1 ½C 3;y g þ C 2;y n þ C 1;y gn þ C 4;y  þ C 4;y  1  1  R

ec2

 j½C A n þ C B g þ C C j

yrc ðyÞ dxdy ¼

Z T

j det½D gðn; gÞjdndg

4 X Ni ðn; gÞyi i¼1

!

rc

4 X N i ðn; gÞyi

ð27Þ

Z

yrc ðyÞ dx dy ¼

M¼ X

Z T

4 X

! Ni ðn; gÞyi f cd j det½D gðn; gÞjdndg

i¼1

ð28Þ Using the results from Eqs. (16) and (17), the following expression is obtained:

M¼

1 1

Z

1

1

½C 3;y g þ C 2;y n þ C 1;y gn þ C 4;y f cd j½C A n þ C B g þ C C j dgdn ð29Þ

The numerical evaluation of this expression is obtained using 2  2 integration points and using Eq. (19) (with M instead of N), but taking f(n, g) as:

f ðn; gÞ ¼ ½C 3;y g þ C 2;y n þ C 1;y gn þ C 4;y f cd j½C A n þ C B g þ C C j

3.2. Bending moment due to concrete in compression

Z

rffiffiffi rffiffiffi! rffiffiffi ! rffiffiffiffiffi rffiffiffi! 55 3 3 58 3 55 3 3 ; þ ;0 þ h  h  h  ; M¼ 99 5 5 99 5 99 5 5 rffiffiffi! rffiffiffi! 85 3 88 85 3 þ h 0;  hð0; 0Þ þ h 0; þ 99 5 99 99 5 rffiffiffi rffiffiffi! rffiffiffi ! rffiffiffi rffiffiffi! 55 3 3 58 3 55 3 3 ; þ ;0 þ ; ð26Þ h h h þ 99 5 5 99 5 99 5 5

Z

The numerical evaluation of this expression is obtained using 2  2 integration points and using Eq. (19), but taking f(n, g) as:

M¼

In order to numerically evaluate the double integral, Gauss–Legendre quadrature is used. Points defined with 3  3 grid are necessary, which gives exact results for polynomial expressions up to degree of five. Therefore, the final expression of the resultant bending moment in a trapezoidal element is given by:

T 1

ð25Þ

Similarly, the explicit expression for the resulting bending moment, corresponding to the second range of Eq. (8), results in:

ec2

j½C A n þ C B g þ C C j

 j½C A n þ C B g þ C C jdndg

!

i¼1

ð24Þ

Using the results from Eqs. (16) and (17), the explicit expression for the bending moment results in:

ð30Þ

The position of the resultant axial force is yR = M/N, with M and N obtained from Eqs. (25) and (18) for the first range or Eqs. (29) and (21) for the second range. 4. Subdivision of cross section into trapezoidal elements The subdivision of cross section into trapezoidal elements is based on the Weiler–Atherton algorithm [15]. This algorithm is based on a two-dimensional polygon clipper originally proposed in 1977 by Kevin Weiler and Peter Atherton, in the scope of computer graphics, with the aim of performing hidden surface removal using polygon area sorting. As originally mentioned in [15], ‘‘the output of the method is in the form of polygons, making it useful in a variety of situations including the usual display applications.” One of the advantages of the Weiler–Atherton algorithm is that it preserves the polygon data structure, both in output and input data, which allows recursive use of the algorithm when additional subdivision of the cross-section is required.

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R. Vaz Rodrigues / Engineering Structures 104 (2015) 1–17

In this paper, the algorithm is used to perform polygon clipping of a base polygon after a polygonal window, as shown in Fig. 3. The basic steps of the algorithm are the following: (1) Calculate all the intersections between the base polygon and the clipping window (points A and B in Fig. 3). (2) Compile a list (X, see Table 4) containing all vertices of the base polygon and the intersections points calculated at step (1). (3) Compile a list (Y, see Table 4) containing all vertices of the clipping window and the intersection points calculated at step (1). (4) Compile a list (Z, as indicated in Table 4) containing only the intersection points that ‘‘step in” (when ‘‘walking” along the base polygon) the clipping window. Please note that is required to define a direction associated with the base polygon. In the case of this example (see Fig. 3), counter clockwise direction is chosen, so that vertex A is in this case the only vertex that ‘‘steps in” the clipping window. (5) Select a vertex from list Z (A in the current example) and travel along the list X (starting from this vertex). When an intersection vertex is found on list X (B in this example), one should change the ‘‘walking path” to list Y. Again, when an intersection vertex is found on list Y, one should return to the list X. The procedure arrives to an end when an intersection vertex that has already been selected is found (A, in list Y). The output of this example would then be: {A, P10, P1, P2, B}. Using the exemplified algorithm, it is possible to divide any type of polygon into a group of four (or three) point polygons that can be treated as indicated in the previous section.

5. Computation of internal forces in complete section

W2

P1 Vertex A : “Step-in” vertex

W4 P8

P7

P2 Clipping Window

P10 A

B

Base Polygon

W3

P4

P6

{P1, P2, B, P3, P4, P5, P6, P7, P8, P9, A, P10} {W1, W2, W3, B, A, W4} {A}

the four equations just mentioned and multiplying the value of axial force and bending moment by (1). The contribution of each reinforcing bar is direct: after defining the strains on top and bottom fibers, the curvature becomes known, and therefore the contributions of all reinforcing steel bars are:

Ns ¼

n X 1

R

i¼1

Ms ¼

yi rsi Asi

n X 1 i¼1

R

y2i rsi Asi

ð31Þ

ð32Þ

The interaction diagram point (Ntot, Mtot) is given by:

Ntot ¼

n X Nci þ Ns

ð33Þ

i

Mtot ¼

n X

Mci þ M s

ð34Þ

i

The axial force contribution Nci of trapezoid i is obtained from Eqs. (18) and (21) and moment contribution Mci using Eqs. (25) and (29). One should note that the point where the resultants Mtot and Ntot are calculated could be at any point of the section. In order to allow comparison with the work of other researchers, the gross section’s geometric centroid will be used in this study.

The interaction diagram between bending moments and axial force can be obtained by integration of the stresses that result from the strain states. In order to compute the resultant internal forces (moment and axial force) associated with a particular strain state, it is necessary to define limiting values of strains in steel and concrete. Eurocode 2 [2] states that the compressive strain in concrete is in general limited to ecu2. However, if the section is under uniform compression, the strain value should be limited to ec2. With respect to reinforcement steel, strain values are limited to eud. The implementation of such principles can be formulated in terms of a strain path, defined by means of a diagram where the strains at the top fiber are plotted against the strains at the bottom fiber, as exemplified in Fig. 4(a). The proposed technique allows the definition of the complete failure surface of R/C cross sections, including the points at (Mx–My) plane with N = 0, using a non-iterative procedure. The overall depth d and effective depth h are represented in Fig. 4(b). The strain path key values are listed in Table 5 for an arbitrary section. 6. Computation of failure surface and moment interaction curves

P3

P9

X Y Z

5.1. Possible range of strain distributions

For each imposed strain on top and bottom fibers, the corresponding part of the section that is under compression strains is directly defined. The calculation of a complete section is afterwards performed by treatment of cross section contour, defined by a polygon. By applying the Weiler Atherton algorithm, any polygon is divided into a group trapezoidal elements, or triangular elements. Note that the equations developed for trapezoidal elements are also applicable for triangular elements; two points are in this case coincident. The calculation of the moment and the axial force contributions, for each individual trapezoid situated inside the compression zone, can be performed using the Eqs. (18), (21), (25) and (29). With respect to openings, the contributions of the axial force and the moment for each opening can be obtained using

W1

Table 4 Lists X, Y, Z required for implementing the Weiler–Atherton algorithm.

P5 Fig. 3. Application of Weiler–Atherton algorithm: Base polygon and clipping window.

In order to calculate a point at the failure surface, it is required to calculate the axial force N, and moments Mx and My, which are represented in Fig. 2(a). The axial force and bending moments result from stress integration, for each inclination of neutral axis and strain state defined by imposed strains at top and bottom fibers. The axial force and bending moments are, in a first stage, calculated about the neutral axis, and are afterwards converted to the center of geometries.

7

R. Vaz Rodrigues / Engineering Structures 104 (2015) 1–17

d’ d

(a)

h

(b)

Fig. 4. Strain path for cross sections: (a) strain path for ecu = 2.0‰, ecu2 = 3.5‰, eud = 22.5‰, h = 1.90 m, d = 1.80 m and d0 = 0.05 m; (b) definition of overall depth h, effective depth d and distance d0 in an arbitrary cross section.

My ¼ M x;local sin h þ M y;local cos h þ Ny0;local Table 5 Definition of admissible strain states in cross section. Strain state

esup

einf

1 2 3

ec2 ecu2 ecu2

ec2 0


4

ecu2



5 6

eud 0

d hd0 d0 0 hd

7 8



h ecu2 þ fEsds hd

ecu2 þ e eud ecu2 ecu2

h ecu2 þ eud hd 0 f sd h ecu2 þ Es hd0

d hd



h d ud hd hd

ecu2

0

The axial force and bending moments (about the neutral axis) result from stress integration, for each inclination of neutral axis and strain state defined by imposed strains at top and bottom fibers. The computation of axial force and bending moments about the neutral axis (Mx,local) is done using Eqs. (33) and (34). Additionally, it is required to evaluate the following integral in order to compute the moment about the local y axis. The integration of concrete stresses is expressed in the following equation:

Z

M y;local ¼

ð39Þ

Note that, in this paper, the origin of global axis x and y is located at the geometric centroid of the section. Fig. 5 illustrates the construction of the failure surface for a rectangular cross section with dimensions b = 25 cm  h = 45 cm and total amount of reinforcement 8  2.01 cm2. For a given fixed neutral axis inclination, the cross section is subjected to a series of strain states (Table 5) that generate a non planar curve (Mx, My, N). It can be observed that the values of the bending moments (Mx and My) depend of the strain state of the section, and especially of the geometry of the section. As a consequence the ratio (Mx/My) is not constant along this curve. The computation is performed using neutral axis angle of inclination increments Dh = 3° and 800 strain states for each neutral axis angle of inclination, which results in a total of 48 800 points. Note that the neutral axis angle of inclination ranges between h = 0° and h = 180° (covering all possible strain states, according with strains limits indicated in Table 5) which results in a total of 48 800 points for the complete surface. 7. Examples and verification 7.1. Example 1: Rectangular cross section

xrc ðx; yÞ dx dy

ð35Þ

w

Similarly, the explicit expression for the resulting bending moment, corresponding to the second range of concrete’s constitutive law of Eq. (8), is computed using Eq. (19) (with M instead of N) taking f(n, g) as:

The first example shows the response of a rectangular cross section with dimensions b = 0.30 m  h = 0.70 m, with total reinforcement As,tot = 4  10 cm2, placed at the corners of the cross section, with cover equal to 5 cm, concrete strength fcd = 20 MPa and steel yielding strength fsd = 500 MPa. Analysis is performed using neutral axis angle of inclination increments Dh = 5° and 800 strain states for each neutral axis angle of inclination. The moment interaction curves are shown in Fig. 6 for (compression) axial force values of Nsd = 5000 kN, 1000 kN, 0 kN and (tensile) axial force values of Nsd = 750 kN and 1500 kN. The results of the present study compare very well with those obtained by Papanikolaou [7] using stress integration by Green path integrals with adaptive strain-mapped Gaussian sampling.

f ðn; gÞ ¼ ½C 3;x g þ C 2;x n þ C 1;x gn þ C 4;x f cd j½C A n þ C B g þ C C j

7.2. Example 2: U-shaped cross section

For the first range of Eq. (8), this integral can be evaluated using Eq. (26), where function h(n, g) is given by:

hðn; gÞ ¼ f cd ½C 3;x g þ C 2;x n þ C 1;x gn "  n # 1 ½C 3;y g þ C 2;y n þ C 1;y gn þ C 4;y  R þ C 4;x  1  1 

ec2

 j½C A n þ C B g þ C C j

ð36Þ

ð37Þ

Finally, the bending moments with respect to global axis x and y are obtained by the following equations:

M x ¼ M x;local cos h  My;local sin h þ Nx0;local

ð38Þ

The second example shows the response of a U-shaped cross section, analyzed in a study presented by Rosati et al. [8], with dimensions b = 0.80 m  h = 0.60 m and wall thickness of 20 cm,

8

R. Vaz Rodrigues / Engineering Structures 104 (2015) 1–17

Table 6 Comparison between numerical results and experimental data from Ramamurthy [18], and Pallarès et al. [19].

a (°)

fy (MPa)

Ptest (kN)

Mtest (kN m)

Bars

As (mm2)

b (mm)

d (mm)

d0 (mm)

Mpred (kN m)

Mpred/Mtest

Ramamurthy – series A A-1 52.1 A-2 57.2 A-3 55.2 A-4 47.9 A-5 42.7 A-6 40.3 A-7 44.5 A-8 31.9 A-9 46.2 A-10 55.4 A-11 57.4 A-12 49.6 A-13 20.9 A-14 29.6 A-15 23.7

15 15 15 15 15 14.5 11.3 20 20 30 30 30 33.7 26.57 45

291.9 291.9 291.9 291.9 291.9 291.9 291.9 291.9 291.9 291.9 291.9 291.9 291.9 291.9 291.9

564.9 395.9 378.1 283.6 235.8 171.9 146.8 476.0 280.2 462.6 264.7 170.1 164.6 160.1 266.9

51.7 52.1 49.7 44.7 43.4 36.0 38.0 38.6 45.5 47.5 38.8 34.6 30.1 36.4 33.9

8#4 8#4 8#4 8#4 8#4 8#4 8#4 8#4 8#4 8#4 8#4 8#4 8#4 8#4 8#4

1032 1032 1032 1032 1032 1032 1032 1032 1032 1032 1032 1032 1032 1032 1032

203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2

203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2

31.75 31.75 31.75 31.75 31.75 31.75 31.75 31.75 31.75 31.75 31.75 31.75 31.75 31.75 31.75

54.74 50.34 49.00 42.80 39.30 35.66 35.11 40.53 41.80 51.23 43.34 36.94 28.32 31.94 30.91

1.06 0.97 0.99 0.96 0.91 0.99 0.92 1.05 0.92 1.08 1.12 1.07 0.94 0.88 0.91

Ramamurthy – series B B-1 32.4 B-2 28.6 B-3 37.2 B-4 35.5 B-5 21.7 B-6 30.6 B-7 32.8 B-8 37.9

15 22.5 30 30 45 45 45 45

322.6 322.6 322.6 322.6 322.6 322.6 322.6 322.6

629.0 771.8 533.8 395.9 598.3 500.4 516.0 369.8

51.1 39.2 54.2 50.3 30.4 45.8 52.4 53.1

8#5 8#5 8#5 8#5 8#5 8#5 8#5 8#5

1600 1600 1600 1600 1600 1600 1600 1600

203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2

203.2 203.2 203.2 203.2 203.2 203.2 203.2 203.2

33.3 33.3 33.3 33.3 33.3 33.3 33.3 33.3

49.97 40.41 51.16 49.31 34.23 44.35 45.98 48.87

0.98 1.03 0.94 0.98 1.13 0.97 0.88 0.92

Ramamurthy – series C C-1 34.6 C-1(a) 41.2 C-2 37.7 C-2(a) 48.3 C-3 32.2 C-4 26.8 C-5 27.6 C-6 34.5

15 15 15 15 30 30 45 45

275.8 275.8 275.8 275.8 275.8 275.8 275.8 275.8

464.8 569.4 400.3 489.3 460.4 378.1 506.0 350.3

17.7 21.7 18.3 22.4 17.5 17.3 15.4 21.4

8#4 8#4 8#4 8#4 8#4 8#4 8#4 8#4

1032 1032 1032 1032 1032 1032 1032 1032

152.4 152.4 152.4 152.4 152.4 152.4 152.4 152.4

152.4 152.4 152.4 152.4 152.4 152.4 152.4 152.4

31.75 31.75 31.75 31.75 31.75 31.75 31.75 31.75

18.90 20.53 21.51 25.34 16.92 15.47 13.05 18.73

1.07 0.95 1.18 1.13 0.96 0.90 0.85 0.88

Ramamurthy – series R R-138 25.9 R-238 34.5 R-338 36.2 R-438 29.6

45 45 33.7 33.7

275.8 275.8 275.8 275.8

138.3 160.1 118.8 71.2

14.9 17.3 16.3 13.0

8#3 8#3 8#3 8#3

568 568 568 568

152.4 152.4 152.4 152.4

152.4 152.4 152.4 152.4

30.16 30.16 30.16 30.16

12.30 14.39 13.89 11.69

0.83 0.83 0.85 0.90

Ramamurthy – series D D-1 35.2 D-2 28.3 D-3 27.2 D-4 27.9 D-5 34.3 D-6 26.5

33.7 33.7 33.7 45 45 60

322.6 322.6 322.6 322.6 322.6 322.6

785.1 400.3 311.4 680.6 378.1 400.3

35.9 40.7 42.7 31.1 43.2 36.6

8#5 8#5 8#5 8#5 8#5 8#5

1600 1600 1600 1600 1600 1600

152.4 152.4 152.4 152.4 152.4 152.4

228.6 228.6 228.6 228.6 228.6 228.6

33.3 33.3 33.3 33.3 33.3 33.3

42.78 43.98 43.73 33.13 42.31 31.17

1.19 1.08 1.02 1.06 0.98 0.85

Ramamurthy – series E E-1 26.1 E-2 23.8 E-3 30.8 E-4 27.6

26.57 26.57 45 60

322.6 322.6 322.6 322.6

464.8 311.4 435.9 542.7

59.0 53.1 53.1 41.4

8#5 8#5 8#5 8#5

1600 1600 1600 1600

152.4 152.4 152.4 152.4

304.8 304.8 304.8 304.8

33.3 33.3 33.3 33.3

74.79 70.87 63.83 45.83

1.27 1.34 1.20 1.11

Ramamurthy – series F F-1 32.6 F-2 39.9 F-3 20.0 F-4 28.1 F-5 29.6

15 33.7 33.7 33.7 45

291.9 291.9 291.9 291.9 291.9

600.5 533.8 384.8 266.9 466.0

34.3 30.5 22.0 24.4 26.6

8#4 8#4 8#4 8#4 8#4

1032 1032 1032 1032 1032

152.4 152.4 152.4 152.4 152.4

228.6 228.6 228.6 228.6 228.6

31.75 31.75 31.75 31.75 31.75

40.16 42.86 27.21 34.07 31.18

1.17 1.41 1.24 1.40 1.17

Ramamurthy – series G G-1 36.6 G-2 28.1 G-3 24.2 G-4 41.4 G-5 34.5

15 15 26.57 26.57 45

291.9 291.9 291.9 291.9 291.9

827.4 418.1 507.1 333.6 584.9

63.0 53.1 38.6 50.8 35.7

8#4 8#4 8#4 8#4 8#4

1032 1032 1032 1032 1032

152.4 152.4 152.4 152.4 152.4

304.8 304.8 304.8 304.8 304.8

31.75 31.75 31.75 31.75 31.75

74.71 64.35 54.87 65.47 55.95

1.18 1.21 1.42 1.29 1.57

Pallarès 10_05_3 10_05_4 10_1_4

14.04 14.04 26.56

558 558 558

473.9 175.2 166.1

35.0 25.8 23.0

4/10 4/10 4/10

314 314 314

100 100 100

200 200 200

20 20 20

40.15 28.11 25.84

1.15 1.09 1.12

Test

fcm,cube (MPa)

94.1 94.1 95.1

9

R. Vaz Rodrigues / Engineering Structures 104 (2015) 1–17 Table 6 (continued) Test

fcm,cube (MPa)

a (°)

fy (MPa)

Ptest (kN)

Mtest (kN m)

Bars

10_2_2 10_2_3 10_2_4

92.2 94.1 94.1

45 45 45

558 558 558

899.9 436.5 142.0

22.8 22.1 14.4

4/10 4/10 4/10

As (mm2) 314 314 314

b (mm)

d (mm)

d0 (mm)

Mpred (kN m)

Mpred/Mtest

100 100 100

200 200 200

20 20 20

26.74 27.94 20.64

1.17 1.27 1.44

Average 1.07 Standard deviation 0.17

Secon 25 x 45 cm, cover 5 cm fc = 47.6 MPa EC2 parabolic linear fs = 530 MPa. As,tot = 8 x 2.01 cm2

N [kN]

N [kN]

Curve for fixed neutral axis inclination

7 000 6 000 5 000 4 000 3 000 2 000 1 000 0.000

My [kNm]

-1 000 -500

Mx [kNm]

100

200

300 400

500

Mx [kNm]

(a)

N [kN]

-400 -300 -200 -100 0.00

(b) Curve for fixed neutral axis inclination

7 000 6 000

Curve for fixed neutral axis inclination

-300

My [kNm]

5 000 4 000

-200 -100

3 000 0.00

2 000

100

1 000

200

0.000 -1 000 -300 -200 -100 0.0

100

200 300

300 -500

-400 -300 -200 -100 0.00

100

My [kNm]

Mx [kNm]

(c)

(d)

200 300

400

500

Fig. 5. Computational procedure for obtaining the failure surface of rectangular cross section with dimensions b = 25 cm  h = 45 cm and total amount of reinforcement 8  2.01 cm2: (a) tridimensional view, (b) projected view on the N–Mx plane, (c) projected view on the N–My plane and (d) projected view on the My–Mx plane.

with total reinforcement As,tot = 24  3.14 cm2, cover equal to 3 cm, concrete strength fcd = 0.85  20.75/1.6 MPa and steel yielding strength fsd = 375/1.15 MPa. Analysis is performed using neutral axis angle of inclination increments Dh = 6° and 800 strain states for each neutral axis angle of inclination. The moment interaction

curves are shown in Fig. 7 for (compression) axial force values of Nsd = 4037 kN, 1439 kN, 140 kN and (tensile) axial force values of Nsd = 1809 kN. The results of the present study also compare well with those obtained by Rosati et al. [8] using a method with secant elastic iterations.

10

R. Vaz Rodrigues / Engineering Structures 104 (2015) 1–17 350 Present N=5000 kN

Secon 30 x 70 cm,cover 5 cm fcd = 20 MPa, EC2 parabolic linear fsd = 500 MPa, As, tot = 40 cm2

300

Present N=-750 kN Present N = -1500 kN Present N = 0 kN

My[kNm]

250

Papanikolau Papanikolaou[7]N = 1000 kN Present N = 1000 kN Mxy

200

150

100

50

0 0

100

200

300

400

500

600

700

800

900

1000

Mx [kN m] Fig. 6. Comparison of moment interaction curves for rectangular section, between present method and Papanikolaou [7].

Present N=4037 kN

Present N=1439

Rosati et al. (N = 4037 kN)

800

Present N=140 kN

Rosati et al. (N = 1439 kN)

Present N= -1809 kN

Rosati et al. (N = 140 kN)

Rosati et al. (N = - 1809 kN)

U-shaped Secon,cover 3 cm fcd = 0.85 x 20.75/1.6 MPa EC2 parabolic linear fsd = 375 / 1.15 MPa. Bar diameter 20 mm

My[kN]

300

-200

-700

-1200 -800

-600

-400

-200

0

200

400

600

800

Mx [kNm] Fig. 7. Comparison of moment interaction curves for U-shaped section, between present method and Rosati et al. [8].

Papanikolaou [7] N= -2 657 kN

Present N= -2 657 kN

Papanikolaou [7] N = 6 182 kN

Present N = 6 182 kN

Papanikolaou [7] N = 879 kN

Present N = 879 kN

2000 1500

Gshaped Secon,cover 3 cm fcd = 0.85 x 20.75/1.6 MPa, EC2 parabolic linear fsd = 375 / 1.15 MPa. Bar diameter 24 mm

My [kNm]

1000 500 0 -500 -1000 -1500 -2000 -1000

-800

-600

-400

-200

0

200

400

600

800

1000

M x [kNm] Fig. 8. Comparison of moment interaction curves for G-shaped section, between present method and Papanikolaou [7].

11

R. Vaz Rodrigues / Engineering Structures 104 (2015) 1–17 7000 Rodriguez et al. 1999 [12] Present Reseach

6000

Hollow Circular Column fcd = 27.58 MPa fsd = 413.69 MPa. Bar diameter 7/8''

5000

N [kN]

4000 3000 2000 1000 0 -1000 -2000

0

100

200

300

400

500

600

M [kNm] Fig. 9. Comparison of moment–axial force interaction curve for hollow circular section, between present method and Rodriguez et al. [12].

Papanikolaou [7] N= -5676 kN

Papanikolaou [7] N = 34 129 kN

Papanikolaou [7] N = 10 246 kN

Present N=10 246 kN

Present N=34 129 kN

Present N= -5676 kN

20000 15000

MulcellSecon,cover 3 cm fcd = 0.85 x 20.75 /1.6 MPa fsd = 375 / 1.15 MPa. Bar diameter 32 mm

My [kNm]

10000 5000 0 -5000 -10000 -15000 -20000 -15000

-10000

-5000

0

5000

10000

15000

Mx [kNm] Fig. 10. Comparison of moment interaction curves for multicell section, between present method and Papanikolaou [7].

7.3. Example 3: G-shaped cross section The third example shows the response of a G-shaped cross section analyzed in a study presented by Papanikolaou [7], with dimensions b = 1.00 m  h = 0.60 m and wall thickness of 20 cm, with total reinforcement As,tot = 24  4.52 cm2, cover equal to 3 cm, concrete strength fcd = 0.85  20.75/1.6 MPa and steel yielding strength fsd = 375/1.15 MPa. Analysis is performed using neutral axis angle of inclination increments Dh = 6° and 800 strain states for each neutral axis angle of inclination. The moment interaction curves are shown in Fig. 8 for (compression) axial force values of Nsd = 6182 kN, 879 kN and (tensile) axial force values of Nsd = 2657 kN. The results of the present study are almost coincident with those obtained by Papanikolaou [7] using stress integration by Green path integrals with adaptive strain-mapped Gaussian sampling. It should also be noted that small differences may occur when taking into account the ‘‘bar holes” as if it was of concrete. 7.4. Example 4: Circular hollow section The fourth example shows the response of a circular hollow section analyzed in a study presented by Rodriguez and

Aristizabal-Ochoa [12], with external diameter £ext = 609.6 mm and wall thickness of 254 mm, total reinforcement As,tot = 8  3.87 cm2, placed at radius of 244 mm, concrete strength fcd = 27.6 MPa and steel yielding strength fsd = 413.7 MPa. The moment–axial force interaction curve is shown in Fig. 9. A polygon of 16 points is used to approximate the exterior and interior circular boundaries. The results of the present study compare well with those obtained by Rodriguez and Aristizabal-Ochoa [12].

7.5. Example 5: Multicell cross section The fifth example shows the response of a multicell cross section analyzed in a study presented by Papanikolaou [7], with

Table 7 Execution times for the full interaction surface (48 800 points). Section Time (s)

A 145

B 279

C 307

D 295

E 119

F 220

A: Rectangular section (Fig. 6); B: U-shaped section (Fig. 7); C: G-shaped section (Fig. 8); D: multicell section (Fig. 10); E: design example 1 (Fig. 16); F: design example 2 (Fig. 17).

12

R. Vaz Rodrigues / Engineering Structures 104 (2015) 1–17

Q 1 = 96.1 kN

Q 2 = 18.7 kN

R2

R1 3,00

0,20

2,00

3,00 m

0,20

-20 0 40 80 120 160 200

M [kN·m]

170.9 kN m (Measured at failure)

Fig. 11. Bending moments at failure of beam. Failure caused by tensile failure of reinforcing steel.

dimensions b = 2.50 m  h = 1.40 m and wall thickness of 30 cm, with total reinforcement As,tot = 52  8.04 cm2, cover equal to 3 cm, concrete strength fcd = 0.85  20.75/1.6 MPa and steel yielding strength fsd = 375/1.15 MPa. Analysis is performed using neutral axis angle of inclination increments Dh = 6° and 800 strain states for each neutral axis angle of inclination. The moment interaction curves are shown in Fig. 10 for (compression) axial force values of Nsd = 34 129 kN, 10 246 kN and (tensile) axial force values of Nsd = 5676 kN. The results of the present study are almost coincident with those obtained by Papanikolaou [7]

using stress integration by Green path integrals with adaptive strain-mapped Gaussian sampling. From the comparisons presented in this section, it is concluded that the methodology proposed in the present study (WAGL) provides accurate results for symmetric and non-symmetric sections of complex shapes, including consideration of hollow cross sections. Execution times for obtaining the full interaction surface with 48 800 points are indicated in Table 7, obtained using a system with processor Intel Core i5 CPU M460 @ 2.53 GHz and installed

4 Ø16

α

0.404 m

4 Ø16

0.020

d

0.20 m

0.45 m

2Ø 10

M 0.203

8Ø 12.5 0.025

2Ø 10

0.10 m

b 0.203

0.25 m

(a)

(b)

(c)

Fig. 12. Cross section and reinforcement layout of evaluated tests: (a) Vaz Rodrigues et al. [17]; (b) Ramamurthy [18], A-series; (c) Pallarès et al. [19].

13

R. Vaz Rodrigues / Engineering Structures 104 (2015) 1–17 7000

M 6000 5000

N [kN]

4000 3000 2000 1000 0 -1000 -2000 -500

Experimental point with Vaz Rodrigues et al. 2010, N=0; M = 170.9 kNm

-400

-300

-200

-100

0

100

200

300

400

500

M [kN·m] Fig. 13. Interaction M–N diagrams for rectangular section, for various inclinations of neutral axis h.

indicated in Fig. 14 – a representation of moment interaction curves of the cross section for N = 0 kN. Secondly, the results from the WAGL technique are compared with the results provided by Ramamurthy [18]. The cross sections tested by this author are illustrated in Fig. 12(b) and the test results are indicated in Table 6. Note that Ramamurthy evaluated the concrete compressive strength in cubes. Therefore the values to adopt for cylinders are considered to be fck = 0.85 fck,cube. The angle a is defined in Fig. 12(b), along with the column width and overall depth. The total area of reinforcing steel, the number of provided bars and distance to concrete surface d0 are indicated in Table 6, along with the theoretical bending strengths, showing good agreement with measured values. This fact also results from Fig. 15, which shows the agreement for each series. It can also be observed that for the columns of Series E, F, G the agreement is less effective. However this lack of agreement was also confirmed by other authors, as in Pallarès et al. [13]. Thirdly, the results from the WAGL technique are also compared with the experimental results provided by Pallarès et al. [19]. The results of Table 6 also confirm good agreement with the experimental results provided by these authors. The average ratio (for all 61 tests) of the bending moment obtained through the WAGL technique and the experimental

memory RAM 4.00 GB, and implemented using Visual Basic for Applications (VBA), as mentioned before.

8. Comparison with experimental data Firstly, the results from the WAGL technique are compared with the experimental values for simple bending. The simple cross section tested by Vaz Rodrigues et al. [17] is analyzed. The beam was tested in two point bending with material properties for compressive concrete strength of fc = 47.6 MPa and reinforcing steel yield strength fs = 530 MPa. The experimental bending strength of the beam is of 170.9 kN m. The spans and bending moment diagram at failure of the beam are shown in Fig. 11. The cross section of the beam is shown in Fig. 12(a). The evaluation of the theoretical ultimate strength of the beam can be done after the interaction diagram (M–N) using N = 0 kN for input. The interaction diagram is shown in Fig. 13, corresponding to the curve indicated with h = 0°. The theoretical ultimate bending strength is 164 kN m using the procedures described in this paper. The steel hardening is not considered in the analysis, therefore k = 1.00. The analysis of the cross section is also performed using a different angle of inclination of neutral axis h. The experimental point is represented in the failure surface, as

Present N = 0

Present N = -600 kN

Present N = 600 kN

Present N=1 200 kN

180 Secon 25 x 45 cm,cover 3.30 cm fc = 47,6 MPa, EC2 parabolic linear fy = 530 MPa, As, tot = 8 x 2,01 cm2

160 140

My [kNm]

120 100 80 60 40

Experimental point Vaz Rodrigues et al. 2010 (Mx = 170.9 kNm)

20 0

0

50

100

150

200

250

300

350

400

Mx [kNm] Fig. 14. Moment interaction curves for rectangular section and comparison with experimental results.

14

R. Vaz Rodrigues / Engineering Structures 104 (2015) 1–17 80.0

60.0

50.0 40.0 30.0

20.3 x 20.3 cm 8 #5 fy = 322 MPa

40.0 30.0

40.0 30.0 20.0

10.0

10.0

10.0

20.00

40.00

60.00

0.0 0.00

80.00

20.00

0.0 0.00

80.00

M test [kNm]

60.0

50.0 40.0 30.0

70.0 60.0

50.0 40.0 30.0

30.0 20.0

10.0

10.0

10.0

60.00

0.0 0.00

80.00

20.00

M pred [kNm]

0.0 0.00

80.00

70.0

M test [kNm]

60.0

50.0 40.0 30.0

70.0 60.0

50.0 40.0 30.0

30.0 20.0

10.0

10.0

80.00

0.0 0.00

20.00

40.00

60.00

M pred [kNm]

M pred [kNm]

(g) Ramamurthy – series F

(h) Ramamurthy – series G

80.00

10 x 20 cm 4 Ø10 fy = 558 MPa

40.0

10.0 60.00

80.00

50.0

20.0

40.00

60.00

80.0

15.2 x 30.5 cm 8 #4 fy = 292 MPa

20.0

20.00

40.00

(f) Ramamurthy – series E

80.0

15.2 x 22.9 cm 8 #4 f y = 292 MPa

0.0 0.00

20.00

M pred [kNm]

(e) Ramamurthy – series D

80.0

60.0

60.00

M pred [kNm]

(d) Ramamurthy – series R

70.0

40.00

15.2 x 30.5 cm 8 #5 fy = 322MPa

40.0

20.0

40.00

80.00

50.0

20.0

20.00

60.00

80.0

15.2 x 22.9 cm 8 #5 fy = 322 MPa

M test [kNm]

70.0

40.00

(c) Ramamurthy – series C

80.0

15.2 x 15.4 cm 8 #3 fy = 276 MPa

0.0 0.00

20.00

M pred [kNm]

(b) Ramamurthy – series B

80.0

60.0

60.00

M pred [kNm]

(a) Ramamurthy – series A

70.0

40.00

15.2 x 15.4 cm 8 #4 fy = 276 MPa

50.0

20.0

M pred [kNm]

M test [kNm]

60.0

50.0

20.0

0.0 0.00

M test [kNm]

70.0

M test [kNm]

70.0

M test [kNm]

M test [kNm]

60.0

80.0

80.0

20.3 x 20.3 cm 8 #4 fy = 292 MPa

M test [kNm]

70.0

0.0 0.00

20.00

40.00

60.00

80.00

M pred [kNm]

(i) Pallarès

Fig. 15. Comparison between numerical results and experimental data (square and rectangular cross sections) from Ramamurthy [18] and Pallarès et al. [19].

bending moment is 1.07, as indicated in Table 6, which confirm good agreement. 9. Design examples The application to design is illustrated by means of two examples. The first example consists if a rectangular cross section under biaxial bending moments and second example consists of a hollow cross section. In both examples the designer can plot a series of curves, each curve for a given amount of bending reinforcement. 9.1. Rectangular cross section under biaxial loading Consider the reinforced concrete column of a powerhouse, shown in Fig. 16, with overall depth of 2.20 m and width of 0.80 m. Assume that applied bending moments are Mxd = 6000 kN m, Mysd = 1700 kN m and that the axial compression

force is Nd = 5000 kN. The material properties at design level are the compressive strength fcd = 14.57 MPa and the yield strength fsd = 435 MPa (horizontal yield plateau considered). From the analysis of Fig. 16, it can be seen that the total amount steel, 105.6 cm2, is adequate to provide the required bending strength. Finally, the detailing of the bending reinforcement is shown in Fig. 16 and the element itself during the construction of the column.

9.2. Hollow cross section under biaxial loading Consider in Fig. 17 a square box section (2.0  2.0 m) with wall thickness of 0.30 m. The applied bending moments are Msd,x = 10 400 kN m and Msd,y = 10 400 kN m, combined with axial compression force of Nd = 10 000 kN. The principal bending moment is therefore Msd = 14 707 kN m, applied on an angle of 45° with respect to the major bending axis. Material properties

15

R. Vaz Rodrigues / Engineering Structures 104 (2015) 1–17 4000

(each face)

84.5 cm2

3000

73.9 cm2 63.6 cm2 52.8 cm2 42,2 cm2 31.7 cm2

2000

9Ø20 2.2

(each face)

21.2 cm2 10.6 cm2 0.0 cm2

1000

My [kNm]

5Ø25

As,tot = 105.6 cm2 95.0cm2

Nsd = 5 000 kN

0 0.80

-1000

-2000

-3000

-4000 -10000

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

10000

Mx [kNm] Fig. 16. Design example for rectangular cross section (Msd,x = 6000 kN m; Msd,y = 1700 kN m; Nsd = 5000 kN): moment interaction Mx–My curves for different amounts of reinforcement, cross section details, and column during construction.

20000

As,tot = 105.6 cm2

95.0 cm2

Nsd = 10 000 kN

84.5 cm2

15000

73.9 cm2 63.6 cm2

10000

52.8 cm2 42,2 cm2 31.7 cm2

My [kNm]

5000

21.2 cm2 10.6 cm2 0.0 cm2

0

-5000

-10000

-15000 without strain hardening (k = 1.00) with strain hardening (k = 1.35)

-20000 -20000

-15000

-10000

-5000

0

5000

10000

15000

20000

Mx [kN·m]

2.00

4x13Ø20 (ext) 4x11Ø20 (int)

Mx M

My

45°

0.30 m 2.00 Fig. 17. Design example for hollow cross section (Msd,x = 10 400 kN m; Msd,y = 10 400 kN m; Nd = 10 000 kN): moment interaction Mx–My curves for different amounts of reinforcement and cross section details.

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R. Vaz Rodrigues / Engineering Structures 104 (2015) 1–17

Stress [MPa]

Strain [‰] -2.00

(1.0;2.5)

20 MPa

fiber (2.0;1.50)

(0.25;0.75)

-0.40 (0.50;0.50)

yLocal

7.2 MPa neutral axis

xLocal Fig. 18. Example used for comparing the WAGL technique with the fiber element approach.

2500

0.50

Number of operaons of Fiber element approach

0.45

Error [‰]

0.35 0.30

1500

Error

0.25 1000

0.20 0.15

Number of operaons of WAGL technique

0.10

Number of operaons

2000

0.40

500

0.05 0.00 0

100

200

300

400

0 500

Number of fibers (along y) Fig. 19. Comparison between the WAGL technique and the fiber element approach.

at design level, are the compressive strength fcd = 16 MPa and the yield strength fsd = 400 MPa (horizontal yield plateau considered). From the analysis of Fig. 17, it can be seen that the total amount steel, 301.4 cm2, resulting from 4  13 + 4  11, 20 mm diameter bars, is adequate to provide the required inclined bending strength. The effect of reinforcing steel strain hardening is also analyzed (with k = 1.15) showing a very marginal influence at this level of axial compression. 10. Conclusions A new procedure (WAGL) is developed in this study to evaluate the bending strength of polygonal sections of arbitrary shape, including holes, and arbitrary locations of reinforcing bars. The following conclusions are achieved:  The proposed section subdivision and stress integration technique (WAGL), based on the change of variables theorem, is an effective formulation to provide exact results for the biaxial bending strength (combined with axial force) of normal strength (fck 6 50 MPa) reinforced concrete cross sections.

 The WAGL technique uses a polygon clipping algorithm, originally developed in the scope of computer graphics, by Weiler and Atherton [15]. This was found to be an efficient formulation for dividing any cross section into trapezoidal elements.  The results from the WAGL technique (using the material properties of concrete and steel from Eurocode 2), show a good agreement with the available experimental data.  The WAGL technique provides a method without need of iteration, achieved by evaluating a finite number of algebraic expressions that are derived in this paper, which constitutes an indisputable advantage. The findings of this paper confirm those identified by Bonet et al. [20], i.e., integration methods based on the Gauss–Legendre quadrature (with decomposition of the compression zone into ‘‘thick” layers) are very effective for computing interaction diagrams of concrete section. An example is included in Appendix A, comparing results of the WAGL technique, with the results of fiber element approach, which confirms these findings.  The WAGL technique provides an efficient method in the laborious task of decomposing the section into trapezoidal elements. As a consequence, high computational efficiency is obtained, which is considered to improve both flexibility and productivity in the engineering task of reinforced concrete cross section design.

Appendix A. Comparison between the WAGL technique and the fiber element approach Given the concrete section shown in Fig. 18, the resultant axial force N under the indicated strains is calculated considering the proposed technique (WAGL) and the fiber element approach. The exact value for the axial force NWAGL is (24 822.92 kN). The integral is evaluated using 51 operations of sum and 46 multiplications, which results in 97 operations. The same integral can be evaluated using the fiber method approach; in this case the accuracy of the result depends on the mesh division. The error of the fiber method is therefore defined as |NWAGL  NFiber|/NWAGL, and is represented in Fig. 19. It can be seen that the fiber method approach converge to the exact value. The required operations for the fiber method are also indicated in Fig. 19. By comparing both techniques, it can be seen that, for the same number of operations (97), the WAGL technique leads to the exact value of the integral, whereas the fiber method approach leads to a relative error of about 0.5‰, in the present numerical example.

R. Vaz Rodrigues / Engineering Structures 104 (2015) 1–17

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[12] Rodriguez JA, Aristizabal-Ochoa JD. Biaxial interaction diagrams for short RC columns of any cross section. J Struct Eng 1999;125(6):672–83. [13] Pallarès L, Miguel PF, Fernández-Prada MA. A numerical method to design reinforced concrete sections subjected to axial forces and biaxial bending based on ultimate strain limits. Eng Struct 2009;31:3065–71. [14] De Vivo L, Rosati L. Ultimate strength analysis of reinforced concrete sections subject to axial force and biaxial bending. Comput Methods Appl Mech Eng 1980;166:261–87. [15] Weiler K, Atherton P. Hidden surface removal using polygon area sorting. Comput Graph 1977;11(2):214–22. [16] Lam N, Wilson J, Lumantarna E. Force–deformation behavior modeling of cracked reinforced concrete by EXCEL spreadsheets. Comput Concr 2011;8 (1):43–57. [17] Vaz Rodrigues R, Muttoni A, Fernández Ruiz M. Influence of shear on rotation capacity of reinforced concrete members without shear reinforcement. ACI Struct J 2010;107(50):516–25. [18] Ramamurthy LN. Investigation of the ultimate strength of square and rectangular columns under biaxial eccentric loads. American Concrete Institute; 1966. p. 263–98 [Publ. 13]. [19] Pallarès L, Bonet JL, Miguel PF, Fernández Prada MA. Experimental research on high strength concrete slender columns subjected to compression and biaxial bending forces. Eng Struct 2008;30:1879–94. [20] Bonet JL, Barros MHFM, Romero ML. Comparative study of analytical and numerical algorithms for designing reinforced concrete sections under biaxial bending. Comput Struct 2006;84:2184–93.