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aluminum thin-walled sections of arbitrary shape
Engineering Structures 100 (2015) 57–65
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Automatic cross-section classification and collapse load evaluation for steel/aluminum thin-walled sections of arbitrary shape Francesco Marmo ⇑, Luciano Rosati Department of Structures in Engineering and Architecture, University of Naples Federico II, Naples, Italy
a r t i c l e
i n f o
Article history: Received 21 February 2015 Revised 11 May 2015 Accepted 27 May 2015
Keywords: Cross-section classification Local buckling Thin walled section
a b s t r a c t We present a computational procedure for evaluating the collapse load and assessing the cross-section classification of thin-walled sections of arbitrary shape on the basis of Eurocode prescriptions. The procedure is based on two algorithms which address separately the rigid-plastic model adopted by the Eurocode for ordinary steel cross-sections and arbitrary uniaxial constitutive laws typically used for stainless steel and aluminum. Both algorithms are based on a polygonal description of the cross section boundary so that integrals extended to the section domain are conveniently expressed as algebraic sums depending upon the coordinates of the section vertices. Accordingly, a further algorithm is illustrated in order to automatically convert the plate and node model adopted by Eurocode to a polygonal description of the section geometry. The numerical effectiveness of both algorithms is assessed with reference to an I-shaped, a Z-shaped and a RHS cross sections. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Both the design and the verification of a structure have to take into account not only the attainment of yield or fracture conditions, but also the occurrence of buckling phenomena, as they can cause collapse of the structure for stress levels lower than the strength of material supposed to act on the whole section. In particular, buckling analysis is undoubtedly fundamental for thin-walled beams, as the buckling load is inversely proportional to the slenderness of the beam. Despite the high values of yield and ultimate stress guaranteed by steel or aluminum and their iso-resistant behavior, compression on the plates composing the cross-section of a thin-walled steel/aluminum beam is likely to determine the attainment of local buckling conditions for relatively low values of the applied load. Due to the extraordinarily large employment of structural thin-walled steel elements subject to axial load and bi-axial bending, thus likely to undergo local buckling, it is easy to conclude that a thorough study on this issue is particularly relevant in order to ensure the fulfillment of safety and reliability requirements. Among several strategies proposed in the past for nonlinear analysis of thin walled beams [1–3] we mention the Generalized Beam Theory (GBT) since it has been recently object of a renewed interest [4–6]. Complementary researches regard the analysis of ⇑ Corresponding author. E-mail address: [email protected] (F. Marmo). http://dx.doi.org/10.1016/j.engstruct.2015.05.037 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.
the sectional behavior although they are usually focused on applications to specific cross-section shapes. For example, the analysis of H-sections behavior underlines the relevance of interactive effects, especially for complex load patterns [7,8]. Interaction effects on constituent plates have also been considered by Zhou et al. [9] in order to determine enhanced class 3 slenderness limits for square and hollow sections in compression. Gardner and Theofanous [10] have shown the advantages associated with the application of a new approach, called Continuous Strength Method, based upon the adoption of an experimentally determined curve, relating the strain at which local buckling occurs to the slenderness of the cross section. In spite of their theoretical reliability, these approaches may prove to be hardly applicable to practical design necessities, especially when a large number of different elements has to be taken into account. Furthermore, as previously discussed, some of the existing methodologies, though accurate and sophisticated, refer to cross-sections of specific shapes, whereas it would be clearly preferable to set up a unique strategy able to encompass an arbitrary cross-section geometry. A practical answer to the aforementioned necessities is provided by design regulations. Most of them adopt a cross-section classification which is fundamentally based upon the capacity of the cross section to fully develop a plastic hinge before local buckling occurs. In practice, the section classes are evaluated by comparing the length-to-thickness ratios of the single plate composing the cross-section with suitable functions, which
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depend on the material properties, on the constraint conditions of the plates and on the normal stress distribution acting on the cross-section [11]. Such approaches are followed, with limited differences, both by American regulations, e.g. ANSI/AISC 360-10 [12], and by Eurocodes [13,14]. Though substantially based upon the same approach, Eurocodes seem to provide a more detailed and versatile description of the phenomenon under examination. In particular, unlike ANSI/AISC 360-10, the classification procedure suggested by Eurocodes does not refer to specific cross-section shapes [15,16]. Actually, the section is conceived as an arbitrary collection of rectangular plates so that, on the basis of their mutual constraint conditions, the procedure adopted by Eurocodes can be applied to any cross section and will be addressed in below. Actually, the intent of our work is not to investigate on the reliability of the procedure provided by building regulations, but rather to implement an automatic procedure for thin-walled cross-section classification useful for design purposes. A similar study has been conducted by Rugarli [17]; though limited to the classification of I- or H-shaped cross-sections, it is computationally very efficient in case a high number of strength checks has to be conducted for each section. On the contrary, the procedure described in the present paper has been formulated with the specific intent of being applicable to cross-sections of arbitrary shape and capable of detecting the collapse load for any cross-section class. Specifically, the nonlinear analysis preliminary to class 1 or 2 grading is carried out in the present paper on the basis of two separate algorithms depending on material constitution. Actually, a rigid-perfectly plastic behavior is prescribed by Eurocodes for ordinary steel so that the Nelder–Mead simplex method [18] has been adopted. Conversely, for stainless steel and aluminum, nonlinear constitutive laws are suggested in the literature; for this reason the secant method [19,20] enhanced with the fiber-free approach [21,22] has been adopted. The same method has also been used for the elastic analysis required for class 3 and 4 sections. The domain integrals required by the secant approach are computed analytically, thanks to the fiber-free approach, by considering a boundary representation of the section. On the contrary Eurocode addresses sections by a plate and node model, i.e. as a discrete collection of nodes connected by plates which represent webs and flanges of the section. For this reason, in order to obtain a fully automated cross-section classification procedure, an algorithm which allows one to obtain the polygonal description of the cross-section starting from the plate and node model has been developed. The paper is organized as follows: in Section 2 we formulate the equilibrium problem to be solved and motivate the adoption of two different procedures to solve the sectional equilibrium equation. In Section 3 a different formulation of the sectional equilibrium is described so as to properly employ the simplex method. In Section 4 we present the automatic procedure that is used to switch from the plate and node model of the section to a polygonal representation of its boundary. Finally, three numerical examples are reported in Section 5 for classifying an I-shaped, a Z-shaped and a rectangular hollow section. While for the I-shaped section a comparison can be performed with available results in the literature [17], the other examples have been considered intentionally to show the applicability of the proposed approach to more general cases.
2. The sectional analysis procedure The Eurocodes rules for cross-section classification require the evaluation of the normal stresses r attained at the end points of each plate of the section subject to its ultimate load. Since the
cross-section is subject to axial force N and bending moments M x and M y , sectional ultimate load is not unique but depends on the combination of internal forces acting on the section. In order to define the ultimate load of the section, a load path is defined as follows: the internal forces which act on the section are collected in the vector f ¼ ðN; M y ; Mx Þt . It is assumed that f can be additively decomposed as sum of two components, f d and f l , which respectively denote the internal forces associated with dead and live loads. In this way one is free to decide which part of the internal forces need to be amplified according to a load parameter k. For instance, a load path characterized by the amplification of the internal forces f l is defined as:
fðkÞ ¼ f d þ kf l
ð1Þ
The components of the internal force vector f are evaluated as a function of the normal stresses rðrÞ acting at the points r ¼ ðx; yÞt of the cross-section X by means of the integrals:
N¼
Z X
rðrÞ dA M?r ¼ ðMy ; Mx Þt ¼
Z
rðrÞ r dA
ð2Þ
X
Introducing the vector q ¼ ð1; x; yÞt to simplify the notation, equilibrium of the section is formulated as:
fðkÞ ¼
Z
rðrÞ q dA ¼ f d þ kf l
ð3Þ
X
Due to the nonlinear constitution of the material which composes the section, Eq. (3) is nonlinear and its solution for a given value of the vector f requires an iterative procedure. Some algorithms for solving Eq. (3) in the case of ultimate strength analysis of ordinary and prestressed reinforced concrete sections can be found in [19,20,23,24]. In this case one has to determine k in (3) so that assigned ultimate values of strain are attained in the section. Differently from the constitutive assumptions commonly adopted for the limit state analysis of sections, the procedure described in EC3 for cross section classification is based on a rigid-plastic material, since this is commonly used in the context of limit analysis. This assumption implies that the material is considered to be indefinitely ductile so that no ultimate strain is assigned. Clearly, the use of a tangent approach [23] to the solution of (3) is precluded since the indefinite flat branch of such constitutive law produces a singular cross-section stiffness matrix. Also, the rigid portion of such stress–strain law makes inapplicable the secant algorithms described in [19,20,24], since an infinite value of secant stiffness is associated with the points along the neutral axis. Consequently, a different iterative algorithm needs to be applied in presence of rigid-plastic constitutive assumptions. As a matter of fact the rigid-plastic constitutive law is suggested by EC3 only for steel cross sections of class 1 and 2 since steel sections are assumed to belong to classes 3 and 4 on the basis of the elastic stress limit. However, for stainless steel and aluminum, the yield strength and the elastic limit are not always clearly defined so that constitutive laws more refined than the classical elastic-perfectly plastic or the rigid-plastic ones are usually adopted for these materials. In this case the yield strength and the elastic limit are conventionally assigned by EC3 though the key role played by constitutive modeling of stainless steel and aluminum towards their classification has been recently established [10]. On account of the previous considerations two alternative algorithms are employed in the proposed automatic cross-section classification: (i) the Nelder–Mead simplex method [18] is addressed for the rigid-plastic constitution assumed by EC3 for steel cross sections; (ii) the secant approach formulated in [19,20] and enhanced with the integration formulas presented in [21,22] is considered for an arbitrary nonlinear constitutive law. Being this
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last procedure already documented in previous papers, we directly present the simplex algorithm.
rðxÞ ¼
jf d þ kf l f r ðd; /Þj jf r ðd; /Þj
ð6Þ
t
3. The simplex method If a rigid-plastic constitutive law is assigned to the cross-section the secant algorithms referred to above cannot be applied since the secant modulus Es becomes infinite on the neutral axis. For this reason an alternative approach, based on the application of the simplex method, has been exploited in order to solve the nonlinear equilibrium Eq. (3). Indicating respectively by Xþ and X the stretched and compressed subdomains of a cross-section X endowed with a rigid-plastic constitutive law, the stress resultants (2) are evaluated as:
Nr ¼ f y ðAXþ AX Þ M?r ¼ f y ðsXþ sX Þ
ð4Þ
where f y is the yielding stress, AðÞ and sðÞ are the area and the first area moment of the domain ðÞ. In order to identify the partitions Xþ and X it is necessary to determine the position of the neutral axis, given as a function of the following parameters, see e.g. Fig. 1:
where x ¼ ðd; /; kÞ is the unknown vector. Such a minimization problem can be efficiently solved, e.g., by applying the Nelder– Mead simplex method [18] which is based on the iterative update of the residuals evaluate at four points. 3.1. Summary of the Nelder–Mead simplex method At the generic iteration of Nelder–Mead algorithm, four estimates of xk ; k ¼ 1; . . . ; 4, are available and the relevant residuals rk ¼ rðxk Þ are evaluated. At the subsequent iteration a new set of four estimates (a 3D simplex) is generated by means of the following procedure, which is briefly sketched in Fig. 2: 1. Order: The four estimates xk ; k ¼ 1; . . . ; 4, are ordered in such a way that r1 6 r 2 6 r3 6 r4 so that x1 is the best estimate, while x4 is the worst one. The procedure terminates when the variance of the best three residuals is lower than a given tolerance: 3 1X 2 ðrk r Þ2 6 tol 3 k¼1
with r ¼
r1 þ r2 þ r3 3
d is the distance between the neutral axis and the origin of the reference system. A positive value indicates that the origin belongs to X . / represents the angle between x and the neutral axis, assumed positive if counter-clockwise. In this regard we specify that the neutral axis is oriented in such a way that Xþ lies on its left-side.
the centroid of the best three estimates: 2. Reflect: Denoting by x
Accordingly, the stress resultants (4) become function of d and / and the equilibrium Eq. (3) can be rewritten as:
If r1 < rr < r 3 the estimate x4 and the relevant residual r 4 are replaced by xr and r r and the procedure continues to step 1; otherwise step 3 is executed. 3. Expand or contract: Depending on the value of rr one of the following new estimates of x is performed:
f r ðd; /Þ ¼ f d þ kf l
ð5Þ
Further details on the determination of AðÞ and sðÞ in (4) as function of d and / are given in Section 3.2. Owing to the rigid-plastic constitution, any position of the neutral axis yields a stress resultant f r which lies on the surface of the plastic domain of the cross-section. This allows one to find the solution of Eq. (5) as the unconstrained minimum of the residual defined by:
y
n
¼ x
x1 þ x2 þ x3 3
a new estimate of x is obtained by reflection and the relevant residual is computed:
x4 xr ¼ 2x
8 2x4 > < 3x 0:5x4 xe ¼ 1:5x > : 0:5x þ 0:5x4
if r r < r 1
ðexpansionÞ
if r 3 < r r < r 4 if r 4 < r r
ðoutside contractionÞ ðinside contractionÞ
If the relevant residual r e ¼ rðxe Þ is lower then r r , then x4 and r4 are replaced by xe and re respectively and the procedure continues to step 1; otherwise step 4 is executed. 4. Shrink: The points x2 ; x3 and x4 and the relevant residuals r2 ; r 3 and r 4 , are replaced by the new estimates:
yj ¼ x1 þ
Ω+
r r ¼ rðxr Þ
1 xj x1 j ¼ 2; 3; 4 2
and by the corresponding residuals rðyj Þ. The procedure continues to step 1.
φ
3.2. Partition of the cross section
d O
Ω−
x
Eq. (4) is used to evaluate the stress resultants on the cross section in presence of a rigid-plastic constitutive law. To this end the section X has to be partitioned into it compressed and stretched parts, as a function of the position of the neutral axis, determined by means of its distance d from the origin of the reference frame and its inclination / with respect to the x axis. Adopting such parameters it is possible to parameterize the neutral axis as:
nðnn Þ ¼ nn Fig. 1. Stretched and compressed subdomains of a cross-section X.
cos / sin /
þd
sin / cos /
^ þ dn ^? ¼ nn n
ð7Þ
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xr
x3
new simplex
x1
Expansion
xe
Reflection
new simplex
xr
x
original simplex
x3
x1
x
original simplex
x4
x2
x4
x2 Outside contraction
xr x1
xe
new simplex
x x2
Inside contraction
x3 original simplex
xr
new simplex
x1
x xe
x4
x2
Shrink
x3
new simplex
original simplex
y3
x1
y2
x4
y4 x2
x3 original simplex
x4
Fig. 2. Graphical representation of the new set of estimates in the three-dimensional space d; /; k, obtained at the generic iteration of the simplex method.
Given the polygon X, whose vertices pv ; v ¼ 1; . . . ; n are ordered in a counter-clockwise sense, the procedure described below can be adopted to evaluate the coordinate of the vertices of Xþ and X . The resulting sets of vertices V þ and V are automatically ordered in the same fashion as the vertices of X. The procedure starts by initializing a vertex counter v 1 and the following three steps are executed for all sides of X: 1. Check if pv belongs to Xþ and/or to X :
8 þ > < > 0 ! add pv to V ? ^ d < 0 ! add pv to V pv n > : ¼ 0 ! add pv to both V þ and V
2. Check if the neutral axis intersects the vth side of X: The points of the vth side of X are defined by lv ðnc Þ ¼ pv þ nc pv þ1 pv being ? ^ ¼ 0 there is no intersection 0 6 nc 1. If pv þ1 pv n between lv and n. Otherwise the intersection is found by setting lv ðnc Þ ¼ nðnn Þ and solving for nc :
^? nc ¼ d pv n ^? pv þ1 pv n If 0 6 nc 1 the intersection lv ð nc Þ ¼ pv þ nc pv þ1 pv is interþ nal to the side lv and is added to both V and V . 3. If v < n the side counter is incremented v v þ 1 and the procedure continues to step 1, otherwise the procedure is terminated. Once the position vectors of all vertices of Xþ and X are known, it is possible to evaluate their area AðÞ and first area moment sðÞ by means of the well-known formulas: nðÞ
AðÞ ¼
1X p p? 2 v ¼1 ðÞv ðÞv þ1
1X pðÞv p?ðÞv þ1 pðÞv þ pðÞv þ1 6 v ¼1 nðÞ
sðÞ ¼
ð8Þ
t where p? ðÞv þ1 ¼ ðyðÞv þ1 ; xðÞv þ1 Þ .
domain integrals to boundary integrals. To this end it is mandatory to dispose of the boundary representation of X intended as an ordered collection of segments connecting vertices which are assumed to be ordered in a counter-clockwise sense. For this reason the automatic classification procedure described in the paper requires the adoption of two different geometrical descriptions of the cross-section. Actually the plate and node model adopted by the Eurocodes has been chosen as the standard input for our procedure since it is both simple and intuitive for assigning the geometry of slender sections. Instead the polygonal description of the section is automatically generated by means of the procedure described hereafter. 4.1. Automatic generation of the polygonal boundary of the section The plate and node model describes the cross section as a composition of one-dimensional elements, which resemble the mid-line of a generic rectangular plate (or stiffener) which composes the section; accordingly a thickness is associated with each plate. Plates are defined by two end nodes which can be categorized as internal nodes, if they are shared by two or more plates, or external nodes, if they belong to just one plate. Correspondingly, internal plates are those connecting two internal nodes, while external plates are those connecting an internal node with an external node. Plates cannot connect two external nodes. Starting from this simplified input of the cross-section geometry, a procedure has been developed so as to automatically produce a polygonal model of the section having vertices which are ordered counter-clockwise. The procedure can be summarized as follows: 1. First any of the external plates is selected, namely Rij , defined by an external node N i and an internal node N j . Now two indices, a and b, are used to point to i and j respectively, i.e. a i and b j. In this way, the coordinates of the nodes of Rij , which can be also referred to as Rab , are indicated by the vectors na and nb respectively. The coordinates of the first vertex of the polygon are set to:
4. Determination of the polygonal boundary of the section
?
p 1 ¼ na EC prescriptions are based on a very simple representation of the cross-section, conceived as a composition of rectangular plates. Conversely, evaluation of the integrals (2) and computation of AðÞ and sðÞ in (4) are based on the transformation of the original
lab tab lab 2
ð9Þ
where lab ¼ nb na ; lab ¼ jnb na j and t ab is the thickness of Rab . At this stage the polygon has only one vertex, thus the vertices counter is set to v ¼ 1.
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2. The plate Rbc is selected as the one which shares the node of position nb and forms the smallest counter-clockwise angle with Rab . Accordingly the parameter c points to the index of the node N c of Rbc with coordinates nc . Whether the two plates Rab and Rbc are parallel or not and depending on their thickness, one of the following operations is executed: ?
(a) If lab lbc – 0 the two plates Rab and Rbc are not parallel, hence the ðv þ 1Þth vertex of the polygon is determined as the intersection between two lines parallel to lab and lbc , whose position is obtained by solving for n and g the following equation: ?
pv þ lab n ¼ nb
lbc t bc þ lbc g lbc 2
ð10Þ
4.2. Determination of the effective part of class 4 sections According to EC3 [13] an effective cross section has to be addressed for the analysis of a class 4 section. To this end a plate and node model of the effective section is determined by modifying the original plate and node model used to describe the entire cross section. The relevant procedure is based on the substitution of all plates of class 4 by three new plates whose length is set according to Eurocode’s prescriptions. In particular, for a generic class 4 internal plate, Eurocode defines two effective lengths, namely be1 and be2 , which define the portion of the plate capable to withstand normal stresses, see, e.g., Fig. 4. Accordingly, denoting by Rab the class 4 plate whose effective part has to be determined, two new nodes defined by the position vectors:
?
Multiplying both terms by lbc one has:
pv
? lbc
þ lab
? lbc n
t bc ¼ nb n?c lbc 2
nv ¼ na þ ð11Þ
so that: ?
? n ¼ nb nc lbc tbc pv lbc ? ? ? lab lbc lab lbc 2 lab lbc
ð12Þ
lab be1 ; lab
nw ¼ nb
lab be2 lab
ð16Þ
are added to the plate and node model of the section and the plate Rab is replaced by the three plates Rav ; Rv w and Rwb , having thickness tav ¼ t wb ¼ tab and t v w ¼ 0. For external flanges Eurocode defines only one effective length; in such a case the second effective length is set to zero and the procedure described above can be used as well.
?
which is well defined since lab lbc – 0 by hypothesis. In conclusion the position of the intersection is:
pv þ1
¼ pv þ lab n
ð13Þ
Now the polygon has v þ 1 vertices, thus the counter v is incremented by 1, i.e. v v þ 1, and the procedure continues to step 3. ?
(b) If lab lbc ¼ 0 and tab ¼ tbc the two plates Rab and Rbc are parallel and have the same thickness, what typically happens for the flanges of a I-shaped section. In this case no new vertices are determined and the procedure continues to step 3. ?
(c) If lab lbc ¼ 0 and t ab – t bc the two plates Rab and Rbc are parallel, but have different thicknesses (such circumstance may occur, for instance, when determining the polygon associated with the effective part of a class 4 cross section). In such a case the following two new vertices are added to the polygon: ?
pv þ1 ¼ nb
lab t ab ; lab 2
?
pv þ2 ¼ nb
lbc t bc lbc 2
ð14Þ
Hence the counter v is incremented by 2, i.e. v v þ 2, and the procedure continues to step 3. 3. (a) If N c is an internal node, the indices a and b are updated to a b and b c, so that the previous nodes N b and N c are now referred to as N a and N b , respectively. The procedure continues to step 2. (b) If N c is an external node, two vertices are added to the polygon: ?
pv þ1 ¼ nc
lbc t bc ; lbc 2
?
pv þ2 ¼ nc þ
lbc t bc lbc 2
ð15Þ
Accordingly, the vertices counter is incremented by 2, i.e. v þ 2. In case pv ¼ p1 the polygon has been entirely determined and the procedure terminates here. Otherwise, the index a is updated to a c and the procedure continues to step 2.
v
In Fig. 3 we show an example of plate and node model of a generic section, the evaluation of the first vertices of the polygonal model of the section and the final result.
5. Numerical examples In order to show the performances of the proposed algorithms we report some numerical examples concerning the evaluation of the strength and class of three profiles. In particular, an I-shaped section has been analyzed and the relevant results are compared with those computed by Rugarli [17]. Furthermore, to show the capability of the presented approach to handle cross sections of arbitrary shape, a Z-shaped section and a rectangular hollow section (RHS) have also been analyzed. The section classes relevant to arbitrary combinations of axial force and bending moments are plotted by adopting the class maps employed by Rugarli [17]. They are obtained by associating a pair of parameters n and h with a generic triplet of internal forces f ¼ ðN; M y ; M x Þt , by means of the following relationships:
My W x p n ¼ arctg Mx W y p
0
1
N Wx p Wy p B C h ¼ arctg@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA A M2x W 2y p þ M 2y W 2x p
ð17Þ
where A; W x p and W y p are the area and the principal plastic moduli of the cross section. Accordingly, in order to represent the section class maps in the plane n h by taking into account the fixed and variable parts of and the section internal forces, we set fd ¼ 0 t
f l ¼ f y ðA tan h; W y p sin n; W x p cos nÞ , while the parameters h and n are assumed to span the intervals ½p=2; p=2 and ½0; 2p, respectively. However, we emphasize that in our approach there is no need to compute the full classes map of the section in order to perform a section classification and the relevant strength checks; rather it is sufficient to define only the fixed and variable parts of the internal forces corresponding to the load conditions of interest. Thus, section class maps and 3D strength domains are reported in the sequel with the sole intent of showing the compliance of the proposed algorithm with that proposed by Rugarli [17]. For completeness we also report in Table 1 a comparison between the computational performances of the secant and the simplex algorithms. Assuming the same value of the yield limit, the former has been used with a continuous constitutive function
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Plate and node model R23, t23 n1 R12, t12 n2
(1) step 1: set a=6, b=4, v=1
n3
R24, t24 n4
R45, t45
n5
R64, t64
n6
R57, t57
nb
lab
lab na tab / 2
p1
n7 (3) step 3(a): set a=4, b=5 step 2(a): set c=5, v=2+1=3
(2) step 2(a): set c=5, v=1+1=2
lbc
lab
lab na
nb
lab
lbc
nc
p2
na tbc / 2
p2
p1
tab / 2
lab
nb p3
lbc tab / 2 lbc
p1
nc (4) step 3(b): set v=3+2
p10
Polygonal model p12 p11 p13
nb
p2
p3
p1 p4
p15 p16
lbc nc
p9 p8
p14
lbc
tbc / 2
p7 p6 p2
p3
p1 p5
p4
p5
tbc / 2 tbc / 2 Fig. 3. First three steps of the evaluation of the polygonal boundary of a generic cross section.
effective part
na
nb
tab
Table 1 Performances of the secant and the simplex algorithms. Algorithm
ineffective part
be1 tav=tab na nv
tvw=0
be2 twb=tab nw
nb
Secant Simplex
Average number of iterations
Average computational
Elastic limit
Plastic limit
Time [s] (plastic limit)
Tolerance
2 –
113 217
0.52 0.12
1E4 1E4
Fig. 4. Plate and node model of the effective part of class 4 plates.
and the latter with a rigid-perfectly plastic constitutive law. It is worth emphasizing that the performance of both algorithms is influenced by the number of vertices of the section only in terms of computational time since the number of numerical operations
is proportional to the number of vertices. Noticeably, the number of iterations required by the simplex algorithm is influenced by the section shape. Actually, for concave sections with very long wings, a sharp modification of the sectional forces can be experienced during iterations since slight variations of the neutral axis position can either cut or not the wings of the section.
F. Marmo, L. Rosati / Engineering Structures 100 (2015) 57–65
5.1. I-shaped section: Comparison with results form the literature The first example refers to an IPE300 profile endowed with S 355 steel whose class map is plotted in Fig.5(a), while the plot in Fig. 5(b) shows the class map computed in [17]. Due to the
63
symmetry of the section, the values of n have been chosen in the interval ½0; p=2, while h spans the whole interval ½p=2; p=2. A 3D representation of the strength domain of the section is reported in Fig. 5(c), where different colors have been used to represent the section classes corresponding to the assumed values of
Fig. 5. IPE 300 profile endowed with S 355 steel.
Fig. 6. Stainless steel Z200 100 50 5 profile.
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F. Marmo, L. Rosati / Engineering Structures 100 (2015) 57–65
Fig. 7. Stainless steel RHS200 100 5 profile.
internal forces. Within this representation of the strength domain the discontinuous portions of the domain in which the section behavior abruptly changes form a class to a different one is apparent. 5.2. Z-shaped and rectangular hollow sections In order to show the capability of the proposed procedure to analyze sections of arbitrary shape we report the results obtained for the stainless steel Z-shaped cross section depicted in Fig. 6(a), and for a rectangular hollow section (RHS) of size 100 mm 200 mm and thickness 5 mm. The material behavior of both sections has been modeled by means of the Abdella’s inverse of the Ramberg–Osgood stress–strain law [25], by assuming E0 ¼ 180 GPa; r0:2 ¼ 600 MPa; epp ¼ 0:002 and n ¼ 16. The 3D representation of the Z-shaped section’s strength domain and the relevant class map are reported in Fig. 6(b) and (c), respectively. Similarly, the class map and the strength domain of the RHS section are reported in Fig. 7(a) and (b), respectively. Due to the symmetry of the section, only a portion of the class map has been plotted. 6. Conclusions Two algorithms have been employed for evaluating the collapse load and providing the cross-section classification of thin-walled sections of arbitrary shape according to Eurocode prescriptions. The first one is based on the secant approach previously developed by the authors and is used for addressing stainless steel and aluminum sections. The second one adopts the simplex method for evaluating the collapse load of sections endowed with a rigid-plastic model. Exploiting the fiber-free approach presented in [21,22] the integrals extended to the section domain, required in the secant algorithm, are replaced by exact algebraic expressions depending upon the coordinate of the polygonal boundary of the section. To this end a further algorithm has been developed in order to transform the plate and node model adopted by the Eurocodes to the polygonal representation of the section boundary. An I section derived from the literature, a Z section and a rectangular hollow section have been modeled in order to show the numerical effectiveness of the proposed procedure. In particular, differently from proposals by other authors, we directly provide the section classification associated with a given triplet of axial force and bending moments acting on the section. Due to the
generality of the proposed approach, the algorithm presented in the paper can be implemented in a computer program for the analysis of steel structures.
Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.engstruct.2015. 05.037. References [1] Genoese A, Genoese A, Bilotta A, Garcea G. A mixed beam model with nonuniform warpings derived from the Saint Venant rod. Comput Struct 2013;121:87–98. [2] Genoese A, Genoese A, Bilotta A, Garcea G. Buckling analysis through a generalized beam model including section distortions. Thin-Walled Struct 2014;85:125–41. [3] Genoese A, Genoese A, Bilotta A, Garcea G. A geometrically exact beam model with non-uniform warping coherently derived from the Saint Venant rod. Eng Struct 2014;68:33–46. [4] Abambres M, Camotim D, Silvestre N, Rasmussen KJR. GBT-based structural analysis of elastic–plastic thin-walled members. Comput Struct 2014;136:1–23. [5] de Miranda S, Gutiérrez A, Miletta R, Ubertini F. A generalized beam theory with shear deformation. J Constr Steel Res 2013;67:88–100. [6] De Miranda S, Madeo A, Miletta R, Ubertini F. On the relationship of the shear deformable generalized beam theory with classical and non-classical theories. Int J Solids Struct 2014;51(21–22):3698–709. [7] Chen Y, Cheng X, Nethercot DA. An overview study on cross-section classification of steel h-sections. J Constr Steel Res 2012;80:386–93. [8] Goczek J, Supel L. Resistance of steel cross-sections subjected to bending, shear and axial forces. Eng Struct 2014;70:271–7. [9] Zhou F, Chen Y, Young B. Cold-formed high strength stainless steel crosssections in compression considering interaction effects of constituent plate elements. J Constr Steel Res 2012;80:32–41. [10] Gardner L, Theofanous M. Discrete and continuous treatment of local buckling in stainless steel elements. J Constr Steel Res 2008;64:1207–16. [11] Loorits K. Classification of cross sections for steel beams in different design codes. Rakenteiden Mekaniikka 1995;28(1):19–33. [12] American Institute of Steel Construction. ANSI/AISC 360-10: specification for structural steel buildings; 2010. [13] European Committee for Standardization. Eurocode 3: design of steel structures; 2005. [14] European Committee for Standardization. Eurocode 9: design of aluminium structures; 2005. [15] Bernuzzi C, Simoncelli M. European design approaches for isolated coldformed thin-walled beam–columns with mono-symmetric cross-section. Eng Struct 2015;86:225–41. [16] Van Long H, Hung ND. Local buckling check according to Eurocode-3 for plastic hinge analysis of 3-D steel frames. Eng Struct 2008;30:3105–13. [17] Rugarli P. Classification of I- or H-shaped cross-sections under mixed internal actions. J Constr Steel Res 2009;65:1597–606.
F. Marmo, L. Rosati / Engineering Structures 100 (2015) 57–65 [18] Nelder JA, Mead R. A simplex method for function minimization. Comput J 1965;7:308–13. [19] Rosati L, Marmo F, Serpieri R. Enhanced solution strategies for the ultimate strength analysis of composite steel-concrete sections subject to axial force and biaxial bending. Comput Methods Appl Mech Eng 2008;197:1033–55. [20] Marmo F, Serpieri R, Rosati L. Ultimate strength analysis of prestressed reinforced concrete sections under axial force and biaxial bending. Comput Struct 2011;89:91–108. [21] Marmo F, Rosati L. Analytical integration of elasto-plastic uniaxial constitutive laws over arbitrary sections. Int J Numer Methods Eng 2012;91:990–1022.
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[22] Marmo F, Rosati L. The fiber-free approach in the evaluation of the tangent stiffness matrix for elastoplastic uniaxial constitutive laws. Int J Numer Methods Eng 2013;94:868–94. [23] Alfano G, Marmo F, Rosati L. An unconditionally convergent algorithm for the evaluation of the ultimate limit state of RC sections subject to axial force and biaxial bending. Int J Numer Methods Eng 2007;72:924–63. [24] De Vivo L, Rosati L. Ultimate strength analysis of reinforced concrete sections subject to axial force and biaxial bending. Comput Methods Appl Mech Eng 1998;166(3–4):261–87. [25] Abdella K. Inversion of a full-range stress–strain relation for stainless steel alloys. Int J Non-Linear Mech 2006;41:456–63.
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Automatic cross-section classification and collapse load evaluation for steel/aluminum thin-walled sections of arbitrary shape Francesco Marmo ⇑, Luciano Rosati Department of Structures in Engineering and Architecture, University of Naples Federico II, Naples, Italy
a r t i c l e
i n f o
Article history: Received 21 February 2015 Revised 11 May 2015 Accepted 27 May 2015
Keywords: Cross-section classification Local buckling Thin walled section
a b s t r a c t We present a computational procedure for evaluating the collapse load and assessing the cross-section classification of thin-walled sections of arbitrary shape on the basis of Eurocode prescriptions. The procedure is based on two algorithms which address separately the rigid-plastic model adopted by the Eurocode for ordinary steel cross-sections and arbitrary uniaxial constitutive laws typically used for stainless steel and aluminum. Both algorithms are based on a polygonal description of the cross section boundary so that integrals extended to the section domain are conveniently expressed as algebraic sums depending upon the coordinates of the section vertices. Accordingly, a further algorithm is illustrated in order to automatically convert the plate and node model adopted by Eurocode to a polygonal description of the section geometry. The numerical effectiveness of both algorithms is assessed with reference to an I-shaped, a Z-shaped and a RHS cross sections. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Both the design and the verification of a structure have to take into account not only the attainment of yield or fracture conditions, but also the occurrence of buckling phenomena, as they can cause collapse of the structure for stress levels lower than the strength of material supposed to act on the whole section. In particular, buckling analysis is undoubtedly fundamental for thin-walled beams, as the buckling load is inversely proportional to the slenderness of the beam. Despite the high values of yield and ultimate stress guaranteed by steel or aluminum and their iso-resistant behavior, compression on the plates composing the cross-section of a thin-walled steel/aluminum beam is likely to determine the attainment of local buckling conditions for relatively low values of the applied load. Due to the extraordinarily large employment of structural thin-walled steel elements subject to axial load and bi-axial bending, thus likely to undergo local buckling, it is easy to conclude that a thorough study on this issue is particularly relevant in order to ensure the fulfillment of safety and reliability requirements. Among several strategies proposed in the past for nonlinear analysis of thin walled beams [1–3] we mention the Generalized Beam Theory (GBT) since it has been recently object of a renewed interest [4–6]. Complementary researches regard the analysis of ⇑ Corresponding author. E-mail address: [email protected] (F. Marmo). http://dx.doi.org/10.1016/j.engstruct.2015.05.037 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.
the sectional behavior although they are usually focused on applications to specific cross-section shapes. For example, the analysis of H-sections behavior underlines the relevance of interactive effects, especially for complex load patterns [7,8]. Interaction effects on constituent plates have also been considered by Zhou et al. [9] in order to determine enhanced class 3 slenderness limits for square and hollow sections in compression. Gardner and Theofanous [10] have shown the advantages associated with the application of a new approach, called Continuous Strength Method, based upon the adoption of an experimentally determined curve, relating the strain at which local buckling occurs to the slenderness of the cross section. In spite of their theoretical reliability, these approaches may prove to be hardly applicable to practical design necessities, especially when a large number of different elements has to be taken into account. Furthermore, as previously discussed, some of the existing methodologies, though accurate and sophisticated, refer to cross-sections of specific shapes, whereas it would be clearly preferable to set up a unique strategy able to encompass an arbitrary cross-section geometry. A practical answer to the aforementioned necessities is provided by design regulations. Most of them adopt a cross-section classification which is fundamentally based upon the capacity of the cross section to fully develop a plastic hinge before local buckling occurs. In practice, the section classes are evaluated by comparing the length-to-thickness ratios of the single plate composing the cross-section with suitable functions, which
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depend on the material properties, on the constraint conditions of the plates and on the normal stress distribution acting on the cross-section [11]. Such approaches are followed, with limited differences, both by American regulations, e.g. ANSI/AISC 360-10 [12], and by Eurocodes [13,14]. Though substantially based upon the same approach, Eurocodes seem to provide a more detailed and versatile description of the phenomenon under examination. In particular, unlike ANSI/AISC 360-10, the classification procedure suggested by Eurocodes does not refer to specific cross-section shapes [15,16]. Actually, the section is conceived as an arbitrary collection of rectangular plates so that, on the basis of their mutual constraint conditions, the procedure adopted by Eurocodes can be applied to any cross section and will be addressed in below. Actually, the intent of our work is not to investigate on the reliability of the procedure provided by building regulations, but rather to implement an automatic procedure for thin-walled cross-section classification useful for design purposes. A similar study has been conducted by Rugarli [17]; though limited to the classification of I- or H-shaped cross-sections, it is computationally very efficient in case a high number of strength checks has to be conducted for each section. On the contrary, the procedure described in the present paper has been formulated with the specific intent of being applicable to cross-sections of arbitrary shape and capable of detecting the collapse load for any cross-section class. Specifically, the nonlinear analysis preliminary to class 1 or 2 grading is carried out in the present paper on the basis of two separate algorithms depending on material constitution. Actually, a rigid-perfectly plastic behavior is prescribed by Eurocodes for ordinary steel so that the Nelder–Mead simplex method [18] has been adopted. Conversely, for stainless steel and aluminum, nonlinear constitutive laws are suggested in the literature; for this reason the secant method [19,20] enhanced with the fiber-free approach [21,22] has been adopted. The same method has also been used for the elastic analysis required for class 3 and 4 sections. The domain integrals required by the secant approach are computed analytically, thanks to the fiber-free approach, by considering a boundary representation of the section. On the contrary Eurocode addresses sections by a plate and node model, i.e. as a discrete collection of nodes connected by plates which represent webs and flanges of the section. For this reason, in order to obtain a fully automated cross-section classification procedure, an algorithm which allows one to obtain the polygonal description of the cross-section starting from the plate and node model has been developed. The paper is organized as follows: in Section 2 we formulate the equilibrium problem to be solved and motivate the adoption of two different procedures to solve the sectional equilibrium equation. In Section 3 a different formulation of the sectional equilibrium is described so as to properly employ the simplex method. In Section 4 we present the automatic procedure that is used to switch from the plate and node model of the section to a polygonal representation of its boundary. Finally, three numerical examples are reported in Section 5 for classifying an I-shaped, a Z-shaped and a rectangular hollow section. While for the I-shaped section a comparison can be performed with available results in the literature [17], the other examples have been considered intentionally to show the applicability of the proposed approach to more general cases.
2. The sectional analysis procedure The Eurocodes rules for cross-section classification require the evaluation of the normal stresses r attained at the end points of each plate of the section subject to its ultimate load. Since the
cross-section is subject to axial force N and bending moments M x and M y , sectional ultimate load is not unique but depends on the combination of internal forces acting on the section. In order to define the ultimate load of the section, a load path is defined as follows: the internal forces which act on the section are collected in the vector f ¼ ðN; M y ; Mx Þt . It is assumed that f can be additively decomposed as sum of two components, f d and f l , which respectively denote the internal forces associated with dead and live loads. In this way one is free to decide which part of the internal forces need to be amplified according to a load parameter k. For instance, a load path characterized by the amplification of the internal forces f l is defined as:
fðkÞ ¼ f d þ kf l
ð1Þ
The components of the internal force vector f are evaluated as a function of the normal stresses rðrÞ acting at the points r ¼ ðx; yÞt of the cross-section X by means of the integrals:
N¼
Z X
rðrÞ dA M?r ¼ ðMy ; Mx Þt ¼
Z
rðrÞ r dA
ð2Þ
X
Introducing the vector q ¼ ð1; x; yÞt to simplify the notation, equilibrium of the section is formulated as:
fðkÞ ¼
Z
rðrÞ q dA ¼ f d þ kf l
ð3Þ
X
Due to the nonlinear constitution of the material which composes the section, Eq. (3) is nonlinear and its solution for a given value of the vector f requires an iterative procedure. Some algorithms for solving Eq. (3) in the case of ultimate strength analysis of ordinary and prestressed reinforced concrete sections can be found in [19,20,23,24]. In this case one has to determine k in (3) so that assigned ultimate values of strain are attained in the section. Differently from the constitutive assumptions commonly adopted for the limit state analysis of sections, the procedure described in EC3 for cross section classification is based on a rigid-plastic material, since this is commonly used in the context of limit analysis. This assumption implies that the material is considered to be indefinitely ductile so that no ultimate strain is assigned. Clearly, the use of a tangent approach [23] to the solution of (3) is precluded since the indefinite flat branch of such constitutive law produces a singular cross-section stiffness matrix. Also, the rigid portion of such stress–strain law makes inapplicable the secant algorithms described in [19,20,24], since an infinite value of secant stiffness is associated with the points along the neutral axis. Consequently, a different iterative algorithm needs to be applied in presence of rigid-plastic constitutive assumptions. As a matter of fact the rigid-plastic constitutive law is suggested by EC3 only for steel cross sections of class 1 and 2 since steel sections are assumed to belong to classes 3 and 4 on the basis of the elastic stress limit. However, for stainless steel and aluminum, the yield strength and the elastic limit are not always clearly defined so that constitutive laws more refined than the classical elastic-perfectly plastic or the rigid-plastic ones are usually adopted for these materials. In this case the yield strength and the elastic limit are conventionally assigned by EC3 though the key role played by constitutive modeling of stainless steel and aluminum towards their classification has been recently established [10]. On account of the previous considerations two alternative algorithms are employed in the proposed automatic cross-section classification: (i) the Nelder–Mead simplex method [18] is addressed for the rigid-plastic constitution assumed by EC3 for steel cross sections; (ii) the secant approach formulated in [19,20] and enhanced with the integration formulas presented in [21,22] is considered for an arbitrary nonlinear constitutive law. Being this
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last procedure already documented in previous papers, we directly present the simplex algorithm.
rðxÞ ¼
jf d þ kf l f r ðd; /Þj jf r ðd; /Þj
ð6Þ
t
3. The simplex method If a rigid-plastic constitutive law is assigned to the cross-section the secant algorithms referred to above cannot be applied since the secant modulus Es becomes infinite on the neutral axis. For this reason an alternative approach, based on the application of the simplex method, has been exploited in order to solve the nonlinear equilibrium Eq. (3). Indicating respectively by Xþ and X the stretched and compressed subdomains of a cross-section X endowed with a rigid-plastic constitutive law, the stress resultants (2) are evaluated as:
Nr ¼ f y ðAXþ AX Þ M?r ¼ f y ðsXþ sX Þ
ð4Þ
where f y is the yielding stress, AðÞ and sðÞ are the area and the first area moment of the domain ðÞ. In order to identify the partitions Xþ and X it is necessary to determine the position of the neutral axis, given as a function of the following parameters, see e.g. Fig. 1:
where x ¼ ðd; /; kÞ is the unknown vector. Such a minimization problem can be efficiently solved, e.g., by applying the Nelder– Mead simplex method [18] which is based on the iterative update of the residuals evaluate at four points. 3.1. Summary of the Nelder–Mead simplex method At the generic iteration of Nelder–Mead algorithm, four estimates of xk ; k ¼ 1; . . . ; 4, are available and the relevant residuals rk ¼ rðxk Þ are evaluated. At the subsequent iteration a new set of four estimates (a 3D simplex) is generated by means of the following procedure, which is briefly sketched in Fig. 2: 1. Order: The four estimates xk ; k ¼ 1; . . . ; 4, are ordered in such a way that r1 6 r 2 6 r3 6 r4 so that x1 is the best estimate, while x4 is the worst one. The procedure terminates when the variance of the best three residuals is lower than a given tolerance: 3 1X 2 ðrk r Þ2 6 tol 3 k¼1
with r ¼
r1 þ r2 þ r3 3
d is the distance between the neutral axis and the origin of the reference system. A positive value indicates that the origin belongs to X . / represents the angle between x and the neutral axis, assumed positive if counter-clockwise. In this regard we specify that the neutral axis is oriented in such a way that Xþ lies on its left-side.
the centroid of the best three estimates: 2. Reflect: Denoting by x
Accordingly, the stress resultants (4) become function of d and / and the equilibrium Eq. (3) can be rewritten as:
If r1 < rr < r 3 the estimate x4 and the relevant residual r 4 are replaced by xr and r r and the procedure continues to step 1; otherwise step 3 is executed. 3. Expand or contract: Depending on the value of rr one of the following new estimates of x is performed:
f r ðd; /Þ ¼ f d þ kf l
ð5Þ
Further details on the determination of AðÞ and sðÞ in (4) as function of d and / are given in Section 3.2. Owing to the rigid-plastic constitution, any position of the neutral axis yields a stress resultant f r which lies on the surface of the plastic domain of the cross-section. This allows one to find the solution of Eq. (5) as the unconstrained minimum of the residual defined by:
y
n
¼ x
x1 þ x2 þ x3 3
a new estimate of x is obtained by reflection and the relevant residual is computed:
x4 xr ¼ 2x
8 2x4 > < 3x 0:5x4 xe ¼ 1:5x > : 0:5x þ 0:5x4
if r r < r 1
ðexpansionÞ
if r 3 < r r < r 4 if r 4 < r r
ðoutside contractionÞ ðinside contractionÞ
If the relevant residual r e ¼ rðxe Þ is lower then r r , then x4 and r4 are replaced by xe and re respectively and the procedure continues to step 1; otherwise step 4 is executed. 4. Shrink: The points x2 ; x3 and x4 and the relevant residuals r2 ; r 3 and r 4 , are replaced by the new estimates:
yj ¼ x1 þ
Ω+
r r ¼ rðxr Þ
1 xj x1 j ¼ 2; 3; 4 2
and by the corresponding residuals rðyj Þ. The procedure continues to step 1.
φ
3.2. Partition of the cross section
d O
Ω−
x
Eq. (4) is used to evaluate the stress resultants on the cross section in presence of a rigid-plastic constitutive law. To this end the section X has to be partitioned into it compressed and stretched parts, as a function of the position of the neutral axis, determined by means of its distance d from the origin of the reference frame and its inclination / with respect to the x axis. Adopting such parameters it is possible to parameterize the neutral axis as:
nðnn Þ ¼ nn Fig. 1. Stretched and compressed subdomains of a cross-section X.
cos / sin /
þd
sin / cos /
^ þ dn ^? ¼ nn n
ð7Þ
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xr
x3
new simplex
x1
Expansion
xe
Reflection
new simplex
xr
x
original simplex
x3
x1
x
original simplex
x4
x2
x4
x2 Outside contraction
xr x1
xe
new simplex
x x2
Inside contraction
x3 original simplex
xr
new simplex
x1
x xe
x4
x2
Shrink
x3
new simplex
original simplex
y3
x1
y2
x4
y4 x2
x3 original simplex
x4
Fig. 2. Graphical representation of the new set of estimates in the three-dimensional space d; /; k, obtained at the generic iteration of the simplex method.
Given the polygon X, whose vertices pv ; v ¼ 1; . . . ; n are ordered in a counter-clockwise sense, the procedure described below can be adopted to evaluate the coordinate of the vertices of Xþ and X . The resulting sets of vertices V þ and V are automatically ordered in the same fashion as the vertices of X. The procedure starts by initializing a vertex counter v 1 and the following three steps are executed for all sides of X: 1. Check if pv belongs to Xþ and/or to X :
8 þ > < > 0 ! add pv to V ? ^ d < 0 ! add pv to V pv n > : ¼ 0 ! add pv to both V þ and V
2. Check if the neutral axis intersects the vth side of X: The points of the vth side of X are defined by lv ðnc Þ ¼ pv þ nc pv þ1 pv being ? ^ ¼ 0 there is no intersection 0 6 nc 1. If pv þ1 pv n between lv and n. Otherwise the intersection is found by setting lv ðnc Þ ¼ nðnn Þ and solving for nc :
^? nc ¼ d pv n ^? pv þ1 pv n If 0 6 nc 1 the intersection lv ð nc Þ ¼ pv þ nc pv þ1 pv is interþ nal to the side lv and is added to both V and V . 3. If v < n the side counter is incremented v v þ 1 and the procedure continues to step 1, otherwise the procedure is terminated. Once the position vectors of all vertices of Xþ and X are known, it is possible to evaluate their area AðÞ and first area moment sðÞ by means of the well-known formulas: nðÞ
AðÞ ¼
1X p p? 2 v ¼1 ðÞv ðÞv þ1
1X pðÞv p?ðÞv þ1 pðÞv þ pðÞv þ1 6 v ¼1 nðÞ
sðÞ ¼
ð8Þ
t where p? ðÞv þ1 ¼ ðyðÞv þ1 ; xðÞv þ1 Þ .
domain integrals to boundary integrals. To this end it is mandatory to dispose of the boundary representation of X intended as an ordered collection of segments connecting vertices which are assumed to be ordered in a counter-clockwise sense. For this reason the automatic classification procedure described in the paper requires the adoption of two different geometrical descriptions of the cross-section. Actually the plate and node model adopted by the Eurocodes has been chosen as the standard input for our procedure since it is both simple and intuitive for assigning the geometry of slender sections. Instead the polygonal description of the section is automatically generated by means of the procedure described hereafter. 4.1. Automatic generation of the polygonal boundary of the section The plate and node model describes the cross section as a composition of one-dimensional elements, which resemble the mid-line of a generic rectangular plate (or stiffener) which composes the section; accordingly a thickness is associated with each plate. Plates are defined by two end nodes which can be categorized as internal nodes, if they are shared by two or more plates, or external nodes, if they belong to just one plate. Correspondingly, internal plates are those connecting two internal nodes, while external plates are those connecting an internal node with an external node. Plates cannot connect two external nodes. Starting from this simplified input of the cross-section geometry, a procedure has been developed so as to automatically produce a polygonal model of the section having vertices which are ordered counter-clockwise. The procedure can be summarized as follows: 1. First any of the external plates is selected, namely Rij , defined by an external node N i and an internal node N j . Now two indices, a and b, are used to point to i and j respectively, i.e. a i and b j. In this way, the coordinates of the nodes of Rij , which can be also referred to as Rab , are indicated by the vectors na and nb respectively. The coordinates of the first vertex of the polygon are set to:
4. Determination of the polygonal boundary of the section
?
p 1 ¼ na EC prescriptions are based on a very simple representation of the cross-section, conceived as a composition of rectangular plates. Conversely, evaluation of the integrals (2) and computation of AðÞ and sðÞ in (4) are based on the transformation of the original
lab tab lab 2
ð9Þ
where lab ¼ nb na ; lab ¼ jnb na j and t ab is the thickness of Rab . At this stage the polygon has only one vertex, thus the vertices counter is set to v ¼ 1.
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2. The plate Rbc is selected as the one which shares the node of position nb and forms the smallest counter-clockwise angle with Rab . Accordingly the parameter c points to the index of the node N c of Rbc with coordinates nc . Whether the two plates Rab and Rbc are parallel or not and depending on their thickness, one of the following operations is executed: ?
(a) If lab lbc – 0 the two plates Rab and Rbc are not parallel, hence the ðv þ 1Þth vertex of the polygon is determined as the intersection between two lines parallel to lab and lbc , whose position is obtained by solving for n and g the following equation: ?
pv þ lab n ¼ nb
lbc t bc þ lbc g lbc 2
ð10Þ
4.2. Determination of the effective part of class 4 sections According to EC3 [13] an effective cross section has to be addressed for the analysis of a class 4 section. To this end a plate and node model of the effective section is determined by modifying the original plate and node model used to describe the entire cross section. The relevant procedure is based on the substitution of all plates of class 4 by three new plates whose length is set according to Eurocode’s prescriptions. In particular, for a generic class 4 internal plate, Eurocode defines two effective lengths, namely be1 and be2 , which define the portion of the plate capable to withstand normal stresses, see, e.g., Fig. 4. Accordingly, denoting by Rab the class 4 plate whose effective part has to be determined, two new nodes defined by the position vectors:
?
Multiplying both terms by lbc one has:
pv
? lbc
þ lab
? lbc n
t bc ¼ nb n?c lbc 2
nv ¼ na þ ð11Þ
so that: ?
? n ¼ nb nc lbc tbc pv lbc ? ? ? lab lbc lab lbc 2 lab lbc
ð12Þ
lab be1 ; lab
nw ¼ nb
lab be2 lab
ð16Þ
are added to the plate and node model of the section and the plate Rab is replaced by the three plates Rav ; Rv w and Rwb , having thickness tav ¼ t wb ¼ tab and t v w ¼ 0. For external flanges Eurocode defines only one effective length; in such a case the second effective length is set to zero and the procedure described above can be used as well.
?
which is well defined since lab lbc – 0 by hypothesis. In conclusion the position of the intersection is:
pv þ1
¼ pv þ lab n
ð13Þ
Now the polygon has v þ 1 vertices, thus the counter v is incremented by 1, i.e. v v þ 1, and the procedure continues to step 3. ?
(b) If lab lbc ¼ 0 and tab ¼ tbc the two plates Rab and Rbc are parallel and have the same thickness, what typically happens for the flanges of a I-shaped section. In this case no new vertices are determined and the procedure continues to step 3. ?
(c) If lab lbc ¼ 0 and t ab – t bc the two plates Rab and Rbc are parallel, but have different thicknesses (such circumstance may occur, for instance, when determining the polygon associated with the effective part of a class 4 cross section). In such a case the following two new vertices are added to the polygon: ?
pv þ1 ¼ nb
lab t ab ; lab 2
?
pv þ2 ¼ nb
lbc t bc lbc 2
ð14Þ
Hence the counter v is incremented by 2, i.e. v v þ 2, and the procedure continues to step 3. 3. (a) If N c is an internal node, the indices a and b are updated to a b and b c, so that the previous nodes N b and N c are now referred to as N a and N b , respectively. The procedure continues to step 2. (b) If N c is an external node, two vertices are added to the polygon: ?
pv þ1 ¼ nc
lbc t bc ; lbc 2
?
pv þ2 ¼ nc þ
lbc t bc lbc 2
ð15Þ
Accordingly, the vertices counter is incremented by 2, i.e. v þ 2. In case pv ¼ p1 the polygon has been entirely determined and the procedure terminates here. Otherwise, the index a is updated to a c and the procedure continues to step 2.
v
In Fig. 3 we show an example of plate and node model of a generic section, the evaluation of the first vertices of the polygonal model of the section and the final result.
5. Numerical examples In order to show the performances of the proposed algorithms we report some numerical examples concerning the evaluation of the strength and class of three profiles. In particular, an I-shaped section has been analyzed and the relevant results are compared with those computed by Rugarli [17]. Furthermore, to show the capability of the presented approach to handle cross sections of arbitrary shape, a Z-shaped section and a rectangular hollow section (RHS) have also been analyzed. The section classes relevant to arbitrary combinations of axial force and bending moments are plotted by adopting the class maps employed by Rugarli [17]. They are obtained by associating a pair of parameters n and h with a generic triplet of internal forces f ¼ ðN; M y ; M x Þt , by means of the following relationships:
My W x p n ¼ arctg Mx W y p
0
1
N Wx p Wy p B C h ¼ arctg@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA A M2x W 2y p þ M 2y W 2x p
ð17Þ
where A; W x p and W y p are the area and the principal plastic moduli of the cross section. Accordingly, in order to represent the section class maps in the plane n h by taking into account the fixed and variable parts of and the section internal forces, we set fd ¼ 0 t
f l ¼ f y ðA tan h; W y p sin n; W x p cos nÞ , while the parameters h and n are assumed to span the intervals ½p=2; p=2 and ½0; 2p, respectively. However, we emphasize that in our approach there is no need to compute the full classes map of the section in order to perform a section classification and the relevant strength checks; rather it is sufficient to define only the fixed and variable parts of the internal forces corresponding to the load conditions of interest. Thus, section class maps and 3D strength domains are reported in the sequel with the sole intent of showing the compliance of the proposed algorithm with that proposed by Rugarli [17]. For completeness we also report in Table 1 a comparison between the computational performances of the secant and the simplex algorithms. Assuming the same value of the yield limit, the former has been used with a continuous constitutive function
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Plate and node model R23, t23 n1 R12, t12 n2
(1) step 1: set a=6, b=4, v=1
n3
R24, t24 n4
R45, t45
n5
R64, t64
n6
R57, t57
nb
lab
lab na tab / 2
p1
n7 (3) step 3(a): set a=4, b=5 step 2(a): set c=5, v=2+1=3
(2) step 2(a): set c=5, v=1+1=2
lbc
lab
lab na
nb
lab
lbc
nc
p2
na tbc / 2
p2
p1
tab / 2
lab
nb p3
lbc tab / 2 lbc
p1
nc (4) step 3(b): set v=3+2
p10
Polygonal model p12 p11 p13
nb
p2
p3
p1 p4
p15 p16
lbc nc
p9 p8
p14
lbc
tbc / 2
p7 p6 p2
p3
p1 p5
p4
p5
tbc / 2 tbc / 2 Fig. 3. First three steps of the evaluation of the polygonal boundary of a generic cross section.
effective part
na
nb
tab
Table 1 Performances of the secant and the simplex algorithms. Algorithm
ineffective part
be1 tav=tab na nv
tvw=0
be2 twb=tab nw
nb
Secant Simplex
Average number of iterations
Average computational
Elastic limit
Plastic limit
Time [s] (plastic limit)
Tolerance
2 –
113 217
0.52 0.12
1E4 1E4
Fig. 4. Plate and node model of the effective part of class 4 plates.
and the latter with a rigid-perfectly plastic constitutive law. It is worth emphasizing that the performance of both algorithms is influenced by the number of vertices of the section only in terms of computational time since the number of numerical operations
is proportional to the number of vertices. Noticeably, the number of iterations required by the simplex algorithm is influenced by the section shape. Actually, for concave sections with very long wings, a sharp modification of the sectional forces can be experienced during iterations since slight variations of the neutral axis position can either cut or not the wings of the section.
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5.1. I-shaped section: Comparison with results form the literature The first example refers to an IPE300 profile endowed with S 355 steel whose class map is plotted in Fig.5(a), while the plot in Fig. 5(b) shows the class map computed in [17]. Due to the
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symmetry of the section, the values of n have been chosen in the interval ½0; p=2, while h spans the whole interval ½p=2; p=2. A 3D representation of the strength domain of the section is reported in Fig. 5(c), where different colors have been used to represent the section classes corresponding to the assumed values of
Fig. 5. IPE 300 profile endowed with S 355 steel.
Fig. 6. Stainless steel Z200 100 50 5 profile.
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Fig. 7. Stainless steel RHS200 100 5 profile.
internal forces. Within this representation of the strength domain the discontinuous portions of the domain in which the section behavior abruptly changes form a class to a different one is apparent. 5.2. Z-shaped and rectangular hollow sections In order to show the capability of the proposed procedure to analyze sections of arbitrary shape we report the results obtained for the stainless steel Z-shaped cross section depicted in Fig. 6(a), and for a rectangular hollow section (RHS) of size 100 mm 200 mm and thickness 5 mm. The material behavior of both sections has been modeled by means of the Abdella’s inverse of the Ramberg–Osgood stress–strain law [25], by assuming E0 ¼ 180 GPa; r0:2 ¼ 600 MPa; epp ¼ 0:002 and n ¼ 16. The 3D representation of the Z-shaped section’s strength domain and the relevant class map are reported in Fig. 6(b) and (c), respectively. Similarly, the class map and the strength domain of the RHS section are reported in Fig. 7(a) and (b), respectively. Due to the symmetry of the section, only a portion of the class map has been plotted. 6. Conclusions Two algorithms have been employed for evaluating the collapse load and providing the cross-section classification of thin-walled sections of arbitrary shape according to Eurocode prescriptions. The first one is based on the secant approach previously developed by the authors and is used for addressing stainless steel and aluminum sections. The second one adopts the simplex method for evaluating the collapse load of sections endowed with a rigid-plastic model. Exploiting the fiber-free approach presented in [21,22] the integrals extended to the section domain, required in the secant algorithm, are replaced by exact algebraic expressions depending upon the coordinate of the polygonal boundary of the section. To this end a further algorithm has been developed in order to transform the plate and node model adopted by the Eurocodes to the polygonal representation of the section boundary. An I section derived from the literature, a Z section and a rectangular hollow section have been modeled in order to show the numerical effectiveness of the proposed procedure. In particular, differently from proposals by other authors, we directly provide the section classification associated with a given triplet of axial force and bending moments acting on the section. Due to the
generality of the proposed approach, the algorithm presented in the paper can be implemented in a computer program for the analysis of steel structures.
Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.engstruct.2015. 05.037. References [1] Genoese A, Genoese A, Bilotta A, Garcea G. A mixed beam model with nonuniform warpings derived from the Saint Venant rod. Comput Struct 2013;121:87–98. [2] Genoese A, Genoese A, Bilotta A, Garcea G. Buckling analysis through a generalized beam model including section distortions. Thin-Walled Struct 2014;85:125–41. [3] Genoese A, Genoese A, Bilotta A, Garcea G. A geometrically exact beam model with non-uniform warping coherently derived from the Saint Venant rod. Eng Struct 2014;68:33–46. [4] Abambres M, Camotim D, Silvestre N, Rasmussen KJR. GBT-based structural analysis of elastic–plastic thin-walled members. Comput Struct 2014;136:1–23. [5] de Miranda S, Gutiérrez A, Miletta R, Ubertini F. A generalized beam theory with shear deformation. J Constr Steel Res 2013;67:88–100. [6] De Miranda S, Madeo A, Miletta R, Ubertini F. On the relationship of the shear deformable generalized beam theory with classical and non-classical theories. Int J Solids Struct 2014;51(21–22):3698–709. [7] Chen Y, Cheng X, Nethercot DA. An overview study on cross-section classification of steel h-sections. J Constr Steel Res 2012;80:386–93. [8] Goczek J, Supel L. Resistance of steel cross-sections subjected to bending, shear and axial forces. Eng Struct 2014;70:271–7. [9] Zhou F, Chen Y, Young B. Cold-formed high strength stainless steel crosssections in compression considering interaction effects of constituent plate elements. J Constr Steel Res 2012;80:32–41. [10] Gardner L, Theofanous M. Discrete and continuous treatment of local buckling in stainless steel elements. J Constr Steel Res 2008;64:1207–16. [11] Loorits K. Classification of cross sections for steel beams in different design codes. Rakenteiden Mekaniikka 1995;28(1):19–33. [12] American Institute of Steel Construction. ANSI/AISC 360-10: specification for structural steel buildings; 2010. [13] European Committee for Standardization. Eurocode 3: design of steel structures; 2005. [14] European Committee for Standardization. Eurocode 9: design of aluminium structures; 2005. [15] Bernuzzi C, Simoncelli M. European design approaches for isolated coldformed thin-walled beam–columns with mono-symmetric cross-section. Eng Struct 2015;86:225–41. [16] Van Long H, Hung ND. Local buckling check according to Eurocode-3 for plastic hinge analysis of 3-D steel frames. Eng Struct 2008;30:3105–13. [17] Rugarli P. Classification of I- or H-shaped cross-sections under mixed internal actions. J Constr Steel Res 2009;65:1597–606.
F. Marmo, L. Rosati / Engineering Structures 100 (2015) 57–65 [18] Nelder JA, Mead R. A simplex method for function minimization. Comput J 1965;7:308–13. [19] Rosati L, Marmo F, Serpieri R. Enhanced solution strategies for the ultimate strength analysis of composite steel-concrete sections subject to axial force and biaxial bending. Comput Methods Appl Mech Eng 2008;197:1033–55. [20] Marmo F, Serpieri R, Rosati L. Ultimate strength analysis of prestressed reinforced concrete sections under axial force and biaxial bending. Comput Struct 2011;89:91–108. [21] Marmo F, Rosati L. Analytical integration of elasto-plastic uniaxial constitutive laws over arbitrary sections. Int J Numer Methods Eng 2012;91:990–1022.
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[22] Marmo F, Rosati L. The fiber-free approach in the evaluation of the tangent stiffness matrix for elastoplastic uniaxial constitutive laws. Int J Numer Methods Eng 2013;94:868–94. [23] Alfano G, Marmo F, Rosati L. An unconditionally convergent algorithm for the evaluation of the ultimate limit state of RC sections subject to axial force and biaxial bending. Int J Numer Methods Eng 2007;72:924–63. [24] De Vivo L, Rosati L. Ultimate strength analysis of reinforced concrete sections subject to axial force and biaxial bending. Comput Methods Appl Mech Eng 1998;166(3–4):261–87. [25] Abdella K. Inversion of a full-range stress–strain relation for stainless steel alloys. Int J Non-Linear Mech 2006;41:456–63.