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. 1.060
U _12(l-AiO/»,
.
- ^12(l-;i0^/ >./
(4)
With the transition conditions defined by Eqn. 5 and 6, the critical stress Cc^a) of an I-section of length L with the same thickness t for flanges and web can be determined iteratively. (5)
(6) The critical buckling stress Ccr is a function of the length-to-width ratio of the plate which buckles. For example, the four-sided simply supported plate reaches the first minimum Ccr(i) at Li/h = 1 with a kw-value of 4.0, while the three-sided simply supported plate has its minimum at Ll^f=oo with kf = 0.425. The decisive length Li corresponds to the half-wave length of the buckling pattern. Thus, a four-sided simply supported plate with a length L twice the decisive length Li buckles in two halfwaves. The same applies to whole I-sections, where the critical buckling stress Ga^u) of the whole member reaches its first minimum at a half-wave length Li. As shown in Figure 2 for tests listed in Table 1 and those reported in Lindner and Rusch (1998), a column buckles in n half-waves Li, if the column is approximately n half-waves long. The half-wave length Li is independent of the plate thickness t. This is also observed in the tests. In analogy to single plates, Eqn. 7 is valid for the critical stress Oci^ii) of whole I-sections. ^cr(ii)i^l)
=
-r^cr{ii){0
(7)
With a known half-wave length Li and critical stress Ccr(u), r can be recalculated by Eqn. 3 (Figure 3a). The restraint degree r has a value of zero at a ratio bf^h of flange width to web height, where the single, simply supported plates (web and flanges) have the same critical buckling stresses_aci^^i^t 1 1 • « 0.25!V).50 0.75 1.00 1.25 1.50 . \ ^ r^ (web) s^
bf/h 1
1.75 1
1
>
web
/
^^ flanges
* will be restrained
M
^
-1 - n u
r"'' h-2 *- -3
Figure 3: a) half-wave-length Li and b) elastic restraint r depending on the bf/h ratio Therefore, replacement of h by (4 bf + h) in Eqn. 2 and some simple transformations lead to Eqn. 10, which defines a k-value k(ii) for the whole I-section with a constant plate thickness t. Figure 4 shows the dependence of the k-value k(ii) on the hflh ratio. The critical buckling stress aci). Determination de la distribution des contraintes r^siduelles de soudage dans les plaques soudees d'epaisseur quelconque, dans les assemblages de plaques et apres soudages multipasses. Revue de la soudure 37:2, 91-112. Rangelov N. (1992). A theoretical approach to the limiting of initial imperfections in steel plates. Stahlbau 61:5, \5\-\56. Rangelov N. and Braham M. (1995). Are the web imperfections in built-up I-beams of any importance? International Colloquium ''Stability of Steel Structures'", European Session, Budapest, Hungary. Preliminary Report, vol. I, 263-270.
Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved
183
Prediction of Buckling Strength of Stiffened Plates by Use of the Optimum Eccentricity Method Gunnar Solland and Morten Rotheim Det Norske Veritas, H0vik, Norway
ABSTRACT The buckling capacity of axially loaded stiffened plates, as found in bridges and marine structures, are often analysed by so-called strut models, ref e.g. Galambos (1987). The strut model is traditionally taken as a single span column with hinged end connections. However, the stiffeners in a real structure are continuos over several spans or to some degree rotational fixed at the ends. Therefore, a new strut model is developed. This new concept is called the optimum eccentricity method. This paper presents the background for the method of analysis and the capacity formulations. Furthermore, the present method is compared with numerical calculations made with non-linear finite element methods as well as full-scale tests. The new formulation shows improved predictability in most cases compared to current codes. The method is valid for stiffened plates loaded with axial compression stress both longitudinal as well as transverse to the stiffener combined with in plane shear and lateral load on the plate. The validation of the method presented in this paper is limited to longitudinal compression stress combined with lateral pressure acting both on the stiffener as well as on the plate side. The new formulation for calculation of buckling strength of stiffened panels is implemented in the Norsok standard N-004 for Design of Steel Structures, which is used for design of offshore structures in Norway. Whilst the capacity checks are developed for stiffened plates the same method may be utilised for check of beam-columns loaded by axial force and lateral load with buckling about the same axis as the moment caused by the lateral load.
KEYWORDS Stiffened plates, buckling, combination of axial load and lateral load, continuous over several spans, strut model, marine structures, design codes.
184 PROBLEM FORMULATION Normally buckling of stiffened plates is checked by a stmt analogy as shown in Figure 1. In this paper buckling due to axial and lateral load will be treated. Presence of normal stress transverse to the direction of the stiffener and shear stress may be included in the formulation (see Carlsen (1977) or Norsok (1998)), but these effects are not deaJt with in this paper.
Nx.Sd=Nx.Sd (O'x.Sd. 7 s d )
.qsd^q(Psd. Po)
Nx.Sd
Psd/
STIFFENED PLATE
BEAM COLUMN
Figure 1 The strut model In a laboratory test of an axial loaded single span stiffened plate panel, the working point of the axial load will be fixed by the test set up. The position of the working point for the axial load in a stiffened panel being part of a real structure will, however, in the general case be indeterminate. When analysing the ultimate load carrying resistance of plated structures, the best estimate of the global strength is found when the position of the working point for the strut is selected so that the maximum resistance of the stiffened panel is obtained. By transferring the stiffened plate-buckling problem to an equivalent strut, it is possible to use ordinary beam column formulas for checking of the capacity. However, traditional design codes do not give explicit guidance on how to calculate the resistance of a beam-colunui with axial and lateral load and having only one axis of symmetry. In general the idealised strut or beam-column to be analysed is therefore a continuous beam-column with different section modulus for the plate and stiffener side of the strut.
DESCRIPTION OF THE METHOD The design formulas for continuous beam column with moment about one axis and buckling about the same axis are developed on the basis of traditional beam-column interaction formulas using linear elastic cross-section properties as well as elastic moment distributions. The cross-section may be nonsymmetric or single symmetric as long as the buckling is about the same axis as the moment caused by the lateral load. Furthermore it is assumed that the beam-column interaction formulas need to be
185 satisfied both at the supports as well as at mid-span and at both the plate and the extreme fibre of the stiffener flange. From this assumption it follows that the interaction of axial force and moment need to be made at four positions in the equivalent beam-column, see Figure 2. Consequently there will be four design equations to be checked.
• HniHHJHI
%d
Figure 2 Check points for interaction equations The check at the four design points is dependent upon the location of the working point of the axial force. The distance from the centroid of the effective cross section to the assumed working point for the axial force is denoted z*, see Figure 3. As described the value for z may be optimised for continuous stiffeners such that the maximum capacity is obtained. This means that the calculations need to be done repeatedly for all four design equations with varying values for z* and selecting the z* value giving the maximum capacity. If only a conservative assessment for the capacity is needed a z value equal to zero may be assumed. For calculation of the forces and moments in the total structure, of which the stiffened panel is a part, the working point for the stiffened panel should correspond to the assumed value of z*. In most cases, the influence of variations in z* on global forces and moments will be negligible.
Nsd
HiiHHiitH
%d J^sd
" *
N|
Neutral axis Figure 3 Definition of z
186 DESIGN EQUATIONS The following design equations are developed from traditional formulas for buckling of beamcolumns. Separate formulas are developed dependent upon whether the lateral load acts on the stiffener or the plate side. Lateral pressure on plate side: Stress point 1 Interaction equation number. See Figure 2 Nsd
Equation 1 number
, M^-Ns,-z
^^ (1)
1
Nsd 2
Nkp,Rd
Ns, ^ M ^ - N s , - z * NRJ
(I
^^ (2)
^sd 1
(3)
3
Nsd
1 M,+Ns,-z'
^^ (4)
4
Lateral pressure on stiffener side: Nsd
Ns, ^ M , + N s , - z
N
N
^^ ks,Rd
^^ Rd
Ms,Rd
N
Nsd , M , + N s , - z N kp.Rd M p.Rd l_Nsd
^^ '
^^ (6)
N. N.
M , - N o . -z / M.
N, N
M , - N ^Sd H-Z* N. -, kh,p, np), which are based on experimental evidence. Then, it can be assumed that the generic point along the transition curve describing the pinching is given by a curve expression, which is determined by setting intermediate parameters between the ones related to upper and lower bound curves. In particular, three transition parameters (^i, ti, X) are used, ruling the real deformation path between lower and upper boimd curves.
218
The can^mrisim wHk experimeniai i^suiti Ih^ ^h&m luiikiyticid model has been calibmled on tlie basis of previously pitesemed e}!Cperimenlal resiulfs. bi patticulur,, m order to assess the niodel reliability to well inteipiiet the oonncclion i^ponse in caise of dififeireat defbrmaiioEi histories, comparisons have been carried out for both conslaiit amplitude deJOormation and ECCS histories, mtti feta:ence to speeimen typology with intesmediate eolunm Mze (BCC7). In ofder to geit the best approximation^ model behavioural parameters faa\'e beeo set up differently on the ba^s of each test Tikis is due to the feet that experimeniai test$ show slight difO^tenoes between e^h othef, not allowing for a unique set of parnmeters. Hie avail^le plaslie defbrmation on the virgin curve v^ has been assumed equal to the ultimate detbrmation obtained ftom mcmolonic cxpciimental curve. Parameter P^, ruling the deterioration iate» has been slightly varied, being incr^ing with defbrmaiion amplitude. MOtMn)
M(P(Nn4
BCC7-D
»CC7^
m
•50 •IQO
^m^^
i^tf^m) -150 '] CTn
G = ao+a[l-exp(-ys^)]
(1)
where: GQ is the stress at proportionality limit in an uniaxial test, which can be conventionally assumed equal to the elastic limit strength^oiJ £p is the accumulated plastic strain; the material parameters a and P determine the magnitude of the strain hardening and the shape of the hardening curve, respectively. According to experimental tests, a suitable value for y is 10, whereas the factor a varies depending on the alloy. In particular, a values equal to 80 N/mm^ (low hardening), 140 N/mm^ (medium hardening), 200 N/mm^ (high hardening), 300 N/mm^ (very high hardening) can be practically be assumed for aluminium alloys. For evaluating the deformability of a general basic component of the joint, in general the same formulas as adopted for steel connections could be used. Particular attention must be paid to the assumption of the Young's modulus of the basic material, owing to the effect of welding, as well as to the plate-to-bolt stiffness ratio, by including the possibility to use steel bolts.
232
As far as the ultimate resistance is concerned, the conceptual approach of ECS may be still used. For each basic component, a possible collapse mechanism must be therefore established. Nevertheless, it is important to observe that, while for steel structures the collapse load corresponding to the limit analysis developed with plastic hinge method leads generally to a satisfactory degree of accuracy in the prediction of the ultimate resistance, this ultimate load for roimd-house type materials could be less significant. In order to account for the post-elastic behaviour of structures made of such materials is possible to define suitable correction factors of the conventional yield stress assessing the effects due to hardening and reduced ductility. The correction factor r\ considered in Eurocode 9 (ENV 1999, 1998) represents a first attempt to apply such a procedure, where r| was determined on the basis of an equivalence in terms of load bearing capacity corresponding to a given ductility level between the concentrated plasticity model and the spread plasticity one (Mandara & Mazzolani, 1995). A different approach, which is somewhat complementary to that proposed in EC9, is presented herein for the evaluation of the ultimate resistance of T-stub joint components, allowing the same EC3-Annex J formulas to be applied by introducing a correction factor (k), which accounts for all material peculiar mechanical features.
FRAMING OF THE STUDY Many connection typologies, such as end plate and angle cleat joints, can be represented by simple tensioned equivalent T-stub components, whose behaviour is ductile in case of steel, provided that boh resistance is sufficient to avoid their premature failure. According to EC3-Annex J (ENV 1993, 1997), three possible failure modes for steel equivalent T-stub components are recognised. The complete yielding of the flange takes place in the type-1 mechanism, with the onset of four plastic hinges at ultimate limit state, two of which are located at bolt axes. The type-2 mechanism occurs when bolt failure takes place together with yielding of the flange at the sections corresponding to flange-to-web connection. The type-3 mechanism is due to the sole bolt failure with the overall up-lift of the flange. In this case, neither prying forces nor flange yielding take place. The examination of this component in case of aluminium joint is particularly interesting due to the fact that collapse mechanisms can be deeply different from the ones related to steel, being the possibility to develop completely the collapse mechanisms strongly influenced by the mechanical features of the alloy. In particular, previous studies (De Matteis et al, 1998) pointed out that the type-2 mechanism, i.e. failure caused by attainment of plastic deformation in both flange and bolt material, can not be so clearly defined as for steel. When the T-stub at the flange-to-web connection is in plastic range and while bolts are developing their axial deformation, the collapse may be attained in either flange or bolt depending on their ultimate deformation capacity as well as on their deformation gradient. Therefore, at least three different situations can take place, depending on which element rules the collapse. The first case (type-2a mechanism) is related to a flange plastic deformation higher than that arising in the bolt; the third case (type-2c mechanism) reflects the bolt failure with limited plastic deformation occurred in the flange, whereas an ideal boundary of the behavioural response may be assumed to be correspondent to a situation where both flange and bolts govern the problems together (type-2b mechanism). 150
.30. 45
o1 o1
1 ^ 1 ^'
Figure 2 Geometry of analysed T-stub model
Several numerical analyses have been performed, emphasising all these possible collapse mechanisms. To this purpose, a detailed threedimensional finite element model developed by means of the explicit non-linear code Abaqus has been set up. The model has been calibrated on the basis of experimental results available for steel (Bursi & Jaspart, 1997), showing an
233
excellent accuracy of the FEM model in predicting the joint response up to failure. The typical double T-stub symmetrical arrangement shown in Figure 2 has been considered in the study, where M12 bolts are used to connect the 10.7mm thick flange. In order to analyse cases with different flange-to-bolt resistance ratios, several alloys, including steel for bolts, have been considered. Details on the assumed material behaviour are given in De Matteis et al (1998), where the obtained results are also outlined. The corresponding force-displacement relationships for T-stub joints under consideration have allowed the ultimate resistance of the structural component to be determined. Collapse loads may be now evaluated as a function of the conventional ultimate elongation of either T-tub flange or bolt materials. For aluminium alloys, reference to uniform strain (s„) instead of ultimate strain (s/) should be made. Uniform strain, which corresponds of the attainment of maximum stress in engineering stress-strain experimental curves, may been approximately assumed equal to half the ultimate elongation. Since the ultimate load is determined by the true collapse of the material, i.e. when the conventional ultimate strain is attained at least in one point over the section, this assumption makes the method proposed safer for design purposes.
- i ^ - A W 6082-T6 - • — A W 6082-14 - X - A W 6061-T6
Steel 6.8
AW 6062-T6
Steel 4.6
AW 7075-T6
AW 5063
Bolt type
Figure3: Nimierical and codified (Annex J) ultimate resistance (F„//) values The numerical assessment of collapse loads for examined T-stubs are compared in Figure 3 with the design plastic resistance suggested by EC3-Annex J (1997). The latter has been therefore determined by considering an equivalent four-support beam, whose effective width (beff) can be evaluated according to code provisions, as a function of the geometry of the actual scheme. In the analysed case, for each bolt row, ^^^has been therewith assimied equal to 40 mm, which represent the geometrical width (b) as well. Codified formula has been applied to aluminium material by replacing the steel yielding stress Jj, with aluminiimi conventional limit elastic stress ^.2- It is clearly evident that values suggested by Annex J are largely conservative, leading to an underestimation of collapse load up to 50%. So high scatters are mainly due to the fact that the codified procedure does not include strain hardening of the material (this can be accepted for steel only, but not for alimiiniimi as well). In fact, errors are greater for T-stubs flange materials characterised by stronger hardening values, i.e for AW 6061-T6 and AW 6082-T4 alloys. Besides, the codified procedure is generally inadequate for predicting T-stub mechanism types (De Matteis et al, 1998), confirming the need to establish different rules in predicting the failure mode and the corresponding ultimate load for aluminium joints. THE PARAMETRICAL STUDY The examined cases In order to investigate to a greater extent the effect of hardening and ductility of aluminium alloys on the ultimate resistance of T-stub joints, several numerical analyses have been carried out. Since the
234
effect of hardening is especially exploited in case of collapse mechanism type 1, where plastic hinges forming in T-stub flanges rule the development of post-elastic behaviour, the analysis has been firstly focused on strong bolt-to-weak flanges cases. Parameters considered in the study are the ones mostly influencing T-sub collapsing behaviour, i.e. hardening of the alloy (a), uniform elongation (SM), flange-to-bolt resistance ratio and T-stub geometry. The influence of both hardening and ductility of T-stub flange material has been inspected by analysing conventional alloys. Therefore, with reference to some fixed value of elastic stress, namely y&.2=l 20 and 240 N/mm^, values of a=0, 80,140, 200 and 300 N/mm^ and values of e„ = 3, 4, 6, 8 % have been considered, thus practically covering all realistic cases. For a conventional yielding stress yb.2=240 N/mm^, stress-strain curves of flange material are depicted in Figure 4. The case a=0, corresponding to an ideal elastic-plastic material, has been also included for the sake of comparison. As far as bolts are concerning, bolt steel grade 8.8 have been considered for all flange materials, these flange-to-bolt combinations allowing for mechanism types 1 to be almost invariably involved. Three T-stub geometry has been considered. The first one is related to the geometry depicted in Figure 1. In the following it will be labelled as T-stub type 1. The second one (T-stub type 2) practically coincides with the previous one, but with a different thickness for flange, namely 14 mm instead of 10.7 mm a=0 [N/mm2) (Figure 5a). This allows the influence of a=80[N/mm2l flange thickness on the T-stub collapse a=140 [N/mmri a=200 [N/mm2]| behaviour to be investigated. Because of the a=300 (N/mm2]| close relative distance between bolt rows, ^ 0 according to Annex J both the above cases 0 2 4 6 8 10 12 correspond to T-stub collapse in the situation Figure 4 Analysed conventional alloys of combined bolt patterns, i.e. collapse mechanism which yielding lines involve both bolt rows. In order to cover the opposite situation, i.e. T-stub collapse in the situation of separate bolt pattern, the geometry depicted in Figure 5b has been considered as well. In the following it will be labelled as T-stub type 3. a [kN/mm^
The obtained results The results obtained from FEM analysis are shown in Figure 6 for all T-stub typologies, where in dashed lines resistance values corresponding to assumed ductility values are indicated. The curves well emphasise the role of strain hardening of the flange alloy, which strongly influences the ultimate resistance of the joint component. This is particularly evident for the we^er flange material (/J).2=120 N/mm^), which always involves a failure mechanism type-1, where bolt is completely excluded by the collapse of the system and the flanges develop their maximum plastic deformations. In the same figure, for each curve, the corresponding number indicates the type of collapse mechanism as it has been deduced by observing specimen yielding patterns and bolt plastic engagement. For the sake of comparison with the codified procedure, it should be observed that for all the cases EC3 annex J suggests to consider a mechanism type 1. The numerical and codified values corresponding to the ultimate resistance of T-stubs under consideration are summarised in Table 1. Scatters are remarkable in all cases, but they increase with the hardening, especially for more ductile alloys. Besides, the importance of the flange ductility should be underlined, which only in case of high hardening, i.e. for a > 140 N/mm^, produces a considerable reduction of T-stub deformation capacity for low 8„ values. In particular, in case of T-stub types 1 and 2, the ultimate resistance values proposed by the code are strongly conservative, while they are unsafe for T-stub type 3, which is characterised by bolts resisting to tensioned forces individually.
235
.39^ 4?.,
n^
Figure 4: Geometry of T-stub-type 2 and 3
CRITICAL EXAMINATION OF RESULTS The codified approach A careful examinations of results demands for other considerations. In fact, one should note that the error due to codified approach is not negligible even in case of flange without any hardening, which practically corresponds to the idealised steel material. Obviously, this is a consequence of the approximated procedure, which is based upon the equivalence between the actual three-dimensional behaviour of the examined joint component and the one related to a monodimensional scheme. The design strength, in fact, is evaluated by means of equilibrium conditions based on a concentrated plastic hinge approach. The conventional resistance is therefore a function of the assumed failure mechanism as well as of the nominal geometrical and mechanical features of the structural component. In case of mechanism type 1, the governing parameter is the bending plastic resistance of the flange rectangular section (M^), and the following relationship is suggested: ^M,l
(2)
m where m = d-0.^r ,dbeing the distance between the bolt axis and the T-stub web and r the root radius of the flange-to-web connection. In particular, Mu is the fully plastic bending moment evaluated with respect to the conventional yield stress^.2 and to the effective width beff, which is defined in such a way that the design tensile resistance of the equivalent T-stub is approximately equal to the actual one. Its determination represents, therefore, the critical point of the procedure. In both cases, when the bolt-row is considered individually and when the bolt row is considered as part of a bolt-group, EC3-Annex J allows the effective width to be determined by assuming the smallest value between the one related to the circular yielding pattern and the one related to other non-circular patterns. In particular, for bolts as a part of a group, ^e^is limited to the T-stub geometric width b. Besides, yielding patterns observedfi*omnumerical results show a substantial difference between the shape and therefore the effective length of the inner plastic hinge, located close to the root of flange-toweb connection, and the outer one, which is close to the inner edge of bolt hole (Figure 7). Yielding lines are, in fact, almost straight in the first position, while are mainly circular in the second one. Consequently, the effective v^dth used in relation (2) accoimts for this difference by assuming an average value for both hinges. Finally, really speaking, another factors influencing results could be due to membrane actions developed by T-stub flanges, which confers a sort of geometrical hardening on joint behaviour. Such complementary actions beneficially affect the actual specimen response, but it is not taken into account by the simplified procedure suggested by the code.
236 T'Stub type 1 Steel bolt grade 8.8
steel bolt grade 8.8
6%
' • a * 8 0 IN/mm^] " * " a » 1 4 0 IN/mm ] — • " a « 2 0 0 [N/mml —•— a = 3 0 0 [N/mml Displacement [mm]
10
12
14
16
18
20
T'Stub type 2 steel bolt grade 8.8
Steel bolt grade 8.8
Displacement [mm]
'1-2a
6%
8% ' a « 0 IN/mm ]
- • - a « 0 {NAnm 1 • "•
a « 8 0 |N/mm^l
-*"
a « 1 4 0 fN/mm^J
a « 8 0 (N/mm^J a«140 [N/mm^j • a « 2 0 0 JN/mm^l 2 • a « 3 0 0 IN/mm J
- * - a « 2 0 0 |N/mm^] - * — a « 3 0 0 JN/mm^l Dispaeemcnt [mm]
Aluminiumflangef 0,2=240 N/mm^
Displacement [mm]
Aluminiumflangefo.2=120 N/mm^
Figure 6: Numerical F-A relationship for analysed cases The evaluation of effective width In order to provide a correction procedure based on the same simplified relationships suggested by EC3-Annex J, accounting for the effects due to peculiar aspects of aluminium, i.e. strain hardening and alloy available ductility, the effective width should be preUminarily determined. This allows the main
237
imprecision due to the approximate method suggested by ECS to be skipped, hampering the remaining difference to the effect of the hardening and ductihty. According to the basic conceptual definition, the effective width beff may assumed as the one which, in case of not hardening material, is able to reproduce through the relation (2) the same ultimate resistance Fuu as predicted by numerical simulation. For each T-stub geometry, the numerical results for a=0 lead to effective width values summarised in Table 2. Conventionally, the ultimate resistance has been evaluated as the one corresponding to the attainment of a global strain in the most developed plastic hinge equal to 8%. This allows to assume that the plastic mechanism is almost completely developed. TABLE 1 NUMERICAL AND CODIFIED ULTIMATE RESISTANCE FUU [kN]
3
Material Hardening g fN/mm^l
8=3 % 89.3
80
93.9
140
97.1
100.5
Numerical 8=6 %
8=8 %
Codified (EC3-Annex J)
90.0
90.6
90.9
62.7
96.2
99.2
101.6
62.7
105.1
109.6
62.7
8=4 %
200
101.6
105.6
112.4
117.7
62.7
300
107.2
113.0
122.8
130.6
62.7
46.1
46.4
46.8
47.2
31.4
80
51.5
53.1
56.0
58.2
31.4
140
55.3
58.0
62.6
66.3
31.4
200
59.0
62.7
68.9
74.2
31.4
300
64.8
70.4
79.5
87.1
31.4
140.6
142.5
145.6
148.1
107.4
80
146.8
151.1
157.9
163.6
107.4
w):z;
140
148.6
153.7
162.3
169.8
107.4
is
200
154.8
161.8
174.1
184.3
107.4
300
158.3
167.3
183.3
197.4
107.4
71.9
72.9
74.4
75.7
53.0
80
78.4
81.4
86.4
90.4
53.0
140
83.3
87.8
95.4
101.6
53.0
200
88.2
94.2
104.5
112.9
53.0
300
96.3
104.8
119.3
128.0
53.0
203.2
206.2
209.1
210.4
256.0
213.2
219.3
226.7
231.6
256.0
80
JS II
l§ II
140
219.5
227.8
238.4
245.1
256.0
200
225.2
235.5
249.4
258.9
256.0
300
234.1
247.9
266.3
279.9
256.0
107.0
108.0
109.0
110.0
128.0
80
117.8
121.2
126.2
130.1
128.0
140
124.2
129.6
137.7
143.7
128.0
200
130.7
137.4
148.3
157.4
128.0
140.4
149.8
165.8
179.5
128.0
300
238
hinge line
hinge line
Figure 7: Observed yielding lines for plastic hinges (T-stubs type 1 and 2) It can be seen that effective vsddths are slightly dependent on the flange-to-bolt resistance ratio, as basically assumed by EC3-Annex J. Evidently, in case of aluminium, the actual values of beff will be also dependent on the hardening of the material, which should allow for a major spreading of plasticity within the structural component. Nevertheless, exact determinations are quite difficult and may be disregarded in a first tentative approach. TABLE 2 EFFECTIVE WIDTH VALUES [nim]
T-stub type
Flange type /o 2=240 N/mm ^ 2=120 N/mm /o.2=240 N/mm ^ 2=120 N/mm /o.2=240 N/nmi /h2=120N/mm
1 TYPE 1
1 TYPE 2 TYPE 3
Effective width be/f [mm] Numerical value Codified value 57 40 40 59 52 40 54 40 109 138 138 115
The proposed correction factor The comparison between numerical results and the ones obtained by applying the codified procedure showed that concentrated plasticity model proposed by Annex J for interpreting T-stub ultimate resistance is inadequate essentially due to the fact that the system hardening of the material is completely disregarded. The values of ultimate resistance deduced according to the effective widths corrected as previously mentioned, allow the effect of hardening and ductility to be easily evaluated. As a first attempt, in order to incorporate such effects, relationship (2) could be applied by considering the ultimate resistance fu instead of the conventional elastic stress yb.2- In turn, fu may be determined through the material c - e relationship (1), as a fiinction of the assumed limit strain. In fact, it should be intended as a conventional ultimate stress, corresponding to the attainment at least in one point of the section of the conventional ultimate elongation, previously assumed as the uniform strain (s„). In such a way, the T-stub conventional ultimate resistance would be overestimated due to the variation of normal stress through the section. A balancing factor k may be therefore introduced in order to reduce the section modulus respect to the one corresponding to fiilly plastic behaviour (Figure 8). The following corrected relationships may be therefore assumed:
\Kfft M u,mod = fuT (3) k 4 m From the theoretical point of view, the Mactor should be related to the possibility to develop uniform stress distribution through the section depth, being essentially dependent on the hardening of the alloy 4M.u,mod
u,\,mod
239 (a) and the conventional ultimate elongation (Su). In particular, it ranges from 1, in case of nothardening material and unlimited ductility, to 1.5, for an indefinitely elastic material. On the other side, A:-factor also accounts for other aspects that are strictly connected to the adopted simplified model based upon a monodimensional scheme, such as the different behaviour of plastic hinges at bolt location and root of flange-to-web connection. In the present study, ^-factors have been estimated by applying relationship (3), backward, where T-stub resistance values F„ have been evaluated by means of the numerical results referring to collapse mechanism type 1. e«
fy
Figure 8: Schematised plastic hinge behaviour Results are depicted in Figure 9, where the variation of ^-factor is diagrammed as a function of both material hardening and ductility supply. It is shown that for all the examined T-stubs typologies, which should cover the majority of realistic cases, the correction factor k ranges between 1 and L45, increasing with both a and s„ values. In particular, ^-factor seems to be slightly influenced by ductility alloy (with an influence not higher than 15%), while it is more remarkably dependent on the hardening, whose influence is up to 40%. Going deeply into details, it is interesting to observe that in case of no hardening material (a=0), ^-factor decreases as far as available ductility increases, but this is due to the geometrical hardening effect which, owing to the spreading of plastic zones, produces an an increase of the actual effective width. This trend is opposite for a>0, showing that the effect of stress redistribution over the flange thickness is predominant as respect to the increase of the effective width. Likewise, it should be noticed that the A:-factor tendencies are also slightly influenced by the T-stub geometry and flange-to-bolt resistance ratio. The latter concern is particularly important, proving that the T-stub behaviour is also affected by other factors, rather than the ones considered by the codified approach. This is especially due to the actual effective width evolution with the increase of plastic deformations. Finally, it is to be observed that these differences seem to be due separately to both the effects of flange-to-bolt resistance ratio and to the thickness of the flange.
CONCLUSIVE REMARKS Within a general research project devoted to the codification of rational methods to be applied to the prediction of aluminium joint structural behaviour, the current study has been focused on the possibility to profitably use the existing approach for steel joints. In particular, in case of simple T-stub joints characterised by strong bolt-weak T-stub flange ratio, a wide numerical analysis has been carried out, aiming at the calibration of a simplified procedure that could allow the ultimate resistance of T-stub aluminium joints to be simply evaluated. In order to account for the effects due to the material strain hardening as a fimction of tiie available ductility, a balancing ^-factor has been therefore introduced in the same simplified formulation provided by EC3-Annex J. The results obtained have shown that the actual behaviour of T-stub joints is rather complex, being influenced by several factors, which are not all covered by the codified method. In particular, effective width values suggested by EC3-Annex J appear to be inadequate; besides, in case of hardening materials, the evolution of the effective width during the loading process, plays an important role, compelling somewhat variation of ^-factor with Tstub geometry and flange-to-bolt resistance ratios. Anyway, the first results are encouraging, demanding for succeeding steps of the study. Further analyses should be therefore addressed to the definitive assessment of the procedure as well as to its extension to different collapse mechanisms.
240 T-stub flange materialfo.2=240 Wmnf
T-stub flange material f0.2^120 N/mm^ 1.50
T-stub type 1
T-stub type 2
T-stub type 3
T-stub type 1 ^
T-stub type 2
3 ^
1.30
A
tS 1.20
Io
O
1.10
^
j
^
A j^P^
\^f^'^, , ^^, 0
80 140 200
300 80 140 200
300
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300
^ 1.30 c o •5 1.20 o
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/ / x
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Hardening a [N/mm^J Eu=3%
T-stub type 3
1.40
^
%—1—)—1 300 80 140 200
1 300
i
Hardening a [N/mm^] eu=4%
8u=6%
eu=8%
Figure 9: ^-factor variation for examined cases (collapse mechanism type 1)
ACKNOWLEDGEMENTS This work has been developed in the framework of activity of the research project "Methods of Behaviour Prediction for Aluminium Alloy Joints", sponsored by the Italian Ministry of University and Scientific and Technological Research (MURST). The helpful co-operation of the graduate student Marco Sciarra is also gratefully acknowledged.
REFERENCES Bursi, O. S. and Jaspart, J.P. (1997). Benchmarks for Finite Element Modelling of Bolted Steel Connections. Journal of Constructional Steel Research, 43, (Issue: 1-3), pp. 17-42. De Matteis, G. and Mandara, A. (1997). A Concentrated Plasticity Model for Hardening Material Structures. XVI Congresso C T.A., Ancona, 281 -92. De Matteis, G. Mandara, A. and Mazzolani, F.M. (1998). Numerical Analysis for T-stub Aluminium Joints. 4^^ Int Conf. on "ComputationalStructures Technology", CST'98, Edinburgh. De Matteis, G. Mandara, A. and Mazzolani, F.M. (1999). Interpretative Models for Aluminium Alloy Connections. International Conference on Steel and Aluminium Structures (ICSAS'99), Espoo (Finland) June 20^^-23"^. ENV 1993-1.1 - Eurocode 3 - Annex J, (1997), Joints in Building Frames, CEN/TC250/SC3-PT9. ENV 1999-1.1 - Eurocode 9, (1998), Design ofAluminium Alloy Structures, CEN/TC250/SC9. Hopperstad, O.S.,(1993), Modelling of Cyclic Plasticity with Application to Steel and Aluminium Structures, Dr. Ing. Dissertation, Division of Structural Mechanics, The Norvegian Institute of Technology, Trondheim, Norway. Mandara, A and Mazzolani, F.M. (1995). Behavioural Aspects and Ductility Demand of Aluminium Alloy Structures. International Conference on Steel and Aluminium Structures (ICSAS'95), Istanbul. Mazzolani F.M. (1995), Aluminium Alloy Structures, E & FN SPON, London. Mazzolani, F.M., De Matteis, G. and Mandara, A. (1996). Classification System for Aluminium Alloy Connections. Z4M^M Colloq. on "Semi-RigidStructural Connections", Istanbul
Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved
241
IMPROVED BEAM-TO-COLUMN JOINTS FOR MOMENT-RESISTING FRAMES AN EXPERIMENTAL STUDY P. Sotirov, N. Rangelov, O. Ganchev, Tz. Georgiev, Z.B. Petkov and J. Milev Faculty of Civil Engineering, UACEG, Boul. Hristo Smimenski 1, 1421 Sofia, Bulgaria
ABSTRACT The good performance of steel moment-resisting frames under seismic motions depends strongly on the behaviour of their beam-to-column connections. The Northridge earthquake generated an urgent need to increase the ductility of the connections normally used in practice. This paper presents the study of beam-to-column joints in column-tree configuration of moment-resisting frames. Tapered flanges are used to improve the seismic behaviour of beams. Two types of joints are proposed and tested: type A, considered as an improved pre-Northridge joint, and type B, as a post-Northridge detail. Experimental investigation on six (3+3) full-scale specimens was carried out. The tests were performed in the Steel Structures Research Laboratory of the University of Architecture, Civil Engineering and Geodesy (UACEG), Sofia. Both specimen types exhibited stable hysteretic behaviour with adequate ductility and energy dissipation capacity. Though both variants are quantitatively comparable, the most significant advantage of type B is found in its predictable behaviour.
KEYWORDS Beam-to-column connections, moment-resisting frames, column-tree configuration, ductility, cyclic hysteretic response, plastic rotation, energy dissipation, dog bone, experimental study. INTRODUCTION Steel moment-resisting frames have been popular in many regions of high seismicity not only for their architectural versatility, but also for their good seismic performance as highly ductile systems. However, the seismic response of a ductile moment frame will be satisfactory only if the connections between the framing members have sufficient strength, stiffness and plastic deformation capacity. The traditional strong-column — weak-beam design concept implies that the input energy is absorbed and dissipated primarily by plastic hinges formed at the beam-to-column connections. Experimental
242
work carried out by Popov et al. (1986), demonstrated a lack of deformation capacity of the welded flange — bolted web connections, normally used in practice. The Northridge earthquake definitely proved that conclusion and generated an urgent need for improved detailing. Two key strategies have been developed: (i) strengthening the connection, and (ii) weakening the beam that frame into the connection (Bruneau et aL, 1998). Within the framework of (i) cover plates, upstanding ribs, side plates, and haimches are implemented. The weakening strategies (ii) are based on the idea of shaving beam flanges to intentionally weaken the beam at a predetermined location, originally proposed by Plumier (1990). Besides the classical "dog bone" profile and circular cuts, Chen et al. (1996) proposed to taper flanges according to a linear profile so as to approximately follow the varying bending moment diagram. Popov et al. (1996) suggested a combination between the two strategies, namely a cover plates connection with circular cuts in the beam flanges. Both strategies aim at effectively moving the plastic hinge away from the column face, thus avoiding the problem of poor ductile behaviour and potential fragility of the welds. All the proposed and tested connections have some merits and disadvantages. It seems that no detail can be assumed to be perfect and new solutions are still worth exploring. In Bulgarian practice, there are two features regarding the steel moment-resisting frames: built-up members which have the advantage to be proportioned more flexibly to attain a better balance between internal forces and resistance, and the column-tree configuration witib fully shop-welded rigid beam-tocolumn connections and field-welded erection splices in the beams. High ductility can be achieved by several tj'pes of connections, however fully welded rigid connections are still reckoned to possess the highest dissipation capacity (Mazzolani & Piluso, 1996). Two different beam-to-column joints are considered in the present study. In both of them tapered flanges are used for improvement of the seismic behaviour of the beams. An experimental programme is reported, carried out on six full-scale specimens. Shop-weided connection
i«aa»m^KP««|KP»l4**^«wBcsag&sjk9i;-«4V9»K«o«i|j»jK«an^
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Figure 1 : Beam-to-colounm joints considered
Q
I
243
BEAM-TO-COLUMN JOINTS UNDER CONSIDERATION The two types of joints are shown in Figure 1. In type A the flanges are tapered to follow approximately the varying bending moment diagram. The solution is more economical, and, under an earthquake load, the yielded zones in flanges will be larger, thus dissipating more energy than in the conventional non-tapered flanges. However, the expected dissipative zone is mostly at the column face. Therefore type A is to be regarded as an improved pre-Northridge joint. Detail B is of post-Northridge type. The shifting of the plastic hinge from the column flange is achieved by a specific flange profile, following the idea illustrated in Figure 1. Thus both the abovementioned strategies are incorporated in one detail. On the one hand, the connection is strengthened without additional cover plates, ribs or haunches, keeping a constant beam depth, and, on the other hand, the beam is weakened at a predetermined location to provoke the development of a plastic hinge. TEST SPECIMENS, EXPERIMENTAL SETUP AND LOADING HISTORY The test beam-to-column assemblies were fiill-scale, "extracted"fi-oman especially designed two-bay four-storey regular moment-resisting fi-ame. The specimens were fabricated by a Bulgarian industrial manufacturer. The column sections were produced by automatic welding under flux, whereas the beam sections, small in size for the welding equipment, were built-up by semiautomatic welding under CO2 shield. ThefiiUpenetration welds between beam and column flanges were also made under gas shield. The erection splices were hand-welded with covered electrodes in the laboratory, thus simulating the field conditions. The steel grade used corresponds to S235. A total of 6 specimens were tested: 3 identical ones (labeled A-1, A-2 and A-3) of type A, and 3 identical specimens (B-1, B-2 and B-3) of type B. The layout and main dimensions are shown in Figure 2.
Beam sections: Flanges'. 12x120 to 12x220 for A 12x120 to 12x280 for B Web: 8x350 next to column 6x350 in the span Column section: Flanges'. 16x300 Web'. 10x300
hydraulic
Figure 2: Test specimens and experimental setup
244
The test setup was designed to accommodate specimens in a horizontal position as shown in the figure. The load was applied to the cantilever beam end by a hydraulic actuator with a displacement range of ±200mm and a capacity of ±200kN. No axial load was applied to the column. To prevent out of plane motion of the beam, lateral bracing was provided near the beam end and near the splice. A total of 14 inductive displacement transducers were installed (Figure 2) to measure the global behaviour of the specimens: beam end displacement, joint rotation, panel zone shear deformation, and possible movement of the supports. Strain gauges were used to investigate the local response of the beam flanges including the splice plates. All the instruments were connected to a PC-based data acquisition system with software developed by the laboratory staff. A Keithly DAC-02 card was used to send the displacement command signal to the actuator servovalve. The testing programme was based on the recommendations of ECCS (1986). The specimens were tested under displacement control, following a loading history consisting of stepwise increasing deformation cycles. Initially, four cycles in elastic range were applied with amplitudes of ±0.25vy, ±0.50vy, ±0.75vy, and ±Vy, where Vy is the expected first-yield displacement of the beam end. Eventually, correction of Vy was made. Then the testing continued in the plastic range with three full cycles at each amplitude level ±2vy, ±3vy, and so on (instead of the recommended by ECCS (1986) ±(2+2«)vy; the latter was only applied to specimen A-3). For specimens A-3 and B-3 an initial displacement corresponding to approximately 30% of the plastic bending capacity was applied to simulate the effect of the gravity loading. A typical loading history and the corresponding actuator force response are shown in Figure 3. 200
I 150 ^ 100
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200 400 600 800 Number of reading {'time')
-200
1000
(b)
200 400 600 800 Number of reading ('time')
1000
Figure 3: Typical loading history (a) and force response (b)
EXPERIMENTAL RESULTS AND DISCUSSION In type A specimens plastic deformations started to develop first in the panel zone and later in the flanges near the column. Satisfactory stable hysteretic behaviour was observed. The first tested specimen, A-1, failed by lateral-torsional buckling due to improper bracing. In specimen A-2 flange local buckling occurred, followed by web buckling. Shear deformations of the panel zone were apparent, demonstrating its very significant contribution. This was more pronounced in A-3, where no local buckling appeared. Both A-2 and A-3 failed due tofiractureof the upper flange splice plate. The datafi*omthe strain gauges showed that considerable plastic deformations had developed there. Due to some lack of overstrength and different areas of the two splice plates, the top one became overloaded. This fact, together with the stress concentration, led to low-cycle fatigue. However, no other cracks
245 450
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0.025
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450
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0
0.025
Plastic rotation [rad]
0.05
0.075
-450 -0.075 -0.05 -0.025
0
0.025
Plastic rotation [rad]
Figure 4: Hysteretic behaviour in terms of moment versus total ( ) and beam (—) plastic rotation (moment and rotation determined for the reference cross-section: at the column face for type A, and at the theoretically expected location of the plastic hinge for type B)
246 were observed elsewhere. The good behaviour of the shop-welded full penetration welds was clearly proven. All type B specimens exhibited a very stable cyclic hysteretic response, and behaved exactly as predicted. Plastic zones developed at the designed location, accompanied by flange local buckling in the three specimens. Finally, the resulting secondary bending stresses at the crest of the buckles led to a low-cycle fatigue rupture of the buckled flange. No other cracks were detected either in the web-toflangefilletwelds in that region, or elsewhere. Both in A-3 and B-3, the effect of initial gravity loading was negligible and vanished well within the plastic range. The cyclic response of the joints is illustrated in Figure 4 in terms of moment versus plastic rotation. The latter is obtained using the elastic stiffness determined from the experimental curve by a linear fit. The plastic rotation in the beam is separated to assess the relative contribution of the plastic zone in the beam and the panel zone. Excluding A-1, all joints exhibit adequate ductility. In both types A and B, total plastic rotations of 0,05 rad were reached without a brittle fracture, which is well above the demand of 0.03 rad adopted recently (Bruneau et al., 1998). However, due to the different contribution of the panel zone, beam plastic rotations reached in type A were 0.02-0.025 rad, whereas in type B they were up to 0.03-0.04rad. A very important parameter characterizing the structural behaviour of a joint under seismic loading is its energy dissipation capacity. Herein, it is determined as the plastic work based on the hysteretic area of the experimental loops (Figure 4), and is plotted in Figure 5 versus the relevant cumulative beam end displacement. Regarding type A specimens, one can see that the contribution of the plastic zones in the beams is only about 30%. However, the check of the local response showed that theflangeshad
g400
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\
^
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Figure 8: Plastic collapse mechanisms IScalded ace: Kobe, X=1.18, a=0.35g
[Scaled acciKobe, X=0.85, a=0.25g
1 6nec(rad) Onec (rad) 0
0,03
0,06
0,09
0,12
0,15
0,03
0,06
0,09
0,12
Figure 9: Ductility demands for the normalized Kobe earthquake
0.15
267 jScaled ace: Banat Long, X=l 1.6, a=0.35g Scaled ace: Banat Long, A,=8.33, a=0.25g ^^^•^"T!^?;
3 i
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.' y-^l^-:^':;:'-:'^^ ^r'"-\^.~ l[:J-':'
rK,:'^-;:'
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T(sec)
Figure 11: Time histor>^ moment demands at beam-to-coioumn interface of the first storey IKobe,X=1.18,a=0.35g i Vrancea, X=1.6, a =0.35g B Banat Long, X=l 1.66, a=0.35g
SMRF(T=0,909sec)
OMRF (T=1.05sec)
DBF (T=1.14sec)
T- Natural period of the structure
Figure 12: Shear base index of effectiveness
RF (T=0.92sec)
268 CONCLUSIONS Analytical investigations were performed on both local and global performance of different momentresisting jframe typologies, using DUCTROT'96 and Drain-2D computer programs. The study concludes that the modified moment-resisting firames, DBF, RF, can help to control the seismic response avoiding the formation of undesirable collapse mechanisms, as storey mechanism. In the same time, these solutions transform OMRFs, in ductile moment-resisting firames, obtaining a predetermined failure mode and the ductility control through concentrated rotation requirements only at the beams, far from joints. Li some cases this effect seems to be unreliable; in this way it is need to consider the specific seismicity of the site. Further research work are planned to lighten these aspects.
REFERENCES Anastasiadis A. (1999). Ductility problems of steel MR frames. Ph. D Thesis, Politehnica University Timisoara, Romania Anastasiadis A. and Gioncu V. (1998). Influence of joint details on the local ductility of steel momentresisting frames, Greek National Conference on Steel Structures, Thessaloniki, Greece (in print) Chen S.J., Chu J.M. and Chou Z.L. (1997). Dynamic behavior of steel frames with beam flanges saved around connection. J. Construct. Steel Research 42:1, 49-70. Gioncu V. and Mazzolani F.M. (1999). Ductility of Seismic Resistant Steel Structures, manuscript, E& FN Spon, U.K Gioncu V. and Petcu D. (1996). DUCTROT'96: Plastic rotation of steel beams and beam-columns, Guide for Users, INCERC Timisoara, Romania Gioncu V. and Petcu D. (1997). Available rotation capacity of wide-flange beams and beam-columns: Part I:Theoretical approaches. J. Construct. Steel Research 43:1-3, 161-217. Gioncu V. and Petcu D. (1997). Available rotation capacity of wide-flange beams and beam-columns: Part n: Experimental and numerical tests. J. Construct. Steel Research 43:1-3, 219-244. Mazzolani F.M. and Piluso V. (1993). Design of Steel Structures in Seismic Areas, ECCS Document Mazzolani F.M. (1998). Design of steel structures in seismic regions: The paramount influence of connections, COST 98, Liege Conference, 11-24. Plumier A. (1996). Reduced beam section; a safety concept for structures in seismic areas, Buletin Stiintific, Ser. Constructii, Arhitectura, Tom 41(55), Fasc. 2, 46-60. SAC (1995). Interim Guidelines: Evaluation, Repair, Modification and Design of Welded Steel Moment Frame Structures, Report FEMA 267/SAC-95-02, SAC Joint Venture, California, U.S.A.
Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Iv^nyi, editors © 1999 Elsevier Science Ltd. All rights reserved
269
FACTORS INFLUENCING DUCTILITY IN HIGH PERFORMANCE STEEL I-SHAPED BEAMS C. J. Earls^ Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261-2294, USA
ABSTRACT In general, compactness and bracing provisions associated with ultimate strength design of steel beams are formulated so as to ensure that the resulting beam designs exhibit adequate structural ductility. The specification of such compactness and bracing requirements oftentimes involves assumptions about the constitutive nature of the structural steel being used in construction. Historically, mild carbon steel has been the most common type of steel used in buildings and thus the characteristic constitutive properties of this steel are usually assumed to be the norm. While such material property assumptions form a good basis for compactness and bracing provisions of mild carbon steel structural elements, the same is not true when they are applied to structural members made from newer high performance steels. These new high performance steels have constitutive properties that differ significantly from those of mild carbon steel. It appears from the research reported herein that separate bracing and compactness requirements may not be valid when used to evaluate the ductility of I-shaped beams made from some new high performance steel grades. Results from this same research point to the fact that I-shaped beams made from high performance steel, and subjected to moment gradient loading, display inelastic mode shapes which do not lend themselves to a notional de-coupling of so called local buckling and lateral-torsional buckling phenomena. Rather, the inelastic mode shapes of these high performance steel beams display two distinct types of geometry at failure; both of which possess localized and global buckling components. The structural ductility of the beams is very much dependent upon which of the two inelastic mode shapes govern at failure. Cross sectional proportions, bracing configuration, and geometric imperfections all play a role in influencing which mode governs in the beam at failure. Currently held views as to the impact of cross sectional compactness and bracing on structural ductility often do not apply to I-shaped beams made from these high performance steels. Hence current design specifications will not be able to predict the response of these high performance steel flexural members. This paper will present results from numerical studies of high strength steel I-shaped beams conducted using the nonlinear finite element analysis technique. These studies will focus on the impacts which cross sectional proportions have on the manifestation of the inelastic mode shape and
270
subsequent structural ductility under a moment gradient type loading. In addition, the validity of two methods for predicting flexural ductility in I-shaped members, as obtained from the literature, will be compared with current research results.
KEYWORDS High Performance Steel, Flexural Ductility, Local Buckling, Lateral Torsional Buckling, Non-Linear Finite Element Method, Inelastic Buckling, Unbraced length, Rotation Capacity, Compactness, Bracing Stiffness, HSLA80 Steel.
INTRODUCTION The American (AISC 1994) and European (ECS 1992) steel design specifications simplify the flexural design of I-shaped members such that cross sectional proportions are considered independently from unbraced length when computing design capacity, thus assuming that cross sectional instabilities of constituent plate elements may be considered independently from global instabilities such as lateral-torsional buckling. It is clear that this simplification is made out of necessity due to the substantial complexity of treating the true interactive nature of the local and global modes as outlined by the plastic design commentary of the American Society of Civil Engineers (ASCE 1971): "Even though local and lateral-torsional buckling in the inelastic range are manifestations of the same phenomena, namely, the development of large crosssectional distortions at large strains, they have been treated as independent problems in the literature dealing with these subjects. This is mainly due to the complexity of the problem." Researchers such as Climenhaga et. al (1972) and Gioncu et. al (1996) have addressed the complicated coupled flexural instabilities of I-shaped beams by employing the energy method in conjunction with a yield line mechanism simplification. In this technique, after a predetermined cross-sectional rotation capacity is achieved, the descending portion of the moment rotation curve is traced by plotting the flexural response emanating from two different types of buckling geometry; one dominated by a so called "in-plane mechanism", and the other dominated by an "out-of-plane mechanism". The descending portion of the moment-rotation curve resulting from the mechanism which produces the least capacity is assumed to correspond to the governing failure mode. Gioncu et. al (1996) have also considered interactions that arise between the in-plane and out-of-plane modes. These yield line mechanism methods have produced satisfactory results as compared with experimental data reported in certain studies (Gioncu et. al 1996, Kuhlmann 1989). Other researchers have treated the complicated coupled flexural response of I-shaped beams in terms of simplifications involving cross-sectional geometry alone (Climenhaga et. al 1972, White et. al 1997). Kemp (Kemp et. al 1991, Kemp 1996) addresses the complicated global and local buckling interaction through the consideration of both cross-sectional geometry and unbraced length of the member. Methods for estimating rotation capacity of I-shaped beams have been given by Kemp (Kemp et. al 1991, Kemp 1996) as well as White (White et. al 1997, White et. al 1998). These predictive methods can perform adequately in certain restricted ranges of applicability as is shown later in this paper.
271 VALIDATION OF MODELING TECHNIQUES The commercial multipurpose finite element software package ABAQUS (ABAQUS 1998) is employed in this research. The finite element models described herein consider both geometric and material nonlinearities. These nonlinearities tend to create formidable computing requirements since dense meshes of shell finite elements must be used to properly model localized instabilities that occur in conjunction with global instabilities, both of which are inelastic in nature. Incremental solution strategies are necessary to properly trace the nonlinear equilibrium path of the inelastic Ishaped beams. The ABAQUS modified Riks-Wempner strategy is used. All analyses reported here are carried out in parallel using sixteen processors of a CRAY T90 supercomputer. The ABAQUS S9R5 shell finite elements, used to model the beams, are oriented along the planes of the middle surfaces corresponding to the constituent plate components of the members. A uniaxial representation of the constitutive law given in terms of true stress and logarithmic strain is recorded in the ABAQUS input deck. ABAQUS then uses the von Mises yield criterion to extrapolate a yield surface in three-dimensional principal stress space from the uniaxial material response given in the input deck. The corresponding ABAQUS metal plasticity model is characterized as an associated flow plasticity model with isotropic hardening being used as the default hardening rule. Three finite element models were constructed so as to compare finite element results with the experimental results obtained by Lay and Galambos (Lay et al. 1965). The subjects of the comparison study are B8xl3 wide flange beams of varying length and subjected to three point bending resulting in a moment gradient loading. Both the experimental specimens and the finite element models used fiill depth stiffeners on both sides of the beam web, at the mid-span loading point and supports, so as to control cross sectional distortions under the action of the load. Similarly, out-of-plane translation and rotation were constrained at the supports and load point of the physical specimens and the finite element models. The experimental tests were carried out at Lehigh University in 1965 as part of a study aimed at extending the applicability of plastic analysis and design methods to steel with up to a 345 MPa yield stress. The experimental specimens were made from ASTM A441 steel. A piece-wise linear uniaxial representation of the material model used in the finite element comparison studies has a yield stress of 354.4 MPa. The ratios f^ / fy = 1.49, s„ / 8y = 72.4, and s^^ / Sy = 9.66 fiirther characterize the material model used in the finite element comparison studies. A relatively good agreement can be observed in the comparison plots of the moment-rotation responses obtained from the experiments and finite element models (see Figures 1,2, and 3). In all three cases, the finite element models displayed slightly higher ultimate moment capacities as compared with the experiments. This could be due to necessary differences in the material properties of the experiments and finite element models. In the report of Lay and Galambos (Lay et al. 1965), only the yield strength, ultimate strength, and percent elongation were given. Assumptions as to the strain hardening strain, ultimate strain, and strain hardening modulus had to be made in the finite element modeling. The finite element models did not exhibit the same degree of "roundness" in the portion of the moment-rotation response corresponding to the transition from elastic to inelastic behavior. This discrepancy is most likely due to the fact that residual stresses were not incorporated into the finite element models and thus the onset of first yield was delayed slightly in comparison to the experimental tests which undoubtedly possessed residual stresses.
272
INFLUENCE OF SLENDERNESS ON DUCTILTY Finite element studies of HSALSO beams subjected to a moment gradient loading have been conducted (Earls 1999) so as to develop a notional understanding of potential underlying mechanisms associated with the ultimate response of I-shaped beams. The results of these early studies are seen to run counter to expectations concerning the influence which cross-sectional proportioning has on structural ductility as quantified by plastic hinge rotation capacity. The definition of rotation capacity, used below, is consistent with that given by ASCE (ASCE 1971) in which R = {(Gu / Op) -1} where 0^ is the rotation when the moment capacity drops below Mp on the unloading branch of the M-0 plot and 0p is the theoretical rotation at which the full plastic capacity is achieved based on elastic beam stiffness. The influence of flange compactness on overall rotation capacity is addressed by Table 1. The results presented in this table seem to contradict current practical notions regarding the role of flange compactness in wide flange ultimate response. It is seen from Table 1 that an increase in flange slendemess increases overall rotation capacity for a bf / 2tf increasing from 3 to 6. However a further increase in bf / 2tf from 6 to 7 results in a significant decrease in overall section rotation capacity. The influence of brace spacing on beam rotation capacity is also addressed in this earlier study. Tables 2 provides a summary of results relevant to a discussion of the impact that unbraced length has on such beams. The results outlined in Table 2 contradict practical notions concerning the role of beam slendemess on the rotation capacity of a flexural element governed by lateral torsional buckling. It is seen from these results that both increasing and decreasing the unbraced length can lead to substantial improvements in the rotation capacity exhibited by a beam.
DOMINANT FAILURE MODES Based on the Author's examination of the inelastic mode shapes accompanying the finite element studies of HSLA80 I-shaped beams subjected to a moment gradient loading (as reported in Earls 1999) two distinct failure modes are identified. These two modes will be referred to respectively as Mode 1 and Mode 2. Figure 4 displays a typical Mode 1 geometry, while Figure 5 shows the Mode 2 geometry. Mode 1 is characterized by a local instability of the flange, either with or without substantial web participation, which occurs in close proximity to the mid-span stiffener. The plastic hinge region of Mode 1 is well defined and proximal to the mid-span stiffener region Mode 2 is characterized as a highly asynmietrical inelastic mode whereby local and global buckling are highly coupled. As can be seen in Figure 5, the flange buckling components, or flange-web buckling components, occur at a substantial distance from the mid-span. This distance from the stiffener to the center of the flange buckling wave is in general different for each half-span, but on average this distance is roughly equal to d/2. Similarly, the degree to which the flange-web buckling component of Mode 2 manifests itself varies significantly between the half-spans. Generally speaking, substantial out-of-plane deflections between brace points occur in the Mode 2 failures. The Mode 2 "plastic hinge" is better described as being a "zone of plastification" thereby not implying the usual connotation of a tightly formed concentrated region of plastification. On the contrary, the zone of plasticity in Mode 2 is very ill-defined and quite distributed in nature. Another characteristic feature of the Mode 2 inelastic mode shape is the formation of a mechanism in the compression flange of the beam. An example of this type of compression flange mechanism can be
273
seen in Figure 6 which displays a typical top view of the compression side of an I-shaped beam. From this figure, it can be seen that the compression flange behaves somewhat like a three-barlinkage; the kinematics of which are driven by the location of the mid-span stiffener and the linkage articulations. These articulations coincide with the locations of the flange buckling component (or flange-web buckling components) within the overall Mode 2 manifestation. Beyond the pronounced geometric differences between the inelastic mode shapes of Mode 1 and Mode 2, there are other more quantifiable differences in response. Mode 1 failures achieve a higher ultimate moment capacity and exhibit larger cross-sectional rotation capacities as compared with Mode 2 failures. It has also been found (Earls 1999) that bracing location, bracing stiffness, and the presence of cross-sectional imperfections all play a role in determining whether a Mode 1 or Mode 2 manifestation exists at failure. Climenhaga (Climenhaga et. al 1972) and Gioncu (Gioncu et. al 1996) have also observed that two distinct failure mode manifestations exist in I-shaped flexural members. These researchers categorize them as "in-plane" and "out-of-plane" mechanisms which respectively correspond to Mode 1 and Mode 2 as described previously. Clear examples of Mode 2 failures have also been reported in the experimental work of Schilling (Schilling 1988) and Azizinamini (Azizinamini 1998) as well as in the finite element studies of White (White 1994). Kemp and Dekker have made similar distinctions between mode shapes (Kemp et. al 1991). The categorization of Mode 1 and Mode 2, as defined in their paper, is opposite to that outlined above and given elsewhere (Earls 1999). For clarity of discussion, the definitions of Mode 1 and Mode 2 given earlier in this section of the current paper will be used in the subsequent discussion. Despite their two distinct failure mode categorizations, Kemp and Dekker believe that a single cause results in a reduction of observed moment capacity, or load shedding. This cause is a phenomenon which they term "strain weakening". They describe "strain weakening" as the displacement of the compression flange, due to lateral buckling, inducing a transverse strain gradient across the width of the flange. Kemp and Dekker conclude that "such lateral displacement of the member will lead to loss of moment resistance unless considerable strain hardening occurs on the opposite edge of the flange" (Kemp et. al 1991). Kemp and Dekker also describe their version of the Mode 2 failure as being predominantly elastic in nature with only a short segment adjacent to the hinge region yielding. This characterization is not supported by the Author's own research which shows very clearly that the lateral motion of the compression flange only occurs after very significant yielding is achieved along large portions of the beam longitudinal axis. Schilling (Schilling 1998) also observes that any lateral motion of the compression flange, as observed in his experimental tests, follows fi-om extensive yielding of the beam adjacent to the hinge region.
COMPARISON OF ROTATION CAPACITY PREDICTIVE EQUATIONS This portion of the paper is focused on comparing the rotation capacities observed in several test beams, modeled with validated finite element techniques, with those obtainedfirompredictive equations found in the literature (Kemp 1996, Kemp et. al 1991, White et. al 1997, White et. al 1998). These predictive equations have various geometrical limitations concerning their proposed range of validity. These equations also include scaling parameters to address the differences associated with different steel grades. It is interesting to note however that these scaling parameters consider yield stress only.
274
Kemp's predictive equation (Kemp 1996, Kemp et. al 1991) is based on the use of several basic parameters. The first is a "yield stress factor" for the flange or web given as
7 =
250
for F in MPa. The second parameter is a "slendemess ratio in lateral-torsional buckling" given as 'L^ 7f in which Lj is the length from the section of maximum moment to the adjacent point of inflection, rye is the radius of gyration of the portion of the elastic section in compression. The "flange slendemess factor in local buckling" is the third factor and is of the form 'b^ ry K. _Vj_2 subject to 0.7 < Kf < 1.5 as the range of applicability. Similarly, the "web slendemess factor in local buckling" is
y.
K =
70
subject to 0.7 < K^ < 1.5 as the range of applicability. Kemp also defines a distortional restraint factor to account for the effects of a concrete slab in negative moment regions of a beam. However, since the current comparison study is only applied to bare steel beams this value is taken to be unity as per Kemp (Kemp 1996). The above parameters are then combined to form the "effective lateral slendemess ratio" of the form ' L ^ e
J
^
\^y^ J
valid in the range 25 < X^ < 140. Kemp goes on to define an empirical expression for the relationship between the effective lateral slendemess ratio and the plastic length of the member at maximum moment as
= 0.067
r6o_
i.
The rotation capacity is then predicted with the following equation
if
=45
275 It is noted that all rotation capacities, both from predictions and finite element models, are given with respect to the same definition as outlined by ASCE (ASCE 1971) and described earlier. In the case of White's predictive model (White et. al 1997, White et. al 1998), a deterministic equation, fit to finite element data, for the plastic rotation at which the descending portion of the moment rotation curve passes below the fiill plastic capacity is given as
Oj^ =0.128-0.0119
•0.0216
'D^
+ 0.002
K^fcJ
White recommends that the following restrictions apply to the use of the above equation
D
* ^ ^ •
Figure 2: Double-cantilever test arrangement
Bradford and Kemp (1999) have reviewed briefly the results of 47 of these double-cantilever tests as well as 22 tests on continuous composite beams. The dominant modes of failure observed are local flange, local web and lateral-distortional buckling, although vertical shear failure occurred in a limited number of tests. Kemp et al (1995) and Kemp (1996) have described the interaction between local flange, local web and lateral buckling and have emphasised the importance of inelastic lateral buckling in inducing strain-weakening behaviour of both composite and steel beams. Inelastic lateral distortional buckling is less likely in fully loaded composite beams using hot-rolled steel sections than in plain steel beams due to the restraint from the slab and shorter buckling lengths, although the required inelastic rotation is greater. Kemp and Dekker (1991) have shown that in composite beams large areas of longitudinal
295 reinforcement in the slab relative to the area of the steel section contribute to a significant reduction in the available inelastic rotation for the following reasons: 1. Increased depth of the web in compression induces earlier web buckling and reduces strainhardening in the reinforcement. 2. Reduced curvature, caused by the increased height of the neutral axis when the critical strain is reached in the compression flange, reduces the inelastic rotation at maximum moment. Kemp and Dekker (1991) have analysed all of these aspects and concluded that in a double-cantilever test (Figure 2), if the proportion of web depth in compression is the same, a particular steel section in a composite beam can be assumed to achieve at least the same inelastic rotation as a plain steel section prior to failure. Kemp (1996) has proposed and Couchman and Lebet (1996) have independently evaluated the following empirical expression for this available inelastic rotation 6'ayt in such tests: e',yt = 1.5(0.5 M, Li / El3)(60 / A,)' /a
(6)
in which 0.5 M^ L^ / EI^ = elastic rotation of steel section in double-cantilever specimen at M^ Ms = ultimate moment resistance of the steel section Lj = length between sections of zero and maximum negative moment in Figure la. EIs= flexural rigidity of steel section a = ratio of web depth in compression to total web depth Ag = effective lateral slendemess ratio K^K^K^iL^ /izcV^r Kf = (b/tf)/20ef is the flange factor for width b and thickness t^ K^ = (ad/t^)/35€^ is the web factor for depth d and thickness t^ Kd = distortional restraint factor (1 for steel section, 0.71 for composite section) i^ = radius of gyration of the elastic portion of the web and flange in compression € = JlSOJf , and fy = yield stress of the flange or web. The observed momentcurvature curve in negative bending (solid line) may be represented by the dotted bilinear relationship in Figure 3 based on an effective yield moment M'y shown in this figure. The corresponding distribution of inelastic negative curvature in the yielded length L'y adjacent to the supports in the central span of the beam is shown in Figure 4a. The yield moment M'y in Figure 3 is selected to give the same inelastic rotation as the test value predicted by Eqn.6 for a linear moment gradient:
actual design idealised (bi-linear)
Curvature
Figure 3: Moment-curvature relationship (j);^ = 2 0;^/LV
(7)
Specimens in positive bending {inelastic rotation Q and curvature (|)„„)
296 Inelastic curvatures in positive (sagging) regions of continuous beams which occur during the development of the ultimate moment resistance, increase the required inelastic rotations in the negative moment regions for a particular level of moment redistribution. If the moment-curvature curve is represented in the same way as for negative bending by the idealised bi-linear relationship in Figure 3, a parabolic distribution of inelastic positive curvature is obtained as shown in Figure 4b. The curvature at the ultimate moment ^^y is calculated from the change in strain over a depth d^ between the maximum compressive strain on the upper surface of the concrete of 0.0035 and the plastic neutral axis, as follows: (t),y = ( 0.0035/d„) -(Mp/EI)
(8)
Rotations in End Connection, d^^^ Increasingly, semi-rigid connections have been investigated with a moment resistance similar to or less than that of the adjacent composite beam in negative bending for the purpose of providing significant ductility in the connection rather than in the adjacent member. As these rotations are located at the notional plastic hinge, the available inelastic rotation Q^^^ in Eqn.5 is equated to the rotation in the connection at the ultimate moment M'p. A review of these composite connections has recently been provided by Leon (1998) in which a range of appropriate details are described. If the moment resistance of the connection is less than the adjacent yield moment in the composite beam, the available inelastic rotation due to yielding will be negligible. Nethercot et al (1995) have evaluated the required rotations in such cases.
itnf»»niMiiMriiiiiiiiinmnt»inimiiiiinniriiiinmnnmiiiimiiimirimm AS BS C S"5
I
(a)
(c)
i /
^
Figure 4: Inelastic curvatures (a) and (b) yielding in negative and positive bending; (c) cracking in composite beams
If a partial height, end-plate connection is adopted which does not provide for the transfer of tension force in the upper region of the steel section, the ultimate moment is obtained from the couple formed by the tensile force in the slab reinforcement and the compressive force in the bottom flange and adjacent web of the steel section. Kemp et al (1995) have demonstrated how local and distortional buckling is limited by reducing the proportion of the steel section in compression in this way. Excellent ductility has been achieved in tests in which the rotation capacity was increased more than sixfold over the normal rigid endplate connection. As yielding of the reinforcement is common to the behaviour of both the connection and the adjacent beam, it is proposed that the combined inelastic distortion be considered as part of the yielding of the adjacent beam in accordance with Eqns. 6 and 7. Cracking of the concrete, 0acr in composite beams If the uncracked and cracked elastic flexural rigidities of the composite beam in negative bending are EI and EI' respectively and the moment at which cracking of the concrete effectively occurs is M'^, the associated distribution of inelastic curvature in composite beams is shown in Figure 4c.
297 Calculation of available inelastic rotation The preceding evaluation of the four components of available rotation defined in Eqn. 5 is based on measurements made in a wide range of double-cantilever tests. To assess the limit state of ductility (Eqn. 1) for the representative three-span structure in Figure 1, it is necessary to know whether these test results are applicable to the notional plastic hinge model and on what basis the different components of available rotation should be added together as shown in Eqn.5. The virtual work approach of Eqn.2 may be extended as follows to assess the available rotation Q^ at the notional plastic hinges at B and D in the three-span structure of Figure 1, based on the inelastic curvature distributions shown in Figures 4a to 4c and the connection rotations:
p*ea = e.=LJM*6 + i:M*„„e_
(9)
For this purpose P* is a pair of unit virtual moments in the directions of the change in slope on either side of any notional plastic hinge, M* is the resulting distribution of virtual moments in Figures Id and le, 6 is the distribution of available inelastic curvature in the central span in Figures 4a to 4c, M^^on is the virtual moment at each end connection and Q^^^^^ is the concentrated rotation in the end connection. If the distribution of curvature in Figures 4a and 4b is simplified by replacing the curved relationship by the linear function shown by the dashed line, the following expression is obtained for 6^^ or d^ from Eqn.9: 6aB = 9aD = ^'ay " ^ay + 9acon+ ^acr
= 0.5 (j)',y,LV - 0.333 (|),yLy + 6,,,„ + 0.5 L'„ [(l/EI) - (1/ED]( M^ + M'„)
(10)
in which all the terms are defined in the preceding descriptions of available rotation or in Figures 4a to 4c. The overestimate of O^B caused by replacing the curved distributions of curvature (shown by the solid lines in Figures 4a and 4b) by the dashed linear distributions is less than 2% over a wide range of practical parameters. Due to the symmetrical distribution of inelastic curvatures in the central span, the integral JM* 5 is directly related to the area of each curvature distribution irrespective of the length over which the curvatures are distributed. The distribution of total curvature in the central span is given by the elastic distribution in Figure Ic from which the notional required inelastic rotations are derived, plus the inelastic distributions in Figures 4a to 4c from which the notional available inelastic rotations are determined. These two rotations form the basis of the limits-state criterion of ductility in Eqn. 1 and are compatible as they both refer to the rotation of notional hinges at the supports. Non-Dimensional Form The limit-states criterion in Eqn.l can be expressed in non-dimensional form by considering the ratio of available to required rotation Q^/Qr obtained from Eqns. 4 and 5: ra = Qa/Q'e = (QjQXV&e) = [2Ymd m{ 1 + 0.667(W L,) / (Ely Ey}] / [ ( l - m ) ( l - l / y r ^ ) ]
(11)
in which r^^sO^^ /O'^ is the non-dimensional available rotation capacity obtained by dividing the total available rotation 6^ in Eqn. 10 by the elastic rotation 6'^= 0.5 Mp Lj / EI^. in the test specimen in Figure 2 at moment Mp; Lj = 0.5Lc(l - l/\/l +n) is the length between the section of maximum negative moment Mp and the adjacent point of inflection; EI^, is the flexural rigidity in the central span; n is the ratio of moment resistances Mp / Mp and m is the proportion of moment redistribution defined in Eqn. 4. This equation identifies the factors influencing the amount of rotation capacity which should be available
298 to satisfy the limit-states Eqn. 1 for a specified partial material factor Ymd- Thus, for example, for m=0.3 representing 30% redistribution, n=0.7 and constant ratio of length to flexural rigidity (L/EI) in all spans, a partial material factor of 1.3 is achieved by providing an available rotation capacity of ra=8.0. Conclusions A consistent limit-states equation for ductility in continuous composite and steel beams is satisfied if the available inelastic rotation at each notional plastic hinge (the resistance) sufficiently exceeds the required inelastic rotation (action effect) at the same hinge due to moment redistribution from the elastic condition. The Principle of Virtual Work provides a consistent basis for determining both the required and available rotations at the notional plastic hinges. This principle also enables the components of rotation due to yielding of the steel section, rotation of the end connection and cracking of the concrete to be appropriately combined at each notional hinge on a rational basis. For symmetrical arrangements the rotation at each notional hinge is equal to the area under the inelastic curvature distribution and is therefore compatible with rotations measured in double-cantilever tests. References Bradford M.A. and Kemp A.R. (1999). Buckling in Continuous Composite Beams. Accepted for publication in Progress in Structural Engineering and Materials. Couchman G. and Lebet J.-P. (1996). A New Design Method for Continuous Composite Beams. Structural Engineering International 6:2, 96-101. Gioncu V. and Peteu D. (1997). Available Rotation Capacity of Wide-Flange Beams and Beam-Colunms. Journal of Constructional Steel Research 43:1-3, 161-218 and 219-244. Johnson R.P. and Hope-Gill M.C. (1976). Applicability of Simple Plastic Theory to Continuous Composite Beams. Proceedings of the Institution of Civil Engineers, London Part 2 61:3, 127-143. Kemp A.R. and Dekker N.W. (1991). Available Rotation Capacity in Steel and Composite Beams. The Structural Engineer 69:5, 88-97. Kemp A.R., Trinchero P. and Dekker N.W. (1995). Ductility Effects of End Details in Composite Beams. Journal of Constructional Steelwork 34:2, 187-206. Kemp A.R.. (1996). Inelastic Local and Lateral Buckling in Design Codes. Journal of Structural Engineering, ASCE 122:4, 374-382. Leon R. (1998). Composite Connections. Progress in Structural Engineering andMaterials 111, 159-169. Li T.Q., Choo B.S. and Nethercot D. A. (1995). Determination of Rotation Capacity Requirements for Steel and Composite Beams. Journal of Constructional Steel Research 32:3, 303-332. Nethercot D.A., Li T.Q. and Choo B.S. (1995). Required Rotations and Moment Redistribution for Composite Frames and Continuous Beams. Journal of Constructional Steel Research 35:2, 121-164. Rotter J.M. and Ansourian P. (1979). Cross-Sectional Behaviour and Ductility in Composite Beams. Proceedings of the Institution of Civil Engineers, London. Part 2. 67:3, 111-\ A3.
Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved
299
DUCTILITY OF THIN-WALLED MEMBERS A. Moldovan \ D. Petcu ^ and V. Gioncu ^ * Building Research Institute, 1900 Timisoara, T. Lalescu 2, Romania West University, 1900 Timi§oara, V. Parvan 4, Romania ^ Polytechnical University, 1900 Timi§oara, T. Lalescu 2, Romania
ABSTRACT The study of ductility for thin-walled members finally reduces to the analysis of local plastic mechanisms for steel plates subjected to in-plane compression. The type of mechanism determines the rigid-plastic curve shape of the plate, which is important for estimating the ductility of thin-walled members. In evaluating the plastic rotation capacity three types of possible mechanisms are studied: pyramid, roof and flip-disc shapes mechanisms. A computer program named DUCTROT TWM was elaborated by the authors for rapidly determine the ultimate rotation and the ductility of a thin-walled section. Comparison with the experimental results showed that the obtained theoretical rigid-plastic curves can be used to draw the framework aroimd the actual behaviour of thin-walled members. A rotation capacity of about 1.7 can be used for seismic design.
KEYWORDS Thin-walled members, plastic mechanism, rigid-plastic curve, ductility, reduced plastic moment, computer program. INTRODUCTION Thin-walled steel structural members may lead to a more economic design as a result of ease of construction and of their superior strength to weight ratios. In seismic areas the use of cold-formed profiles has not attained the same importance as hot-rolled ones because their bending behaviour is strongly affected by the local buckling of the compressed part. The evaluation of the complete loaddeformation history pointed out that the exclusion of thin-walled profiles in dissipative zones of seismic resistant structures could be revised. For one or two levels structures and for low or medium earthquake intensities, seismic loads have a lower influence and thin-walled members can be used. The paper has the aim to provide useful data on the use of these profiles in seismic resistant structures. The
300
ductility of thin-walled sections in the plastic range is studied by evaluating the plastic moment, the ultimate moment and the rotation capacity of thin-walled members. PLASTIC MECHANISMS FOR THIN-WALLED MEMBERS Thin-walled steel members often develop local buckling and local plastic mechanisms, both of which are essential features determining their behaviour. The study of plastic mechanisms for thin-walled members is governed by the local plastic mechanisms for thin-walled steel plates subjected to in-plane compression. These plates can develop three types of local plastic mechanism, namely the pyramidshaped, the roof-shaped and the flip-disc mechanism. For doubly supported plates they are observed in Fig.l (a - pyramid-shaped, b - roof-shaped, c - flip-disc).
u '
a)
Figure 1: Types of plastic mechanisms for doubly supported thin-walled plates The type of mechanism determines the rigid-plastic curve shape of the plate. The intersection of elastic and rigid-plastic curves can be used to estimate the ultimate load and the framework around the actual behaviour of thin-walled members can be drawn. For the idealised roof mechanism, the average axial stress a is given by eqn.(l), derived by Mahendran and Murray (1991):
(l-r(fj-l
~(l^r)^K (1)
c —s b
4(l + r)YA
+1
2(1+ r) A k t
-In 2(l + r)A
4(l + r ) V A
+1
2(1+ r) A
where k=cosec^ a + cosec^ p, A is the deflection of the flange and fy the yield stress. The mechanism was considered in two parts, the inner region of width b-2c and the two identical edge regions each of width c. In the above mentioned paper, the basic geometric parameters a, c and r were assumed to be 30°, 0.2b and 0.6, respectively, to obtain the rigid-plastic curve. For the idealised flip-disc mechanism of parabolic shape, the average axial stress a was obtained by eqn.(2). Post-plastic behaviour for the pyramid mechanism was studied by Feldman (1994), who obtained its characteristic equation (3). It was concluded that the value given by eqn.(3) remains practically unchanged for a = 45°.. .55°, so a = 45° may be considered.
301
— = 1.
^1.1
-+4
+1
.2\
1 + 4-
— = 1. /."2
2A
+1
2A
/:/ ,
+—\ni kt 2A
(2)
1+4 b'
2A|
(3)
kt
where k=l+cosec a. The eqns.(l)...(3) determined for doubly supported plates, can be used also for simply supported plates if half of the mechanism shape is used, as Gioncu and Mazzolani (1999) have mentioned. So these mechanism types can describe the collapse shapes of beam flanges. Theoretical and experimental research developed by Batista (1989) showed that for simply supported thin-walled plates, flip-disc mechanisms are formed. The authors of the present paper observed that for the flip-disc mechanism, the variation of ratio a/b from 0.1 to 0.5 gives small differences for the value of plastic moment, so the ratio a/b was assumed to be 0.1. In evaluating the ductility of thin-walled members, a type of section commonly used for beams and columns was considered: I section formed of two U profiles. Since the web of this section has the thickness of 2t, where t is the thickness of the U profile, the local buckling will only affect the compressed flanges. The three types of plastic mechanisms discussed above were taken into account. The local buckling of the compressed flange causes the decrease of its capacity, which is expressed by: C(0) = C,
(4)
fy where Cp is the load at which flange is fiilly plasticised. a/fy depends on the type of the mechanism. The reduction in the width of the compressed flange because of local buckling causes neutral axes to shift from the midpoint of the web. Distance z from the compressed flange to the neutral axes is determined from internal forces equilibrium (Fig.2) and is given by eqn.(5):
C(0 )
T ^ •
2
m i
\
rr^/ ' . — Figure 2: Intemal forces equilibrium for I section
1
J
--^1
302
h z = —1 +- 1 - ^ 2h V 2 Jy)
(5)
and the rotation 9 from eqn (6): (9 = 2
A t h t
1
(6) a
2hy
where b is the width of the flange and h is the depth of web for the I section. Plastic moments become: M = Cz +I(h-z)+fytz^ +fyt(h-z)^
(7)
where C, I and z are functions of rotation 9. Hence, the rigid-plastic curve can be drawn. The ductility of thin-walled beams can be established by intersection of the rigid-plastic curve and the reduced plastic moment curve affected by the safety factor YM. Reduced plastic moment is the plastic bending moment for the effective section of the member. DUCTROT TWM COMPUTER PROGRAM Since the method presented above for evaluating the plastic rotation capacity is complicated enough, a computer program named DUCTROT TWM ( DUCTilitv RQJation of Thin-Walled Members) was elaborated by the authors of the present paper. The ultimate rotation 0r and the ductility of a section can be determined with its contribution. The program consists of 8 sheets. Some of the most important of them are presented in Fig.3. Sheet 1. Types of sections considered: built-up U section, built-up C section, cold-rolled box section, welded I section. Sheet 2. Material characteristics. The user introduces: yield limit, uhimate tensile strength, yield strain, hardening strain, ultimate strain, modulus of elasticity, tangent modulus. Sheet 3. Geometrical characteristics. The user introduces: half-width of the flange, depth of the web, thickness of the section, radius of comers. DUCTROT TWM gives: cross section area, effective cross section area for bending and for compression, position of centroid, gross and effective second moment of area, elastic and plastic section modulus, effective elastic and plastic section modulus. Sheet 4. Loading system. Standard beam may be acted by one or two equal concentrated loads, with or without axial load. Type of mechanism can be in-plane (symmetric or asymmetric) and out of plane. Sheet 5. Mechanical sectional characteristics. The program gives: plastic moment and reduced plastic moment, plastic and reduced plastic axial forces. Sheet 6. Geometrical member characteristics. DUCTROT TWM gives: slendemess of the element, plastic rotation and ultimate rotation.
303 Ahooi DUCTROT r\i/M 98
DUOTROT-TWM
•0®f»:-^Tar, yki^^^^.Mim&imsmm
IDUCTHOT 37
gSi^^^^N^^^^^ S;tem&&rCfi!9i*i^^ ?J^«M\ bending moment in Unear behaviour
312
As a synthesis of the curvihnear M - Or analytical models, the EUROCODE 3 recommends, EUROCODE 3 (1993), two models for unbraced and braced planar frames (Table 2). TABLE 2 RECOMENDED A/- Or MODELS BY EUROCODE 3
Analytical model EUROCODE 3 model 1 for frames: * unbraced
Analytical expression
Remarks
|
_ M m= S _ 256^^4 2 _ ,^ /w = '- , -<m)=0
Nf-Mint (u,(j>)^0
(2)
where A^/„/ (u, (j)) is the internal axial force and Mint (u, (j>) is the internal bending moment in the section. They can be expressed in the following form: N,,,{uJ)^\(j{u,(j>)dA A
(3)
Mi„,(«,(*)=Ja(»,^)yrf/(
The curvature cj) and the axial strain u are the solutions of Eqs. (2) that may be rewritten as follows: (4)
M{u,(l>)=M,,,{u,(l>)-M'=(^
The Eqs. 4 are solved numerically using the modified Newton-Raphson method, and results in two recurrence relationships to obtain the unknowns u and (p. The stress-strain relation of the material is assumed as elastic-perfectly plastic, and a tri-linear type of curve such as shown in Figure 2 is used in the analysis. The elastic modulus E = 29000 ksi and fy = 32.4 ksi. Two types of residual stress pattern was considered: type 1, European linear distribution (Fig.l.a) and type 2, US constant distribution (Fig. l.b). Using these data and the recurrence relationships obtained from Eqs.4 the M-N-O curves of a W8x31 section were computed for different N/Np ratio considering, both compression and tension, with and without residual stresses, as they are shown in Figure 4.a and b. A detailed layout of the procedure used to consider the development of plastification, second-order effects and non-linear behavior of the connections, is presented by Barsan and Chiorean (1999). These curves compare favorably to those computed by Kanchanalai (1977) presented by Chen and Toma (1994). b. Compression
a. Tension
0
0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 Curvature
0
0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004
Figure 4: M-N-O relationship of section W8x31
CurvMure
320
NUMERICAL EXAMPLES Based on the mathematical formulation above described an object-oriented Turbo-Pascal program, NILDFA (Non-linear Inelastic Large Deflection Frame Analysis) has been developed to study the influence of residual stresses on the behavior of steel-framed structures. It combines the structural analysis routine with a graphic routine to display the results, as deformed shape of the structure, M, T, N diagrams, percentages of section areas yielded at ultimate stage, in compression or tension, loaddeflection response curves and the development of plastic zones in the cross-sections and along the members of the structure. This program was used to study the influence of residual stresses on the behavior of steel frames. As a standard for comparison three types of frames were selected: a portal frame first analyzed by El-Zanaty (1980), a six-story two-bay frame (Vogel , 1985) and a two-story frame that has been proposed by Ziemian (1992). Three residual stress patterns, constant, linear and parabolic distribution in the web has been considered for calibration frames. The differences between the linear and parabolic distribution was irrelevant so only the linear (type 1) and constant (type 2) distribution have been represented. Example 1: Portal frame The El-Zanaty (1980) portal frame, member and material properties, and applied loading are shown in Figure 5. The plastic zone developments are represented in the Figure 6. In Figure 7 the lateral-load displacement curves are represented for type 1, type 2 distributions of the residual stress, and without residual stress. Example 2: Six-story frame The Vogel's six-story frame, member and material properties, and the applied loading are shown in Figure 8. In this example the influence of residual stress on the inelastic behavior of the structure and a comparison of computer time necessary to carry out the analysis using NEFCAD and NILDFA programs are presented. Comparable lateral load-displacement curves for the sixth-stories are shown in Figure 9. The NEFCAD program (a^l and n=35) results are in close agreement with NILDFA program where the effects of residual stresses are explicitly taken. Computing time for NEFCAD program was 100 times shorter than the time taken to complete the same analysis with NILDFA program. Percentages of section-areas yielded (Fig. lO.a) and spread of plastic zones in the characteristic cross-sections (Fig. lO.b and Fig. 1 l.a,b,c) are represented. Example 3: Two-story frame The Ziemian's two-story frame, member and material properties, and applied loading are shown in Figure 12. The percentages of section-areas yielded are represented in Figure 13 for all three cases considered and the lateral load-displacement curves, for the same three cases, are shown in Figure 14. CONCLUSIONS In order to assess explicitly the influence of residual stresses on the inelastic behavior of the structures, and in particular, on the carrying-capacity of the steel frames, and to compare this to results obtained
321 by NEFCAD program, a new computer program NILDFA (Non-linear Inelastic Large Deflection Frame Analysis) was elaborated by the authors.
ih r-i
HEA340
With residual stress, type2, Rim=1.03 With residual stress, type 1. Rim=1.03 Without residual stress, Rim=1.06
%=i/4UU
nfn
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
D(m)
Figure 5: Example 1. Portalframedescription
W
45.6 (38.3) [0.0]
51.2 (48.8) [45.6]
Figure 7: Lateral load-displacement curves for portal frame
52.0 (45.2) [41.1]
62.6 (57.6) [57.5]
a. Percentages of section areas yielded at ultimate for the portal frame Type 1 distribution. (Type 2 distribution.) [Without residual stress]
T T b. Type 1
c. Type 2
Figure 6: Spread of plastic zones for portal frame
d. Without residual stress
322
23 .-K
1M j M i ii M 1 1 i i MM iPE240
.4
Kh.
lilillMI
o
IPE300
H,
1 i
i
IMIMIM
oj
MIMMII
1 1 1 ! 1 1 IPE300 I
x|
! 1 1 M!1 M
H,
Ml
c
IFE330
-,
: M :MM
H,
i
1 ; 1 1 i
1 1 1
X
ol
1
l|
1 1 i 1 1 i! ! 1 i ! i i l 1 IPE400
U - c':
o
/
¥
Figure 9: Lateral load-displacement curves for sixstory frame Figure 8: Example 2. Six-story frame description 89.6
[90.2|
,-1
yflTV
96.6
'^-^J ^
t
75.7 (74.8) (77.01
-1 yf[)
86.2 (86.6) |86.6|
Value indicates total % of section area yielded, Type 1 distribution (Type 2 distribution) [Without residual stress]
r Al
- Tension
Cross section 1-1
T
Cross section 2-2
T
Compression
Cross section 3-3
3 3 Figure lO.a: Percentages of section-areas yielded
Figure lO.b: Spread of plastic-zones
323
a. Type 1 distribution
b. Type 2 distribution
c. Without residual stress
} \^mc n I he spread of pla 0.15, it is checked
that:
Ns/Npi,R + U5X = 0.22 +1.35 x 0.46 = 0.84 < 1.0.
0,716m H 0,468m H
0,468m
. 0,223m
\
mM
W W
MM
m iLk X& m
m m
M
V(p.i)=1609kN
V*'2> = 3606kN
V*''>=1292kN
V(P'4) = 4242kN
Figure 8 : Checking of base shear force corresponding to various local plastic mechanisms (1) b. To illustrate also the procedure in the case of partially dissipative structure, the same structure is considered with partial resistant joints whose characteristics are rotational stiffiiess Sj = 137300kNm, resistant moment Mj^ =0.8Mpj^ = 177kNm and rotation capacity
