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The remaining load-bearing capacity of corroded steel angle compression members
Journal of Constructional Steel Research 120 (2016) 188–198
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Journal of Constructional Steel Research
The remaining load-bearing capacity of corroded steel angle compression members Katalin Oszvald ⁎, Pál Tomka, László Dunai Budapest University of Technology and Economics, Department of Structural Engineering, H-1111 Budapest, Műegyetem rkp. 3, Hungary
a r t i c l e
i n f o
Article history: Received 23 April 2015 Received in revised form 14 December 2015 Accepted 1 January 2016 Available online 15 January 2016 Keywords: Corrosion Steel Angle-section Compression member Stability Remaining capacity
a b s t r a c t The paper presents a study on the buckling of corroded equal-leg angle-section members. The remaining loadbearing capacity and the prospective behaviour modes are analysed by experimental and numerical research. The diversity of the corrosion is taken into consideration in the analysis; the corrosion is modelled by thickness reduction. Compressive buckling tests, finite element and analytical studies are completed to analyse the modified buckling behaviour and the ultimate load. On the basis of the results simplified design method is developed for the prediction of remaining compressive resistance. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction 1.1. General Generally, the corrosion is one of the major problems during the lifetime of steel structures. The corrosion can be appeared on every structure irrespective of where those can be found. The following structures are considered particularly sensitive to corrosion: bridges, towers, transmission line columns, offshore structures and pipelines. The unfavourable weather conditions and the lack of maintenance are leading to corrosion damages. There is no standardized method or process on how to consider the effect of corrosion in the analyses of the elements [1]. The application of an average cross-section is typically proposed by the standards, but in general this estimation is not accurate enough due to the diversity of the corrosion. The corrosion appears on many structures where the conditions are suitable for its development. Every structure has a typical and a possible type of corrosion depending on its location and the operating conditions. The lattice towers are the most common structural types which are exposed to the effects of the environment. Their typical corrosion types are as follows [2]: (i) uniform corrosion, (ii) pitting corrosion and (iii) crevice corrosion. The current research focuses on these types of structures and the first two corrosion types are considered. According to [3] these do not cause changes in the material properties.
⁎ Corresponding author. E-mail address: [email protected] (K. Oszvald).
http://dx.doi.org/10.1016/j.jcsr.2016.01.003 0143-974X/© 2016 Elsevier Ltd. All rights reserved.
The corrosion damage of the lattice towers is an existing problem all over the world, as it is shown by the different fields of research studies about this problem. Some years ago the Hungarian Electricity Co. (MVM) had published a report on the corrosion of the structural elements and connections of these towers, emphasizing the importance of this problem [4]. The Réseau de Transport d'Électricité (RTE in France) also reported that the corrosion on the lattice towers is also significant in France. Reiner introduced a visual inspection process of lattice towers by a flying machine [5]. Fig. 1 shows the corroded lattice towers which are taken by this device. Based on the observations on the existing corroded lattice towers it can be stated, that both chord and bracing angle members are damaged, and generally, the horizontal members are more sensitive to this effect (Fig. 1). Previously only few experimental studies were completed on corroded angle section members. Beaulieu et al. [1] investigated steel angle members, simulating the corrosion by galvanic process. The main studied parameters in their work were the slenderness, the width-to-thickness ratio and the extent of corrosion. The specimens were experimentally investigated in a truss structure under eccentric compression. The failure modes and the compressive forces were determined and compared to analytical results. The observed failure modes were the (i) global buckling, (ii) the local buckling near to the connection, (iii) the local buckling near to the centre, and in some cases it was not clearly identified in the test. On the basis of the results of the tests and the related analytical calculations the authors proposed a method using an average residual thickness of corroded member to estimate the buckling capacity according to the ASCE 10-97 code. The proposed method, however, is not able to consider the localized corrosion.
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Fig. 1. UAV flying machine and photographs of lattice towers taken by UAV [5].
Therefore, further studies are necessary. Recently Japanese researcher published a study about severely corroded angle members [6], where 17 corroded angle and 10 corroded channel section specimens taken from real structures were analysed. The support condition was fixed on both ends and axial compression was applied through end plates. The failure modes were detailed and a prediction method is proposed. Based on the previous studies it is clear that the analysis of the corroded angle members under compression is incomplete. The diversity of the corrosion is not taken into consideration, and only one corrosion parameter – the thickness reduction – was applied in the previous research studies. 1.2. Purpose and scopes The purpose of the research is to extend the knowledge about the corroded angles under compression by a more complex analysis of the damages. It is aimed to develop an applicable design procedure
for the remaining load-bearing capacity of these members. The proposed checking procedure should predict the ultimate behaviour mode and the ratio of the ultimate buckling forces of corroded to the non-corroded members. In the research strategy the typical corrosion patterns and the characterizing parameters are defined at first. Then in the next step experimental, numerical and analytical analyses are carried out considering the diversity of corrosion by considering the following parameters: thickness reduction, extension of corrosion, location of corrosion along the length and corrosion pattern. Finally design methods are derived for the remaining buckling resistance. Chord and bracing angle members of a lattice tower are shown in Fig. 2. The effect of loading and supporting conditions on the stability behaviour are considered according to the two structural components. In the case of chord members the axial compression is applied in the centre of the gravity, while in the case of bracing members the load is applied through bolted connection to one leg of the chord.
Fig. 2. Members of lattice towers with double circuits.
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Table 1 Corrosion parameters. Ext = Lcorr / L [%] pc = Lpc / (L/2) Tred = tcorr / t [%]
2. Experimental and numerical analyses 2.1. Experimental study In the first phase of the research experimental tests are carried out. In the test programme the investigated specimens can be divided into two main groups according to the end support. In the first group the support is hinged in the centre of the gravity of the non-corroded cross-section (hereinafter called chord members). The second group contains the eccentric loading, the specimens are connected by one or two bolts to one leg of the chord (bracing members). Altogether 22 and 24 specimens are tested; in each group non-corroded members are also analysed, as reference elements. The original cross-section is 40 × 40 × 4 mm and the total length of the specimens is 840 mm. The corrosion is modelled as thickness reduction, which is worked out by the mechanical milling process. The main corrosion parameters are changed during the preparation of the specimens. These parameters are as follows (see Table 1): the extension (Ext) and the location of corrosion (pc) along the length, thickness reduction (Tred) and the corrosion pattern: one (“A”) or both legs (“B”) corroded. The exact corrosion parameters, the geometrical properties and the results are published in [7,8]. The test set-up of the two different supporting cases is shown in Fig. 3. The main conclusions and the results of the tests can be summarized as follows. The type of the failure mode highly depends on the measure of corrosion. If the thickness reduction is less than around 50%, the failure mode is dominant flexural buckling (the same as for non-corroded members). If the thickness reduction is higher local plate buckling is observed. These statements are valid irrespective of the loading type, as centric or eccentric. In the case of bracing elements beside the dominant
flexural buckling mode, some torsion also occurred due to eccentricity; actually it is called as a general case of torsional–flexural buckling mode. In Fig. 3 all types of failure modes and element types are presented. During the tests horizontal and vertical displacements are measured. Typical load–horizontal displacement curves are presented in Fig. 4. In every case, when the ultimate failure mode is local plate buckling a sudden decrease of load is observed (Fig. 4, specimens L1, PI-8 and PII-8). The load-bearing capacity is determined in all cases, and the ratio of the corroded and non-corroded elements (Nb,m/Nb0,m) is plotted in the function of the maximal cross-section reduction (Mcorr) in Fig. 5 (the Mcorr is interpreted as the ratio of the maximal cross-section reduction along with the member compared to the gross area). The continuous line supposes the same reduction of load-bearing capacity of the corroded members as the Mcorr. This is a reference line which shows that in some cases the difference from this approximation is about 20%. This result clearly shows that in general one parameter alone is not enough to characterize the corrosion and to take into consideration the effect of it on the remaining load bearing capacity.
2.2. Numerical study The experimental results are extended by a numerical parametric study to analyse the effect of the joint influence of the corrosion parameters on the buckling phenomena and to predict the prospective ultimate failure mode in the function of the corrosion. In the numerical study basically shell finite element model is developed in Ansys programme environment [9]. Furthermore in the case of bracing members the bolted connection is modelled by beam finite elements. The corrosion in the models is considered by thickness reduction. The model of the bracing member is shown in Fig. 6(a). In the model a 4-node shell element (SHELL 181 element of Ansys) is applied, which can model thin and moderately thick plated structures and it is well-suited for large strain nonlinear applications. The steel is modelled by linear elastic-perfectly plastic material (modulus of elasticity 210,000 N/mm2; Poisson's ratio 0.3). In the numerical model the geometry of the corroded test specimens and the measured yield strength (fy = 345 N/mm2) is used. At the end of the element rigid region (CERIG — Ansys command) is defined and in the centre of the gravity of non-corroded cross-section hinge connection as a support is applied.
Fig. 3. Test set-up and the typical ultimate behaviour modes; (a), (d) — local plate buckling; (b), (c) — general case of torsional–flexural buckling.
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Fig. 4. Load–displacement curves: (a) centric loading; (b) eccentric loading.
Fig. 5. Experimental results in the function of Mcorr reduction: (a) centric loading; (b) eccentric loading.
In the case of bracing members the components of bolts, as head, stud, and nut are modelled by beam elements with equivalent properties. In the case of the stud the beam element (BEAM188) has the same cross-section properties as the applied bolts in the tests. This element is based on Timoshenko beam theory and shear deformation effects are included. In the case of the nut and head BEAM44 [9] element is applied, which is a uniaxial element with tension, compression, torsion, and bending capabilities. These beam elements are connected by joint end to the shell elements and by fix end to the stud. The parameters of the cross-section are determined to get the same cross-section
properties as the real bolt. The model also contains the leg of the T support element. In the frame of the numerical study programme bifurcation stability analysis (geometrically non-linear buckling analysis — GNB) and geometrically and materially non-linear imperfect analysis (GMNI) are carried out. As a first step, the test specimens are analysed by the GNB. The results of the GNB analyses provided the buckling load together with the buckled shape to define the initial geometrical imperfections for the nonlinear analysis. The amplitude of the initial geometrical
Fig. 6. (a) Detail of numerical model and (b) typical load–displacement curves of bracing element specimen.
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Table 2 Results of preliminary numerical study on chord members. ID
Location of corrosion
Mcorr [%]
Nb,m [kN]
Nb,NUM [kN]
Nb,NUM /Nb,m
O2 A1
–
0.0 29.0
50.33 33.2
48.73 32.44
0.97 0.98
A2
12.8
51.02
44.6
0.87
A3
11.7
41
40.9
0.99
A4
14.7
50.95
43.60
0.85
A5
12.7
35.5
34.4
0.97
A6
18.8
33.6
33.6
1.00
A7
25
43.8
42.81
0.98
A8
25
37.7
37.03
0.98
A9
13
43.68
42.16
0.97
A10
13
38.5
38.6
1.00
A11
13
36.93
37.04
1.02
L1
70
14
12.88
0.92
L2
73.5
9.6
9.15
0.95
L3
76
4.9
4.99
1.02
L4
49.8
28.3
27.9
0.98
L5
27.8
34
33.98
1.00
Mean value Average deviation [%]
imperfection is calibrated by the test results. It means that the applied magnitude is determined to get the same load-bearing capacity in the numerical analysis as in the test. Thus obtained values were various on the different elements. In the case of chord members, the interval of applied magnitudes of geometrical imperfection is between L/400– L/2000 (2.1–0.42 mm), where L is the length of the member. In the case of bracing member, the interval of magnitude values is between L/200–L/1500. Seventeen investigated members of the preliminary study of chord member are illustrated in Table 2. In the schematic figure of the specimens, both legs are illustrated and the black colour marks the corrosion Table 3 Corrosion patterns and parameters applied for chord members. Pattern
Tred [%]
Ext [%]
Corrosion position (pc)
20 30 40 50 60 70 80
20 30 40 50 70 100
0.20, 0.47, 0.73, 1.00 0.30, 0.53, 0.77, 1.00 0.40, 0.70, 1.00 0.50, 0.75, 1.00 0.70, 1.00 1.00
0.97 4.1
as in previously. The position of the corrosion is detailed on the figures. The cross-section reduction of the members is given in Mcorr column. Table 2 contains Nb,m, the measured load-bearing capacity of the test and the of the numerical models: Nb,NUM. The ratios of the numerical analysis to the test results are also presented in Table 2. In the GMNI analyses the ultimate behaviour and load-bearing capacity are determined. The results showed good agreement with the test in the prediction of the ultimate behaviour and in the loadbearing capacity, as it can be seen on the typical load–horizontal displacement curves in Fig. 6(b) and in Table 2. More details of the models and its verification are published in [10,11].
Table 4 Corrosion patterns and parameters applied for bracing members. Pattern
Tred [%]
Ext [%]
Corrosion position (pc)
20 30 40 50 60 70
20 30 40 50 60 75
0.45, 0.63, 0.80, 1.00 0.55, 0.70, 0.85, 1.00 0.65, 0.82, 1.00 0.75, 0.87, 1.00 1.00 1.00
K. Oszvald et al. / Journal of Constructional Steel Research 120 (2016) 188–198 Table 5 Geometric properties in the parametric study. Type
Cross-section [mm × mm × mm]
Relative slenderness
Chord member
40 × 40 × 4 60 × 60 × 8 100 × 100 × 12 40 × 40 × 4
0.7, 0.9, 1.15, 1.3, 1.5 0.7, 1.15, 1.5 0.7, 1.15, 1.5 1.15, 1.5, 1.92
Bracing member
In the numerical study four different corrosion parameters are applied (Table 1): (i) thickness reduction (Tred), (ii) extension (Ext), (iii) location (pc), and (iv) corrosion pattern. Table 3 contains the corrosion parameter of chord members. In the case of bracing members, Table 4 summarizes the parameter values in detail, referring to one and two bolted end connections. Further parameters of the study are the gross cross-section and the relative slenderness of the members as detailed in Table 5. Note that in the case of bracing elements the connections are considered as not corroded. In the case of chord members altogether 2618, while in the case of bracing members 1152 cases are analysed in the parametric study. The GNB analyses define the typical buckling modes, as follows: (i) flexural buckling (FB), (ii) torsional–flexural buckling (TFB) in the corroded zone, (iii) local plate buckling (LPB) and (iv) general case of torsional–flexural buckling (GTFB). Based on the results three zones (marked by I, II and III) are determined according to the buckling modes. The zones are defined by the width to reduced thickness ratio (b=t) of the corroded zone. In the cases of zones I and III the buckling mode is clearly specified, as shown in Figs. 7–9. The b=t is the horizontal axis and the vertical marks the different relative slenderness ratios (λ) in
193
the figures. The zones are plotted for both corrosion patterns and marked by different colours. If the members belong to zone II, the buckling mode depends not only on b=t parameter but also on other corrosion parameters, such as the position and the extension. The limit values of zones depend on the initial relative slenderness ratio of noncorroded member, on the corrosion pattern and on the type of the support as well. In the GMNI analyses initial geometric imperfections are applied using the shape of the first buckling mode. The magnitude was taken as L/200, following the proposal of the standard [12]. The applied yield strength is fy = 235 N/mm2. The ultimate behaviours correspond to the previously identified stability modes. The results of the GMNI analyses are plotted in the function of the b=t parameter (Fig. 10). The vertical axis is the remaining load-bearing capacity compared to the non-corroded reference results, which can be separated according to the buckling modes (Fig. 10). Note that this observation is the basis of the simplified design method development. 3. Analytical study of the measured resistance In the case of equal-leg angle chord members if the buckling mode is FB (flexural buckling) the measured buckling resistance can easily be obtained using analytical methods. This has to be calculated for Class 3 cross-section by Eq. (1) of the Eurocode 3 [12] recommendations:
Nb;Rd
χ A fy ¼ ;λ ¼ γM1
sffiffiffiffiffiffiffiffiffiffiffiffi A fy N cr
Fig. 7. Definition of I–III zones according to the relevant buckling modes — Chord members.
Fig. 8. Definition of I–III zones according to the relevant buckling modes — 1 bolted connection.
ð1Þ
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Fig. 9. Definition of I–III zones according to the relevant buckling modes — 2 bolted connection.
Fig. 10. Results of GMNI analyses.
where fy = 345 N/mm2 (measured yield stress of the material used at the tests) in order to compare the calculated resistances to the measured ones, A the relevant cross-sectional area, χ reduction factor for buckling, γM1 partial safety factor and Ncr critical force according to the relevant value of [13]. The schematic test set-up is shown in Fig. 11. The buckling length is l = 840 mm, the profile is L40.40.4.
Table 6 contains the remaining resistance of the specimens showed in Fig. 12 (O denotes the non-corroded elements). The notations used are: Ncorr calculated resistance of the corroded specimens, N0 calculated resistance of the non-corroded specimens, Nb,m measured resistance of the corroded specimens Nb0,m measured resistance of the non-corroded specimens (average value).
3.1. The remaining resistance of columns with constant cross-section
If the corrosion is not extended to the whole length, but it is symmetrical about the middle plain of the member and the corrosion pattern is “B” (both legs are equally corroded) Eq. (2) by Timoshenko and Gere [13] can be applied to determine the critical force. Table 7 contains the analysed specimens and the figure on the right side gives the notations.
The critical force Ncr can be obtained: - if the element (O, A1. A5) has a symmetry axis from the quadratic equations of [13] (5-38) or (5-39), otherwise - in the case of elements A3 and A6 using the cubic equations of [13] (5.31) or (5.32).
3.2. Columns with varying cross-sections
Ncr ¼
π 2 EI2 4l
2
1 a l a I2 1 I2 πa þ 1 sin l l I1 π I1 l
ð2Þ
where: I1 moment of inertia of corroded section, I2 moment of inertia of non-corroded section, l buckling length of the element. Table 6 Critical forces, calculated and measured reduction of the resistance due to corrosion of A1, A3, A5 and A6 specimens with uniform profile.
Fig. 11. Schematic test set-up.
ID
Ncr [kN]
Ncorr/N0
Nb,m [kN]
Nb,m/Nb0,m
O A1 A3 A5 A6
54.63 36.95 48.76 44.06 41.99
1.00 0.68 0.89 0.81 0.77
50.5 33.2 41 35.5 33.6
1.00 0.65 0.81 0.70 0.66
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195
Fig. 12. Columns with constant cross-section.
Note: Eq. (2) is valid also for specimen A7, where the corroded zones are at the end of the specimen.
remaining load-bearing capacity of members with moderate or even medium penetration of corrosion.
3.3. The remaining resistance of members with varying cross-sections
3.5. Prediction of the ultimate behaviour mode
The buckling resistance here can also be calculated by the Eurocode 3 [12] recommendation Eq. (1) with the measured yield stress of fy = 345 N/mm2. The only difference is that in the formula for the relative slenderness ratio:
The results presented in Sections 3.1–3.4 are only valid if the buckling mode is flexural buckling. This section shows how to use the analytical methods of [13] to calculate the torsional–flexural buckling of equal-leg angles, providing that the corrosion extends to the whole length and the corrosion pattern is “B”. In this case the possible stability modes are the in-plane buckling (FB) and the torsional–flexural buckling (TFB).
sffiffiffiffiffiffiffiffiffiffiffiffi A fy : λ¼ N cr
ð3Þ
The cross-sectional area A is not constant. Therefore, as an approximation, the authors suggest the use of an equivalent cross-sectional area Aequ with the notation of Table 7 in the form of Aequ ¼ ðA1 ðl−aÞ þ A2 aÞ=l;
ð4Þ ð5; 6Þ
where A1 and A2 are the relevant cross-sectional areas. Table 8 contains the remaining resistance of the specimens from Table 7 (O denotes the non-corroded elements). The notations used are the same as above. 3.4. Analytical study, conclusions The previously calculated and measured results are given for the remaining resistance of the corroded members with constant- (Table 6) and varying cross-sections (Table 8). The calculations are made using the Eurocode provision [12], while the governing quantities (critical forces Ncr) were obtained according to the fundamental theoretical study of [13]. The calculated and measured results show an acceptable agreement: the max. deviation – as a basis of the measured ones – is approximately +10%. As the final conclusion, it can be said that the analytical approach of this chapter is able to give easy to handle procedures to calculate the
Table 7 Critical forces, calculated and measured reduction of the resistance due to corrosion of A7, A8, L4 and L5 specimens with varying cross-sections. Model by Timoshenko [13]. ID A7
Location of corrosion
where, Nu, Nv Euler critical loads for buckling about the u and v axes, Nϕ critical load for pure torsional buckling, IC and IO polar moment of inertia with respect to centroid and shear centre. The transition between TFB and FB can be easily determined by equating Ncr,TFB (Eq. (5)) with Ncr,FB (Eq. (6)). The limit of the transition is the Lcr,TFB value. If the analysed element is shorter than this value, the stability mode is torsional–flexural buckling (TFB), dominantly torsional. In other cases the stability mode is flexural buckling (FB). In the case of equal-leg angles the cross-section may be replaced with good accuracy by two intersection perpendicular flats, where the length bmod = h-tcorr / 2 and the thickness is obviously tcorr. Solving the previously mentioned equality of Eq. (7) is obtained. 2
Lcr;TFB ¼ 1:097b mod =tcorr
ð7Þ
The critical forces obtained by using FE model based on GNB analysis (Ncr,FE) and the analytical methods show good agreement and the critical length Lcr,TFB also fits to the results referring to the stability mode (Table 9). In the table the stability (buckling) mode, the numerical Ncr,FE results, the calculated Ncr,TFB (Eq. (5)), Ncr,FB (Eq. (6)) and Lcr,TFB are presented in the function of the assumed corrosion (Tred). The non-corroded profile is a uniform L40.40.4, the corrosion pattern is “B” through the whole length of the specimens. In the case of 60–80% thickness reduction the stability mode is TFB because Lcr,TFB N L which are marked by dark grey colour in Table 9. In
A8
L4
L5
Table 8 Calculated and measured reduction of the resistance due to corrosion of A7, A8, L4 and L5 specimens with varying cross-sections. ID
Ncorr
Ncorr/N0
Nb,m
Nb,m/Nb0,m
O A7 A8 L4 L5
41.14 36.04 31.97 25.96 30.89
1.00 0.88 0.78 0.63 0.75
50.5 43.8 37.7 28.3 34
1.00 0.87 0.75 0.56 0.67
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Table 9 Determination of the behaviour mode in the function of critical length (Lcr,TFB) in the function of Tred. Tred [%]
L [mm]
Stability mode
Pcr,FE
Pcr,FB
Pcr,TFB
Lcr,TFB
0
840
FB
54.5
53.50
148.28
395.9
20
840
FB
43.52
44.17
95.75
505.5
30
840
FB
38.00
39.25
70.49
583.6
40
840
FB
32.52
34.17
47.917
687.9
50
840
FB
27.06
28.92
29.40
834.0
60
840
TFB
17.1
23.49
15.70
1053.3
70
840
TFB
7.54
17.89
6.82
1418.7
80
840
TFB
2.304
12.11
2.05
2149.8
Table 10 Design method for elements with FB and GTFB behaviour modes. Ncorr/N0
Pattern “A”
Pattern “B”
1 bolt 2 bolts
1 - 0.01 ∙ (Tred/2) 1 - 0.01 ∙ (Tred/2) 1 - 0.01 ∙ (Tred/2)
1 - 0.01 ∙ Tred 1 - 0.01 ∙ Tred 1 - 0.01 ∙ Tred
Chord member Bracing member
Table 11 Design method for corroded chord angle members — TFB and LPB behaviour modes. Ncorr/N0
λ ¼ 0:7 λ ¼ 1:15 λ ¼ 1:5
Pattern “A”
Pattern “B”
0.995 - 0.023∙Mcorr 0.73 - 0.016∙Mcorr 1 - 0.0225∙Mcorr
1.098 - 0.013∙Mcorr 1.53 - 0.018∙Mcorr 0.79 - 0.009∙Mcorr
Table 12 Design method for corroded bracing angle members – TFB and LPB behaviour modes. Ncorr/N0 Pattern
Bolts
λ ¼ 1:15
λ ¼ 1:5
“A”
1 2 1 2
0.931 - 0.0101∙Tred 0.759 - 0.0087 ∙Tred 0.931 - 0.0112 ∙Tred 0.72 - 0.008 ∙Tred
0.662 - 0.0069 ∙ Tred 0.642 - 0.0067 ∙ Tred 1.965 - 0.0259 ∙ Tred 1.36 - 0.0174 ∙Tred
“B”
λ ¼ 1:92 0.804 - 0.0085 ∙ Tred
Fig. 13. Definition of the modified length parameter.
case of less than 60% thickness reduction the failure mode is flexural buckling, marked by light grey colour. Note again that this prediction is not applicable in the case of partial corrosion and “A” corrosion patterns. 4. Design method development On the basis of the numerical parametric studies, design methods are developed to calculate the remaining load-bearing capacity of corroded members. The members are separated according to the ultimate behaviour mode using the previously defined zones. If the ultimate behaviour mode is FB or GTFB the proposed design equation in Table 10 is applicable irrespective to the relative slenderness. If the ultimate behaviour mode is local plate or torsional–flexural buckling, Tables 11–12 contain the design equations. It shows that in these cases the equations depend on the corrosion pattern, on the slenderness ratio and on the type of the elements, as chord or bracing member (one or two bolts). These equations do not take into account the extension and the location of the corrosion, therefore it is accurate only in special cases. To consider the general corrosion parameters a more accurate estimation is determined by applying an Lmod parameter, as shown in Fig. 13 and calculated by Eq. (8). The improved design equations are Eqs. (9) and (10): in the cases of FB and GTFB modes (zone I) Eq. (9), and in the cases of LPB and TFB (zone III) Eq. (10) to be used. L L L mod ¼ L pc þ Ext 0:01 2 2
ð8Þ
Ncorr =N0 ¼ a þ b T red þ c T red L mod =L þ d T red ðL mod =LÞ2
ð9Þ
Ncorr =N0 ¼ a þ b T red þ c L mod =L:
ð10Þ
Some of the calculated remaining load-bearing capacity results are shown in Fig. 14, in the function of the Lmod/L ratio; the effect of the various extension and corrosion position can be considered by this parameter. The constants of the equations are determined by surface fitting method; these depend on the element type, the corrosion patterns and relative slenderness. Fig. 15 presents the detailed results of the comparison of the proposed method to the numerical results. The results belong to flexural and general case of torsional–flexural buckling, it means elements that are in Zone I. The notation of the horizontal axis's legends is the following X–Y–Z, where X — Bracing (with bolt numbers) and Chord member; Y — corrosion pattern; and Z — relative slenderness. The mean value, the standard deviation and the maximal difference of the Ncorr/N are presented. The mean value of the difference between the numerical and the estimated values by Eq. (9) is 2%, the maximal difference is 7–9%. The standard deviation is small despite the fact that sometimes the maximum value is higher, usually in the
Fig. 14. Improved design method for the remaining capacity applying Lmod parameter.
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Fig. 15. Difference of Ncorr/N of proposed method compare to numerical results.
Fig. 16. Steps of the checking procedure.
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cases of “B” pattern corrosion. Referring to local failure modes the results are similar to above mentioned. 4.1. Checking procedure Based on the presented analyses and results the following checking procedure is proposed to determine the ultimate behaviour mode and the remaining load-bearing capacity. The first step is the visual inspection to describe and determine the rate of the corrosion. The type of the corrosion and the type of the corrosion pattern also can be determined by visual inspection. The area of the corroded surface can be measured simply by tape-measure and the extension of corrosion can be calculated from the measured data. The main part is the measurement of the reduced thickness by devices such as the mortise gauge and the thickness gauge using ultrasound (ultrasonic thickness gauge). Having the measured geometry, the checking is to be completed by the algorithm shown in Fig. 16. 5. Conclusions The subject of this paper is the remaining buckling resistance of corroded chord and bracing angle elements of lattice towers. On the basis of the executed experimental, the numerical and analytical analyses, the following conclusions can be done. The diversity of the corrosion can be described by four corrosion parameters, like the thickness reduction, the extension of corrosion, the location of corrosion along the length and the corrosion pattern. These parameters are selected and determined by real structural examples. The proposed parameters can describe the damages of the corroded element in the cases of the most common corrosion types. Experimental and numerical analyses are completed on corroded chord and bracing members supposing various measures of corrosion using the defined corrosion parameters under compressive loading. The ultimate failure mode of corroded elements is determined with the calculated b=t ration and the corrosion parameters. The corroded angle members can be classified based on the prospective behaviour mode by b=t to predict the remaining load-bearing capacity. Simplified analytical methods can be used to predict the critical force of corroded elements if the initial assumptions are met with the
corrosion parameters. The applicability and accuracy of different methods are discussed in the paper. Approximate and improved design equations are developed to calculate the remaining load-bearing capacity of compression angle members. The L mod /L corrosion parameter is introduced, which is applicable together with the thickness reduction to get practically accurate estimation for the ratio of the load-bearing capacities depending on the ultimate failure modes. The proposed method can be used for the corroded members, which are described by the previously defined corrosion parameters. Having the measured corrosion parameters the checking can be done by the algorithm of Fig. 16.
References [1] L.V. Beaulieu, F. Legeron, S. Langlois, Compression strength of corroded steel angle members, J. Constr. Steel Res. 66 (2010) 1366–1373. [2] http://www.nace.org/ [3] L. Szunyogh, Welding Technology Handbook, Scientific Association of Mechanical Engineering, Műegyetem Publisher, Budapest, 2007 (in Hungarian). [4] J. Szabó, S. Simon, Corrosion damages of lattice towers on the MVM network, A Technical Reports of the Hungarian Electric Co., vol. 2000/3 2000, pp. 47–49 (in Hungarian). [5] F. Reiner, Lattice Tower Corrosion Inspection With UAV, CIGRE, 2012 B2–303. [6] K. Hisazumi, R. Kanno, T. Tominaga, K. Imafaku, Axial compressive strength of severely corroded channel and angle members used in truss structures, 7th European Conference on Steel and Composite Structures, EUROSTEEL’2014, Naples, Italy, vol. A 2014, pp. 393–398. [7] K. Oszvald, L. Dunai, Effect of corrosion on the buckling of steel angle members — experimental study, Period. Polytech. Civ. Eng. 56 (2) (2012) 175–183. [8] K. Oszvald, L. Dunai, Experiments of corroded angle-section compressive members with bolted connections, International Conference on the Design Fabrication and Economy of Metal Structures, April 24–26, 2012, Miskolc, Hungary, Proceedings 2013, pp. 216–222. [9] ANSYS® v11.0, Canonsburg, Pennsylvania, USA. [10] K. Oszvald, L. Dunai, Behaviour of corroded steel angle compression members — numerical study, Period. Polytech. Civ. Eng. 57 (1) (2013) 63–75. [11] K. Oszvald, L. Dunai, Analysis 1of corroded steel angles under eccentric loading, Second Conference of Junior Researchers in Civil Engineering at BME, June 17–18, 2013, Budapest, Hungary, Proceedings 2013, pp. 148–154. [12] EN 1993-1-1:2005, Eurocode 3: design of steel structures. Part 1–1: general rules and rules for buildings. [13] S.P. Timoshenko, J.M. Gere, “Theory of Elastic Stability”, Chapter 5, McGraw-Hill International Book Company, 1963.
Contents lists available at ScienceDirect
Journal of Constructional Steel Research
The remaining load-bearing capacity of corroded steel angle compression members Katalin Oszvald ⁎, Pál Tomka, László Dunai Budapest University of Technology and Economics, Department of Structural Engineering, H-1111 Budapest, Műegyetem rkp. 3, Hungary
a r t i c l e
i n f o
Article history: Received 23 April 2015 Received in revised form 14 December 2015 Accepted 1 January 2016 Available online 15 January 2016 Keywords: Corrosion Steel Angle-section Compression member Stability Remaining capacity
a b s t r a c t The paper presents a study on the buckling of corroded equal-leg angle-section members. The remaining loadbearing capacity and the prospective behaviour modes are analysed by experimental and numerical research. The diversity of the corrosion is taken into consideration in the analysis; the corrosion is modelled by thickness reduction. Compressive buckling tests, finite element and analytical studies are completed to analyse the modified buckling behaviour and the ultimate load. On the basis of the results simplified design method is developed for the prediction of remaining compressive resistance. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction 1.1. General Generally, the corrosion is one of the major problems during the lifetime of steel structures. The corrosion can be appeared on every structure irrespective of where those can be found. The following structures are considered particularly sensitive to corrosion: bridges, towers, transmission line columns, offshore structures and pipelines. The unfavourable weather conditions and the lack of maintenance are leading to corrosion damages. There is no standardized method or process on how to consider the effect of corrosion in the analyses of the elements [1]. The application of an average cross-section is typically proposed by the standards, but in general this estimation is not accurate enough due to the diversity of the corrosion. The corrosion appears on many structures where the conditions are suitable for its development. Every structure has a typical and a possible type of corrosion depending on its location and the operating conditions. The lattice towers are the most common structural types which are exposed to the effects of the environment. Their typical corrosion types are as follows [2]: (i) uniform corrosion, (ii) pitting corrosion and (iii) crevice corrosion. The current research focuses on these types of structures and the first two corrosion types are considered. According to [3] these do not cause changes in the material properties.
⁎ Corresponding author. E-mail address: [email protected] (K. Oszvald).
http://dx.doi.org/10.1016/j.jcsr.2016.01.003 0143-974X/© 2016 Elsevier Ltd. All rights reserved.
The corrosion damage of the lattice towers is an existing problem all over the world, as it is shown by the different fields of research studies about this problem. Some years ago the Hungarian Electricity Co. (MVM) had published a report on the corrosion of the structural elements and connections of these towers, emphasizing the importance of this problem [4]. The Réseau de Transport d'Électricité (RTE in France) also reported that the corrosion on the lattice towers is also significant in France. Reiner introduced a visual inspection process of lattice towers by a flying machine [5]. Fig. 1 shows the corroded lattice towers which are taken by this device. Based on the observations on the existing corroded lattice towers it can be stated, that both chord and bracing angle members are damaged, and generally, the horizontal members are more sensitive to this effect (Fig. 1). Previously only few experimental studies were completed on corroded angle section members. Beaulieu et al. [1] investigated steel angle members, simulating the corrosion by galvanic process. The main studied parameters in their work were the slenderness, the width-to-thickness ratio and the extent of corrosion. The specimens were experimentally investigated in a truss structure under eccentric compression. The failure modes and the compressive forces were determined and compared to analytical results. The observed failure modes were the (i) global buckling, (ii) the local buckling near to the connection, (iii) the local buckling near to the centre, and in some cases it was not clearly identified in the test. On the basis of the results of the tests and the related analytical calculations the authors proposed a method using an average residual thickness of corroded member to estimate the buckling capacity according to the ASCE 10-97 code. The proposed method, however, is not able to consider the localized corrosion.
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189
Fig. 1. UAV flying machine and photographs of lattice towers taken by UAV [5].
Therefore, further studies are necessary. Recently Japanese researcher published a study about severely corroded angle members [6], where 17 corroded angle and 10 corroded channel section specimens taken from real structures were analysed. The support condition was fixed on both ends and axial compression was applied through end plates. The failure modes were detailed and a prediction method is proposed. Based on the previous studies it is clear that the analysis of the corroded angle members under compression is incomplete. The diversity of the corrosion is not taken into consideration, and only one corrosion parameter – the thickness reduction – was applied in the previous research studies. 1.2. Purpose and scopes The purpose of the research is to extend the knowledge about the corroded angles under compression by a more complex analysis of the damages. It is aimed to develop an applicable design procedure
for the remaining load-bearing capacity of these members. The proposed checking procedure should predict the ultimate behaviour mode and the ratio of the ultimate buckling forces of corroded to the non-corroded members. In the research strategy the typical corrosion patterns and the characterizing parameters are defined at first. Then in the next step experimental, numerical and analytical analyses are carried out considering the diversity of corrosion by considering the following parameters: thickness reduction, extension of corrosion, location of corrosion along the length and corrosion pattern. Finally design methods are derived for the remaining buckling resistance. Chord and bracing angle members of a lattice tower are shown in Fig. 2. The effect of loading and supporting conditions on the stability behaviour are considered according to the two structural components. In the case of chord members the axial compression is applied in the centre of the gravity, while in the case of bracing members the load is applied through bolted connection to one leg of the chord.
Fig. 2. Members of lattice towers with double circuits.
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Table 1 Corrosion parameters. Ext = Lcorr / L [%] pc = Lpc / (L/2) Tred = tcorr / t [%]
2. Experimental and numerical analyses 2.1. Experimental study In the first phase of the research experimental tests are carried out. In the test programme the investigated specimens can be divided into two main groups according to the end support. In the first group the support is hinged in the centre of the gravity of the non-corroded cross-section (hereinafter called chord members). The second group contains the eccentric loading, the specimens are connected by one or two bolts to one leg of the chord (bracing members). Altogether 22 and 24 specimens are tested; in each group non-corroded members are also analysed, as reference elements. The original cross-section is 40 × 40 × 4 mm and the total length of the specimens is 840 mm. The corrosion is modelled as thickness reduction, which is worked out by the mechanical milling process. The main corrosion parameters are changed during the preparation of the specimens. These parameters are as follows (see Table 1): the extension (Ext) and the location of corrosion (pc) along the length, thickness reduction (Tred) and the corrosion pattern: one (“A”) or both legs (“B”) corroded. The exact corrosion parameters, the geometrical properties and the results are published in [7,8]. The test set-up of the two different supporting cases is shown in Fig. 3. The main conclusions and the results of the tests can be summarized as follows. The type of the failure mode highly depends on the measure of corrosion. If the thickness reduction is less than around 50%, the failure mode is dominant flexural buckling (the same as for non-corroded members). If the thickness reduction is higher local plate buckling is observed. These statements are valid irrespective of the loading type, as centric or eccentric. In the case of bracing elements beside the dominant
flexural buckling mode, some torsion also occurred due to eccentricity; actually it is called as a general case of torsional–flexural buckling mode. In Fig. 3 all types of failure modes and element types are presented. During the tests horizontal and vertical displacements are measured. Typical load–horizontal displacement curves are presented in Fig. 4. In every case, when the ultimate failure mode is local plate buckling a sudden decrease of load is observed (Fig. 4, specimens L1, PI-8 and PII-8). The load-bearing capacity is determined in all cases, and the ratio of the corroded and non-corroded elements (Nb,m/Nb0,m) is plotted in the function of the maximal cross-section reduction (Mcorr) in Fig. 5 (the Mcorr is interpreted as the ratio of the maximal cross-section reduction along with the member compared to the gross area). The continuous line supposes the same reduction of load-bearing capacity of the corroded members as the Mcorr. This is a reference line which shows that in some cases the difference from this approximation is about 20%. This result clearly shows that in general one parameter alone is not enough to characterize the corrosion and to take into consideration the effect of it on the remaining load bearing capacity.
2.2. Numerical study The experimental results are extended by a numerical parametric study to analyse the effect of the joint influence of the corrosion parameters on the buckling phenomena and to predict the prospective ultimate failure mode in the function of the corrosion. In the numerical study basically shell finite element model is developed in Ansys programme environment [9]. Furthermore in the case of bracing members the bolted connection is modelled by beam finite elements. The corrosion in the models is considered by thickness reduction. The model of the bracing member is shown in Fig. 6(a). In the model a 4-node shell element (SHELL 181 element of Ansys) is applied, which can model thin and moderately thick plated structures and it is well-suited for large strain nonlinear applications. The steel is modelled by linear elastic-perfectly plastic material (modulus of elasticity 210,000 N/mm2; Poisson's ratio 0.3). In the numerical model the geometry of the corroded test specimens and the measured yield strength (fy = 345 N/mm2) is used. At the end of the element rigid region (CERIG — Ansys command) is defined and in the centre of the gravity of non-corroded cross-section hinge connection as a support is applied.
Fig. 3. Test set-up and the typical ultimate behaviour modes; (a), (d) — local plate buckling; (b), (c) — general case of torsional–flexural buckling.
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Fig. 4. Load–displacement curves: (a) centric loading; (b) eccentric loading.
Fig. 5. Experimental results in the function of Mcorr reduction: (a) centric loading; (b) eccentric loading.
In the case of bracing members the components of bolts, as head, stud, and nut are modelled by beam elements with equivalent properties. In the case of the stud the beam element (BEAM188) has the same cross-section properties as the applied bolts in the tests. This element is based on Timoshenko beam theory and shear deformation effects are included. In the case of the nut and head BEAM44 [9] element is applied, which is a uniaxial element with tension, compression, torsion, and bending capabilities. These beam elements are connected by joint end to the shell elements and by fix end to the stud. The parameters of the cross-section are determined to get the same cross-section
properties as the real bolt. The model also contains the leg of the T support element. In the frame of the numerical study programme bifurcation stability analysis (geometrically non-linear buckling analysis — GNB) and geometrically and materially non-linear imperfect analysis (GMNI) are carried out. As a first step, the test specimens are analysed by the GNB. The results of the GNB analyses provided the buckling load together with the buckled shape to define the initial geometrical imperfections for the nonlinear analysis. The amplitude of the initial geometrical
Fig. 6. (a) Detail of numerical model and (b) typical load–displacement curves of bracing element specimen.
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Table 2 Results of preliminary numerical study on chord members. ID
Location of corrosion
Mcorr [%]
Nb,m [kN]
Nb,NUM [kN]
Nb,NUM /Nb,m
O2 A1
–
0.0 29.0
50.33 33.2
48.73 32.44
0.97 0.98
A2
12.8
51.02
44.6
0.87
A3
11.7
41
40.9
0.99
A4
14.7
50.95
43.60
0.85
A5
12.7
35.5
34.4
0.97
A6
18.8
33.6
33.6
1.00
A7
25
43.8
42.81
0.98
A8
25
37.7
37.03
0.98
A9
13
43.68
42.16
0.97
A10
13
38.5
38.6
1.00
A11
13
36.93
37.04
1.02
L1
70
14
12.88
0.92
L2
73.5
9.6
9.15
0.95
L3
76
4.9
4.99
1.02
L4
49.8
28.3
27.9
0.98
L5
27.8
34
33.98
1.00
Mean value Average deviation [%]
imperfection is calibrated by the test results. It means that the applied magnitude is determined to get the same load-bearing capacity in the numerical analysis as in the test. Thus obtained values were various on the different elements. In the case of chord members, the interval of applied magnitudes of geometrical imperfection is between L/400– L/2000 (2.1–0.42 mm), where L is the length of the member. In the case of bracing member, the interval of magnitude values is between L/200–L/1500. Seventeen investigated members of the preliminary study of chord member are illustrated in Table 2. In the schematic figure of the specimens, both legs are illustrated and the black colour marks the corrosion Table 3 Corrosion patterns and parameters applied for chord members. Pattern
Tred [%]
Ext [%]
Corrosion position (pc)
20 30 40 50 60 70 80
20 30 40 50 70 100
0.20, 0.47, 0.73, 1.00 0.30, 0.53, 0.77, 1.00 0.40, 0.70, 1.00 0.50, 0.75, 1.00 0.70, 1.00 1.00
0.97 4.1
as in previously. The position of the corrosion is detailed on the figures. The cross-section reduction of the members is given in Mcorr column. Table 2 contains Nb,m, the measured load-bearing capacity of the test and the of the numerical models: Nb,NUM. The ratios of the numerical analysis to the test results are also presented in Table 2. In the GMNI analyses the ultimate behaviour and load-bearing capacity are determined. The results showed good agreement with the test in the prediction of the ultimate behaviour and in the loadbearing capacity, as it can be seen on the typical load–horizontal displacement curves in Fig. 6(b) and in Table 2. More details of the models and its verification are published in [10,11].
Table 4 Corrosion patterns and parameters applied for bracing members. Pattern
Tred [%]
Ext [%]
Corrosion position (pc)
20 30 40 50 60 70
20 30 40 50 60 75
0.45, 0.63, 0.80, 1.00 0.55, 0.70, 0.85, 1.00 0.65, 0.82, 1.00 0.75, 0.87, 1.00 1.00 1.00
K. Oszvald et al. / Journal of Constructional Steel Research 120 (2016) 188–198 Table 5 Geometric properties in the parametric study. Type
Cross-section [mm × mm × mm]
Relative slenderness
Chord member
40 × 40 × 4 60 × 60 × 8 100 × 100 × 12 40 × 40 × 4
0.7, 0.9, 1.15, 1.3, 1.5 0.7, 1.15, 1.5 0.7, 1.15, 1.5 1.15, 1.5, 1.92
Bracing member
In the numerical study four different corrosion parameters are applied (Table 1): (i) thickness reduction (Tred), (ii) extension (Ext), (iii) location (pc), and (iv) corrosion pattern. Table 3 contains the corrosion parameter of chord members. In the case of bracing members, Table 4 summarizes the parameter values in detail, referring to one and two bolted end connections. Further parameters of the study are the gross cross-section and the relative slenderness of the members as detailed in Table 5. Note that in the case of bracing elements the connections are considered as not corroded. In the case of chord members altogether 2618, while in the case of bracing members 1152 cases are analysed in the parametric study. The GNB analyses define the typical buckling modes, as follows: (i) flexural buckling (FB), (ii) torsional–flexural buckling (TFB) in the corroded zone, (iii) local plate buckling (LPB) and (iv) general case of torsional–flexural buckling (GTFB). Based on the results three zones (marked by I, II and III) are determined according to the buckling modes. The zones are defined by the width to reduced thickness ratio (b=t) of the corroded zone. In the cases of zones I and III the buckling mode is clearly specified, as shown in Figs. 7–9. The b=t is the horizontal axis and the vertical marks the different relative slenderness ratios (λ) in
193
the figures. The zones are plotted for both corrosion patterns and marked by different colours. If the members belong to zone II, the buckling mode depends not only on b=t parameter but also on other corrosion parameters, such as the position and the extension. The limit values of zones depend on the initial relative slenderness ratio of noncorroded member, on the corrosion pattern and on the type of the support as well. In the GMNI analyses initial geometric imperfections are applied using the shape of the first buckling mode. The magnitude was taken as L/200, following the proposal of the standard [12]. The applied yield strength is fy = 235 N/mm2. The ultimate behaviours correspond to the previously identified stability modes. The results of the GMNI analyses are plotted in the function of the b=t parameter (Fig. 10). The vertical axis is the remaining load-bearing capacity compared to the non-corroded reference results, which can be separated according to the buckling modes (Fig. 10). Note that this observation is the basis of the simplified design method development. 3. Analytical study of the measured resistance In the case of equal-leg angle chord members if the buckling mode is FB (flexural buckling) the measured buckling resistance can easily be obtained using analytical methods. This has to be calculated for Class 3 cross-section by Eq. (1) of the Eurocode 3 [12] recommendations:
Nb;Rd
χ A fy ¼ ;λ ¼ γM1
sffiffiffiffiffiffiffiffiffiffiffiffi A fy N cr
Fig. 7. Definition of I–III zones according to the relevant buckling modes — Chord members.
Fig. 8. Definition of I–III zones according to the relevant buckling modes — 1 bolted connection.
ð1Þ
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Fig. 9. Definition of I–III zones according to the relevant buckling modes — 2 bolted connection.
Fig. 10. Results of GMNI analyses.
where fy = 345 N/mm2 (measured yield stress of the material used at the tests) in order to compare the calculated resistances to the measured ones, A the relevant cross-sectional area, χ reduction factor for buckling, γM1 partial safety factor and Ncr critical force according to the relevant value of [13]. The schematic test set-up is shown in Fig. 11. The buckling length is l = 840 mm, the profile is L40.40.4.
Table 6 contains the remaining resistance of the specimens showed in Fig. 12 (O denotes the non-corroded elements). The notations used are: Ncorr calculated resistance of the corroded specimens, N0 calculated resistance of the non-corroded specimens, Nb,m measured resistance of the corroded specimens Nb0,m measured resistance of the non-corroded specimens (average value).
3.1. The remaining resistance of columns with constant cross-section
If the corrosion is not extended to the whole length, but it is symmetrical about the middle plain of the member and the corrosion pattern is “B” (both legs are equally corroded) Eq. (2) by Timoshenko and Gere [13] can be applied to determine the critical force. Table 7 contains the analysed specimens and the figure on the right side gives the notations.
The critical force Ncr can be obtained: - if the element (O, A1. A5) has a symmetry axis from the quadratic equations of [13] (5-38) or (5-39), otherwise - in the case of elements A3 and A6 using the cubic equations of [13] (5.31) or (5.32).
3.2. Columns with varying cross-sections
Ncr ¼
π 2 EI2 4l
2
1 a l a I2 1 I2 πa þ 1 sin l l I1 π I1 l
ð2Þ
where: I1 moment of inertia of corroded section, I2 moment of inertia of non-corroded section, l buckling length of the element. Table 6 Critical forces, calculated and measured reduction of the resistance due to corrosion of A1, A3, A5 and A6 specimens with uniform profile.
Fig. 11. Schematic test set-up.
ID
Ncr [kN]
Ncorr/N0
Nb,m [kN]
Nb,m/Nb0,m
O A1 A3 A5 A6
54.63 36.95 48.76 44.06 41.99
1.00 0.68 0.89 0.81 0.77
50.5 33.2 41 35.5 33.6
1.00 0.65 0.81 0.70 0.66
K. Oszvald et al. / Journal of Constructional Steel Research 120 (2016) 188–198
195
Fig. 12. Columns with constant cross-section.
Note: Eq. (2) is valid also for specimen A7, where the corroded zones are at the end of the specimen.
remaining load-bearing capacity of members with moderate or even medium penetration of corrosion.
3.3. The remaining resistance of members with varying cross-sections
3.5. Prediction of the ultimate behaviour mode
The buckling resistance here can also be calculated by the Eurocode 3 [12] recommendation Eq. (1) with the measured yield stress of fy = 345 N/mm2. The only difference is that in the formula for the relative slenderness ratio:
The results presented in Sections 3.1–3.4 are only valid if the buckling mode is flexural buckling. This section shows how to use the analytical methods of [13] to calculate the torsional–flexural buckling of equal-leg angles, providing that the corrosion extends to the whole length and the corrosion pattern is “B”. In this case the possible stability modes are the in-plane buckling (FB) and the torsional–flexural buckling (TFB).
sffiffiffiffiffiffiffiffiffiffiffiffi A fy : λ¼ N cr
ð3Þ
The cross-sectional area A is not constant. Therefore, as an approximation, the authors suggest the use of an equivalent cross-sectional area Aequ with the notation of Table 7 in the form of Aequ ¼ ðA1 ðl−aÞ þ A2 aÞ=l;
ð4Þ ð5; 6Þ
where A1 and A2 are the relevant cross-sectional areas. Table 8 contains the remaining resistance of the specimens from Table 7 (O denotes the non-corroded elements). The notations used are the same as above. 3.4. Analytical study, conclusions The previously calculated and measured results are given for the remaining resistance of the corroded members with constant- (Table 6) and varying cross-sections (Table 8). The calculations are made using the Eurocode provision [12], while the governing quantities (critical forces Ncr) were obtained according to the fundamental theoretical study of [13]. The calculated and measured results show an acceptable agreement: the max. deviation – as a basis of the measured ones – is approximately +10%. As the final conclusion, it can be said that the analytical approach of this chapter is able to give easy to handle procedures to calculate the
Table 7 Critical forces, calculated and measured reduction of the resistance due to corrosion of A7, A8, L4 and L5 specimens with varying cross-sections. Model by Timoshenko [13]. ID A7
Location of corrosion
where, Nu, Nv Euler critical loads for buckling about the u and v axes, Nϕ critical load for pure torsional buckling, IC and IO polar moment of inertia with respect to centroid and shear centre. The transition between TFB and FB can be easily determined by equating Ncr,TFB (Eq. (5)) with Ncr,FB (Eq. (6)). The limit of the transition is the Lcr,TFB value. If the analysed element is shorter than this value, the stability mode is torsional–flexural buckling (TFB), dominantly torsional. In other cases the stability mode is flexural buckling (FB). In the case of equal-leg angles the cross-section may be replaced with good accuracy by two intersection perpendicular flats, where the length bmod = h-tcorr / 2 and the thickness is obviously tcorr. Solving the previously mentioned equality of Eq. (7) is obtained. 2
Lcr;TFB ¼ 1:097b mod =tcorr
ð7Þ
The critical forces obtained by using FE model based on GNB analysis (Ncr,FE) and the analytical methods show good agreement and the critical length Lcr,TFB also fits to the results referring to the stability mode (Table 9). In the table the stability (buckling) mode, the numerical Ncr,FE results, the calculated Ncr,TFB (Eq. (5)), Ncr,FB (Eq. (6)) and Lcr,TFB are presented in the function of the assumed corrosion (Tred). The non-corroded profile is a uniform L40.40.4, the corrosion pattern is “B” through the whole length of the specimens. In the case of 60–80% thickness reduction the stability mode is TFB because Lcr,TFB N L which are marked by dark grey colour in Table 9. In
A8
L4
L5
Table 8 Calculated and measured reduction of the resistance due to corrosion of A7, A8, L4 and L5 specimens with varying cross-sections. ID
Ncorr
Ncorr/N0
Nb,m
Nb,m/Nb0,m
O A7 A8 L4 L5
41.14 36.04 31.97 25.96 30.89
1.00 0.88 0.78 0.63 0.75
50.5 43.8 37.7 28.3 34
1.00 0.87 0.75 0.56 0.67
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Table 9 Determination of the behaviour mode in the function of critical length (Lcr,TFB) in the function of Tred. Tred [%]
L [mm]
Stability mode
Pcr,FE
Pcr,FB
Pcr,TFB
Lcr,TFB
0
840
FB
54.5
53.50
148.28
395.9
20
840
FB
43.52
44.17
95.75
505.5
30
840
FB
38.00
39.25
70.49
583.6
40
840
FB
32.52
34.17
47.917
687.9
50
840
FB
27.06
28.92
29.40
834.0
60
840
TFB
17.1
23.49
15.70
1053.3
70
840
TFB
7.54
17.89
6.82
1418.7
80
840
TFB
2.304
12.11
2.05
2149.8
Table 10 Design method for elements with FB and GTFB behaviour modes. Ncorr/N0
Pattern “A”
Pattern “B”
1 bolt 2 bolts
1 - 0.01 ∙ (Tred/2) 1 - 0.01 ∙ (Tred/2) 1 - 0.01 ∙ (Tred/2)
1 - 0.01 ∙ Tred 1 - 0.01 ∙ Tred 1 - 0.01 ∙ Tred
Chord member Bracing member
Table 11 Design method for corroded chord angle members — TFB and LPB behaviour modes. Ncorr/N0
λ ¼ 0:7 λ ¼ 1:15 λ ¼ 1:5
Pattern “A”
Pattern “B”
0.995 - 0.023∙Mcorr 0.73 - 0.016∙Mcorr 1 - 0.0225∙Mcorr
1.098 - 0.013∙Mcorr 1.53 - 0.018∙Mcorr 0.79 - 0.009∙Mcorr
Table 12 Design method for corroded bracing angle members – TFB and LPB behaviour modes. Ncorr/N0 Pattern
Bolts
λ ¼ 1:15
λ ¼ 1:5
“A”
1 2 1 2
0.931 - 0.0101∙Tred 0.759 - 0.0087 ∙Tred 0.931 - 0.0112 ∙Tred 0.72 - 0.008 ∙Tred
0.662 - 0.0069 ∙ Tred 0.642 - 0.0067 ∙ Tred 1.965 - 0.0259 ∙ Tred 1.36 - 0.0174 ∙Tred
“B”
λ ¼ 1:92 0.804 - 0.0085 ∙ Tred
Fig. 13. Definition of the modified length parameter.
case of less than 60% thickness reduction the failure mode is flexural buckling, marked by light grey colour. Note again that this prediction is not applicable in the case of partial corrosion and “A” corrosion patterns. 4. Design method development On the basis of the numerical parametric studies, design methods are developed to calculate the remaining load-bearing capacity of corroded members. The members are separated according to the ultimate behaviour mode using the previously defined zones. If the ultimate behaviour mode is FB or GTFB the proposed design equation in Table 10 is applicable irrespective to the relative slenderness. If the ultimate behaviour mode is local plate or torsional–flexural buckling, Tables 11–12 contain the design equations. It shows that in these cases the equations depend on the corrosion pattern, on the slenderness ratio and on the type of the elements, as chord or bracing member (one or two bolts). These equations do not take into account the extension and the location of the corrosion, therefore it is accurate only in special cases. To consider the general corrosion parameters a more accurate estimation is determined by applying an Lmod parameter, as shown in Fig. 13 and calculated by Eq. (8). The improved design equations are Eqs. (9) and (10): in the cases of FB and GTFB modes (zone I) Eq. (9), and in the cases of LPB and TFB (zone III) Eq. (10) to be used. L L L mod ¼ L pc þ Ext 0:01 2 2
ð8Þ
Ncorr =N0 ¼ a þ b T red þ c T red L mod =L þ d T red ðL mod =LÞ2
ð9Þ
Ncorr =N0 ¼ a þ b T red þ c L mod =L:
ð10Þ
Some of the calculated remaining load-bearing capacity results are shown in Fig. 14, in the function of the Lmod/L ratio; the effect of the various extension and corrosion position can be considered by this parameter. The constants of the equations are determined by surface fitting method; these depend on the element type, the corrosion patterns and relative slenderness. Fig. 15 presents the detailed results of the comparison of the proposed method to the numerical results. The results belong to flexural and general case of torsional–flexural buckling, it means elements that are in Zone I. The notation of the horizontal axis's legends is the following X–Y–Z, where X — Bracing (with bolt numbers) and Chord member; Y — corrosion pattern; and Z — relative slenderness. The mean value, the standard deviation and the maximal difference of the Ncorr/N are presented. The mean value of the difference between the numerical and the estimated values by Eq. (9) is 2%, the maximal difference is 7–9%. The standard deviation is small despite the fact that sometimes the maximum value is higher, usually in the
Fig. 14. Improved design method for the remaining capacity applying Lmod parameter.
K. Oszvald et al. / Journal of Constructional Steel Research 120 (2016) 188–198
Fig. 15. Difference of Ncorr/N of proposed method compare to numerical results.
Fig. 16. Steps of the checking procedure.
197
198
K. Oszvald et al. / Journal of Constructional Steel Research 120 (2016) 188–198
cases of “B” pattern corrosion. Referring to local failure modes the results are similar to above mentioned. 4.1. Checking procedure Based on the presented analyses and results the following checking procedure is proposed to determine the ultimate behaviour mode and the remaining load-bearing capacity. The first step is the visual inspection to describe and determine the rate of the corrosion. The type of the corrosion and the type of the corrosion pattern also can be determined by visual inspection. The area of the corroded surface can be measured simply by tape-measure and the extension of corrosion can be calculated from the measured data. The main part is the measurement of the reduced thickness by devices such as the mortise gauge and the thickness gauge using ultrasound (ultrasonic thickness gauge). Having the measured geometry, the checking is to be completed by the algorithm shown in Fig. 16. 5. Conclusions The subject of this paper is the remaining buckling resistance of corroded chord and bracing angle elements of lattice towers. On the basis of the executed experimental, the numerical and analytical analyses, the following conclusions can be done. The diversity of the corrosion can be described by four corrosion parameters, like the thickness reduction, the extension of corrosion, the location of corrosion along the length and the corrosion pattern. These parameters are selected and determined by real structural examples. The proposed parameters can describe the damages of the corroded element in the cases of the most common corrosion types. Experimental and numerical analyses are completed on corroded chord and bracing members supposing various measures of corrosion using the defined corrosion parameters under compressive loading. The ultimate failure mode of corroded elements is determined with the calculated b=t ration and the corrosion parameters. The corroded angle members can be classified based on the prospective behaviour mode by b=t to predict the remaining load-bearing capacity. Simplified analytical methods can be used to predict the critical force of corroded elements if the initial assumptions are met with the
corrosion parameters. The applicability and accuracy of different methods are discussed in the paper. Approximate and improved design equations are developed to calculate the remaining load-bearing capacity of compression angle members. The L mod /L corrosion parameter is introduced, which is applicable together with the thickness reduction to get practically accurate estimation for the ratio of the load-bearing capacities depending on the ultimate failure modes. The proposed method can be used for the corroded members, which are described by the previously defined corrosion parameters. Having the measured corrosion parameters the checking can be done by the algorithm of Fig. 16.
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