Experimental determination of critical loads in thin-walled bars with Z-section subjected to warping torsion

Thin-Walled Structures 75 (2014) 87–102

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Experimental determination of critical loads in thin-walled bars with Z-section subjected to warping torsion Zbigniew Kowal, Andrzej Szychowski n Faculty of Civil Engineering and Architecture, Kielce University of Technology, Al. Tysiąclecia Państwa Polskiego 7, 25-314 Kielce, Poland

art ic l e i nf o

a b s t r a c t

Article history: Received 5 July 2013 Received in revised form 22 October 2013 Accepted 22 October 2013 Available online 28 November 2013

Critical loads were determined experimentally from the condition of the local buckling of thin-walled bars with Z-section subjected to warping torsion. The experimental investigations were carried out using simply supported models, loaded with a concentrated torsional moment at the mid-span. A method of determining the so-called “local ordered deflection interval” was developed. In this interval, the modeled deflection of the component plates (walls) of a thin-walled bar with random wall geometrical imperfections is compliant with the local buckling mode. The “local ordered deflection interval” makes it possible to adjust the known experimental methods so that they could be used to determine critical torsional loads and local critical bimoments. Experimental investigations showed the occurrence of two local critical bimoments in bars with Z-section. The bimoments are different in their absolute values, depending on the sense (sign) of the torsional load. Experimentally determined critical loads were compared with theoretical results. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Thin-walled Z-section Warping torsion Local buckling Critical load Local critical bimoment Experimental methods

1. Introduction In cold formed thin-walled members with open cross-section, cases are found, in which the form of stability loss is determined by the local buckling of the section components walls. After the critical load is reached, the component plates (walls of the section) undergo local deflections that are transverse to the stress direction. The inevitable random geometric (general and local) imperfections affect the load-carrying capacity and stiffness, and also the stress distribution in the open cross-sections of thin-walled bars. Experimental investigations into the local buckling of thinwalled bars with open cross-sections described in the literature concern mainly members that are axially and eccentrically compressed, or bent and sheared. Warping torsion is generally disregarded. Only a few studies, e.g. [1–4], describe the experimental investigations and provide theoretical analyses of open thinwalled bars subjected to transverse bending, in which the plane of bending does not pass through the shear center of the section. Such a load causes additional torsion of bars. In those investigations, the impact of warping normal stresses on the local buckling of thin-walled members was observed. In study [5], the occurrence of the local stability loss in open thin-walled bars subjected to warping torsion, without the participation of other components of the section load was proved

n

Corresponding author. Tel.: þ 48 41 342 4575; fax: þ 48 41 344 3784. E-mail address: [email protected] (A. Szychowski).

0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2013.10.020

experimentally and confirmed theoretically. In [5], the local critical bimoment inducing the local buckling of a thin-walled bar was defined. The theoretical method of determining local critical bimoments in a thin-walled bar, subjected to warping torsion, with an arbitrary open cross-section built from flat walls (thin plates) was described in [6]. The Southwell method is most often used for experimentally determining critical loads in bars, e.g. [7], plates, e.g. [8,9], or thinwalled bars composed of flat walls (thin plates), e.g. [10]. In the classical approach, the method was used to experimentally determine the buckling load in compressed bars with geometrical imperfections, the form of which was close to the buckling mode. When the Southwell method is used in experimental investigations into the local buckling of isolated plates or thin-walled bars made from plates, the accuracy of results may depend on the form of initial geometrical imperfections of the plate (or walls of the section). If the form of the plate imperfections is similar to the buckling mode in accordance with the least critical load (the first eigenvalue), the deflections that have shapes corresponding to the plate buckling mode become dominant from the beginning of the loading process. Then, the Southwell graph, determined for even small loads, makes it possible to estimate the critical load in a satisfactory manner [9]. Shortcomings of the Southwell method, applied to the experimental investigations into plates or thin-walled bars built from plates, result from the fact that the form of random geometrical imperfections in plates or walls of the thin-walled section that actually occur can differ considerably from the “ordered” shape of buckling of such elements. Kowal and Szychowski [11] conducted

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Nomenclature width, thickness of the plate (walls s) bs , t s B bimoment Bcr , Bcr;L , Bcr;R local critical bimoment, (“left” – positive; “right” – negative) Becr , BTcr experimental value, theoretical value of the local critical bimoment B1;2;3;4 successive eigenvalues of the local critical bimoment cr By first yield bimoment E, E Young's modulus, mean value determined experimentally G shear modulus of elasticity i, j index, natural number Iω warping section constant It St-Venant torsion constant kω , kω;L , kω;R coefficients of critical warping stresses L length of a thin-walled bar ls length of a bar segment, length of a plate (wall s) M t , M t;L , M t;R load of a concentrated torsional moment (“left”, “right”)

numerical investigations and showed the causes of discrepancies between the experimental results for plate critical loads obtained with the classical Southwell method and the theoretical results. Tereszkowski [12] proposed a method, different from the classical Southwell approach, for determining the critical load in uniformly compressed plates within the elastic range. The point of departure for the Tereszkowski method is von Karman nonlinear plate theory [13]. In order to determine the critical load with this method, it is necessary to take into consideration the coordinates ðui N i Þ (deflection-load) of the three “measurement points” of the plate static equilibrium path determined in the experiment. An advantage of the Tereszkowski method is the possibility of using deflections from the pre-buckling and post-buckling load range. The author [12] obtained the solution to the problem for two variants of the shape of geometrical imperfections. In the first variant, the form of initial imperfections corresponds to the buckling mode, and can be described by the same function. Formulas derived for this case make it possible to experimentally determine the critical load. The other variant of the method assumes that the form of initial imperfections is specified by a function different from the one describing the plate buckling mode. In this instance, the procedure proposed in [12] makes it possible to estimate the lower and the upper limits, within which the experimental critical load of the plate is contained. In the experimental investigations into thin-walled bars, composed of thin plates [5], subjected to warping torsion, it was found that the form of geometrical imperfections significantly affects plate deflections in the load pre-buckling range. In many cases, the form of imperfections differed much from the “ordered” (theoretical) shape of the local buckling in the examined thin-walled bars. For very close successive eigenvalues of critical loads, that can result in the least buckling load being undetected with classical methods because every experimental model renders an individual set of random geometrical imperfections. The analysis of such experimental situations demonstrate the significance of the phenomenon of the so-called “ordering of deflections” [5,11]. The present study gives the results of experimentally determined critical loads from the condition of the local buckling of thin-walled bars with Z-section, subjected to warping torsion. These bars showed wall random geometrical imperfections. Only loads which generate normal stresses from warping torsion were

M t;cr , M et;cr , M Tt;cr critical torsional moment (external) from the condition of the local buckling, experimental value, theoretical value ui local deflection of a plate u0 pointer of the initial deflection of a plate xs ,ys ,zs Cartesian coordinates of a plate (wall s) ϕ angle of twist rotation ν, ν Poisson's ratio, mean value determined experimentally sω;cr critical warping stress in accordance with [6] sE;s Euler's stress for a plate (wall s) xi , x, sx;n  1 ith measurement, mean value of measurement, standard deviation. ω, ωc sectorial coordinate, a sectorial coordinate corresponding to the critical stress sω;cr pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi κ ¼ GIt =EI ω flexural – torsional coefficient of a cross-section ReH , ReL yield stress of the steel (basic material): upper, lower Rm ultimate tensile strength

considered. Assuming such a mode of loading allowed the experimental determination of the critical torsional load and the local critical bimoment.

2. The experimental investigations 2.1. The test stand and test models A simply supported thin-walled bar with Z-section was used for the experimental investigations. The model was loaded with a concentrated torsional moment at the mid-span, with opposite senses in succession, in accordance with the diagram shown in Fig. 1a. Z-section is characterized by different behavior in the local critical state, depending on the sense of the torsional load [5,6]. In order to facilitate a comparison of experimental results with the theoretical ones presented in [6], a simplified bimoment notation was used (Fig. 1a). The bimoment, which generated warping compressive (positive) stresses in the Z-section on the free edges of the plates b1 and b3 (flanges), is denoted as BL – “left” (with the sign “þ”). The bimoment inducing the compression of the plate b2 (web) in the Z-section is denoted as BR – “right” (with the sign “–”). The bimoment BL (or BR) is produced by loading the thin-walled bar with the concentrated torsional moment M t;L (with the sign “þ”), or M t;R (with the sign “–”). Cold-formed galvanized sheet (Z-275-I-1  1000  2000) of nominal thickness tn ¼1 mm, was used to construct the models (mZ1– mZ6). The average steel core thickness, after deduction for the zinc coating, was t cor ¼ 0:97 mm. Computational dimensions of the model composed of component plates: b1, b2 and b3 (Fig. 1b) were assumed for the midline of the cross-section. Warping (Fig. 1d) and torsional characteristics, determined on this basis, are listed in Table 1. The test stand (Fig. 1c) made it possible to load each model with concentrated torsion moment M t with alternately opposite sense, following the diagram shown in Fig. 1a. Six models with Zsection were examined altogether in more than 18 experiments in accordance with the following program: 1. Loading with the “left” torsional moment ðM t;L Þ – experiment 1L. Measurements were taken for the loads exceeding approx.

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Fig. 1. The diagram of support and load of the model of a thin-walled bar with Z-section: (a) warping normal stresses for local critical bimoments (Bcr;L , Bcr;R ); (Note: compressive stresses with the sign “ þ”); (b) division of the section into the component plates b1, b2, b3; (c) a general view of the test stand; (d) sectorial coordinates; (e) notation of deflections of component plates; and (f) systems of forces that load the model.

Table 1 Dimensions of the midline of the cross-sections; warping and torsional characteristics of models (t ¼0.97 mm). Test model

b1 [mm]

b2 [mm]

b3 [mm]

ω1 [cm2]

ω2 [cm2]

ω3 [cm2]

ω4 [cm2]

Iω [cm6]

It [cm4]

κ [1/cm]

mZ1

60.68

119.32

60.49

26.95

 9.07

 9.14

27.13

318.4

0.00732

2:96 U 10  3

mZ2

61.06

119.00

61.06

27.13

 9.20

 9.20

27.13

323.2

0.00734

2:95 U 10  3

mZ3

61.07

119.41

60.75

27.03

 9.12

 9.24

27.33

323.6

0.00734

2:94 U 10  3

mZ4

61.58

119.86

59.93

26.46

 8.34

 9.48

28.01

323.9

0.00734

2:94 U 10  3

mZ5

61.48

119.49

60.70

26.91

 9.08

 9.37

27.63

326.6

0.00735

2:93 U 10  3

mZ6

60.72

119.46

61.15

27.41

 9.27

 9.11

27.00

324.2

0.00734

2:94 U 10  3

10% of the critical value M t;cr;L , after which the model was gradually unloaded. 2. Loading with the “right” torsional moment ðM t;R Þ – experiment 1R. Measurement was taken for the load exceeding approx. 10% of the critical value M t;cr;R , after which the model was gradually unloaded. 3. A subsequent change in the test stand setup and tests repeated in accordance with point 1 (experiment 2L). The stress generated by the warping torsion applied to the models in accordance with the loading program presented above did not exceed the linear elastic range ðsω; max o 0:75ReH Þ. In order to maintain the assumed boundary conditions of support eliminating rotation ðϕ ¼ 0Þ, with a simultaneous freedom II of section warping on the supports ðϕ ¼ 0Þ, linear fork handles (see Fig. 1a) were used, separately for each component plate. The theoretical span of the models positioned in the system of support

handles of the test stand was L ¼720 mm. The use of relatively short models ðL  12b1  6b2 Þ minimized the effect of additional displacements produced by the rotation angle of the twisted thinwalled bar on the local critical bimoment determined in the experiment. The loading of the model with the concentrated torsional moment was performed by means of steel lines of ø 4 mm, attached to a special steel diaphragm (see Fig. 1f) which was fastened to the model at its mid-span. The task of the diaphragm was to transfer the external load to the model without distorting the central crosssection, and to maintain the “rigid cross-section contour” [6] at the site of load application. In this way, the thin-walled bar was divided into two segments, symmetrical in respect of geometry and load (“A” and “B”), which differed only in random geometrical imperfections of walls. The experimental investigations were preceded by the measurement of imperfections resulting from the geometrical properties of the model and initial deflections caused by the model

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assembly in the support handles and the central diaphragm. The measurement of imperfections was reduced to determining the inaccuracy of linearity of the free edges of the plates b1 and b3 (in segments “A” and “B”). In the experimental investigations, the following measurements were taken: (1) vertical displacements of the component plates b1 and b3 (see Fig. 1e); (2) the angle of the section twist rotation at the model mid-span, (3) horizontal displacements of plate b2 (Fig. 1e), (4) the rotation angles of sections in the support handles, (5) the model strains in the central cross-section below the stiffening diaphragm at 8 measurement points. 2.2. The investigations into the material properties The investigations on the properties of steel (basic material) were carried out using samples collected from metal sheet, from which the models were made. The samples of the geometry shown in Fig. 2 were executed in compliance with the Polish Standard [14]. The experimental estimation of Young's modulus E and Poisson's coefficient ν was performed using samples, on which four electric resistant wire strain gauges TFs-5/120 (Fig. 2a) were stuck on both sides. The upper and the lower yield stresses (ReH and ReL ) and ultimate tensile strength (Rm) were experimentally estimated for samples with marked measurement lengths (as per Fig. 2b). Six measurements of Young's modulus and Poisson's coefficient and five measurements of the steel strength were performed. Mean values, standard deviations and coefficients of variations of steel mechanical characteristic are specified in Table 2. 2.3. The experiment and experimental results The results of the measurements of displacements and strains obtained from the experimental investigations made it possible to analyze the phenomena occurring in thin-walled bars, subjected to warping torsion, with the unsymmetrical Z-section sensitive to

Fig. 2. Geometry of samples: (a) for experimental determination of Young's modulus (E) and Poisson's ratio (ν); and (b) for examination of yield stress and ultimate tensile strength of steel in models.

Table 2 Mechanical characteristic of steel (basic material) used in models. Yield stresses [MPa]

Upper Lower

ReH 331.86 ReL 318.17

Ultimate tensile strength [MPa]

Rm 362.10

Young's modulus [MPa]

E 200352 ν 0.3086

Poisson's coefficient

sReH

vReH

7.33 sReL

0.022 vReL

5.56 sRm

0.017 vRm

5.22 sE

0.014 vE

2917 sν 0.0021

0.0145 vν 0.0068

local stability loss within the elastic range. Results are presented below, separately for experiments 1L and 1R, in which a different behavior of the models was observed in the local critical state.

2.3.1. Experiment 1L When the models were loaded with the “left” torsional moment M t;L ¼ SL U e (Fig. 1f), local displacements of the free edges of the plates b1 and b3 with random distribution of maximum deflections were observed to increase at first. Plate deflections were caused by an increase in the amplitudes of initial geometric imperfections. With an increase in the torsional load, a gradual change in the shape, and frequently, in the sign of local deflections occurred. In the vicinity of the critical load ðM t;cr;L Þ, a considerable, non-linear increase in both local deflections of all component plates, and in the angle of the model twist rotation was observed. The exception was the model mZ1 (the first one to be tested), in which the phenomenon of “deflection snap-through” of component plates in both segments was observed in the experiment 1L. Fig. 3 presents experimental results for the model mZ1-1L. Fig. 3a shows the dependence between the torsional load and the angle of twist rotation at the model mid-span ðM t;L  ϕL=2 Þ, whereas in Fig. 3b, the dependence between the torsional load and local deflections ðM t;L  ui Þ of the free edges of the plates b1 and b3 (cf. Fig. 1e) can be seen. Local deflections ui were determined in relation to the “rigid cross-section contour”, after taking into account displacements induced by the model twist rotation angle. For the sake of comparison, the dependence M t;L  ϕL=2 for the model mZ1 determined on the basis of the Vlasov theory [15] was shown in Fig. 3a. Local deflections of component plates (b1, b3) were determined at sites of extreme deflections for the theoretical first buckling mode corresponding to B1cr;L (cf. Fig. 12, Section 5). The analysis of local deflections of the model mZ1-1L component plates indicates that within the pre-buckling range, deflection forms shows point symmetry with respect to the crosssection centroid. The symmetry is characteristic of the 2nd and 4th buckling mode corresponding to the 2nd and 4th eigenvalue of the local critical bimoment B2;4 cr;L (cf. Fig. 12, Section 5). Under the load Mt,L ¼ 5.44 þ0.147 [kNcm], a rapid “deflection snap-through” of component plates occurred. It was accompanied by the acoustic effect, i.e. a short sound, after which local deflections of a thinwalled bar were suddenly reconfigured to the buckling mode characteristic of the first (least) local critical bimoment ðB1cr;L Þ. The effect of “deflection snap-through” of the model mZ1 component plates is illustrated, in a diagram form, in Fig. 4. The dashed line shows local deflections of segments “A” and “B”, shaped in the pre-buckling range, prior to the “snap-through”, and the thicker, continuous line represents the deflection configuration after the “snap-through”. Fig. 5. compares the experimental dependencies: M t;L  ϕL=2 for models mZ2, mZ3, mZ5, and mZ6 with the Vlasov theory. Section rotation angles at the model mid-spans were determined after taking into account the rotational flexibility of the support sections. Fig. 6 presents the experimental dependencies holding between the torsional load and local deflections ðM t;L  ui Þ of the free edges of the plates b1 and b3 (cf. Fig. 1e) in models mZ2, mZ3, mZ5, and mZ6. A comparison of graphs of local deflections presented in Fig. 3b and Fig. 6 shows their dependence on the forms of random geometric imperfections that affect, in a different way, the stiffness of individual walls. A continuous reversal of the sign of local deflections of component plates was frequently observed. Due to initial imperfections, the deflections, enhanced at the first stage of pre-buckling load, did not adjust to the local buckling mode of a thin-walled bar. When the torsional load increased, local deflections

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Fig. 3. Results for the model mZ1 obtained in the experiment 1L with “deflection snap-through”: (a) dependence: torsional load – angle of twist rotation at the model midspan ðM t;L  ϕL=2 Þ; and (b) dependencies: torsional load – local deflections ðM t;L  ui Þ of the free edges of plates b1 and b3 (cf. Fig.1e); tV – angle of twist rotation determined in accordance with the Vlasov theory.

Fig. 4. The diagram of “deflection snap-through” of component plates in the model mZ1-1L (experiment 1L); (a) deflections in the segment “A”; and (b) deflections in the segment “B”.

of component plates “returned to their original place”. As a result, the buckling mode became similar to the one determined theoretically (e.g. deflections:u4 ,u6 – model mZ1 – Fig. 3b; u5 – model mZ3; u3 ,u4 – model mZ6 – Fig. 6). Antisymmetry of the form of local deflections for both component segments “A” and “B” was found in each model (in the critical state), however, their amplitudes generally differed randomly. The greatest differences were noted for the model mZ3. The impact of geometric and material imperfections (including residual stresses) was manifested when the model mZ6 was examined. The model wall b3 was made from the external (longitudinal) edge of sheet metal of the basic material. A substantial decrease in the local stiffness of the plate b3 (cf. deflections u3 ,u5 , Fig. 6) in both component segments reduced the torsional rigidity of this model (cf. Fig. 5). Fig. 7 compares warping normal stresses, determined on the basis of experimental measurements of mZ2 and mZ5 models, with the ones calculated in accordance with the Vlasov theory [15].

2.3.2. Experiment 1R When the models were loaded with the “right” torsional moment M t;R ¼ SR e (Fig. 1f), in the pre-buckling range of the section

load, local displacements of tensioned free edges of plates b1 and b3 were observed. The displacements had the form of elastic “straightening” of initial imperfections. A considerable, non-linear increase in local deflections of compressed plate b2 occurred in the vicinity of the critical load ðM t;cr;R Þ. The angle of the model twist rotation of, as the function of the torsional moment load, showed linear behavior up to the critical load. After the critical load was exceeded, a slight reduction in the model torsional stiffness was observed. That was also confirmed by the nonlinear decrease in compressive stress measured in the axis of the plate b2. Fig. 8 compares the experimental dependencies: M t;R  ϕL=2 for models mZ2, mZ3, mZ5 and mZ6 with the Vlasov theory. Section rotation angles at model mid-spans were determined after taking into account the rotational flexibility of support sections. Fig. 9 presents experimental dependencies holding between the torsional load and local deflections ðM t;R  ui Þ of the compressed plate b2 (cf. Fig. 1e) in models mZ3 and mZ5. Local deflections of the plate b2 in both component segments showed an antisymmetric form in relation to the diaphragm located at the model mid-span. Fig. 10 compares the warping normal stresses determined on the basis of measurements of the mZ3 model with the ones calculated in accordance with the Vlasov theory [15].

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3. The analytical model As per Fig. 1a, the constructional system of a thin-walled bar consists of two symmetrical segments “A” and “B”, each ls ¼ 360 mm in length. Experimental investigations show that deflections generated by the local buckling of the correspondig component plates of the adjacent segments are antisymmetric in relation to the central diaphragm. The external torsion moment M t;L or M t;R acts in the plane of the diaphragm. The method of theoretical determination of local critical bimoments and of the thin-walled

bar local buckling modes using the construction and load symmetry was shown in study [6]. As the cross-sections of models (mZ1 to mZ6) are characterized by a small value of coefficient κ ¼ 2:94  10  3 cm  1 (cf. Table 1), the bimoment function was estimated in accordance with [6] neglecting the residual St-Venant torsion rigidity (GI t ffi 0) in the known differential equation for nonuniform torsion [15]. For the thin-walled bar static scheme, in accordance with Fig. 1a, with κ -0, a linear distribution of the bimoment is obtained in the component segments (“A” and “B”), which is expressed by the formula   M t sinhðκxÞ Mt x BðxÞ ¼ lim ¼ ð1Þ 2 κ-0 2κ coshðκL=2Þ In this case, the maximum absolute value of bimoment at the member mid-span is BðL=2Þ ¼ M t L=4

ð2Þ

For the models of concern, the error of conservative calculation, resulting from neglecting the St-Venant torsion rigidity (GI t ¼ 0) does not exceed 0.4%. The study [6] presents a method of theoretical determination of the local critical bimoment under critical warping stress using the formula (3) Bcr ¼

Fig. 5. Experimental dependence: torsional load – rotation angle ðM t;L  ϕL=2 Þ for models mZ2, mZ3, mZ5 and mZ6 determined in experiments 1L. A comparison with the Vlasov theory (tV).

sω;cr I ω

ωc

ð3Þ

where, sω;cr ¼ kω UsE;s in accordance with [6]. The computational Z_LBcr.nb program [6] was developed in the environment of the Mathematica package [16] to analyze the local buckling of the thin-walled bars segments with the Z-section. The program makes it possible to determine critical warping stresses and local buckling modes for a segment with linear or nonlinear (in accordance with the 2nd degree parabola) bimoment distribution (BL or BR ) over its length. The theoretical calculations of critical warping stress for models (mZ1–mZ6) were performed using the Z_LBcr.nb program. The geometry of the cross-section midlines was assumed in accordance with Table 1 and the properties of the basic material were determined experimentally (Table 2). A linear distribution of

Fig. 6. Experimental dependencies: torsional load – local deflections ðM t;L  ui Þ for the free edges of plates b1 and b3 (cf. Fig. 1e) in models mZ2, mZ3, mZ5 and mZ6, determined in experiments 1L.

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Fig. 7. Stress state in the measured section of models mZ2 and mZ5 as the function of the external torsional moment M t;L – experiment 1L; Note: n1, n2 and n3 – normal stresses determined on the basis of experimental measurements, s1 ¼s2 and s3 – stresses determined on the basis of the Vlasov theory [15]; (compressive stresses with the sign “þ”).

2

and Bcr;R ¼  290:5 kNcm , respectively. The critical torsional moments (the external ones), from the condition of the thinwalled bar local buckling (depending on the bimoment sign), calculated by transforming the formula (2) are M t;cr;L ¼ 5:03 kNcm and M t;cr;R ¼  16:14 kNcm, respectively. The results of theoretical calculations, performed with the Z_LBcr.nb program for models mZ1 to mZ6 are shown in Tables 5–8 in Section 5.

4. Experimental methods of the critical load determination 4.1. The impact of geometrical imperfections

Fig. 8. Experimental dependence: torsional load – angle of twist rotation ðM t;R  ϕL=2 Þ in models mZ2, mZ3, mZ5 and mZ6, determined in experiments 1R. A comparison with the Vlasov theory (tV).

bimoment (BL or BR ) along the length of component segments was assumed in accordance with Fig. 1a. For example, the calculation results for the model mZ2 are as follows: for ls =b1 ¼ 5:9 coefficients of the critical warping stresses are kω;L ¼ 1:654 and kω;R ¼ 6:833; hence the following was obtained: sω;cr;L ¼ kω;L sE;1 ¼ 76 MPa and sω;cr;R ¼ kω;R sE;2 ¼ 82:7 MPa. The theoretical values of local critical 2 bimoments determined from formula (3) are Bcr;L ¼ 90:6 kNcm

In thin-walled steel members composed of thin plates, subjected to warping torsion, the form of initial random imperfections (as shown in experimental investigations [5]), may differ considerably from the “ordered” shape of the local buckling. Fig. 11 presents the form of initial geometrical imperfections in the free edges of the plates b1 and b3 (dashed line) of the model mZ5. The continuous line marks the theoretical (“ordered”) first local buckling mode in the free edges of the plates b1 and b3, which corresponds to the first eigenvalue of the local critical bimoment B1cr;L (in accordance with Z_LBcr.nb program [6]). The comparison of graphs indicates that the shape of the geometrical imperfections considerably deviates from the theoretical first local buckling mode. Attempts, made in this study, to experimentally determine local critical bimoments with the Southwell classical method [7], or with the Tereszkowski method [12] in the universally applied load range show a distinct lack of congruence between experimental and theoretical results. That results from the random geometrical imperfections of component plates, which deviate from the “ordered” shape of the local buckling. To explain why substantial errors occur in the estimation of the first local critical bimoment ðB1cr Þ with known experimental methods, the hypothesis of the “disturbances of expected deflections” in the thin-walled bar local buckling can be employed. Deflections corresponding to the buckling mode in accordance with B1cr are disturbed by the pre-buckling displacement enhancement along the directions of initial imperfections, especially if the latter are similar to the buckling mode corresponding to higher eigenvalues of the local critical bimoments (Bjcr , j ¼ 2; 3; 4:::). A numerical experiment was carried out to examine the impact of local geometrical imperfections that are similar in shape to

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Fig. 9. Experimental dependencies: torsional load – local deflections ðM t;R  ui Þ for the compressed plate b2 (cf. Fig. 1e) in models mZ3 and mZ5 determined in experiments 1R.

Table 4 Experimental coordinate of the local static equilibrium path ðjM t;R j  u2 Þ for plate b2A in model mZ5-1R used to estimate the absolute value of “right” critical torsional moment ðM e;II t;cr;R Þ.

Fig. 10. Stress state in the measured section of model mZ3 as the function of the external torsional moment M t;R – experiment 1R; Note: notation as per Fig. 7. Table 3 Experimental coordinate of the local static equilibrium path ðM t;L  u6 Þ for plate b1A in model mZ5-1L used to estimate the “left” critical torsional moment ðM e;II t;cr;L Þ. Torsional jth “Computational load M t;L;j point” (cf. Fig. 14 [kNcm] (a))

Pointer of the local M t;L;j deflection of the plate u6;j [mm]

1 2 3

0.094 0.513 1.551

4.634 5.168 5.898

u6;j

M t;L;j =u6;j

1 1 1 1.115 5.488 0.2032 1.273 16.574 0.0768

deflections corresponding to higher eigenvalues of the local critical bimoments. For the exemplary model mZ2, loaded in accordance with the diagram in Fig. 1a, theoretical local buckling modes for several eigenvalues of the local critical bimoments (Bcr;L and Bcr;R ) were determined using the Z_LBcr.nb program [6]. Fig. 12 shows the 1st, 2nd, 3rd and 4th local buckling modes in the component plates of the symmetrical segment “B” (cf. Fig. 1a) in the model mZ2-1L, which correspond to the successive eigenvalues of the “left” local critical bimoment ðB1;2;3;4 Þ. Eigenvectors of cr;L the successive buckling modes were normalized so that maximum deflection of plate b1 was 1. Fig. 13 shows the 1st, 2nd, 3rd, and 4th local buckling modes in the component plates of the symmetrical segment “B” (cf. Fig. 1a) in the model mZ2-1R, which correspond to the successive eigenvalues of the “right” local critical bimoment ðB1;2;3;4 cr;R Þ. In this

Torsional jth   “Computational load M t;R;j  point” (cf. Fig. 16 [kNcm] (a))

Pointer of the local deflection of the plate u2;j [mm]

M t;R;j

1 2 3

0.048 0.420 0.671

1 1 1 1.185 8.750 0.1354 1.304 13.979 0.0933

14.710 17.425 19.177

u2;j

M t;R;j =u2;j

instance, eigenvectors of the successive buckling modes were normalized so that maximum deflection of plate b2 was 1. Theoretical investigations show that in the model mZ2-1L (under load M t;L ), the first two local critical bimoments (B1cr;L and B2cr;L – Fig. 12) differ from each other only by approx. 5.8%. The buckling modes are characterized by the opposite direction of extreme deflections of the free edge of the plate b3 with a simultaneous deflection of the plate b2 in accordance with one or two “halfwaves” along its width. The two successive eigenvalues (B3cr;L , B4cr;L – Fig. 12) are greater than B1cr;L by approx. 40.5% and 48.6%, respectively, whereas the difference between them is only just 5.75%. In this instance, the local deflection extremes on the plate free edges are moved towards the center of the segment length. There, due to boundary conditions, the pointer of local initial imperfections of component plates can reach extreme values (cf. Fig. 11). In all the situations presented above, the form of deflections indicates that eccentrically compressed plates with a free edge make the greatest impact on the local stability loss in a thin-walled bar loaded with the torsional moment M t;L . Deflections of the plates b1 and b3 enforce a symmetric or antisymmetric (in the transverse direction) form of deflection of the plate b2 in tension. If the model mZ2-1R is loaded with the torsional moment M t;R , the successive eigenvalues of local critical bimoments are more diverse (B2cr;R ffi 1:34 U B1cr;R ; B3cr;R ffi1:69 U B1cr;R ; B4cr;R ffi 2:17 UB1cr;R ), and in contrast to Bcr;L , they do not occur in pairs. The plate b2 in compression, which is elastically restrained on the edges of joints with the plates b1 and b3, has a decisive influence on the local buckling in this instance. The buckling mode is characterized by the deflection of the plate b2 in accordance with one “half-wave” in the width direction, and by a variable amplitude of deflections in the length direction. Maximum deflections in the successive buckling modes move towards the center of the segment length. There, because of boundary conditions, the pointer of local initial imperfections of component plates can reach extreme values. Characteristic deflections of the plates b1 and b3 result from displacements forced by the buckled plate b2. Towards free edges, deflections are limited due to the “straightening” effect of warping tensile stresses.

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95

Table 5 A comparison of experimental “left” critical torsional moments ðM et;cr;L Þ for models mZ1 – mZ6 with theoretical results. Model-experiment Local deflection Experimental “left” critical torsional moment of plates (b1, b3) Method I M e;I t;cr;L [kNcm]

Theoretical “left” critical torsional moment

Method II M e;II t;cr;L

M Tt;cr;L [kNcm]

[kNcm]

1

2

xi 3

x,(sx;n  1 ) 4

xi 5

x,(sx;n  1 ) 6

mZ1-1L

u3 u4 u5 u6 u3 u4 u5 u6 u3 u4 u5 u6 u3 u4 u5 u6 u3 u4 u5 u6 u3 u4 u5 u6

lack of effective “measurement points” 5.757 5.295 5.302 5.424 5.219 5.322 5.195 5.370 5.356 5.220 5.240 5.495 5.382 5.511 5.489 5.275 4.612 5.286 4.847 5.287

–

5.366 5.290 5.166 5.206 5.238 5.254 5.208 5.390 5.159 5.270 5.153 5.276 5.262 5.259 5.213 5.352 5.277 5.302 5.286 5.198 4.556 5.242 4.503 5.065

5.257 (0.089) 5.044

mZ2-1L

mZ3-1L

mZ4-1L

mZ5-1L

mZ6-1L

5.444 (0.217)

5.276 (0.083)

5.328 (0.127)

5.414 (0.109)

5.008 (0.335)

Table 6 A comparison of experimental “left” local critical bimoments (Becr;L ) estimated from formula (4) with theoretical results (BTcr;L ). Experimental “left” local critical bimoment

Theoretical “left” local critical bimoment

Method I Be;I cr;L

Method II Be;II cr;L

BTcr;L [kNcm2]

1

[kNcm2] 2

[kNcm2] 3

4

mZ1-1L mZ2-1L mZ3-1L mZ4-1L mZ5-1L mZ6-1L

– 98.00 94.96 95.90 97.46 90.14

94.63 94.90 93.86 94.88 94.79 87.15

90.79 90.57 90.90 91.24 90.98 90.94

Modelexperiment

5.272 (0.080) 5.032

5.214 (0.068) 5.050

5.271 (0.058) 5.070

5.266 (0.047) 5.054

4.842 (0.368) 5.052

Table 7 A comparison of experimental “right” critical torsional moments (M et;cr;R ) for models mZ1–mZ6 with theoretical results. ModelLocal experiment deflection of plate (b2)

Experimental “right” critical torsional moment

Theoretical “right” critical torsional moment

Method

Method

I M e;I t;cr;R

II M e;II t;cr;R

M Tt;cr;R [kNcm]

[kNcm]

[kNcm]

1

2

xi 3

x 4

mZ1-1R

u1 u2 u1 u2 u1 u2 u1 u2 u1 u2 u1 u2

18.124 17.208 15.712 17.042 17.520 17.704 17.468 17.704 16.321 16.992 16.760 15.994

17.666 17.605 17.493 16.377 16.047 16.589 17.612 17.236 17.416 17.586 17.352 17.470 16.656 16.643 16.573 16.377 16.447 15.899

mZ2-1R

In the initial range of the warping torsion in a thin-walled bar with initial geometrical imperfections, component plates “seek” deflection modes associated with the least (first) local critical bimoment. In this load range, deflections along the “directions” of initial imperfections, similar to buckling modes corresponding to higher eigenvalues of local critical bimoments can have substantial impact. That generates incorrect indications resulting from the classical Southwell graphs [7] or the Tereszkowski formulas [12]. Errors also originate in slight differences between the successive eigenvalues of the local critical bimoments (e.g. Bjcr;L for j ¼ 1; 2; 3; 4 or Bjcr;R for j ¼ 1; 2; 3), associated with different local buckling modes (cf. Figs. 12 and 13). In the experimental investigations on the warping torsion in thin-walled bars [5], a local increase in deflections of component plates was found along the “directions” of initial random imperfections. Only in the range of considerable load, the deflections became ordered in a specific way following the buckling mode

7

mZ3-1R mZ4-1R mZ5-1R mZ6-1R

xi 5

x 6

7

17.549 15.987 16.318 16.137 17.326 16.086 17.411 16.032 16.608 16.142 16.173 16.094

associated with the first local critical bimoment ðB1cr Þ. On the basis of observations made in the experiments [5], a hypothesis was formulated. It states that in thin-walled bars with random geometrical imperfections, subjected to warping torsion, the domination of increases in local deflections ordered in accordance with the buckling mode for the first local critical bimoment ðB1cr Þ occurs in the individually realized load interval, termed a “local ordered deflection interval”. The interval begins in the pre-buckling range of high load and continues in the moderate (elastic) post-buckling

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range. Moderate local post-buckling deflections are as a rule ordered in accordance with the first eigenvalue of the thinwalled local buckling mode. It is necessary to determine the “local ordered deflection interval” every time we want to experimentally estimate critical torsional moments ðM t;cr Þ and local critical bimoments ðBcr Þ from the condition of the local stability loss in the thin-walled bar with random geometrical imperfections. In the interval, the form of local deflections in the section walls tends to the local buckling mode from the condition of the minimum of the total potential energy of the system. 4.2. Method I – pre-buckling “local ordered deflection interval” The method for determining the pre-buckling “local ordered deflection interval” and estimating, on its basis, the critical loads is presented below. It is done on example of the model mZ5, separately for experiments 1L and 1R.

4.2.1. Model mZ5 – experiment 1L – critical loads: M et;cr;L , Becr;L The absolute values of coordinates: M t;L  jui j (load – local deflections) of the plates b1 and b3, determined in experiment 1L (cf. Fig. 6), was used for experimentally determining the “left” critical torsional moment ðM et;cr;L Þ. The experimental “left” local critical bimoment was estimated from the formula Becr;L ¼

M et;cr;L L 4

ð4Þ

Fig. 14a presents the local static equilibrium path: M t;L – u6 (for the plate b1A in segment “A” – cf. Fig. 1e), determined experimentally. Deflection u6 was measured at the site of expected extreme displacements for the theoretical first local buckling mode which corresponds to B1cr;L (cf. Fig. 12). Fig. 14b shows the graph of “measurement points” in the coordinate system (u6 =M t;L , u6 ). Table 8 A comparison of experimental “right” local critical bimoments (Becr;R ) estimated from formula (6) with theoretical results (BTcr;R ). Modelexperiment

Experimental “right” local critical Theoretical “right” local bimoment critical bimoment Method I Be;I cr;R

Method II Be;II cr;R

BTcr;R [kNcm2]

1

[kNcm2] 2

[kNcm2] 3

4

mZ1-1R mZ2-1R mZ3-1R mZ4-1R mZ5-1R mZ6-1R

318.00 294.79 317.02 316.55 299.81 294.79

315.88 293.72 311.87 313.40 298.94 291.11

287.76 290.46 289.55 288.57 290.55 289.68

The corresponding “measurement points” in both graphs were numbered from 1 to 22. The effective execution of the Southwell straight line for the presented graph (Fig. 14b) is impossible due to the lack of linearity of the determined “measurement points”. To experimentally estimate the critical load ðM et;cr;L Þ, the pre-buckling “local ordered deflection interval” was determined. Analyzing the graphs in Fig. 14, it is possible to distinguish three intervals which characterize different stages of the local static equilibrium path for the plate b1A in the model mZ5-1L. Interval I (points 1–9) is characterized by the absence of the ordering of “measurement points” (cf. Fig. 14b), which results from amplification of local deflections along directions of random geometrical imperfections. The interval ends with the “adjustment” of the local deflections of the model (including the component plate b1A) to the “proper” buckling mode for the first critical load (points 8–9). Interval II (points 10-15) is the pre-buckling “local ordered deflection interval”, in which a continuous increase in local deflections of component plates occurs. The increase is compliant with the first local buckling mode. In the coordinate system (u6 =M t;L , u6 ), Fig. 14b, the “measurement points” of interval II determine a straight line. Interval III (points 16–22) determines the post-buckling “local ordered deflection interval” in accordance with the first local buckling mode. The coordinates of “measurement points” of interval III make a nonlinear, increasing function (Fig. 14b). “Measurement points” from the pre-buckling “local ordered deflection interval” (interval II), in which deflections of component plates best correspond to the first local buckling mode of a thin-walled bar, provide the basis to experimentally determine the “left” critical torsional moment with the method I ðM e;I t;cr;L Þ. Due to the relatively narrow extent of the interval II, it is especially important to select the limit points in a proper way. Point 15 is assumed to be the end of the pre-buckling range, i.e. the upper limit of interval II. Point 15 is located below the point of inflexion of the curve approximating those coordinates of the equilibrium path ðu6  M t;L Þ that are in the neighborhood of the critical loads. In Fig. 14a, it is the point of intersection with the abscissa. The lower limit of the interval II was determined from the analysis of the linear regression of the coordinates of “measurement points” (from 15 down). That was done on the basis of the condition of the maximum of points (at least 5 points) that determine the straight line (linear correlation coefficient above 0.999). It was also an advantage to make an analysis of residual errors [16] of the extreme points of the interval of interest. The analysis makes it possible to reject the measurement coordinates that most deviate from the originally determined regression line. The regression line was determined again for a reduced interval. In the example under consideration, the coordinates of “measurement points” 10–15 (Fig. 14) were qualified as belonging to the interval II.

Fig. 11. The form of initial geometrical imperfections of the free edges of plates b1 and b3 (dashed line) in the model mZ5 and the theoretical “ordered” local buckling mode (continuous line) for Bcr;L .

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97

Fig. 12. Successive (1st, 2nd, 3rd and 4th) theoretical modes of the local buckling of component plates of segment “B” in the model mZ2-1L with the “left” local critical bimoment B1;2;3;4 (cf. Fig. 1a,b). cr;L

Fig. 15 shows the computational “measurement points” of the interval II and a straight line determined on this basis using the least squares method (Fig. 15b). This means that coordinates of the interval II “measurement points” (Fig. 15a) satisfy the deflection

amplification equation in accordance with the formula ui ¼

u0 1  M t;L =M t;cr;L

ð5Þ

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Fig. 13. Successive (1st, 2nd, 3rd and 4th) theoretical modes of the local buckling of component plates of segment “B” in the model mZ2-1R with the “right” local critical bimoment B1;2;3;4 (cf. Fig. 1a and b). cr;R

The further procedure is the same as with the Southwell method. The slope of a straight line determines the critical load value. In the example under consideration, the slope of the

straight line determines the experimental value of the “left” critical torsional moment M e;I t;cr;L ¼ 5:275 ½kNcm in accordance with the method I. The difference in relation to the theoretical value

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99

Fig. 14. (a) The experimental local static equilibrium path: torsional moment M t;L – local deflection u6 of plates b1A (cf. Fig.1e); (b) “measurement points” in the coordinate system ðu6 =M t;L ; u6 Þ. Note: circled points – coordinates of “computational points” used in the method II.

Fig. 15. (a) Computational “measurement points” of the pre-buckling “local ordered deflection interval” used to estimate the “left” critical torsional moment M e;I t;cr;L , (b) regression straight line in the coordinate system ðu6 =M t;L ; u6 Þ.

M Tt;cr;L ¼ 5:054 ½kNcm, determined using the Z_LBcr.nb program [6], is approx. 4.4%. The experimental “left” local critical bimoment 2 estimated from the formula (4) was Be;I cr;L ¼ 97:46 kNcm . 4.2.2. Model mZ5 – experiment 1R – critical loads: M et;cr;R , Becr;R The absolute values of the coordinates: jM t;R j  jui j (load – local deflections) of the compressed wall b2, determined in the experiment 1R (cf. Fig. 9), were used to experimentally determine the absolute value of the “right” critical torsional moment ðM et;cr;R Þ. The experimental “right” local critical bimoment was estimated from the formula Becr;R ¼

M et;cr;R U L 4

ð6Þ

Fig. 16a presents the local static equilibrium path jM t;R j  u2 (for the plate b2A in the segment “A” – cf. Fig. 1e) determined experimentally. The deflection u2 was measured at the site of expected extreme displacements for the theoretical first local buckling mode corresponding to B1cr;R (cf. Fig. 13). Fig. 16b shows the graph of the “measurement points” in the coordinate system ðju2 =M t;R j; u2 Þ. The corresponding “measurement points” in both graphs were numbered from 1 to 25. The method for determining the pre-buckling “local ordered deflection interval” of the plate b2 in compression is the same as the one presented in Section 4.2.1. In this instance, the following coordinates of “measurement points”: interval I – points 1–13; interval II – points 14–18; interval III – points 19–25 were qualified to belong to the successive characteristic intervals.

Fig. 17 shows the computational “measurement points” of the interval II, and a straight line determined on this basis using the least squares method (Fig. 17b). It means that also when the system is loaded with the torsional moment M t;R , the coordinates of the interval II “measurement points” (Fig. 17a) satisfy the deflection amplification equation in accordance with the formula ui ¼

u0 1  jM t;R j=M t;cr;R

ð7Þ

As previously, the further procedure is the same as with the Southwell method. The slope of the straight line determines the absolute value of the experimental “right” critical torsional moment jM e;I t;cr;R j ¼ 16:99 ½kNcm in accordance with the method I. The difference in relation to the theoretical value jM Tt;cr;R j ¼ 16:142 ½kNcm determined using the Z_LBcr.nb program [6], is approx. 5.3%. The absolute value of the experimental “right” local critical bimoment, 2 estimated from formula (6), was: jBe;I cr;R j ¼ 299:81 kNcm . The method I was used to experimentally estimate the critical loads in the models mZ1 – mZ6, depending on the torsional load sense. Cumulative results are presented in Section 5 of the present study. 4.3. Method II – the full “local ordered deflection interval” The present study demonstrates that the determination of the “local ordered deflection interval” also makes it possible to apply the Tereszkowski method to the experimental estimation

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Fig. 16. (a) The experimental local static equilibrium path: the absolute value of torsional moment jM t;R j – local deflection u2 of plates b2A (cf. Fig. 1e); (b) “measurement points” in the coordinate system ðju2 =M t;R j; u2 Þ. Note: circled points – coordinates of “computational points” used in the method II.

Fig. 17. (a) Computational “measurement points” of the pre-buckling “local ordered deflection interval” used to estimate the absolute value of “right” critical torsional moment (M e;I t;cr;R ), (b) regression straight line in the coordinate system ðju2 =M t;R j; u2 Þ.

of critical loads in thin-walled bars with open cross-section, subjected to warping torsion. On the basis of the experiments described in the present study, it can be stated that three “computational points” (j¼1, 2, 3) of the method in [12] should be selected from the full “local ordered deflection interval”. The interval contains “ordered” pre-buckling deflections (interval II determined in accordance with Section 4.2.) and moderate (elastic) post-buckling deflections (interval III). Such an approach improves the accuracy of the experimental estimation of the critical load. That particularly concerns the cases, in which a different shape of geometrical imperfections (cf. Fig. 11) “delays” the occurrence of deflections conformable with the local buckling mode within the pre-buckling range. Transforming the formulas obtained in [12] (in accordance with the first variant of the Tereszkowski method) so that they would fit with the critical load investigations based on the condition of the local buckling of thin-walled bars, subjected to warping torsion, we received Eqs. (8–10). Parameters u0 , ξ, μcr , and the experimental critical torsional moment M et;cr are obtained from those equations using the formula (11) 1 þ u0  M t;2  ðM t;2 =u2 Þu0 ð1  u2 Þð1 þ u2 þ 3u0 Þ ¼ 1 þ u0  M t;3  ðM t;3 =u3 Þu0 ð1  u3 Þð1 þ u3 þ 3u0 Þ

ξ¼

1 þu0  M t;2  ðM t;2 =u2 Þu0 ð1  u2 Þð1 þ u2 þ3u0 Þ

μcr ¼ Mt;j þ ðMt;j =uj Þu0  ξðuj þ u0 Þðuj þ 2u0 Þ

ð8Þ

ð9Þ ð10Þ

M et;cr ¼ μcr M t;1

ð11Þ

where M t;j ¼ M t;j =M t;1 ; uj ¼ uj =u1 ; u0 ¼ u0 =u1 ; uj – deflection of a component plate for the jth “computational point”, determined at the site of local extreme deflection for the first local buckling mode, M t;j – external torsional moment for the jth “computational point”. The application of the method II is presented on example of the same model mZ5. The local static equilibrium path M t;L  u6 of the component plate b1A (cf. Fig. 14a) was taken into account when determining the “left” critical torsional moment ðM e;II cr;L Þ. Three “computational points” of the method II were selected from the full “local ordered deflection interval” (points 10–22). The following coordinates: i¼ 11 for j¼1; i¼15 for j¼2; and i¼ 21 for j¼ 3 (cf. Fig. 14a – points marked with a circle) were taken as “computational points” (j¼1, 2, 3). Table 3 shows the experimental coordinates of selected “computational points” (j¼1, 2, 3), and relative values M t;L;j and uj . The experimental critical load determined from the sequence of formulas (8–11) was M e;II t;cr;L ¼ 5:198 ½kNcm (where: u0 ¼ 0:122; ξ ¼ 0:571  10  3 ;μcr ¼ 1:122). The difference in relation to the theoretical value: M Tt;cr;L ¼ 5:054 ½kNcm was about 2.8%. The experimental “left” local critical bimoment estimated from the formula (4) 2 was Be;II cr;L ¼ 94:79 kNcm . The absolute values of coordinates: jM t;R j u2 of the component plate b2A (cf. Fig. 16a) were taken into account to determine the

Z. Kowal, A. Szychowski / Thin-Walled Structures 75 (2014) 87–102

absolute value of “right” critical torsional moment ðM e;II cr;R Þ. Three “computational points” of the method II were selected from the full “local ordered deflection interval” (points 14–25). The following coordinates: i¼ 15 for j ¼1; i¼20 for j¼ 2; and i¼24 for j¼3 (cf. Fig. 16a – points marked with a circle) were taken as “computational points” (j¼1, 2, 3). Table 4 shows the experimental coordinates of the selected “computational points” (j¼1, 2, 3), and relative values M t;R;j and uj . The absolute value of the experimental critical load determined from the sequence of formulas (8–11) was jM e;II t;cr;R j ¼ 16:573 ½kNcm (where: u0 ¼ 0:128; ξ ¼ 0:941 U 10  3 ;μcr ¼ 1:127). The difference with respect to the theoretical value: jM Tt;cr;R j ¼ 16:142 ½kNcm was about 2.7%. The absolute value of the experimental “right” local critical bimoment estimated from the formula (6) was 2 jBe;II cr;R j ¼ 298:94 kNcm . The comparative investigations show a low sensitivity ð r 0:5%Þ of the experimentally estimated critical load to the selection of the three computational points in the method II, provided that they are “widely spaced” to cover the full “local ordered deflection interval”. It was found that in the method II, not a single point located outside of the determined “interval” could be selected because that leads to serious errors in the results. Experimental critical loads for the models mZ1–mZ6 were estimated with the method II, depending on the sense of torsional load. Cumulative results are presented in Section 5 of the present study.

e;II measurements with the methods I ðM e;I t;cr;R Þ and II ðM t;cr;R Þ with theoretical results determined using the Z_LBcr.nb program [6] is presented in Table 7. Column 1 in Table 7 shows the number of the model; Column 2 – deflections of component plates (cf. Fig. 1e); Columns 3 and 5 show the ith critical loads (xi) determined on the basis of ith local deflections of the plate b2 in compression; Columns 4 and 6 show the mean value (x) of the critical torsional moment; while Column 7 shows the value of the critical load determined theoretically. In the investigation on the “right” critical torsional moment, a satisfactory congruence between experimental and theoretical results was also obtained. The maximum difference was 10.5% for the model mZ1-1R. Table 8 compares the experimental “right” local critical bimoments ðBecr;R Þ, estimated from the formula (6) on the basis of the e mean values of the critical torsional moments ðM t;cr;R Þ determined using methods I and II, with the theoretical results. A comparative analysis of the results presented in Tables 6 and 8 indicates that the absolute value of the local critical bimoment in a thin-walled bar with Z-section depends on the sense of the torsional load. For example, the absolute value jBecr;R j for the model mZ2, was about 3.1 times higher than the absolute value jBecr;L j. However, the absolute value of the first yield bimoment By (neglecting local buckling), does not depend on the sense of the torsional load and can be determined from the formula

By ¼ 5. A comparison of experimental and theoretical results Table 5 provides a comparison of the “left” critical torsional moments for the models mZ1–mZ6, estimated on the basis of experimental measurements using the method I ðM e;I t;cr;L Þ and method II ðM e;II Þ with theoretical results determined using the t;cr;L Z_LBcr.nb program [6]. In Table 5, Column 1 shows the number of the model; Column 2 – deflections of component plates (cf. Fig. 1e); Columns 3 and 5 show the ith critical loads (xi) determined on the basis of ith local deflections of the compressed plates b1 and b3; Columns 4 and 6 show the mean value (x) of the critical torsional moment and the standard deviation of results (sx;n  1 ); while Column 7 shows the value of the critical load determined theoretically. It should be noted that in the model mZ1-1L (experiment 1L with so-called “deflection snap-through”), effective “measurement points” are absent for the method I. Critical load was estimated with the method II taking into account only a narrow post-buckling “local ordered deflection interval” (interval III), which occurred after the “deflection snap-through”. e;I;II In most cases, the experimental “left” critical moments (M t;cr;L , Table 5) are slightly greater (max. by 8.2% in model mZ2-1L – in accordance with the method I) than the value M Tt;cr;L determined theoretically. A weak restraint of the model component plates in the central diaphragm (cf. Fig. 1) could be one of the main causes of the above-mentioned differences. As regards the model mZ6-1L, the experimental “left” critical torsional moment determined from an analysis of local deflections (u3, u5) of the plate b3 is smaller (maximum by 10.9% in accordance with the method II) than the theoretical value. The reduction in the critical load and torsion rigidity (cf. Fig. 5) was caused by the impact of physical imperfections (including rolling internal stresses) of the external edge of the basic material sheet. Table 6 presents the comparison of the “left” local critical bimoments ðBecr;L Þ, estimated from the formula (4) on the basis of e the mean values of critical torsional moments ðM t;cr;L Þ determined with methods I and II, with the theoretical results. A comparison of the “right” critical torsional moments in the models mZ1–mZ6, estimated on the basis of experimental

101

f y UI ω

ωmax

:

ð12Þ

For the model mZ2, the maximum (minimum) first yield bimoment, determined on the basis of mean values of lower yield 2 stresses ðReL Þ for steel, is By ¼ 7379:1 kNcm . It is about 4 times e higher than Bcr;L , and about 1.3 times higher than Becr;R . The experimental results presented above confirm the results of theoretical investigations [6], namely, that in thin-walled bars with open, unsymmetrical (in relation to axis) cross-section (e.g. Z-section), subjected to warping torsion, local critical bimoments determine the applicability range of the Vlasov theory in accordance with the relation Bcr;R r B r Bcr;L

ð13Þ

In cold-formed thin-walled members, characterized by considerable slenderness (bi/ti) of component plates, correct stress calculations based on the classical Vlasov theory is limited to the range (13). With loads producing bimoment falling ouside the range (13), local buckling of a thin-walled bar occurs and the Vlasow theory basic assumption [15] concerning the “rigid crosssection contour” loses its validity. In this load range, an analysis should be made of the post-buckling resistance of component plates of a thin-walled bar with open cross-section, subjected to warping torsion.

6. Summary and conclusions Experimental investigations presented in the study show that local buckling can occur in thin-walled bars with open crosssection, subjected to warping torsion, with no other components of section load involved. The critical torsional load is conditioned by the occurrence of local critical bimoment that causes the local buckling of thin-walled bar component plates. After the local critical bimoment is exceeded, a reduction in the longitudinal rigidity of compressed plates (particularly those with a free edge) occurs. That is manifested by a nonlinear decrease in warping compressive stresses and reduction in the torsional rigidity of a thin-walled bar.

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In thin-walled bars with unsymmetrical cross-section (e.g. Z-section), two local critical bimoments (Bcr;L , Bcr;R ) occur. They have different absolute values and a different local buckling mode. The section position in relation to the sense of the external torsional load determines the occurrence of a higher or lower local critical bimoment. Experiments 1L showed a substantial impact of the initial form of geometric imperfections in the free edges of the plates b1 and b3 on local deflections in the pre-buckling range of the section load. With increasing load, initial deflections of component plates became gradually “ordered”, aiming to achieve the buckling mode in accordance with the first eigenvalue of local critical bimoment B1cr;L . The ordering of local deflections in the model mZ1-1L proceeded in a violent way producing the “deflection snapthrough” of the component plates. In the experiments on other models, the ordering of deflections occurred gradually as the load increased. Observation of the above phenomena make it possible to generalize the well-known experimental methods of the critical load estimation. As a result, the case of the local stability loss in thin-walled bars with open cross-section due to warping torsion could be included. In experiments 1R, the compressed plate b2, bilaterally elastically restrained in plates b1 and b3, produced the highest impact on the local buckling of a thin-walled bar. Initial geometric imperfections in the free edges of the plates b1 and b3 did not cause a sharp reduction in torsional rigidity in this case. They became elastically “straightened” due to the occurrences of warping tensile stresses on free edges. In thin-walled bars with open cross-sections, subjected to warping torsion, it is possible to experimentally determine the critical loads from the condition of local buckling (M t;cr ,Bcr ) despite models showing, as a rule, wall random geometrical imperfections. For that reason, it is necessary to determine the “local ordered deflection interval”. In this interval, deflections of the thin-walled bar component plates tend to conform with the local buckling mode conditioned by the first local critical bimoment. In experimental investigations into thin-walled bars, which are members composed of thin plates with random geometrical imperfections, deflections conformable with the form of local buckling mode generally occur under a greater relative prebuckling load than it is the case with single (separated) plates. Errors in the estimation of critical loads determined with wellknown methods [7,12] on the basis of “measurement points” located below the pre-buckling “local ordered deflection interval” result from disturbances in local deflections. That happens due to plate bending along the directions of random initial imperfections which deviate from the “expected” local buckling mode. It was possible to modify the Tereszkowski method [12] and extend its application to include the estimation of critical loads in thin-walled bars under warping torsion due to the plate buckling of component walls in the models. Three computational points of the method II should be selected in the pre-buckling and moderately post-buckling (elastic) “local ordered deflection interval” (intervals I and II). In this load interval, deflections of component plates tend to the first local buckling mode of thin-walled bars. In some cases of the experimental analysis of thin-walled bars, the ordering of local deflections in accordance with the first local buckling mode cannot occur. Then, the local static equilibrium path (load – local deflection) can be performed by means of the

“deflection snap-through” of component plates. That took place in the investigations on the model mZ1-1L. In such cases, interval I of displacement increments along the directions of geometrical imperfections ends with the “snap-through” of deflections and the thin-walled bar passes to the post-buckling stage of “ordered deflections” (interval III). A comparison of results presented in Tables 5–8 shows a good congruence between the critical loads determined on the basis of experimental coordinates of “measurement points” from the “local ordered deflection interval” and the results of theoretical calculations. The method II gives critical loads that are a few percent lower when compared with the method I. However, the lowest local critical bimoments were obtained from the theoretical analysis. The physical form of local stability loss in the models examined in the study was congruent with the local buckling mode determined theoretically. The local deflections of component plates showed antisymmetry with respect to the central diaphragm (load plane), irrespective of the torsional load sense. In thin-walled bars with open cross-section (sensitive to local buckling), local critical bimoments account for a limitation in the applicability of the Vlasov theory in accordance with relation (13). Using the classical Vlasov theory to calculate the load-bearing capacity for this class of thin-walled bars, under the loads that produce bimoments beyond the validity range (13), may lead to substantial errors caused by the local stability loss. After the critical load is reached, a thin-walled bar is subjected to local buckling and the basic assumption of the Vlasov theory on a cross section rigid contour loses its validity [15]. References [1] Murray NW, Lau YC. The behavior of a channel cantilever under combined bending and torsional loads. Thin-Walled Struct 1983;1:55–74. [2] Kavanagh KT, Ellifritt DS. Design strengths of cold-formed channels in bending and torsion. J Struct Eng ASCE 1994;120:5. [3] Put BM, Pi YL, Trahair NS. Bending and torsion of cold-formed channel beams. J Struct Eng ASCE, 125; 1999; 540–6. [4] Gotluru BP, Schafer BW, Peköz T. Torsion in thin-walled cold-formed steel beams. Thin-Walled Struct 2000;37:127–45. [5] Szychowski A. Local critical load capacity of nonuniform torsion of thin-walled bars with open cross-section [Ph.D. thesis]. Kielce: Kielce University of Technology; 2001 [in Polish]. [6] Szychowski A. A theoretical analysis of the local buckling in thin-walled bars with open cross-section subjected to warping torsion. Thin-Walled Structures, http://dx.doi.org/10.1016/j.tws.2013.11.002, this issue. [7] Southwell RV. On the analysis of experimental observations in problems of elastic stability. Proc R Soc Lond Ser A 1932;135:601–16. [8] Timoshenko SP, Gere JM. Theory of elastic stability. Part II. New York, NY: McGraw-Hill; 1961. [9] Jakubowski S. The analysis of the post-buckling state of the rectangular shield subjected the acting of the eccentric compression [Ph.D. thesis]. Technical University of Lodz: Lodz; 1981 [in Polish]. [10] Tomblin J, Barbero E. Local buckling experiments on FRP columns. Thin-Walled Struct. 1994;18:97–116. [11] Kowal Z, Szychowski A. The experimental determination of critical bearing capacity of plates on models with geometrical imperfections. In: Proceedings of the XLVIII Scientific Conference on KILiW PAN and KN PZITB, Opole – Krynica; 2002. II: p. 101–8 [in Polish]. [12] Tereszkowski Z. An experimental method for determining critical loads of plates. Arch Budowy Masz 1970;XVII-Z3:485–93 (in Polish). [13] Von Kármán T. Festigkeitsprobleme im Maschinenbau. Encyklopädie Math Wissenchafsten 1910;V4:349. [14] PN-EN 10002-1. 1998 – Metallic materials – Tensile testing – Part 1: method of test at ambient temperature [in Polish]. [15] Vlasow VZ. Thin-walled elastic beams. Jerusalem: Israel Program for Scientific Translations; 1961. [16] Wolfram S. Mathematica. Cambridge: Cambridge University Press.