Bearing capacity of axially loaded timber members--estimation under uneven fire action/Mediniu centriskai gniuzdomu elementu, netolygiai veikiamu ugnies laikomosios galios, skaiciavimas.

Sauciuvenas, Gintas ; Sapalas, Antanas ; Griskevicius, Mecislovas 等

1. Introduction

Timber structures in a fire situation are commonly rated as more dangerous if compared with steel and concrete structures. Due to the low conductivity of timber, the bearing capacity of timber structures decreases slower in a fire situation if compared with other mentioned types of structures. The behaviour of timber in fire, its charring rate, humidity influence on the charring rate, strength variation at elevated temperatures are investigated profoundly by Jong and Clancy (2004), Frangi et al. (2008). Modification factors of strength in a fire situation are provided in LST EN 1995-1-2:2005. Under a typical fire situation, the charring of timber goes at a constant rate, which depends on the shape and size of the member. Investigations into charring rates depend on different fire scenarios, parameters and fire protection means and were executed by Lipinskas and Ma?iulaitis (2005), Polka (2008). Many investigations were carried out to obtain changes in timber properties at elevated temperatures. They were performed by Konig (2005), Bednarek and Kamocka (2006), Bednarek and Kaliszuk-Wietecka (2007), Bednarek et al. (2009). According to LST EN 1995-1-2:2005 for solid timber with the density of more than 290 kg/[m.sup.3], the design charring rate will be 0,8 mm/min. For members with a higher cross section reduction ratio for a cross section area, the strength and modulus of elasticity decrease due to rising temperature. Temperature distribution in timber members under heating from one or all sides was analyzed by Frangi and Fontana (2003) and Reszka and Torero (2006).

Articles about investigations into the behaviour of compressed timber members under direct flame action without the means of protection are very rare (Sauciuvenas, Griskevicius 2009). More extensive investigations are executed on the studs of the framed wall elements by direct flame action on a wall fragment (Young, Clancy 2001; Clancy 2002; Richardson 2001; Hadvig 1981).

The latest investigations deal with modelling fire conditions (Galaj 2009) and analysis of temperature distribution in a cross section of the members (Reszka, Torero 2006). Up to now, the software of finite elements has not been so widely used for determining temperature fields. Blazevicius and Kvedaras (2007) employed this method for temperature fields of composite sections and Erchinger et al. (2010)--for temperature fields of timber members.

The uneven charring of a timber cross section can induce a bending moment even under axial loading conditions. Streiger and Fontana (2005) pointed to the influence of a bending moment on the bearing capacity of timber columns; however, the results of these investigations were based on data obtained at ambient temperatures and details set out in expressions taken from LST EN 1995-1-1:2005. Data on experimental investigations into timber in a fire situation was collected by Sauciuvenas and Griskevicius (2009) and applied for analysis presented in this article. It can be stressed that investigations into the behaviour of a compressed timber member under the fire conditions of uneven charring are very rare, and therefore it is not possible to make reliable conclusions about this phenomena.

2. Method for calculating timber members under axial force and under axial force and the bending moment

Calculations of a timber member under axial force and combining axial force and the bending moment are performed using the same method of a reduced cross-section. The aim of the made calculations was to obtain the influence of eccentricities induced by uneven charring on the behaviour and bearing capacity of a timber member under fire conditions. The bearing capacity of the reduced fire cross-section can be calculated according to chapter 6.3.2 of LST EN 1995-1-1:2005. The resistance of the member in case axial compression force and the bending moment are acting if [[lambda].sub.rel,z] [less than or equal to] 0.3 and [[lambda].sub.rel,[gamma]] [less than or equal to] 0.3 must be calculated according to the below formulas:

[([[sigma].sub.c,0,d]/[f.sub.c,0,d]).sup.2] + [[sigma].sub.m,y,d]/[f.sub.m,y,d] + [k.sub.m] [[sigma].sub.m,z,d]/ [f.sub.m,z,d] [less than or equal to] 1 (1)

and

[([[sigma].sub.c,0,d]/[f.sub.c,0,d]).sup.2] + [k.sub.m] [[sigma].sub.m,y,d]/[f.sub.m,y,d] + [[sigma].sub.m,z,d]/ [f.sub.m,z,d] [less than or equal to] 1 (2)

In other cases:

[[sigma].sub.c,0,d]/[k.sub.c,y] [f.sub.c,0,d] + [[sigma].sub.m,y,d]/[f.sub.m,y,d] + [k.sub.m] [[sigma].sub.m,z,d]/ [f.sub.m,z,d] [less than or equal to] 1; (3)

[[sigma].sub.c,0,d]/[k.sub.c,z] [f.sub.c,0,d] + [k.sub.m] [[sigma].sub.m,y,d]/[f.sub.m,y,d] + [[sigma].sub.m,z,d]/ [f.sub.m,z,d] [less than or equal to] 1; (4)

where: [k.sub.m] - a factor considering the re-distribution of bending stresses in a cross-section; [f.sub.m,y,d]--design bending strength about the y axis; [f.sub.m,z,d]--design bending strength about the z axis;

where:

[k.sub.c,y] = 1/[k.sub.y] + [square root of [k.sup.2.sub.y] - [[lambda].sup.2.sub.rel,y]; (5)

[k.sub.c,z] = 1/[k.sub.z] + [square root of [k.sup.2.sub.z] - [[lambda].sup.2.sub.rel,z]; (6)

[k.sub.y] = 0,5(1 + [[beta].sub.c]([[lambda].sub.rel,y] -0.3) + [[lambda].sup.2.sub.rel,y]); (7)

[k.sub.z] = 0,5(1 + [[beta].sub.c]([[lambda].sub.rel,z] -0.3) + [[lambda].sup.2.sub.rel,z]); (8)

[[beta].sub.c]--a straightness factor for members the straightness limits of which are described in LST EN 1995-1-1 :2005 chapter 6.3. For solid timber [[beta].sub.c] = 0.2 .

The calculations of the strength of the members loaded with axial force and bending as well as induced by uneven cross-section charring was based on the remaining cross-section method and experimental charring depth obtained for each side of the member. Following LST EN 1995-1-2:2005 chapter 3.4.2, the radius of the roundings of the remaining cross-section corner r = [d.sub.char,0] and eccentricities were calculated. For a more precise estimation of the remaining geometrical parameters of a crosssection-a section area, the second moment of the area, section modulus and eccentricities - the section was divided into four parts (see Fig. 1). The radius of the roundings of the cross-section corner was calculated like the average for adjacent horizontal and vertical section faces ([R.sub.1a][d.sub.char,n,ha1] + [d.sub.char,n,bal]/2). Zero strength layer thickness --[k.sub.0][d.sub.0]--was not rated in calculations. Eccentricities used in calculations consist of the mean value of the measured bow imperfection equal to 8 mm and eccentricities due to reduction in the cross-section induced by charring (see Fig. 2).

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The made calculations have revealed that bending moments in a member can appear due to the nonuniform charring of a cross-section and can be increased by the eccentric action of axial force. The imperfection of the initial bow has not been taken into account. Along with discussed assumptions, the strength of residual cross-sections is presented in Fig. 2. For some specimens (1, 4, 10, 14, 15, 16, 22), the strength of the residual cross section that corresponds to the experimental limit state is higher than experimental strength shown in Fig. 3. The average values of the bearing capacity ratio calculated according to LST EN 1995-1-1:2005 (6.23) are equal to 1.19 and according to LST EN 1995-1-1:2005 (6.24)--to 0.8.

[FIGURE 3 OMITTED]

It can be assumed that the above introduced situation occurred due to the fact that some factors were not considered during calculations. The carried out calculations showed it was not sufficient to take into account only a reduction in the geometry of the cross section, because a reduction in timber strength at elevated temperatures may also considerably influence the bearing capacity of the member. To prove it, a more complex calculation model discussed in LST EN 1995-1-2:2005 Annex B, was applied. In such a case, it was necessary to estimate temperature distribution fields in a cross-section of a timber element. For solving this problem, finite element software SolidWorks Simulation was used.

3. Results of a numerical simulation of the behaviour of slender timber members

Temperature distribution fields in a member cross-section were estimated for each interval of 60 seconds according to approximate temperature curves obtained during tests. Experimental temperature curves are presented in Fig. 4. During simulation, the bottom side of a cross-section (below the horizontal section axis) was affected by temperature according to the approximate temperature curve of the bottom side and the upper cross-section side (above the horizontal section axis)--according to the approximate temperature curve of the top side. It was assumed that:

--the specimen cross-section--50 x 50 mm;

--the buckling length of member--1200 mm;

--timber average density - 580 kg/[m.sup.3].

[FIGURE 4 OMITTED]

For solving the problem of temperature field distribution, as in the case of stress analysis with FEM simulation, it is necessary to divide the model into finite elements. For this particular problem of temperature field distribution, the cross-section was subdivided into 2.5 mm square finite elements.

FEM simulation was used only for temperature distribution in a cross-section. It was caused by the complexity and lack of information related to the relationships of strength and deformations at elevated temperatures. For more complex FEM simulation considering the behaviour of a timber member in a fire situation, it is not enough to know the relationship between strength and temperature as the influence of temperature on deformations must also be taken into account. Thus, for a general case, 3D relationship for strength, deformations and temperature must be used in FEM simulation. EC 1-1-2:2004 and other literature sources do not determine similar relationships. In the absence of the before discussed relationships, the application of popular FEM software SolidWorks Simulation for timber member complex analysis taking into consideration the strength and deformation of timber at elevated temperatures seems to be limited. It was not possible to find publications related to investigation into the stresses and behaviour of timber members in a normal environmental situation employing FEM software. Temperature distribution to the various types of cross-sections of timber wall posts and the charring process of them was investigated by means of software Frangi et al. (2008). Timber members were affected by elevated temperature from three sides. The comparison results of data on experimental and computer simulations were presented. In both cases, a standard fire curve was applied.

4. Thermal properties of timber

For modelling a designed fire situation, on the basis of LST EN 1995-1-2:2005 Annex B recommendations, thermal conductivity, specific heat and density ratio values of soft wood were applied according to curves presented in Figs 5, 6 and 7.

The caused charring layer thermal conductivity values are true, whereas those of timber charcoal conductivity are not measured. While increasing heat transfer due to shrinkage cracks at higher than 500[degrees]C temperature, the timber decay process at approximately 1000 [degrees]C was taken into account. The charring layer cracks increased heat transfer by radiation and convection. It is generally prevalent not to consider these factors in computer models (LST EN 1995-1-2:2005).

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

5. Temperature distribution fields in the cross-section

Let us focus on temperature distribution fields at 840 s (14 min) after fire starts. This corresponds to the average experimental fire resistance. Temperature distribution fields after 14 min are presented in Fig. 8. Temperature in the cross-section considering this situation is higher than that in a normal environment, thus, the strength of timber in a big part of the section is reduced and must be considered for calculations.

[FIGURE 8 OMITTED]

According to temperature distribution in the cross-section, the charring boundary corresponding to 300 [degrees]C isotherm was found. According to the rules of the code for timber behaviour in a fire situation, this isotherm can be considered as the charring boundary. It was estimated that according to simulation results, the thickness of the average charring layer for the bottom was 4.5 mm and according to experimental data - 7.7 mm. For the top side of the cross-section, it made 1.9 mm and 0.9 mm respectively. For the left and right side faces of the cross-section, only computer simulation results were obtained. The picture in Fig. 10 clearly shows that the thickness of the average charring layer will be more than 2.7 mm. The differences between computational and experimental investigation results can be explained by some inaccuracy in the approximation of the relationship between time and temperature assuming that one relation is suitable for all the area of the top part of the section and the other--for all the area of the bottom part of the cross-section.

For comparison, Fig. 9 displays the temperature fields of the same cross-section after 14 min period of fire according to the standard temperature curve when heating is affecting the member from all sides.

6. Bearing capacity of timber members having material properties at elevated temperature

The radius of the charred corner rounding according to temperature distribution fields and geometrical parameters of the areas bounded by different temperature isobars are presented in Table 1.

For each area bounded by the isotherm starting from 300[degrees]C, the basic cross-section properties were calculated and presented in Table 1.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

Taking into account the properties of the bounded areas and the average values of the temperature interval, the weighted reduction factor for average properties related to the 300[degrees]C isotherm bounded area (bright green) of bending and compression strength and the modulus of elasticity was calculated. The values of the reduction factor of compression and bending strength were considered according to experimental relationships presented by Bednarek et al. (2009). The reduction factor of the elasticity modulus was obtained from Eurocode 5 (LST EN 1995-1-2:2005) Annex B because of the lack of experimental data. Relationships with reduction factors used in calculations are presented in Fig. 11.

The weighted average reduction factors of bending and compression strength and the modulus of elasticity for a concerned area according to the average temperatures of the area were calculated by the formula:

[S.sub.p,vid] = [S.sub.A1] + [S.sub.A2] [A.sub.2]/[A.sub.1] + ... + [S.sub.Ai] [A.sub.i]/[A.sub.1], (9)

where: [S.sub.p,vid]--the weighted average reduction factor value of the strength or modulus of elasticity; [S.sub.Ai]--the reduction factor of the strength or modulus of elasticity depending on mean temperature for a concerned area;

[A.sub.i]--the concerned area.

Table 2 presents the values of reduction factors calculated according to expression (9).

The bearing capacity of the residual design crosssection (without the charring layer) was calculated according to the rules imposed in Eurocode 5 Part 1-2 (LST EN 1995-1-2:2005) and formulas (1)-(8). The calculations used the following initial mean values of experimental timber properties:

--compression strength--43.3 MPa;

--bending strength--87.1 MPa;

--the modulus of elasticity--16355 MPa.

[FIGURE 11 OMITTED]

The bearing capacity of a timber member in a fire situation was calculated two times. The member was calculated like a column subjected to compression and like a column subjected to combined bending and compression. Initial geometrical imperfections were not taken into account. An additional bending moment that caused axial load eccentricity due to uneven charring was evaluated.

The results of calculating bearing capacity considering code LST EN 1995-1-1:2005 may be observed in Table 3.

Data presented in Table 3 disclose that taking into account the experimental values of timber strength and reduction factors for the modulus of elasticity in compression according to Eurocode 5 Part 1-2 (LST EN 1995-1-2:2005), the calculated values of bearing capacity are conservative in comparison with experimental data. As for all strength properties, Eurocode 5 (LST EN 19951-1:2005) gives smaller reduction factors; if compared with experimental data, it seems to be that the factor for the modulus of elasticity is also conservative. To determine the impact of the modulus of elasticity, reduction factors for the results of bearing capacity calculations and the tension modulus coefficient of elasticity (which is higher) were used. In such a case, the factor for utilizing the bearing capacity of a slender timber member in a fire situation was:

--buckling about the horizontal axis--1.07;

--buckling about the vertical axis--1.06;

--combined bending and compression--1.04.

It can be concluded that the rules of Eurocode 5 part 1-2 (LST EN 1995-1-2:2005) can be applied not only in the case of a standard fire curve situation, but also for other time-temperature relationships. When reduction in strength and deformation properties at elevated temperatures is taken into account, the results of capacity calculations are on the safe side if compared with experimental data.

7. Conclusions

1. The analysis of experimental data and calculation results of axially compressed timber members show that some additional factors can be significant and the influence of these factors must be investigated more thoroughly:

--uneven temperature redistribution in furnace;

--local flame effect;

--additional eccentricity induced by a higher charring layer in the bottom zone of the member and timber exposed to reduction in higher temperature strength.

2. Mean experimental fire resistance of a timber member with a cross-section of 50*50 mm is equal to 15 min for the acting external force of 6 kN and 14 min for the external force of 7.5 kN. A relatively small increase in external force does not expose to a big influence on fire resistance.

3. Design fire resistance of a 50*50 mm cross-section member calculated applying the method of the reduced cross-section with the mean timber strength of specimens is 9.1 min for the external compressing force of 6 kN and 8.2 min for the external compressive force of 7.5 kN.

4. The influence of a horizontal position of specimens in the furnace during the experiment has not been accepted because of a small weight of test specimens.

5. A typical collapse mode of the members having one heated side was identified. Buckling a member appears in the direction of the heated side of specimens.

6. Measuring the residual section dimensions of the heated members shows the loss of a section depending on drying and charring.

7. The analysis of experimental data and calculation results shows that the rules established by Eurocode 5 Part 1-2 (LST EN 1995-1-2:2005) can be applied not only in case of a standard fire curve situation, but also for other time-temperature relationships (Konig 2005).

8. If a residual cross-section of a member affected by fire is known, it is possible to precisely calculate the reserve of the bearing capacity of such member using Eurocode 5 (LST EN 1995-1-1:2005) expressions for members under combined bending and compression, which is important for planning building rehabilitation and cleaning measures.

9. When small members of a cross-section, not covered by Eurocode 5 Part 1-2 (LST EN 1995-1-2:2005) are calculated, to figure out bearing capacity, it is reasonable to determine temperature distribution in the cross-section according to the Annex B of this code using weighted reduction factors for timber mechanical properties. If only axial force is acting, calculations can be done as for a simple axially loaded member.

10. Investigations reveal the weakness of FEM software in case of complex stress and deformation analysis of timber members at normal and elevated temperatures.

http://dx.doi.org/ 10.3846/13923730.2011.625655

References

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Bednarek, Z.; Kaliszuk-Wietecka, A. 2007. Analysis of the fire-protection impregnation influence on wood strength, Journal of Civil Engineering and Management 13(2): 79-85. 10. doi:1080/13923730.2007.9636423

Bednarek, Z.; Kamocka, R. 2006. The heating rate impact on parameters characteristic of steel behaviour under fire condition, Journal of Civil Engineering and Management 12(4): 269-275. doi:10.1080/13923730.2006.9636403

Blazevicius, Z.; Kvedaras, A. K. 2007. Experimental investigation into fire resistance of HC-FST columns under axial compression, Journal of Civil Engineering and Management 13(1): 1-10. doi:10.1080/13923730.2007.9636413

Clancy, P. 2002. A parametric study on the time-to-failure of wood framed walls in fire, Journal of Fire Technology 38(3): 243-269. doi:10.1023/A:1019882131985

Erchinger, C.; Frangi, A.; Fontana, M. 2010. Fire design of steel-to-timber dowelled connections, Engineering Structures 32(2): 580-589. doi:10.1016/j.engstruct.2009.11.004

Frangi, A.; Erchinger, C.; Fontana, M. 2008. Charring model for timber frame floor assemblies with void cavities, Fire Safety Journal 43(8): 551-564. doi:10.1016/j.firesaf.2007.12.009

Frangi, A.; Fontana, M. 2003. Charring rates and temperature profiles of wood sections, Fire and Materials 27(2): 91-102. doi:10.1002/fam.819

Galaj, J. 2009. A general concept of fire hybrid modelling in copart ents, Journal of Civil Engineering and Management 15(3): 237-245. doi:10.3846/1392-3730.2009.15.237-245

Hadvig, S. 1981. Charring of Wood in Building Fires: practice, theory, instrumentation, measurements. Denmark: Technical University of Denmark. 238 p.

Jong, F.; Clancy, P. 2004. Compression properties of wood as functions of moisture, stress and temperature, Fire and Materials 28(2-1): 209-225. doi:10.1002/fam.859

Konig, J. 2005. Structural fire design according to Eurocode 5-design rules and their background, Fire and Materials 29(3): 147-163. doi:10.1002/fam.873

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LST EN 1995-1-1:2005 Eurokodas 5. Mediniu konstrukciju projektavimas. 1-1 dalis. Bendrosios nuostatos. Bendrosios ir pastatu taisykles. Vilnius: LSD. 2005. 130 p.

LST EN 1995-1-2:2005 Eurokodas 5. Mediniu konstrukciju projektavimas. 1-2 dalis. Bendrosios nuostatos. Konstrukciju elgsenos ugnyje skaiciavimas. ilnius: S. 73 p.

Polka, M. 2008. The influence of flame retardant additives on fire properties of epoxy materials, Journal of Civil Engineering and Management 14(1): 45--48. doi:10.3846/1392-3730.2008.14.45-48

Reszka, P.; Torero, J. L. 2006. In-depth temperature measurements of timber in fires, in Proc. of the 4th International Workshop Structures in Fire, Aveiro, Portugal, May, 2006, 921-930.

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Sauciuvenas, G.; Griskevicius, M. 2009. Mediniu centriskai gniuzdomu elementu elgsena ugnyje, Statybines konstrukcijos ir technologijos [Engineering structures and technologies] 1(1): 50-57.doi:10.3846/skt.2009.06

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Gintas Sauciuvenas (1), Antanas Sapalas (2), Mecislovas Griskevicius (3)

Vilnius Gediminas Technical University, Saul?tekio al. 11, LT-10223 Vilnius, Lithuania

E-mails: (1) gintas.sauciuvenas@vgtu.lt (corresponding author); (2) antanas.shapalas@vgtu.lt; (3) mecislovas.griskevicius@smm.lt

Received 05 Jan. 2011; accepted 05 Jul. 2011

Gintas SAUCIUVENAS. Assoc. Prof. Dr at the Department of Steel and Timber Structures, Vilnius Gediminas Technical University. Research interests: evaluation of the existing steel and timber structures, evaluation of fire resistance of steel and timber structures.

Antanas SAPALAS. Prof. Dr, head of the Department of Steel and Timber Structures, Vilnius Gediminas Technical University. Research interests: evaluation of the existing steel and timber structures, evaluation of fire resistance of steel and timber structures.

Mecislovas GRISKEVICIUS. Assoc. Prof. Dr at the Department of Labour Safety and Fire Protection, Vilnius Gediminas Technical University. Research interests: activities of fire prevention services, fire prevention, organization and implementation of life safety measures.

Table 1. The parameters of cross sections circumscribed by
temperature isotherms displayed in Fig. 10

Bright green
(the isotherm at a temperature of 300[degrees]C)--A1

Shape and dimensions, mm

             Geometrical properties of the area                Values

Area, [mm.sup.2]                                               1748.2
Perimeter, mm                                                   153.9
The second moment of the area about the horizontal axis,       252028
  [mm.sup.4]
The second moment of the area about the vertical axis,         251327
  [mm.sup.4]
The radius of gyration about the horizontal axis, mm            12.01
The radius of gyration about the vertical axis, mm              11.99

Blue--A2

Shape and dimensions, mm

Geometrical properties of the area                             Values

Area, [mm.sup.2]                                               1631.0
Perimeter, mm                                                   148.0
The second moment of the area about the horizontal axis,       218957
  [mm.sup.4]
The second moment of the area about the vertical axis,         217427
  [mm.sup.4]
The radius of gyration about the horizontal axis, mm            11.59
The radius of gyration about the vertical axis, mm              11.55

Green--A3

Shape and dimensions, mm

Geometrical properties of the area                             Values

Area, [mm.sup.2]                                               1346.2
Perimeter, mm                                                   133.9
The second moment of the area about the horizontal axis,       148334
  [mm.sup.4]
The second moment of the area about the vertical axis,         147769
  [mm.sup.4]
The radius of gyration about the horizontal axis, mm            10.50
The radius of gyration about the vertical axis, mm              10.48

Violet--A4

Shape and dimensions, mm

Geometrical properties of the area                             Values
Area, [mm.sup.2]                                               1004.9
Perimeter, mm                                                  116.30
The second moment of the area about the horizontal axis,        83009
  [mm.sup.4]
The second moment of the area about the vertical axis,          81812
  [mm.sup.4]
The radius of gyration about the horizontal axis, mm            9.09
The radius of gyration about the vertical axis, mm              9.02

Small blue--A5

Shape and dimensions, mm

Geometrical properties of the area                             Values

Area, mm                                                        93.10
Perimeter, mm                                                   34.4
The second moment of the area about the horizontal axis,         736
  [mm.sup.4]
The second moment of the area about the vertical axis,           698
  [mm.sup.4]
The radius of gyration about the horizontal axis, mm            2.81
The radius of gyration about the vertical axis, mm              2.74

Note. Sketches of the areas are rotated about the horizontal
axis.

Table 2. The results of calculating reduction factors of timber
parameters

 Area       Tem-          Mean          Area,
symbol    perature    temperature    [mm.sup.2]

 A1-A2       299         283.57        117.24
 A2-A3     268.14        237.07        284.74
 A3-A4       206        175.035        341.31
 A4-A5     144.07       113.0525       911.81
  A5       82.035       51.0175         93.10

Weighted values of reduction
factors for the whole area A1:

       Area           Reduction      Reduction      Reduction factor
      symbol          factor of        factor      of the modulus of
                     compression     of bending        elasticity
                                                     (compression)

      A1-A2            0.007143       0.004286          0.00175
      A2-A3            0.227571       0.136543          0.055755
      A3-A4              0.58           0.38             0.1295
      A4-A5            0.71186        0.61186           0.197878
        A5             0.83593        0.77186           0.395966

Weighted values of      0.573          0.461             0.160
reduction factors
for the whole area
A1:

Table 3. Calculation results

                                            Values of
                 Compression               experimental
                                              loads

                                        6.0 kN    7.5 kN

Equation                Design load,    Experimental and
                             kN         design load ratio

(3) buckling about          4.25         1.412     1.765
the horizontal axis

(4) buckling about          4.26         1.408     1.761
the vertical axis

Combined bending about the horizontal axis
and compression

(3)                         4.25         1.446     1.810
                        Eccentricity
                        e = 1.973 mm