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Behaviour of cold-formed steel shear walls under horizontal and vertical loads
ARTICLE IN PRESS
Thin-Walled Structures 44 (2006) 1214–1222 www.elsevier.com/locate/tws
Behaviour of cold-formed steel shear walls under horizontal and vertical loads Jo¨rg Langea,, Bernd Naujoksb a
Technische Universita¨t Darmstadt, Institut fu¨r Stahlbau und Werkstoffmechanik, Petersenstrasse 12, 64287 Darmstadt, Germany b Donges Stahlbau GmbH, Mainzer Strasse 55, 64293 Darmstadt, Germany Available online 28 February 2007
Abstract Shear walls made of cold-formed steel with sheathing on one or both sides are composite structures. They can carry horizontal and vertical loads. Based on the results of a large series of tests a design procedure was developed that allows for the design of walls carrying horizontal and vertical loads. In addition to this a model is introduced by which the stabilising effect of the sheeting for the cold-formed sections can be assessed. r 2007 Elsevier Ltd. All rights reserved. Keywords: Cold-formed steel sections; Composite; Walls; Screws; Shear walls
1. Introduction Constructions with cold-formed steel sections have become very successful in Scandinavia and North America during the last couple of years. The construction differs from timber frame structures. Floor panels and shear walls are made of bearing cold-formed steel sections, which are arranged in a pattern of 400–800 mm. In addition the construction includes sheathing on both sides. As a rule the plate thickness of the sections varies between 1.0 and 2.0 mm. An elastic composite construction is made by the connection of the sections and the sheathing. Analogue to timber frame construction the stiffening of a building can be carried out with help of these floor panels or shear walls. A shear wall is then stressed by both vertical and horizontal loads from the floor panels. The assembly of a shear wall is identical to that of a light separating wall. The studs are made of C-sections which are placed into the floor- and ceiling-U-sections. The sheathing is attached to the steel substructure by self-drilling screws. To examine the structural behaviour of the shear walls an analysis with several test runs has been carried out at the Institute for Steel Construction and Material Mechanics. Corresponding author. Tel.: +49 6151 16 2145; fax: +49 6151 16 3245.
E-mail address: [email protected] (J. Lange). 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.01.007
The maximum horizontal load of shear walls with 1.25 m width and 2.60 m height and different types of sheathing was established during the first test run. The structural behaviour with additional vertical loads was examined in the following test run. The most important parameter in the analysis of the structural behaviour of shear walls is the connection with self-drilling screws between the cold-formed steel section and the sheathing. Therefore, a third test run was carried out on detail specimens of screwed connections. The maximum load as well as the load–displacement curves were determined in this test run. 2. State of the art The design in the US American practice is carried out with load tables. This simplification is justified on the grounds that the structural behaviour of shear walls under horizontal loads is to complex for a simple mechanical analysis. The tables contain the ultimate capacities for static and seismic stress for different wall assemblies. The structural properties were determined with tests from Serette at Santa Clara University [1,2]. In addition the behaviour of shear walls under periodic horizontal loads with alternating stressing was examined. The American tests prove that the stiffening with shear walls is qualified
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for quasi-static loading, which means wind loads, as well as for seismic loading. But with the records of the measured values of the ‘‘head displacement’’ and the ‘‘related horizontal load’’ there can only be drawn conclusions on the ductility of this stiffening system. An accurate description of the load transfer is not possible. The stabilising properties of the sheathing on the compressed cold-formed steel sections have already been examined by Winter [3] at Cornell University. Simaan [4] has continued this work. The design in the US AISI Specifications [5] is based on these research results. However, the model to include the stabilising properties of the sheathing by means of the shear stiffness of the shear wall could not be verified with later tests [6]. Similar design methods are missing in Germany. The intention is therefore to develop an universal design method for shear walls under horizontal loads based on the now completed research project [7]. Furthermore the influence of additional effective vertical loads has been examined. Based on the test results an analysis model could be developed in which the stabilising properties of the sheathing during the structural safety verification of the cold-formed steel posts can be included realistic. 3. Research results 3.1. Shear walls under horizontal loads During this test-run the parameters of the sheathing material and the spacing of the screws sr along the edge of the shear panel has been varied. C-sections with the dimensions h b c tn ¼ 97 50 8 1.5 mm and a yield strength of fy ¼ 320 N/mm2 have been used as studs.
The sheathing was attached with self drilling screws + 4.2 mm (4.2 38 WD–O–H from the Company EJOT). The edge distance was constant er ¼ 20 mm. The horizontal load (Fig. 1) was applied after a load cycle which was based on the standards for testing shear walls in timber frame construction, DIN EN 594/07 96. During 120 s the shear walls were stressed with a load up to 40 per cent of the estimated collapse load Hu. This stress was held for 30 s after which the shear wall was unloaded again. After a break of 120 s the load was increased with the same velocity to Hu. The following types of sheathing were used:
Chipboard according to DIN 68763, V100-13 mm, Livingboard from the company Kuntz, with polyurethane, polyurea glued panels. Gypsum fibreboard, Fermacell 0 G 05 from the company Fels with 12.5 mm thickness. Cement-bonded fibreboard Eterplan N from the company Eternit with 8 mm thickness. Trapezoidal sheet 8/172 with 0.88 mm thickness from EKO Stahl.
In the German timber frame construction chipboards and gypsum fibreboards are mainly used. The cementbonded fibreboard has the highest resistance among the offered sheathings. It is possible, according to DIN 18807 part 3/06.87, that cold-formed steel sections with trapezoidal sheeting transfer the loads which act in their plane as diaphragm. The specimens with trapezoidal sheeting and gypsum fibreboards were primarily used to prove that the theory of diaphragm was also true for the problem examined within this project. The collapse of the shear walls with non-metal sheathing starts with the fracture of the edge under tension at the hold-down as shown in Fig. 2.
vertical load
load distribution horizontal load
y
stud 1
stud 2
stud 3
load introduction
tension restrain (hold down) x Fig. 1. Test rig for walls under horizontal and vertical load.
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Fig. 2. Fracture of the edge at the hold-down.
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After that the screws start to tilt in direction of the respective shearing stress. The screw-heads carve themselves into the sheathing. There are three forms of collapse to be observed once the collapse load is reached 1. Fracture of the sheathing edges in the area of the holddown and extraction of the screws through immersion of the sheathing at the bottom of the compression stud, as shown in Fig. 3. 2. Local buckling of the cold-formed steel section at the bottom of the compression stud, as shown in Fig. 4. 3. Local buckling of the trapezoidal sheeting by means of shear stress. The individual test-results are listed in Table 1. The measured load–displacement curves for three different sheathings are given in Fig. 5. In addition to the displacement kv and the horizontal load two other things were measured, the relative displacement of the sheathing compared to the substructure and the normal force distribution along the height of the studs. Independent of the sheathing material the following structural behaviour was examined. The substructure of the shear wall which is composed of studs,
Fig. 4. Buckling of the cold-formed profile at the bottom of the compression stud.
Fig. 3. Fracture of the edge (tension).
floor and ceiling profile deforms under a horizontal load into a parallelogram. The sheathing rotates in its plane like an almost rigid panel as illustrated in Fig. 6. The predominant part of the inserted energy is thus dissipated in the connection between sheathing and substructure. The mathematical model from McCutcheon [8] is based on the geometric interrelation between the distortion of the sheathing and the deflection of the connections which are pictured in Fig. 6. With diaphragms it is assumed that all connections are stressed with the same shear force parallel to the edge of the diaphragm. With shear walls the shear forces perpendicular to the edge of the diaphragm also have to be considered. These shear forces are the result of a rotation of the sheathing on the substructure. Through this effect the screws in the corner areas are being stressed noticeable more than the screws in the field. The shear force which is introduced into the studs longitudinally causes a linear increasing compression, respectively, tension and a corresponding extension of the stud. The corner screws on the base of the shear wall are much more stressed than the screws on the head of the shear wall because of this extension. Due to the components of screw forces which are directed towards the edge, the corner screws on the bottom of the tension stressed stud are governing. Above a certain critical deformation of the corner screw, the screw is no longer able to withstand additional force and will collapse. Subsequently the necessary redistribution of the loads causes a zipper-like collapse of the remaining connectors.
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Table 1 Test results of walls subjected to horizontal loads Test number
Sheathing I
Sheathing II
Screw interval sr (mm)
Collapse load Hu in (kN) (form of collapse)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Chipboard Chipboard Chipboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Chipboard Chipboard Chipboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Chipboard Chipboard Chipboard Cement-bonded fibre Cement-bonded fibre Cement-bonded fibre Cement-bonded fibre Cement-bonded fibre Cement-bonded fibre Chipboard Chipboard Chipboard Trapezoidal sheeting Trapezoidal sheeting Trapezoidal sheeting
Chipboard Chipboard Chipboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Chipboard Chipboard Chipboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Chipboard Chipboard Chipboard Cement-bonded fibre Cement-bonded fibre Cement-bonded fibre Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard
100 100 100 100 100 100 150 150 150 150 150 150
45.79 44.93 43.50 36.85 39.93 42.70 40.50 40.40 39.14 34.46 32.72 32.06 43.97 43.56 54.34 50.29 50.70 53.78 41.69 45.40 43.79 42.10 36.59 40.89 36.53 39.83 40.79
150 150 150 150 150 150 150 150 150 172/150 172/150 172/150
2
Displacement kv (mm) 26.39 28.32 26.70 27.06 28.49 27.48 30.83 29.69 27.99 29.98 34.52 37.94 43.85 37.83 44.73 23.34 23.64 22.10 23.63 31.32 26.63 50.10 42.99 50.21 42.83 27.14 36.60
2 2 2 2 2 2 2 2 2 2 1 1 1 2 2 2 2 1 2 2 1 1 1 3 2 3
Horizontal Load [kN]
Form of collapse: 1 ¼ collapse of connection material, 2 ¼ buckling of cold formed sections, 3 ¼ sheeting buckling. Mixed screw intervals, high density in the corners.
55 50 45 40 35 30 25 20 15 10 5 0
V16 V8 V12
V8, Chipboard, s= 150 mm V12, Gypsum fibre, s = 150 mm V16, Cement fibre, s = 150 mm
0
5
10
15 20 25 Head displacement [mm]
30
35
40
Fig. 5. Load–deflection curve of walls (sr ¼ 150 mm).
3.2. Connection of cold-formed steel section to sheathing with self-drilling screws The basis of the verification method for shear walls is the load–deflection behaviour of the connection between sheathing and cold-formed steel section. This was determined with a test rig which is shown in Fig. 7. It was also possible to investigate alternating loading with this test rig.
The results of the tests on connections with 10 samples each are given in Table 2. The statistical interpretation was carried out according to EC 1, [9]. The specimens were tested force controlled with the same measuring routine as the shear walls under horizontal loads. The connection has to carry in addition to shear forces also a tensile force in direction of the shank due to wind suction. Therefore an ultimate deformation (dult) of 3 mm has been set for the analysis of the collapse load.
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kv
With higher deformations it is possible that the screw head is pulled through the sheathing under inclined tension without any resistance due to the widened hole. The average value of the modulus of sub grade reaction for the tested connection is calculated according to DIN EN 383/10.93 as modified secant stiffness:
ceiling profile d
H
H: horizontal load kv: head displacement d: displacement of the corner-screws stud
0:4V¯ u N=mm . 4=3ðw0:4 w0:1 Þ
In addition the secant stiffness for the range between 0.4 Vu up to Vu is calculated with the following expression: kS2 ¼
H floor profile
kS1 ¼
V¯ u 0:4V¯ u N=mm , ¯ wu 0:4V u =kS1
where w0.1 is the displacement with 0.1 Vu; w0.4 the displacement with 0.4 Vu; wu the displacement with Vu.
sheathing
Fig. 6. Deflection of walls under horizontal load.
3.3. Shear walls under horizontal and vertical loads
P
P
P displacement gauge roller bearing U-profile 15/50/15 sheathing Roller bearing
U-profile 15/50/15
Displacement gauge
Fig. 7. Test rig for tests on connections under cyclic load.
The specimens under horizontal and vertical loads were sheathed with cement-bonded chipboards. Because of its resistance against moisture this sheathing material is used increasingly in prefabricated house construction. It is also biological unobjectionable because of the use of cement as binding agent. Shear walls with sheathings on one side only were also examined in the context of this test run. From consideration of building physics the use of this wall type is sensible. For example in rooms with high sound protection requirements the studs are connected alternately to one sheathing only to decouple the shells. The sheathing can be fixed in spring tracks as an alternative. Both variations are from static point of view one-sided sheathed walls. In the tests it was assessed, that the influence of the deflection forces does not reduce the shear stiffness of the diaphragm of two-sided sheathed shear walls. With onesided sheathed walls lateral torsional buckling of the studs was expected. The stiffness of the connection, as an elastic
Table 2 Stiffness and characteristic values of the resistance of the connection between cold formed section and sheathing using self drilling screws Sheathing material, thickness (mm)
Screw + (mm)
Sheet thickness (mm)
k¯ S1 (N/mm)
k¯ S2 (N/mm)
0.4 V¯ u (N)
V¯ u (N)
dult (mm)
Vu (N)
Chipboard, 13 Gypsum fibreboard, 12.5 Cement bonded fibre, 8 Gypsum wallboard, 12.5 Cement bonded chipb., 12 OSB 3, 12 Plywood board, 13 Trapezoidal sheet, 0.88 Chipboard, 13 Chipboard, 13 Chipboard, 19 Gypsum fibreboard, 12.5
4.2 4.2 4.2 4.2 4.2
1.5 1.5 1.5 1.5 1.5
1927 1517 19 281 635 3811
670 451 3301 121 804
1088 753 1580 215 1410
2721 1884 3951 536 3524
3 3 0.8 3 3
2147 1541 3443 504 2879
4.2 4.2 4.2 4.2 3.5 4.2 4.2
1.5 1.5 1.5 2.0 2.0 2.0 2.0
3174 1004 18 239 1745 1073 1912 1843
434 462 738 693 718 589 496
795 708 1437 1096 993 977 842
1988 1769 3593 2740 2483 2442 2104
3 3 3 3 3 3 3
1363 1504 2983 2325 2122 1986 1895
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Fig. 8. Failure of one-sided sheathed shear walls under vertical loads.
support of the studs, did not sink to the level that this or buckling around the weak axis occurred. Several specimens were loaded with vertical loads only. The observed failure patterns of one-sided sheathed shear walls in this case show, that the section deformation cannot be disregarded for the determination of the load-carrying capacity of C-sections supported in this way, see also Fig. 8.
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it is recommended to use winged-screws with a diameter of 4.2 mm. These screws are robust and can be used for a plate thickness of up to 2.5 mm. The wings prevent an unwanted uplift of the sheathing during the drill process through the steel sheet. The edge distance of the screws should add up to at least 20 mm within the sheathing. Consequential the flange width of the C-sections has to be 50 mm. This dimension should also be maintained for higher static requirements to ensure that the influence of the studs as a thermal bridge and sound bridge is minimised. For the same reasons the grid dimension of 625 mm should be kept, even with additional loads for example near wall openings. The grid dimension is determined by the width of the sheathing materials. The plate thickness of the C-sections should be changed rather than changing the grid dimensions. To ensure a constant load introduction of the normal forces into the studs the bottom and ceiling sections should meet several demands. These demands are that the sections have to be moulded or slotted before folding in such a way that the studs have full contact with the sections. The practice to produce U-sections which have an additional width of twice the plate thickness compared to the stud sections is copied from dry construction. This method causes that the C-sections are bedded on the bending radii of the U-sections (Fig. 9). Because of this the load is only introduced into the flanges. The buckling load of the C-sections sinks in the area of load introduction correspondingly. The minimum thickness of the sheeting should be 12 mm (8 mm with cement-bonded fibreboards). A failure of the sheathing due to shear buckling can be excluded with this thickness.
b + 2.t U-section for load introduction
4. Design concept The design-method for shear walls with cold-formed studs under vertical und horizontal loads can be divided into the following steps:
Structural requirements. Load–deflection behaviour of the connections. Determination of the horizontal collapse load HR. Normal force stress on the studs. Design of the tension anchorage. Stabilisation of the studs.
Concentrated load introduction
Stud
4.1. Structural requirements
b If self-drilling screws are used for the connection between the cold-formed steel section and the sheathing,
Fig. 9. Concentrated load introduction in bending.
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Table 3 Expression for the load deflection relation of the connection between section and sheathing using self-drilling screws, + 4.2 mm and a plate thickness of 1.5 mm
Table 4 Mean value of S for 0.3obo0.7, see also Table 3 Dimensions of the shear wall, h b (mm)
S ¼ f (sr) sr in (mm)
V ¼ a wb
2600 1250
100 S ¼ 60 sr 100 S ¼ 68 sr 100 S ¼ 78 sr
Sheathing
Thickness (mm)
a (N/mmb)
b (dimensionless)
3000 1250
Chipboard Gypsum fibreboard Gypsum wall board Cement bonded fibreboard OSB Plywood board
13 12.5 12.5 12
1486 1077 338 2173
0.55 0.51 0.42 0.44
3500 1250
12 13
1324 885
0.37 0.63
Step 3: Determination of the horizontal ultimate load HR: sin a , 2 with n the number of sheathing, one-sided n ¼ 1, two-sided n ¼ 2, (dimensionless); EA the extensional stiffness of the cold-formed steel studs (N); Vu the mean value of the collapse load of the connection according to Table 2; sr the distance between the screws at the outer edge of the shear wall; dult the ultimate displacement of the connection according to Table 2; a the coefficient of the load–deflection relation of the connection in (N/mmb); b the coefficient of the load–deflection relation of the connection, (dimensionless); S the auxiliary coefficient according to Table 4, (dimensionless); a the diagonal angle of the screw outline of the shear wall; aE arctan (b/h), h ¼ height, b ¼ width of the shear wall; d1 the resisting ultimate deformation of the edge screw in consideration of the extension of the studs, d1pdult. If two different sheathings are used on the two sides of the shear wall, the smaller of the two ultimate deflections should be used. The respective collapse loads HR may be added. Sheathing in the second layer (e.g. because of sound insulation) should not be included. An enhancement of the load bearing capacity cannot be expected due to the considerably higher lever arm of the forces on the screws. H R ¼ n a S d b1
4.2. Load–deflection behaviour of the connections The values for self-drilling screws with 4.2 mm diameter were determined experimentally. The load–deflection behaviour of these connections can be determined by good approximation with the expression in Table 3. 4.3. Determination of the horizontal ultimate load HR The ultimate load HR for shear walls under horizontal loads is defined through an ultimate displacement of the edge screws dult. The foundation of the design model is the assumption that the energy which is introduced through the head of the wall is only dissipated through the deformation of the connectors. All of the other deformations due to the flexural and extensional stiffness of the cold-formed sections as well as the shear stiffness of the sheathing are assumed to be zero in the calculation. For this case, ‘‘rigid board rotates on hinged studs’’, a geometric correlation between head displacement of the shear wall and the individual displacement of the connectors was derived. With this correlation the energy theorem of McCutcheon [8] can be formulated. For the solution of the energy theorem the expression from Table 3 is used which describes the deflection behaviour of the connection between cold-formed section and sheathing. Governing are the edge screws at the bottom of the shear wall. There the influence of the extension of the studs has to be considered in the form of a correction factor. The result of the design method which is described thoroughly in [7] can be outlined in the following steps: Step 1: Determination of the correction factor Z, which accounts for the influence of the extension of the studs on the loading of the edge screws: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V un hl l¼ . !Z¼ sr d ult EA tanhðlhÞ Step 2: Determination of the resisting ultimate deformation d1: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 2ult . d1 ¼ ðZ cos aÞ2 þ sin a2
4.4. Normal force on the studs By means of a horizontal load which is applied at the head of the shear wall the studs are stressed with a normal force which increases linear over the height of the stud. Hh ðat the bottom of the shear wallÞ. b This load has to be superposed with the support normal forces in consequence of the floor and vertical loading from the upper floors. The measured normal forces of the internal studs due to horizontal loads are negligibly small. max N ¼
4.5. Dimensioning of the tension anchorage The tension anchorage of the shear wall has to be proportioned to the maximum of the normal force, Z ¼ H h/b. Permanent compression loads on the edge studs of the shear walls can be included given that they unload the
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tension anchorage. Analogue to timber frame construction a tension anchorage is only necessary at the outer edge if several shear walls are placed in a row. At the butt joint of the walls the normal forces of the studs cancel each other. 4.6. Stabilisation of two-sided sheathed studs The stabilising effect of the sheathing differs noticeable from that of a diaphragm. The deviation forces of a predeformed stud cause only local deformations at the screw connections. The displacements due to shear strain of the sheathing are negligible small. The bifurcation load in direction of the wall plane as well as the screw stressing due to deviation forces can therefore be established by means of a model of an elastic bedded beam, Figs. 10 and 11. The kvalues have to be determined by tests [7].
kϑ
kv2
It has been shown [7] that with a chosen distance between the screws and the examined well-established sheathing materials, a collapse of a two-sided sheathed cold-formed section in direction of the wall plane can be excluded. C-sections with the dimensions as commonly used in domestic construction must therefore only be verified for lateral buckling transverse to the direction of the wall plane. Starting from the idealised model with linear elastic springs it is apparent that the connections will behave plastic rather than elastic with very small deformations. The load bearing capacity of the connectors is determinative for the design. If thick-walled cold-formed sections or more flexible sheathing, such as gypsum wall boards, are used it must therefore be examined whether the connectors are able to resist the deviation forces. If necessary the distance between the individual screws has to be reduced, Fig. 11. The analysis method for one-sided sheathed C-sections with regard to the deformation of the section and local buckling is derived thoroughly in [7]. An inclination of the studs causes an additional load on the diaphragm due to the P–D-effect. In case of a high normal force the influence according to the analysis of second order should be included with help of an amplification factor: H II ¼ H I
5. Summary
Fig. 10. Model for the support condition of the sections.
V
V
V
sr
Pre-camber
1 P , 1 N=N Ki
with HI the sum of Phorizontal load due to the inclination of the studs H 0 ¼ Nc0 and the outer horizontal load; c0 ¼ 1/200, pre-rotation of the P stud according to DIN 18800-2 section 2.3; N Ki ¼ ks1 L; N ¼ sum of the normal forces on the diaphragm; ks1 ¼ 0:4H u =0:4kv stiffness of the diaphragm (Table 1).
kϑ
kv1
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Pre-deflection
Fig. 11. Forces acting on the connection due to the imperfection of the studs.
The basis for the previous described design-method for shear walls made of cold-formed steel sections and different sheathing is a test series. The test series was carried out at the Institute of Steel Construction and Materials Mechanics at the Technische Universita¨t Darmstadt. In contrast to the known analysis for shear walls in timber frame constructions the method is based on a permitted displacement of the corner screws at the outlining edge of the shear wall. With regard to the stabilising behaviour of the sheathing a calculation model for the stabilisation verification of the compression stressed C-section-studs has been introduced in extracts. Two-sided sheathed C-sections with a plate thickness of up to 2.5 mm are stabilised sufficiently with the examined sheathings. The sections have to be verified with regard to buckling transverse to the direction of the wall plane only. With thick-walled sections (42.5 mm) or flexible sheathing materials (e.g. gypsum board) it is necessary to determine whether the connectors are able to resist the deviation
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forces according second order theory also in direction of the wall plane. The analysis method for one-sided sheathed C-sections with regard to the deformation of the section and local buckling is derived thoroughly in [7]. References [1] Serrette R, Hall G, Ngyen H. Shear wall values for light weight steel framing. Final report, AISI, Washington, January 1996. [2] Serrette R, Hall G, Encalada J. Additional shear wall values for light weight steel framing. Final report, AISI, Washington, March 1997. [3] Green GG, Winter G, Cuykendall TR. Light gage steel columns in wall-braced panels. Bulletin no. 35/2, Cornell University, Ithaca NY, 1947.
[4] Simaan A. Buckling of diaphragm-braced columns of unsymmetrical sections and application to wall studs design. Report no. 353, Cornell University, Ithaca NY, 1973. [5] AISI. Specification for the design of cold-formed steel structural members. Washington, 1996 Edition. [6] Miller TH, Peko¨z T. Behavior of gypsum-sheathed cold-formed steel wall studs. J Struct Eng, ASCE 1994. [7] Naujoks B. Tragverhalten von Wandtafeln mit Kaltprofilen unter horizontalen und vertikalen Lasten. PhD thesis, TU Darmstadt, November 2002. [8] McCutcheon WJ. Racking deformations in wood shear walls. J Struct Eng, ASCE 1985. [9] Eurocode 1 — EN1991-1. Actions on Structures, Appendix D, 2001.
Thin-Walled Structures 44 (2006) 1214–1222 www.elsevier.com/locate/tws
Behaviour of cold-formed steel shear walls under horizontal and vertical loads Jo¨rg Langea,, Bernd Naujoksb a
Technische Universita¨t Darmstadt, Institut fu¨r Stahlbau und Werkstoffmechanik, Petersenstrasse 12, 64287 Darmstadt, Germany b Donges Stahlbau GmbH, Mainzer Strasse 55, 64293 Darmstadt, Germany Available online 28 February 2007
Abstract Shear walls made of cold-formed steel with sheathing on one or both sides are composite structures. They can carry horizontal and vertical loads. Based on the results of a large series of tests a design procedure was developed that allows for the design of walls carrying horizontal and vertical loads. In addition to this a model is introduced by which the stabilising effect of the sheeting for the cold-formed sections can be assessed. r 2007 Elsevier Ltd. All rights reserved. Keywords: Cold-formed steel sections; Composite; Walls; Screws; Shear walls
1. Introduction Constructions with cold-formed steel sections have become very successful in Scandinavia and North America during the last couple of years. The construction differs from timber frame structures. Floor panels and shear walls are made of bearing cold-formed steel sections, which are arranged in a pattern of 400–800 mm. In addition the construction includes sheathing on both sides. As a rule the plate thickness of the sections varies between 1.0 and 2.0 mm. An elastic composite construction is made by the connection of the sections and the sheathing. Analogue to timber frame construction the stiffening of a building can be carried out with help of these floor panels or shear walls. A shear wall is then stressed by both vertical and horizontal loads from the floor panels. The assembly of a shear wall is identical to that of a light separating wall. The studs are made of C-sections which are placed into the floor- and ceiling-U-sections. The sheathing is attached to the steel substructure by self-drilling screws. To examine the structural behaviour of the shear walls an analysis with several test runs has been carried out at the Institute for Steel Construction and Material Mechanics. Corresponding author. Tel.: +49 6151 16 2145; fax: +49 6151 16 3245.
E-mail address: [email protected] (J. Lange). 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.01.007
The maximum horizontal load of shear walls with 1.25 m width and 2.60 m height and different types of sheathing was established during the first test run. The structural behaviour with additional vertical loads was examined in the following test run. The most important parameter in the analysis of the structural behaviour of shear walls is the connection with self-drilling screws between the cold-formed steel section and the sheathing. Therefore, a third test run was carried out on detail specimens of screwed connections. The maximum load as well as the load–displacement curves were determined in this test run. 2. State of the art The design in the US American practice is carried out with load tables. This simplification is justified on the grounds that the structural behaviour of shear walls under horizontal loads is to complex for a simple mechanical analysis. The tables contain the ultimate capacities for static and seismic stress for different wall assemblies. The structural properties were determined with tests from Serette at Santa Clara University [1,2]. In addition the behaviour of shear walls under periodic horizontal loads with alternating stressing was examined. The American tests prove that the stiffening with shear walls is qualified
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for quasi-static loading, which means wind loads, as well as for seismic loading. But with the records of the measured values of the ‘‘head displacement’’ and the ‘‘related horizontal load’’ there can only be drawn conclusions on the ductility of this stiffening system. An accurate description of the load transfer is not possible. The stabilising properties of the sheathing on the compressed cold-formed steel sections have already been examined by Winter [3] at Cornell University. Simaan [4] has continued this work. The design in the US AISI Specifications [5] is based on these research results. However, the model to include the stabilising properties of the sheathing by means of the shear stiffness of the shear wall could not be verified with later tests [6]. Similar design methods are missing in Germany. The intention is therefore to develop an universal design method for shear walls under horizontal loads based on the now completed research project [7]. Furthermore the influence of additional effective vertical loads has been examined. Based on the test results an analysis model could be developed in which the stabilising properties of the sheathing during the structural safety verification of the cold-formed steel posts can be included realistic. 3. Research results 3.1. Shear walls under horizontal loads During this test-run the parameters of the sheathing material and the spacing of the screws sr along the edge of the shear panel has been varied. C-sections with the dimensions h b c tn ¼ 97 50 8 1.5 mm and a yield strength of fy ¼ 320 N/mm2 have been used as studs.
The sheathing was attached with self drilling screws + 4.2 mm (4.2 38 WD–O–H from the Company EJOT). The edge distance was constant er ¼ 20 mm. The horizontal load (Fig. 1) was applied after a load cycle which was based on the standards for testing shear walls in timber frame construction, DIN EN 594/07 96. During 120 s the shear walls were stressed with a load up to 40 per cent of the estimated collapse load Hu. This stress was held for 30 s after which the shear wall was unloaded again. After a break of 120 s the load was increased with the same velocity to Hu. The following types of sheathing were used:
Chipboard according to DIN 68763, V100-13 mm, Livingboard from the company Kuntz, with polyurethane, polyurea glued panels. Gypsum fibreboard, Fermacell 0 G 05 from the company Fels with 12.5 mm thickness. Cement-bonded fibreboard Eterplan N from the company Eternit with 8 mm thickness. Trapezoidal sheet 8/172 with 0.88 mm thickness from EKO Stahl.
In the German timber frame construction chipboards and gypsum fibreboards are mainly used. The cementbonded fibreboard has the highest resistance among the offered sheathings. It is possible, according to DIN 18807 part 3/06.87, that cold-formed steel sections with trapezoidal sheeting transfer the loads which act in their plane as diaphragm. The specimens with trapezoidal sheeting and gypsum fibreboards were primarily used to prove that the theory of diaphragm was also true for the problem examined within this project. The collapse of the shear walls with non-metal sheathing starts with the fracture of the edge under tension at the hold-down as shown in Fig. 2.
vertical load
load distribution horizontal load
y
stud 1
stud 2
stud 3
load introduction
tension restrain (hold down) x Fig. 1. Test rig for walls under horizontal and vertical load.
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Fig. 2. Fracture of the edge at the hold-down.
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After that the screws start to tilt in direction of the respective shearing stress. The screw-heads carve themselves into the sheathing. There are three forms of collapse to be observed once the collapse load is reached 1. Fracture of the sheathing edges in the area of the holddown and extraction of the screws through immersion of the sheathing at the bottom of the compression stud, as shown in Fig. 3. 2. Local buckling of the cold-formed steel section at the bottom of the compression stud, as shown in Fig. 4. 3. Local buckling of the trapezoidal sheeting by means of shear stress. The individual test-results are listed in Table 1. The measured load–displacement curves for three different sheathings are given in Fig. 5. In addition to the displacement kv and the horizontal load two other things were measured, the relative displacement of the sheathing compared to the substructure and the normal force distribution along the height of the studs. Independent of the sheathing material the following structural behaviour was examined. The substructure of the shear wall which is composed of studs,
Fig. 4. Buckling of the cold-formed profile at the bottom of the compression stud.
Fig. 3. Fracture of the edge (tension).
floor and ceiling profile deforms under a horizontal load into a parallelogram. The sheathing rotates in its plane like an almost rigid panel as illustrated in Fig. 6. The predominant part of the inserted energy is thus dissipated in the connection between sheathing and substructure. The mathematical model from McCutcheon [8] is based on the geometric interrelation between the distortion of the sheathing and the deflection of the connections which are pictured in Fig. 6. With diaphragms it is assumed that all connections are stressed with the same shear force parallel to the edge of the diaphragm. With shear walls the shear forces perpendicular to the edge of the diaphragm also have to be considered. These shear forces are the result of a rotation of the sheathing on the substructure. Through this effect the screws in the corner areas are being stressed noticeable more than the screws in the field. The shear force which is introduced into the studs longitudinally causes a linear increasing compression, respectively, tension and a corresponding extension of the stud. The corner screws on the base of the shear wall are much more stressed than the screws on the head of the shear wall because of this extension. Due to the components of screw forces which are directed towards the edge, the corner screws on the bottom of the tension stressed stud are governing. Above a certain critical deformation of the corner screw, the screw is no longer able to withstand additional force and will collapse. Subsequently the necessary redistribution of the loads causes a zipper-like collapse of the remaining connectors.
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Table 1 Test results of walls subjected to horizontal loads Test number
Sheathing I
Sheathing II
Screw interval sr (mm)
Collapse load Hu in (kN) (form of collapse)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Chipboard Chipboard Chipboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Chipboard Chipboard Chipboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Chipboard Chipboard Chipboard Cement-bonded fibre Cement-bonded fibre Cement-bonded fibre Cement-bonded fibre Cement-bonded fibre Cement-bonded fibre Chipboard Chipboard Chipboard Trapezoidal sheeting Trapezoidal sheeting Trapezoidal sheeting
Chipboard Chipboard Chipboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Chipboard Chipboard Chipboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Chipboard Chipboard Chipboard Cement-bonded fibre Cement-bonded fibre Cement-bonded fibre Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard Gypsum fibreboard
100 100 100 100 100 100 150 150 150 150 150 150
45.79 44.93 43.50 36.85 39.93 42.70 40.50 40.40 39.14 34.46 32.72 32.06 43.97 43.56 54.34 50.29 50.70 53.78 41.69 45.40 43.79 42.10 36.59 40.89 36.53 39.83 40.79
150 150 150 150 150 150 150 150 150 172/150 172/150 172/150
2
Displacement kv (mm) 26.39 28.32 26.70 27.06 28.49 27.48 30.83 29.69 27.99 29.98 34.52 37.94 43.85 37.83 44.73 23.34 23.64 22.10 23.63 31.32 26.63 50.10 42.99 50.21 42.83 27.14 36.60
2 2 2 2 2 2 2 2 2 2 1 1 1 2 2 2 2 1 2 2 1 1 1 3 2 3
Horizontal Load [kN]
Form of collapse: 1 ¼ collapse of connection material, 2 ¼ buckling of cold formed sections, 3 ¼ sheeting buckling. Mixed screw intervals, high density in the corners.
55 50 45 40 35 30 25 20 15 10 5 0
V16 V8 V12
V8, Chipboard, s= 150 mm V12, Gypsum fibre, s = 150 mm V16, Cement fibre, s = 150 mm
0
5
10
15 20 25 Head displacement [mm]
30
35
40
Fig. 5. Load–deflection curve of walls (sr ¼ 150 mm).
3.2. Connection of cold-formed steel section to sheathing with self-drilling screws The basis of the verification method for shear walls is the load–deflection behaviour of the connection between sheathing and cold-formed steel section. This was determined with a test rig which is shown in Fig. 7. It was also possible to investigate alternating loading with this test rig.
The results of the tests on connections with 10 samples each are given in Table 2. The statistical interpretation was carried out according to EC 1, [9]. The specimens were tested force controlled with the same measuring routine as the shear walls under horizontal loads. The connection has to carry in addition to shear forces also a tensile force in direction of the shank due to wind suction. Therefore an ultimate deformation (dult) of 3 mm has been set for the analysis of the collapse load.
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kv
With higher deformations it is possible that the screw head is pulled through the sheathing under inclined tension without any resistance due to the widened hole. The average value of the modulus of sub grade reaction for the tested connection is calculated according to DIN EN 383/10.93 as modified secant stiffness:
ceiling profile d
H
H: horizontal load kv: head displacement d: displacement of the corner-screws stud
0:4V¯ u N=mm . 4=3ðw0:4 w0:1 Þ
In addition the secant stiffness for the range between 0.4 Vu up to Vu is calculated with the following expression: kS2 ¼
H floor profile
kS1 ¼
V¯ u 0:4V¯ u N=mm , ¯ wu 0:4V u =kS1
where w0.1 is the displacement with 0.1 Vu; w0.4 the displacement with 0.4 Vu; wu the displacement with Vu.
sheathing
Fig. 6. Deflection of walls under horizontal load.
3.3. Shear walls under horizontal and vertical loads
P
P
P displacement gauge roller bearing U-profile 15/50/15 sheathing Roller bearing
U-profile 15/50/15
Displacement gauge
Fig. 7. Test rig for tests on connections under cyclic load.
The specimens under horizontal and vertical loads were sheathed with cement-bonded chipboards. Because of its resistance against moisture this sheathing material is used increasingly in prefabricated house construction. It is also biological unobjectionable because of the use of cement as binding agent. Shear walls with sheathings on one side only were also examined in the context of this test run. From consideration of building physics the use of this wall type is sensible. For example in rooms with high sound protection requirements the studs are connected alternately to one sheathing only to decouple the shells. The sheathing can be fixed in spring tracks as an alternative. Both variations are from static point of view one-sided sheathed walls. In the tests it was assessed, that the influence of the deflection forces does not reduce the shear stiffness of the diaphragm of two-sided sheathed shear walls. With onesided sheathed walls lateral torsional buckling of the studs was expected. The stiffness of the connection, as an elastic
Table 2 Stiffness and characteristic values of the resistance of the connection between cold formed section and sheathing using self drilling screws Sheathing material, thickness (mm)
Screw + (mm)
Sheet thickness (mm)
k¯ S1 (N/mm)
k¯ S2 (N/mm)
0.4 V¯ u (N)
V¯ u (N)
dult (mm)
Vu (N)
Chipboard, 13 Gypsum fibreboard, 12.5 Cement bonded fibre, 8 Gypsum wallboard, 12.5 Cement bonded chipb., 12 OSB 3, 12 Plywood board, 13 Trapezoidal sheet, 0.88 Chipboard, 13 Chipboard, 13 Chipboard, 19 Gypsum fibreboard, 12.5
4.2 4.2 4.2 4.2 4.2
1.5 1.5 1.5 1.5 1.5
1927 1517 19 281 635 3811
670 451 3301 121 804
1088 753 1580 215 1410
2721 1884 3951 536 3524
3 3 0.8 3 3
2147 1541 3443 504 2879
4.2 4.2 4.2 4.2 3.5 4.2 4.2
1.5 1.5 1.5 2.0 2.0 2.0 2.0
3174 1004 18 239 1745 1073 1912 1843
434 462 738 693 718 589 496
795 708 1437 1096 993 977 842
1988 1769 3593 2740 2483 2442 2104
3 3 3 3 3 3 3
1363 1504 2983 2325 2122 1986 1895
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Fig. 8. Failure of one-sided sheathed shear walls under vertical loads.
support of the studs, did not sink to the level that this or buckling around the weak axis occurred. Several specimens were loaded with vertical loads only. The observed failure patterns of one-sided sheathed shear walls in this case show, that the section deformation cannot be disregarded for the determination of the load-carrying capacity of C-sections supported in this way, see also Fig. 8.
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it is recommended to use winged-screws with a diameter of 4.2 mm. These screws are robust and can be used for a plate thickness of up to 2.5 mm. The wings prevent an unwanted uplift of the sheathing during the drill process through the steel sheet. The edge distance of the screws should add up to at least 20 mm within the sheathing. Consequential the flange width of the C-sections has to be 50 mm. This dimension should also be maintained for higher static requirements to ensure that the influence of the studs as a thermal bridge and sound bridge is minimised. For the same reasons the grid dimension of 625 mm should be kept, even with additional loads for example near wall openings. The grid dimension is determined by the width of the sheathing materials. The plate thickness of the C-sections should be changed rather than changing the grid dimensions. To ensure a constant load introduction of the normal forces into the studs the bottom and ceiling sections should meet several demands. These demands are that the sections have to be moulded or slotted before folding in such a way that the studs have full contact with the sections. The practice to produce U-sections which have an additional width of twice the plate thickness compared to the stud sections is copied from dry construction. This method causes that the C-sections are bedded on the bending radii of the U-sections (Fig. 9). Because of this the load is only introduced into the flanges. The buckling load of the C-sections sinks in the area of load introduction correspondingly. The minimum thickness of the sheeting should be 12 mm (8 mm with cement-bonded fibreboards). A failure of the sheathing due to shear buckling can be excluded with this thickness.
b + 2.t U-section for load introduction
4. Design concept The design-method for shear walls with cold-formed studs under vertical und horizontal loads can be divided into the following steps:
Structural requirements. Load–deflection behaviour of the connections. Determination of the horizontal collapse load HR. Normal force stress on the studs. Design of the tension anchorage. Stabilisation of the studs.
Concentrated load introduction
Stud
4.1. Structural requirements
b If self-drilling screws are used for the connection between the cold-formed steel section and the sheathing,
Fig. 9. Concentrated load introduction in bending.
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Table 3 Expression for the load deflection relation of the connection between section and sheathing using self-drilling screws, + 4.2 mm and a plate thickness of 1.5 mm
Table 4 Mean value of S for 0.3obo0.7, see also Table 3 Dimensions of the shear wall, h b (mm)
S ¼ f (sr) sr in (mm)
V ¼ a wb
2600 1250
100 S ¼ 60 sr 100 S ¼ 68 sr 100 S ¼ 78 sr
Sheathing
Thickness (mm)
a (N/mmb)
b (dimensionless)
3000 1250
Chipboard Gypsum fibreboard Gypsum wall board Cement bonded fibreboard OSB Plywood board
13 12.5 12.5 12
1486 1077 338 2173
0.55 0.51 0.42 0.44
3500 1250
12 13
1324 885
0.37 0.63
Step 3: Determination of the horizontal ultimate load HR: sin a , 2 with n the number of sheathing, one-sided n ¼ 1, two-sided n ¼ 2, (dimensionless); EA the extensional stiffness of the cold-formed steel studs (N); Vu the mean value of the collapse load of the connection according to Table 2; sr the distance between the screws at the outer edge of the shear wall; dult the ultimate displacement of the connection according to Table 2; a the coefficient of the load–deflection relation of the connection in (N/mmb); b the coefficient of the load–deflection relation of the connection, (dimensionless); S the auxiliary coefficient according to Table 4, (dimensionless); a the diagonal angle of the screw outline of the shear wall; aE arctan (b/h), h ¼ height, b ¼ width of the shear wall; d1 the resisting ultimate deformation of the edge screw in consideration of the extension of the studs, d1pdult. If two different sheathings are used on the two sides of the shear wall, the smaller of the two ultimate deflections should be used. The respective collapse loads HR may be added. Sheathing in the second layer (e.g. because of sound insulation) should not be included. An enhancement of the load bearing capacity cannot be expected due to the considerably higher lever arm of the forces on the screws. H R ¼ n a S d b1
4.2. Load–deflection behaviour of the connections The values for self-drilling screws with 4.2 mm diameter were determined experimentally. The load–deflection behaviour of these connections can be determined by good approximation with the expression in Table 3. 4.3. Determination of the horizontal ultimate load HR The ultimate load HR for shear walls under horizontal loads is defined through an ultimate displacement of the edge screws dult. The foundation of the design model is the assumption that the energy which is introduced through the head of the wall is only dissipated through the deformation of the connectors. All of the other deformations due to the flexural and extensional stiffness of the cold-formed sections as well as the shear stiffness of the sheathing are assumed to be zero in the calculation. For this case, ‘‘rigid board rotates on hinged studs’’, a geometric correlation between head displacement of the shear wall and the individual displacement of the connectors was derived. With this correlation the energy theorem of McCutcheon [8] can be formulated. For the solution of the energy theorem the expression from Table 3 is used which describes the deflection behaviour of the connection between cold-formed section and sheathing. Governing are the edge screws at the bottom of the shear wall. There the influence of the extension of the studs has to be considered in the form of a correction factor. The result of the design method which is described thoroughly in [7] can be outlined in the following steps: Step 1: Determination of the correction factor Z, which accounts for the influence of the extension of the studs on the loading of the edge screws: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V un hl l¼ . !Z¼ sr d ult EA tanhðlhÞ Step 2: Determination of the resisting ultimate deformation d1: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 2ult . d1 ¼ ðZ cos aÞ2 þ sin a2
4.4. Normal force on the studs By means of a horizontal load which is applied at the head of the shear wall the studs are stressed with a normal force which increases linear over the height of the stud. Hh ðat the bottom of the shear wallÞ. b This load has to be superposed with the support normal forces in consequence of the floor and vertical loading from the upper floors. The measured normal forces of the internal studs due to horizontal loads are negligibly small. max N ¼
4.5. Dimensioning of the tension anchorage The tension anchorage of the shear wall has to be proportioned to the maximum of the normal force, Z ¼ H h/b. Permanent compression loads on the edge studs of the shear walls can be included given that they unload the
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tension anchorage. Analogue to timber frame construction a tension anchorage is only necessary at the outer edge if several shear walls are placed in a row. At the butt joint of the walls the normal forces of the studs cancel each other. 4.6. Stabilisation of two-sided sheathed studs The stabilising effect of the sheathing differs noticeable from that of a diaphragm. The deviation forces of a predeformed stud cause only local deformations at the screw connections. The displacements due to shear strain of the sheathing are negligible small. The bifurcation load in direction of the wall plane as well as the screw stressing due to deviation forces can therefore be established by means of a model of an elastic bedded beam, Figs. 10 and 11. The kvalues have to be determined by tests [7].
kϑ
kv2
It has been shown [7] that with a chosen distance between the screws and the examined well-established sheathing materials, a collapse of a two-sided sheathed cold-formed section in direction of the wall plane can be excluded. C-sections with the dimensions as commonly used in domestic construction must therefore only be verified for lateral buckling transverse to the direction of the wall plane. Starting from the idealised model with linear elastic springs it is apparent that the connections will behave plastic rather than elastic with very small deformations. The load bearing capacity of the connectors is determinative for the design. If thick-walled cold-formed sections or more flexible sheathing, such as gypsum wall boards, are used it must therefore be examined whether the connectors are able to resist the deviation forces. If necessary the distance between the individual screws has to be reduced, Fig. 11. The analysis method for one-sided sheathed C-sections with regard to the deformation of the section and local buckling is derived thoroughly in [7]. An inclination of the studs causes an additional load on the diaphragm due to the P–D-effect. In case of a high normal force the influence according to the analysis of second order should be included with help of an amplification factor: H II ¼ H I
5. Summary
Fig. 10. Model for the support condition of the sections.
V
V
V
sr
Pre-camber
1 P , 1 N=N Ki
with HI the sum of Phorizontal load due to the inclination of the studs H 0 ¼ Nc0 and the outer horizontal load; c0 ¼ 1/200, pre-rotation of the P stud according to DIN 18800-2 section 2.3; N Ki ¼ ks1 L; N ¼ sum of the normal forces on the diaphragm; ks1 ¼ 0:4H u =0:4kv stiffness of the diaphragm (Table 1).
kϑ
kv1
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Pre-deflection
Fig. 11. Forces acting on the connection due to the imperfection of the studs.
The basis for the previous described design-method for shear walls made of cold-formed steel sections and different sheathing is a test series. The test series was carried out at the Institute of Steel Construction and Materials Mechanics at the Technische Universita¨t Darmstadt. In contrast to the known analysis for shear walls in timber frame constructions the method is based on a permitted displacement of the corner screws at the outlining edge of the shear wall. With regard to the stabilising behaviour of the sheathing a calculation model for the stabilisation verification of the compression stressed C-section-studs has been introduced in extracts. Two-sided sheathed C-sections with a plate thickness of up to 2.5 mm are stabilised sufficiently with the examined sheathings. The sections have to be verified with regard to buckling transverse to the direction of the wall plane only. With thick-walled sections (42.5 mm) or flexible sheathing materials (e.g. gypsum board) it is necessary to determine whether the connectors are able to resist the deviation
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forces according second order theory also in direction of the wall plane. The analysis method for one-sided sheathed C-sections with regard to the deformation of the section and local buckling is derived thoroughly in [7]. References [1] Serrette R, Hall G, Ngyen H. Shear wall values for light weight steel framing. Final report, AISI, Washington, January 1996. [2] Serrette R, Hall G, Encalada J. Additional shear wall values for light weight steel framing. Final report, AISI, Washington, March 1997. [3] Green GG, Winter G, Cuykendall TR. Light gage steel columns in wall-braced panels. Bulletin no. 35/2, Cornell University, Ithaca NY, 1947.
[4] Simaan A. Buckling of diaphragm-braced columns of unsymmetrical sections and application to wall studs design. Report no. 353, Cornell University, Ithaca NY, 1973. [5] AISI. Specification for the design of cold-formed steel structural members. Washington, 1996 Edition. [6] Miller TH, Peko¨z T. Behavior of gypsum-sheathed cold-formed steel wall studs. J Struct Eng, ASCE 1994. [7] Naujoks B. Tragverhalten von Wandtafeln mit Kaltprofilen unter horizontalen und vertikalen Lasten. PhD thesis, TU Darmstadt, November 2002. [8] McCutcheon WJ. Racking deformations in wood shear walls. J Struct Eng, ASCE 1985. [9] Eurocode 1 — EN1991-1. Actions on Structures, Appendix D, 2001.