A Plastic Lower Bound Method for Design of Wood-framed Shear Walls

Transcription

1 A Plastic Lower Bound Method for Design of Wood-framed Shear Walls Bo KÄLLSNER Växjö University Sweden 1977 Dr in Civ. Eng. Since 1975 at Trätek - Swedish Institute for Wood Technology Research in Stockholm. Since 2001 also at Växjö University as Adjunct Prof. Ulf Arne GIRAMMAR Umeå University Sweden ulfarne.girhammar@tfe.umu.se 1980 Dr. Eng.; 1981 Assoc. Prof.; 1987 Eng. Consultant, wood and building industry; 1989 Adjunct Prof Senior Specialist, mechanics industry. Presently at Umeå University. Summary Design of shear walls has been a topic of major discussions during the work to develop a common European code for design of timber structures. The main problem has been that shear walls are fastened to the substrate in different ways in different countries and that this fact must be reflected in the code. A plastic lower bound method is proposed for design of shear walls. The analytical model can be used in design of shear walls with different sheet materials, sheathing-to-timber joints, geometric layout, anchoring conditions and load configurations. The method is based on plastic characteristics of the sheathing-to-timber joints. Keywords: Wood-framed shear wall, partially anchored, racking resistance, plastic model. 1. Introduction 1.1 Background Design of shear walls has been a topic of major discussions during the work to develop a common European code for design of timber structures. The main problem has been that shear walls are fastened to the substrate in different ways in different countries and that this fact must be reflected in the code. In a Nordic project, dealing with the design of shear walls, it became obvious that there is a need for developing a general method suitable for calculation by hand that takes different anchoring conditions into account. The principles of such a method have previously been presented [1,2]. In this paper a further generalisation of the method is presented. 1.2 Objective The objective of this paper is to present an analytical method for determining the racking capacity of wood-framed shear walls that: - can be used for design of walls with various geometric layout, sheet materials and fasteners - handles different ways of anchoring the bottom rail and the vertical studs - is able to consider different load configurations - results in economical structures - explains the structural behaviour and - opens up for possibilities to reinforce the structures.

2 1.3 Limitations The proposed model can only be applied on shear walls where the sheet material is fixed by mechanical fasteners to the timber members and where these sheathing-to-timber joints show plastic behaviour. The model covers only static loads in the ultimate limit state. The model can not be used to determine the deformations in the serviceability limit state. In this paper, shear walls with openings are not dealt with. A generalisation of the model to include openings will be addressed in a coming study. 2. Plastic lower bound method 2.1 Basic assumptions For design oartially anchored shear walls a plastic lower bound method [3] is proposed. This means that a force distribution is chosen that fulfils the conditions of force and moment equilibrium for each timber member and sheet. The basic assumptions are identical to those given in [1,2] and are as follows: - the sheathing-to-timber joints, referring to the vertical studs and the top rail, are assumed to transfer shear forces only parallel to the timber members - the sheathing-to-timber joints, referring to the bottom rail, are assumed to transfer forces both parallel and perpendicular to the bottom rail - the framing joints are not assumed to transfer any tensile or shear forces - compressive forces can be transferred via contact between adjacent sheets and in the framing joints. In order to obtain simple expressions for the racking resistance of the shear walls, the fasteners are assumed to be continuously distributed along the timber members. The load-carrying capacity of the sheathing-to-timber joints is consequently given in force per unit length. In all the examples presented below, it is assumed that the fastener spacing around the perimeter of the sheets is constant. 2.2 General solution for partially anchored shear walls By a partially anchored wall is meant a wall where the vertical stud on the tension side is not completely anchored to the substrate against uplift. In order to clarify the structural behaviour of shear walls, the wall configuration according to Figure 1 first will be studied. The wall is subjected to n+1 external vertical loads V 0 to and a racking load. The spacing of the studs is s and the length of the wall is l. This means that l=ns. The width and height of the full format sheets are denoted by b and h, respectively. The bottom rail is assumed to be fully anchored to the substrate. The studs are assumed to have no tie-downs i.e. they are completely free from the bottom rail with respect to tension and shear forces. The forces acting on the wall in the ultimate limit state are assumed to be distributed according to Figure 1. The forces acting along the lower part of the wall are shown in a section immediately above the bottom rail and represent the forces in the sheathing-to-timber joints. These forces are assumed to act either perpendicular or parallel to the bottom rail. The plastic capacity per unit length of these joints is denoted by and it is assumed that plasticity has been attained along the entire bottom rail. The factor opens for the possibility of using reduced strength properties when the fastener forces act perpendicular to the edges of the sheets and the frame members. The notation l eff is used to indicate that this part of the total length of the wall is fully effective for horizontal load transfer. The rest of the wall length, l-l eff, must be used for anchoring the wall to the substrate.

3 V 0 h R n l i l-l eff l eff l s s s s s s s s V 0 R n Figure 1. Forces acting on a shear wall with fully anchored bottom rail and no tie-downs. The plastic capacity has not been attained in the sheathing-to-timber joints along any of the vertical studs. Further, it is assumed that the plastic shear flow has not been attained along any of the vertical studs. Since the stud furthest to the right is subjected to the highest shear force this condition can be formulated as n1 i0 V f ( ll ) f h i p ef Moment equilibrium around the lower right corner of the wall diaphragm gives (1) 1003

4 n1 l leff h Vi( l li) ( l leff ) 0 i0 2 Force equilibrium in horizontal direction gives fpleff We now introduce the non-dimensional parameters l h (2) (3) (4) n1 i0 n i n f h p The denominator of is simply equal to the reaction force of the vertical forces V 0 to considering the shear wall as a simply supported beam on the two end studs i.e. corresponding the equivalent vertical load on the leading stud. The nominator of is equal to the maximum load that can be transferred between a vertical stud and a sheet, i.e. corresponding to the full plastic capacity. The parameter must always be less than unity. Using the equations (2) - (5) the effective wall length can be calculated as leff l 1 Vertical force equilibrium gives the reaction force R n acting on stud n as n n i p eff i0 R V f ( l l ) By studying the force and moment equilibrium of the individual sheets in Figure 1, it is obvious that the proposed model will only work if compressive forces are transferred between the adjacent sheets. If contact forces between adjacent sheets are not accepted the model must be applied on each individual sheet. This will result in a somewhat lower load-carrying capacity for the wall. So far the calculations have been based on condition (1) saying that the plastic shear flow has not been attained along any of the vertical studs. Now we will study the case when this condition is not fulfilled. Consider the wall in Figure 2. We assume that the plastic shear flow has been attained in a vertical section immediately to the right of stud number n, while it has not been attained in a section immediately to the left of the same stud. In this case the wall can be handled as two separate walls as shown in the lower part of Figure 2. We observe that for the left part of the wall, condition (1) is fulfilled. This means that the equations (2)-(6) can be applied on the left part of the wall, observing that l now denotes the length of this part. For the right part of the wall the plastic shear flow has been attained along the edges of all sheets. This means that these sheets can not transfer any more forces and that the external vertical forces +1 to V N are transferred via the vertical studs directly to the bottom rail. Since the length of the left part of the wall is l=ns and the length of the right part of the wall is equal to (N-n)s the total effective length l eff becomes leff ns 1 ( N n) s Vertical force equilibrium gives the reaction force R n acting on stud n as (5) (6) (7) (8)

5 n R V f ( ll ) f h n i p ef i0 and the reaction force R N acting on stud N as R V f h N N p (9) (10) V 0 +1 V N-1 V N h V 0 +1 V N-1 V N l i R n R n+1 R N-1 R N l-l eff l eff l s s s s s s s s s s Figure 2. Forces acting on a shear wall with fully anchored bottom rail and no tie-downs. The plastic capacity has been attained in the sheathing-to-timber joints on the right side of the vertical stud n. 3. Illustrative example The model is applied on a shear wall consisting of four sheet segments on a timber frame. All studs are subjected to vertical load of equal magnitude V. The bottom rail is fully anchored to the substrate. No tie-downs are fastened to any of the studs. All sheets have a height-to-width ratio equal to two and the factor =1. The results of the calculations using equation (8) are presented in Figure 3. The bold lines and blackened areas in the small figures indicate the progress olastification in the wall, when the load s increased. The maximum load is obtained for V/ h=1 corresponding to fully anchored leading stud. In this case there is no influence from the vertical loads acting on the other studs. 1005

6 l eff /l V/ h Figure 3. Relationship between relative effective length l eff /l and relative load V/ h acting on all studs. 4. Discussion At the derivation of the equations (8)-(10) the position of the vertical stud n was chosen along the border between two sheets. It is easy to show that these equations are still valid if the position of stud n is changed to the centre of one of the sheets. The proposed method can be applied on walls containing sheets of different sizes, e.g. sheets of half size without any modifications. If the spacing s between the studs is not constant the expression for the parameter must be modified. The influence of tie-downs in a wall can be handled by considering them as external vertical loads V. They should, however, show a reasonable plastic load-slip behaviour. The three-dimensional influence of transverse walls, with respect to inter component connections between transverse walls and shear walls, can be handled in the same way. Finally it should be mentioned that the method could also serve as a basis for development of diagrams and tables that can facilitate a structural design. 5. Conclusions An analytical plastic model for design of wood-framed shear walls has been presented. The proposed model can only be applied on shear walls where the sheet material is fixed by mechanical fasteners to the timber members and where these sheathing-to-timber joints show plastic behaviour. The model covers only static loads in the ultimate limit state. The model can be used in design of shear walls with different sheet materials, sheathing-to-timber joints, geometric layout, anchoring conditions and load configurations. References [1] Källsner B., Girhammar U.A., Wu L., A Simplified Plastic Model for Design of Partially Anchored Wood-Framed Shear Walls, Proceedings CIB-W18 Meeting, Venice, Italy, [2] Källsner B., Girhammar U.A., Wu L., A Plastic Design Model for Partially Anchored Wood- Framed Shear Walls with Openings, Proceedings CIB-W18 Meeting, Kyoto, Japan, [3] Neal B.G., Plastic Methods of Structural Analysis, 2 nd edition, London, 1978.