7.3 APPROXIMATE THEORY FOR LATERAL LOAD ANALYSIS OF WALLS SUBJECTED TO PRECOMPRESSION WITH AND WITHOUT RETURNS 7.3.1 Wall without returns Having taken into consideration all the factors contributing to the lateral strength of the wall, an approximate analysis (Hendry et al., 1971) can be developed based on the following assumptions: • Elastic deflections of the wall supports are negligible. • Failure occurs by horizontal cracking at the top, centre and bottom of the wall, causing rotation about horizontal lines through A, B and C (Fig. 7.4). The forces acting on the top half of the wall at the point of failure are shown in Fig. 7.4. By taking moments about A (7.1) (7.2) where =precompressive stress, t=thickness of the wall which is subject to precompression (in the case of a cavity wall with inner leaf loaded, thickness should be equal to the thickness of inner leaf only), L=length of wall, h=height of wall, q 0 =transverse or lateral pressure and a=horizontal distance through which centre of the wall has moved. If the compressive stress is assumed constant throughout the uplift of the wall at failure, the maximum pressure resisted by the wall is equal to (7.3) If the precompression increases on the wall with uplift of the building, as explained above, it is possible for the moment of resistance, tL (t-a), to increase, even though the moment arm (t-a) decreases—thus resulting in an increase in the maximum lateral pressure resisted by the wall. 7.3.2 Wall with returns In the case of a wall with returns, part of the lateral pressure is transmitted to the return, thus causing axial and bending stresses in the return. ©2004 Taylor & Francis Now substituting the value of q 0 (wall with no return) from equation (7.3) into equation (7.5) (7.6) Similarly, for a wall with two returns (Fig. 7.5 (a)): (7.7) (7.8) From equation (7.3) (7.9) For various values of ␣ , the q 1 /q 0 and q 2 /q 0 plots have been shown in Fig. 7.6 together with the experimental results. In the British Code of Practice BS 5628 the factors 1/[1-1/(3 ␣ )] and 1/ [1-2/(3 ␣ )] are replaced by a single factor k. Table 7.1 shows the comparison between factor k obtained from the theory and from the code. From Table 7.1 it can be seen that the British code values are in good agreement with the theoretical results. The theoretical values in Fig. 7.5 Simplified failure mechanism for walls with returns. ©2004 Taylor & Francis • Simply supported top and bottom, i.e. vertically spanning panel. • Simply supported on two edges, i.e. horizontally spanning panel. • Simply supported or continuous on three or four sides, i.e. panels supported on more than two sides of various boundary conditions. It will of course be realized that simple supports are an idealization of actual conditions which will usually be capable of developing some degree of moment resistance. 7.5.1 Vertically or horizontally spanning panels The maximum moments per unit width for a wall spanning vertically or horizontally can be calculated from: vertically spanning panel (7.10) horizontally spanning panel (7.11) Fig. 7.7 Effect of wall rotation: (a) basic rotation; (b) modified rotation (with high precompression). =precompression; ⌬=half maximum uplift of wall with no corner deformations; δ h =elastic shortening. ©2004 Taylor & Francis Fig. 7.8 Prccompression versus maximum lateral pressure on 102.5mm wall of storey height. ©2004 Taylor & Francis where w=design pressure, M x and M y =maximum moments per unit width at midspan on strips of unit width and span h and L. Similarly, the moment of resistance per unit width of the panel can be calculated from the known value of the flexural tensile strengths in respective directions as: M y =f ty Z (7.12) M x =f tx Z (7.13) where f ty =allowable tensile strength perpendicular to the bed joint, f tx =allowable tensile strength parallel to the bed joint and Z=sectional modulus for unit width. In case of limit state design, the design bending moments per unit width in two directions will be (7.14) (7.15) where w k =characteristic wind load per unit area and ␥ f =partial safety factor for loads. The moment of resistance of the panel spanning vertically and horizontally will be given by (7.16) (7.17) where f ky and f kx are characteristic tensile strength normal and parallel to bed joints. 7.5.2 Panels supported on more than two sides with various boundary conditions The lateral load analysis of masonry panels of various boundary conditions is very complicated since masonry has different strength and stiffness properties in two orthogonal directions. Some typical values of brickwork moduli of elasticity on which the stiffness depends are given in Table 7.2. The British limit state code BS 5628 recommends bending moment coefficients for the design of laterally loaded panels. The code does not indicate the origin of these coefficients, but they are numerically equal to those given by yield-line analysis as applied to under-reinforced concrete slabs with corresponding boundary conditions. Strictly speaking, yield- line analysis is not applicable to a brittle material like masonry which cannot develop constant-moment hinges as occur in reinforced concrete ©2004 Taylor & Francis of the panel into which it is divided by the fracture lines is in equilibrium under the action of external forces and reactions along the fracture lines and supports. Since it is symmetrical, only parts 1 and 2 need consideration. In case of asymmetry the entire rigid area needs to be considered. Consider triangle AFB: (7.18) and its moment along AB is (7.19) For equilibrium Therefore (7.20) Similarly, for AFED(2) (the left-hand side of equation (7.21) has been obtained by dividing the rigid body 2 into two triangles and one rectangle for simplification of the calculation) 7.21) From equations (7.20) and (7.21), (7.22) or Fig. 7.9 Idealized fracture lines. ©2004 Taylor & Francis therefore (7.23) For minimum collapse load or maximum value of moment d(m/w)/ dß=0, from which (7.24) The value of ß can be substituted in equations to obtain the relationship between the failure moment and the load. For a particular panel, the fracture pattern that gives the lowest collapse load should be taken as failure load. The values of m and ß for various fracture-line patterns for panels of different boundary conditions are given in Table 7.3, and the reader can derive them from first principles as explained above. 7.5.4 How to obtain the bending-moment coefficient of BS 5628 or EC6 from the fracture-line analysis Although the fracture-line method has been suggested for accurate analysis, the designer may prefer to use the BS 5628 coefficients. Hence this section briefly outlines the method to obtain the coefficients from the fracture line. In BS 5628 the bending-moment coefficients are given for horizontal bending (M x ), whereas the analysis presented in this chapter considers the vertical bending (M y ). Similarly, the orthotropy ratio in case of BS 5628 is taken as the ratio Hence the orthotropy is less than 1, whereas in the present analysis the orthotropy is the reciprocal of this ratio. The BS 5628 coefficients can be obtained by putting K=1 in equations (7.24) and (7.23) and also in the equation of Table 7.3, and by multiplying the vertical moment (M y ) by the orthotropy defined as in the fracture-line analysis. The provisions of EC 6 for lateral load design for resistance to wind loads are the same as BS 5628, and hence need no separate explanation. Example Consider the case of a panel similar to Fig. 7.9. We have ©2004 Taylor & Francis (Note that in BS 5628 the symbol ␣ is used for bending moment coefficient.) From equation (7.24) From equation (7.23) vertical moment therefore horizontal moment The bending moment coefficient from BS 5628 for the corresponding case (hlL=0.75) is also 0.035. ©2004 Taylor & Francis 8 Composite action between walls and other elements 8.1 COMPOSITE WALL-BEAMS 8.1.1 Introduction If a wall and the beam on which it is supported can be considered to act as a single composite unit then, for design purposes, the proportion of the load acting on the wall which is carried by the supporting beam must be determined. Prior to 1952 it was common practice to design the beams or lintels so as to be capable of carrying a triangular load of masonry in which the span of the beam represented the base of an equilateral triangle. The method allowed for a proportion of the self- weight of the masonry but ignored any additional superimposed load. Since that period a great deal of research, both practical and theoretical, has been undertaken, and a better understanding of the problem is now possible. Consider the simply supported wall-beam shown in Fig. 8.1. The action of the load introduces tensile forces in the beam due to the bending of the deep composite wall-beam and, since the beam now acts as a tie, the supports are partially restrained horizontally so that an arching action results in the panels. The degree of arching is dependent on the relative stiffness of the wall to the beam, and it will be shown later that both the flexural stiffness and the axial stiffness must be taken into account. In general, the stiffer the beam the greater the beam-bending moment since a larger proportion of the load will be transmitted to the beam. The values of the vertical and horizontal stresses depend on a number of factors, but typical plots of the vertical and horizontal stress distributions along XX and YY of Fig. 8.1 are shown in Fig. 8.2. Note that the maximum vertical stress, along the wall-beam interface, occurs at the supports and that at mid-span the horizontal ©2004 Taylor & Francis frictional forces developed are sufficient to supply the required shear capacity. 8.1.2 Development of design methods For design purposes the quantities which must be determined are: • The maximum vertical stress in the wall. • The axial force in the beam. • The maximum shear stress along the interface. • The central bending moment in the beam. • The maximum bending moment in the beam and its location. Methods which allowed for arching action were developed by Wood (1952) for determining the bending moment and axial force in the beams. The panels were assumed to have a depth/span ratio greater than 0.6 so that the necessary relieving arch action could be developed and moment coefficients were introduced to enable the beam bending moments to be determined. These were: • PL/100 for plain walls or walls with door or window openings occurring at centre span. • PL/50 for walls with door or window openings occurring near the supports. An alternative approach, based on the assumption that the moment arm between the centres of compression and tension was 2/3×overall depth with a limiting value of 0.7×the wall span (Fig. 8.3) was also suggested (Wood and Simms, 1969). Using this assumption, the tensile force in the beam can be calculated using (8.1) and the beam designed to carry this force. Following this early work of Wood and Simms, the composite wall- beam problem was studied by a number of researchers who considered not only the design of the beam but also the stresses in the wall. The characteristic parameter K introduced by Stafford-Smith and Riddington (1977) to express the relative stiffness of the wall and beam was shown to be a useful parameter for the determination of both the compressive stresses in the wall and the bending moments in the beam. The value of K is given by (8.2) where E w , E bm =Young’s moduli of the wall and beam respectively, I b =second moment of area of the beam and t, L=wall thickness and span. The parameter K does not contain the variable h since it was considered ©2004 Taylor & Francis [...]... S=0.5 for Rу7 (e) Example To illustrate the use of the method consider the wall-beam shown in Fig 8.10 Here ©2004 Taylor & Francis These calculations are carried out in terms of design loads and are to be compared with the design strengths of the material in compression and shear The design of the beam would be carried out in accordance with the relevant code of practice 8.2 8.2.1 INTERACTION BETWEEN WALL... the diagonal or shear along the bedding planes The beams and columns of the frame are designed on the basis of a simple static analysis of an equivalent frame with pin-jointed connections in which panels are represented as diagonal pin-jointed bracing struts A description of the design method proposed by Wood is given below 8.2.2 Design method based on plastic failure modes (a) Introduction In the... Wood for design purposes and Mp is the effective plastic moment given by Zσy/ ␥ms For design purposes the design strength must be equal to or greater than the design load as shown in Chapter 4 (c) Example Assume the following dimensions and properties: • • • • • • • • • • Panel height=2m Panel length=4m Panel thickness=110mm Characteristic strength of panel=10N/mm2 Partial safety factor for masonry= 3.1... already obtained from Figs 8.5 and 8 .6 (d) Bending moments in the beam The maximum bending moment in the beam does not occur at the centre, because of the influence of the shear stresses along the interface Both the maximum and central bending moments can, however, be Fig 8 .6 Axial stiffness parameter ©2004 Taylor & Francis obtained from one graph (for a particular range of R) by using the appropriate abscissae... factor to allow for the fact that masonry is not ideally plastic These methods are too cumbersome for practical design purposes, and simplifying assumptions are made for determining acceptable approximate values of the unknowns The basis of the design method proposed by Riddington and StaffordSmith is that the framed panel, in shear, acts as a diagonal strut, and failure of the panel occurs owing to compression... WALL PANELS AND FRAMES Introduction Wall panels built into frameworks of steel or reinforced concrete contribute to the overall stiffness of the structure, and a method is required for predicting modes of failure and calculating stresses and lateral collapse loads The problem has been studied by a number of authors, and although methods of solution have been proposed, work is still continuing and more... relationship for the particular range of R shown To obtain the maximum moment, the lower C1 scale is used and for the central moment the C 1 ×C 2 scale is used In each case use of the appropriate d/L ratio will give the value of MC1/PL where M is either the maximum or the central bending moment Fig 8.7 Moments for cubic stress distribution ©2004 Taylor & Francis (8.11) The location of the maximum moment... height of the wall replaces the span, and the second is an axial stiffness parameter used for determining the axial force in the beam: (8 .6) (8.7) A typical vertical stress distribution at the wall-beam interface is shown in Fig 8.2(a) To simplify the analysis it is assumed that the distribution of this stress can be represented by a straight line, a parabola or a cubic parabola depending on the range of. .. thickness=110mm Characteristic strength of panel=10N/mm2 Partial safety factor for masonry= 3.1 Section modulus for each column =60 0 cm3 Section modulus for each beam=800 cm3 Yield stress of steel=250N/mm2 Partial safety factor for steel=1.15 Effective plastic moment for beam=(800×103)×250/(1.15×1 06) =174kN/m • Effective plastic moment for column=130kN/m • µp=134 • L/h=2 These give From Fig 8.12, δp=0.25 So From... (8.18) Fig 8.13 Design chart for racking loads: optional correction ∆φ added to Φ S (µ=Mpb/Mpc) From Wood (1978) ©2004 Taylor & Francis • If µp у1 (strong beams) use the chart directly • If µp . capable of carrying a triangular load of masonry in which the span of the beam represented the base of an equilateral triangle. The method allowed for a proportion of the self- weight of the masonry. of design loads and are to be compared with the design strengths of the material in compression and shear. The design of the beam would be carried out in accordance with the relevant code of. strength of panel=10N/mm 2 • Partial safety factor for masonry= 3.1 • Section modulus for each column =60 0 cm 3 • Section modulus for each beam=800 cm 3 • Yield stress of steel=250N/mm 2 • Partial