Chapter 2 – Strip Method for Slabs

Chapter 2 – Strip Method for Slabs Dr.-Ing. GirmaZerayohannes Dr.-Ing. AdilZekaria

Chapter 2- Strip Method for Slabs • 2.1 Introduction • Different methods of analysis are allowed by EBCS-2 • One of these is plastic methods • Strip method is a plastic method of analysis Dr.-Ing. GirmaZerayohannes

Chapter 2- Strip Method for Slabs • The upper bound theorem of the theory of plasticity is presented in chapter 1. The YL method of slab analysis is an upper bound approach to determining the capacity of the slab • Disadvantages: • An upper bound analysis if in error will be on the unsafe side. The actual carrying capacity will be less than, or at best equal to the capacity predicted, which is a cause for concern. Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • When applying this method it is necessary to assume that the distribution of reinforcement is known over the whole slab. •  a tool for review. • Can be used for design only in an iterative sense, i.e., trail design until a satisfactory amount is found Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • These circumstances motivated Hillerborg (1956) to develop what is known as stripmethod for slab design • In contrast to yield line analysis, the strip method is a lower bound approach, based on the satisfaction of equilibrium requirements every where in the slab Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • By the strip method, a moment field is first determined that fulfills equilibrium requirements, after which the reinforcements of the slab at each point is designed for this moment field Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • Lower Bound Theorem: If a distribution of moments can be found that satisfies both equilibrium and boundary conditions for a given external loading, and if the yield moment capacity of the slab is nowhere exceeded, then the given external loading will represent a lower bound of the true carrying capacity Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • Advantages: • The strip method gives results on the safe side, which is certainly preferable in practice • The strip method is a design method, by which the needed reinforcement can be calculated Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • 4.2 Basic Principles • The governing equilibrium equation for a small slab element having sides dx and dy is: Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs Figure 1 Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • Where w = the external load per unit area • mx, my = Bending Moments per unit width in the x and y directions and • mxy = the twisting moment Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • So according to the lower bound theorem, any combination of mx, my, and mxy that satisfies the equilibrium at all points in the slab and that meets boundary conditions is a valid solution, provided that the reinforcement is placed to carry these moments Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • The basis for the simple strip method is that the torsional moment is chosen equal to zero; no load is assumed to be resisted by the twisting strength of the slab  mxy = 0 • The equilibrium equation then reduces to: Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • This equation can be split conveniently into 2 parts, representing twist less beam strip action. Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • Where the proportion of load taken by the strips is k in the x-direction and (1-k) in the y-direction (concept of load dispersion) • In many regions in slabs, the value k will be either 0 or 1, i.e., load is dispersed by strips in x or in y direction • In other regions, it may be reasonable to assume that the load is divided equally in the two directions, i.e. k=0.5 Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • 2.3 Choice of load distribution • Theoretically, the load w can be divided arbitrarily b/n x and y directions. • Different divisions will, of course, lead to different patterns of reinforcement, and all will not be equally appropriate. Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • The desired goal is to arrive at an arrangement of steel that is safe and economical and will avoid problems at service load level associated with excessive cracking or deflections. • In general, the designer may be guided by his knowledge of the general distribution of elastic moments. Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • To see an example of the strip method and to illustrate the choices open to the designer, consider the square, simply supported slab shown below, with side length a and a uniformly distributed factored load w per unit area. • The simplest load distribution is obtained by setting k=0.5 over the entire slab, as shown in Figure 2. Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • The load on all strips in each direction is thus w/2 ( with k=0.5), as illustrated by the load dispersion arrows • This gives maximum design moments mx = my= wa2/16, implying a constant curvature for all strips in the x-direction at mid-span corresponding to a constant moment wa2/16 across the width of the slab (see fig. 2) Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • The same applies for y-direction strips • It is recognized however that the curvatures in the strips (say x-direction strips) near the supports, for such a slab, are less than near mid-span. • If the slab were reinforced according to this solution, extensive redistribution of moments would be required, certainly accompanied by much cracking in the highly stressed regions near the middle of the slab Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • So what we need is a type of load distribution (dispersion) which can give a moment distribution that gives rise to greater curvatures in strips near the middle of the slab and less near the ends • Try the alternative, more reasonable distribution shown in Figure 3 next slide. Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • Here the regions of different load dispersion separated by the dash-doted discontinuity lines follow the diagonals, and all of the load on any region is carried in the direction giving the shortest distance to the nearest support. • k=0 or 1 in the different regions Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • The lateral distribution of moments shown in Fig (3) would theoretically require a continuously variable bar spacing  impractical • A practical solution would be to reinforce for the average moment over a certain width, approximating the actual lateral variation in Fig. (4) in a stepwise manner. Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • Hillerborg notes that this is not strictly in accordance with the equilibrium theory and that the design is no longer certainly on the safe side, but other conservative assumptions, e.g., neglect of membrane strength in the slab or strain hardening of the reinforcement, would compensate for the slight reduction in safety margin Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • A third alternative is with discontinuity lines parallel to the edges. • Here again the division is made so that the load is carried to the nearest support, as before, but load near the diagonals is divided with one-half taken in each direction. • Thus k is given the values 0 or 1 along the middle edges and 0.5 in the corners and center of the slab Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • Two different strip loadings are now identified, strip along A-A and along B-B. • This design leads to practical arrangement, one with constant spacing through the center strip of width a/2 and a wider spacing through the outer strips, where the elastic curvatures and moments are known to be less. Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • The averaging of moments necessitated in the second solution is avoided here, and the 3rd (Fig. 4) solution is fully consistent with the equilibrium theory. • The three examples also illustrate the simple way in which moments in the slab can be found by strip method, based on familiar beam analysis. Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • It is important to note too that the load on the supporting beams is easily found because it can be computed from the end reactions of the slab-beam strips in all cases. Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • 2.4 Rectangular slabs with simple support • Discontinuity lines parallel to the edges as shown in the figure • In the x-direction: • Side strips: mx = w/2×b/4×b/8 = wb2/64 • Middle strips: mx = w×b/4×b/8 = wb2/32 • In the y-direction • Side strips: my = wb2/64 • Middle strips: my = wb2/8 Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs Figure 5 Rectangular slab with discontinuity lines originating at the corners.

Chapter 2- Strip Method for Slabs Figure 6 Rectangular slab with discontinuity lines parallel to the edges Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • Design the rectangular slab using the strip method for slabs • Use a=6.0 m, b= 4.5 m, t = 150 mm, C-25 concrete and S-300 reinforcing steel. • Compare the results with the solution using the coefficients in EBCS-2 • Take variable load q= 3.0kN/m2 • Floor finish-30 mm screed and 20mm thick marble Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • 2.5 Fixed Edges and Continuity • Up to now we have dealt with positive moments in strips, where a large amount of flexibility in assigning loads to the various regions of the slab was provided • This same flexibility extends to the assignment of moments b/n negative and positive bending sections of slabs that are fixed or continuous over their supported edges Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • Some attention should be paid to elastic moment ratios to avoid problems with cracking and deflection at service loads • Figure 7 (next slide) shows a uniformly loaded rectangular slab having two adjacent edges fixed and the other two edges simply supported • Let us consider slab strips with one end fixed and one end simply supported as shown in Fig. 7 Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs B A A B Figure 7 Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • In designing by strip method, slab strips carrying loads only near supports and unloaded in the central region are encountered • It is convenient if the unloaded region is subject to a constant moment (and zero shear) because this simplifies the selection of positive reinforcement Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • The discontinuity lines are shifted to account for the greater stiffness of the strips with fixed ends (i.e. bigger reaction at the fixed support) • Their location is defined by a coefficient , with a value less than 0.5, so that the edge strips have widths greater and less than b/4 at the fixed and simple end respectively Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • For a BMD for x-direction middle strips (section A-A) with constant moment over the unloaded part, the following maximum moments are achieved • and Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • The first term is the “cantilever” moment at the left end • So the negative moment at a support plus the span moment = the “cantilever” moment • Now the ratio of negative to positive moments in the x-direction middle strip is: Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • mxs/mxf = 1-2/2 • Hillerborg notes that as a general rule for fixed edges, the support moment should be about 1.5 to 2.5 times the span moment in the same strip. • For mxs/mxf =2.0   = 0.366 • Determine moment in the x-direction edge strips  They are half middle strip values Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • Determine moments in the y-direction middle strip • It is reasonable to choose the same ratio b/n support and span moments in the y-direction as in the x-direction. • To achieve this, choose the distance from the right support to maximum moment section as b Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • myf = wb(b)-wb(b/2)= 2(wb2/2) • The cantilever span = (1-)b • mys =w(1-)b.(1-)b/2 = (1-2)(wb2/2) • So the ratio of negative to positive moment is as before  mys/myf = 1-2/2 • Determine moment in the y-direction edge strips • myf = w(b)2/16 Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • Cantilever moment=(w/2)(1-)(b/2).(1-)(b/4) • mys=(1-)(wb2/16)  1/8 of y-direction middle strip • With the above expressions, all the design moments for the slab can be found once a suitable value for  is chosen Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • 0.350.39 give corresponding ratios of Negative to positive moments from 2.45 to 1.45 • 2.6 Unsupported Edges • The real power of the strip method becomes evident when dealing with nonstandard problems, such as with unsupported edge, slabs with holes, or slabs with reentrant corners (L-shaped) Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • For a slab with one edge unsupported, a reasonable basis for analysis by the simple strip method is that a strip along the unsupported edge takes a greater load per unit area than the actual load acting, i.e., that the strip along the unsupported edge acts as a support for the strips at right angles. Dr.-Ing. Girma Zerayohannes

Chapter 2- Strip Method for Slabs • Such strips have been referred to as “strong bands”. • A strong band is, in effect, an integral beam, usually having the same total depth as the remainder of the slab but containing a concentration of reinforcement. • The strip may be made deeper than the rest of the slab to increase its carrying capacity, but this will not usually be necessary Dr.-Ing. Girma Zerayohannes

Chapter 2 – Strip Method for Slabs