Dr Ing Girma Zerayohannes Dr Ing Adil Zekaria Chapter 2 Strip Method for Slabs 21 Introduction Different methods of analysis are allowed by EBCS2 One of these is plastic methods ID: 1020449
Download Presentation The PPT/PDF document "Chapter 2 – Strip Method for Slabs" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
1. Chapter 2 – Strip Method for SlabsDr.-Ing. Girma ZerayohannesDr.-Ing. Adil Zekaria
2. Chapter 2- Strip Method for Slabs2.1 IntroductionDifferent methods of analysis are allowed by EBCS-2One of these is plastic methodsStrip method is a plastic method of analysis2Dr.-Ing. Girma Zerayohannes
3. Chapter 2- Strip Method for SlabsThe 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 slabDisadvantages: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.3Dr.-Ing. Girma Zerayohannes
4. Chapter 2- Strip Method for SlabsWhen 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 found4Dr.-Ing. Girma Zerayohannes
5. Chapter 2- Strip Method for SlabsThese circumstances motivated Hillerborg (1956) to develop what is known as strip method for slab designIn contrast to yield line analysis, the strip method is a lower bound approach, based on the satisfaction of equilibrium requirements every where in the slab5Dr.-Ing. Girma Zerayohannes
6. Chapter 2- Strip Method for SlabsBy 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 field6Dr.-Ing. Girma Zerayohannes
7. Chapter 2- Strip Method for SlabsLower 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 capacity7Dr.-Ing. Girma Zerayohannes
8. Chapter 2- Strip Method for SlabsAdvantages:The strip method gives results on the safe side, which is certainly preferable in practiceThe strip method is a design method, by which the needed reinforcement can be calculated8Dr.-Ing. Girma Zerayohannes
9. Chapter 2- Strip Method for Slabs4.2 Basic PrinciplesThe governing equilibrium equation for a small slab element having sides dx and dy is:9Dr.-Ing. Girma Zerayohannes
10. Chapter 2- Strip Method for Slabs10Dr.-Ing. Girma ZerayohannesFigure 1
11. Chapter 2- Strip Method for Slabs11Dr.-Ing. Girma Zerayohannes
12. Chapter 2- Strip Method for SlabsWhere w = the external load per unit areamx, my = Bending Moments per unit width in the x and y directions andmxy = the twisting moment12Dr.-Ing. Girma Zerayohannes
13. Chapter 2- Strip Method for SlabsSo 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 moments13Dr.-Ing. Girma Zerayohannes
14. Chapter 2- Strip Method for SlabsThe 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 = 0The equilibrium equation then reduces to:14Dr.-Ing. Girma Zerayohannes
15. Chapter 2- Strip Method for SlabsThis equation can be split conveniently into 2 parts, representing twist less beam strip action.15Dr.-Ing. Girma Zerayohannes
16. Chapter 2- Strip Method for SlabsWhere 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 directionIn other regions, it may be reasonable to assume that the load is divided equally in the two directions, i.e. k=0.516Dr.-Ing. Girma Zerayohannes
17. Chapter 2- Strip Method for Slabs2.3 Choice of load distributionTheoretically, 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.17Dr.-Ing. Girma Zerayohannes
18. Chapter 2- Strip Method for SlabsThe 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.18Dr.-Ing. Girma Zerayohannes
19. Chapter 2- Strip Method for SlabsTo 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 2.19Dr.-Ing. Girma Zerayohannes
20. Chapter 2- Strip Method for Slabs20Dr.-Ing. Girma ZerayohannesFigure 2
21. Chapter 2- Strip Method for SlabsThe load on all strips in each direction is thus w/2 ( with k=0.5), as illustrated by the load dispersion arrowsThis 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)21Dr.-Ing. Girma Zerayohannes
22. Chapter 2- Strip Method for SlabsThe same applies for y-direction stripsIt 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 slab22Dr.-Ing. Girma Zerayohannes
23. Chapter 2- Strip Method for SlabsSo 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 endsTry the alternative, more reasonable distribution shown in Figure 3 next slide.23Dr.-Ing. Girma Zerayohannes
24. Chapter 2- Strip Method for Slabs24Dr.-Ing. Girma ZerayohannesFigure 3
25. Chapter 2- Strip Method for SlabsHere 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 regions25Dr.-Ing. Girma Zerayohannes
26. Chapter 2- Strip Method for SlabsThe lateral distribution of moments shown in Fig (3) would theoretically require a continuously variable bar spacing impracticalA 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.26Dr.-Ing. Girma Zerayohannes
27. Chapter 2- Strip Method for SlabsHillerborg 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 margin27Dr.-Ing. Girma Zerayohannes
28. Chapter 2- Strip Method for SlabsA 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 slab28Dr.-Ing. Girma Zerayohannes
29. Chapter 2- Strip Method for Slabs29Dr.-Ing. Girma ZerayohannesFigure 4
30. Chapter 2- Strip Method for SlabsTwo 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.30Dr.-Ing. Girma Zerayohannes
31. Chapter 2- Strip Method for SlabsThe 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.31Dr.-Ing. Girma Zerayohannes
32. Chapter 2- Strip Method for SlabsIt 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.32Dr.-Ing. Girma Zerayohannes
33. Chapter 2- Strip Method for Slabs2.4 Rectangular slabs with simple supportDiscontinuity lines parallel to the edges as shown in the figureIn the x-direction:Side strips: mx = w/2×b/4×b/8 = wb2/64Middle strips: mx = w×b/4×b/8 = wb2/32In the y-directionSide strips: my = wb2/64Middle strips: my = wb2/833Dr.-Ing. Girma Zerayohannes
34. Figure 5 Rectangular slab with discontinuity lines originating at the corners.Chapter 2- Strip Method for Slabs
35. Chapter 2- Strip Method for Slabs35Dr.-Ing. Girma ZerayohannesFigure 6 Rectangular slab with discontinuity lines parallel to the edges
36. Chapter 2- Strip Method for SlabsDesign the rectangular slab using the strip method for slabsUse 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-2Take variable load q = 3.0 kN/m2Floor finish-30 mm screed and 20mm thick marble36Dr.-Ing. Girma Zerayohannes
37. Chapter 2- Strip Method for Slabs2.5 Fixed Edges and ContinuityUp 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 providedThis 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 edges37Dr.-Ing. Girma Zerayohannes
38. Chapter 2- Strip Method for SlabsSome attention should be paid to elastic moment ratios to avoid problems with cracking and deflection at service loadsFigure 7 (next slide) shows a uniformly loaded rectangular slab having two adjacent edges fixed and the other two edges simply supportedLet us consider slab strips with one end fixed and one end simply supported as shown in Fig. 738Dr.-Ing. Girma Zerayohannes
39. Chapter 2- Strip Method for Slabs39Dr.-Ing. Girma ZerayohannesAABBFigure 7
40. Chapter 2- Strip Method for SlabsIn designing by strip method, slab strips carrying loads only near supports and unloaded in the central region are encounteredIt is convenient if the unloaded region is subject to a constant moment (and zero shear) because this simplifies the selection of positive reinforcement40Dr.-Ing. Girma Zerayohannes
41. Chapter 2- Strip Method for SlabsThe 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 41Dr.-Ing. Girma Zerayohannes
42. Chapter 2- Strip Method for SlabsFor a BMD for x-direction middle strips (section A-A) with constant moment over the unloaded part, the following maximum moments are achievedand42Dr.-Ing. Girma Zerayohannes
43. Chapter 2- Strip Method for SlabsThe first term is the “cantilever” moment at the left endSo the negative moment at a support plus the span moment = the “cantilever” momentNow the ratio of negative to positive moments in the x-direction middle strip is:43Dr.-Ing. Girma Zerayohannes
44. Chapter 2- Strip Method for Slabsmxs/mxf = 1-2/2Hillerborg 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 = 0.366Determine moment in the x-direction edge strips They are half middle strip values44Dr.-Ing. Girma Zerayohannes
45. Chapter 2- Strip Method for SlabsDetermine moments in the y-direction middle stripIt 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 b45Dr.-Ing. Girma Zerayohannes
46. Chapter 2- Strip Method for Slabsmyf = wb(b)- wb(b/2)= 2(wb2/2)The cantilever span = (1-)bmys =w(1-)b.(1-)b/2 = (1-2)(wb2/2)So the ratio of negative to positive moment is as before mys/myf = 1-2/2Determine moment in the y-direction edge stripsmyf = w(b)2/1646Dr.-Ing. Girma Zerayohannes
47. Chapter 2- Strip Method for SlabsCantilever moment=(w/2)(1-)(b/2).(1-)(b/4)mys=(1-)(wb2/16) 1/8 of y-direction middle stripWith the above expressions, all the design moments for the slab can be found once a suitable value for is chosen47Dr.-Ing. Girma Zerayohannes
48. Chapter 2- Strip Method for Slabs0.350.39 give corresponding ratios of Negative to positive moments from 2.45 to 1.452.6 Unsupported EdgesThe 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)48Dr.-Ing. Girma Zerayohannes
49. Chapter 2- Strip Method for SlabsFor 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.49Dr.-Ing. Girma Zerayohannes
50. Chapter 2- Strip Method for SlabsSuch 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 50Dr.-Ing. Girma Zerayohannes
51. Chapter 2- Strip Method for SlabsConsider the rectangular slab carrying a uniformly distributed ultimate load w with fixed edges along three side and no support along one short side, shown in Figure 8.51Dr.-Ing. Girma Zerayohannes
52. Chapter 2- Strip Method for Slabs52Dr.-Ing. Girma ZerayohannesFigure 8
53. Chapter 2- Strip Method for SlabsThe following are observed:Discontinuity lines are chosen as shownThe load on a unit middle strip in the x–direction, includes the downward load in the region adjacent to the fixed left edge and an upward reaction kw in the region adjacent to the free edge53Dr.-Ing. Girma Zerayohannes
54. Chapter 2- Strip Method for SlabsM about the left end (with moments +ve clockwise and with the unknown support moment mxs acting clockwise) mxs+wb2/32-(kwb/4)(a-b/8)=0 k=(1+32mxs /wb2)/(8(a/b)-1) k will be known once mxs is selectedSelection of mxs will depend on the shape of the slab.54Dr.-Ing. Girma Zerayohannes
55. Chapter 2- Strip Method for SlabsIf a is large relative to b, the strong band in the y-direction at the edge will be relatively stiff, and the moment at the left support of the x-direction strips will approach the elastic value for a propped cantilever55Dr.-Ing. Girma Zerayohannes
56. Chapter 2- Strip Method for SlabsIf the slab is nearly square, the deflection of the strong band would tend to increase the support moment; a value about half the free cantilever moment might be selectedWith mxs selected and k calculated from the above equation, the max span moment is determined56Dr.-Ing. Girma Zerayohannes
57. Chapter 2- Strip Method for Slabsmxf = (kwb2/32)((8a/b)-3+k)Determine moments in the x direction edge strips They are one-half those in middle stripIn the y direction middle strip, the cantilever moment is wb2/8Adopting a ratio of support to span moment of 2 results in support and span moments, respectively, of57Dr.-Ing. Girma Zerayohannes
58. Chapter 2- Strip Method for Slabsmyf + mys = myf + 2myf = wb2/8 myf = wb2/24 and mys = wb2/12Determine moments in y-direction strip adjacent to the fixed edge It is one-eighth the middle strip values (check)In the y-dir strip along the free edge, moments can, with slight conservatism, be made equal to (1+k) times y-dir middle strip values58Dr.-Ing. Girma Zerayohannes
59. Chapter 2- Strip Method for Slabs2.7 Slabs with HolesSlabs with small openings can usually be designed as if there were no openings, replacing the interrupted steel with bands of rebar of equivalent area on either side of the opening in each direction.Smaller dimensions are those needed to accommodate heating, plumbing and ventilating risers, etc.59Dr.-Ing. Girma Zerayohannes
60. Chapter 2- Strip Method for SlabsLarger size holes are required by stairways and elevator shaftsSlabs with larger openings must be treated more rigorouslyThe strip method offers a rational and safe basis for design in such cases. Integral load-carrying beams (strong bands) are provided along the edges of the opening, usually having the same depth as the remainder of the slab but with extra reinforcement, t pick up the load from the affected regions and transmit it to the supports60Dr.-Ing. Girma Zerayohannes
61. Chapter 2- Strip Method for SlabsIn general, these integral beams should be chosen so as to carry the loads most directly to the supported edges of the slab.The width of the strong bands should be selected so that the steel ratios are at or below the maximum for beams61Dr.-Ing. Girma Zerayohannes
62. Chapter 2- Strip Method for SlabsExample: Rectangular slab with central openingFigure shows a 5m×8m slab with fixed supports along 4 sides. A central opening 1.2m×2.4m must be accommodated. Estimated slab thickness is 200 mm. The slab is to carry a uniformly distributed factored load of 15kN.m2 including self weight.62Dr.-Ing. Girma Zerayohannes
63. Chapter 2- Strip Method for SlabsDevise an appropriate system of strong bands to reinforce the opening, and determine moments to be resisted at all critical sections of the slabDiscontinuity lines for the basic slab (w/o hole) are first chosen and the moments determined which are used as a guide in selecting moments for the actual slab with hole 63Dr.-Ing. Girma Zerayohannes
64. Chapter 2- Strip Method for SlabsEdge strips are defined having width equal to 5/4 = 1.25mIn the central region, 100% of the load is assigned to the y directionMoments of the basic case w/o hole will be calculated and later used as a guide in selecting moments for the actual slab with hole. A ratio of support to span moments of 2.0 will be used generally64Dr.-Ing. Girma Zerayohannes
65. Chapter 2- Strip Method for SlabsMoments for the slab w/o is: x direction middle strips:Cantilever mx = wb2/32 = 552/32=11.72 kNm/mNegative mxs = 11.722/3 = 7.81 kNm/mpositive mxf = 11.721/3 = 3.91 kNm/mX direction edge strips are ½ middle strips65Dr.-Ing. Girma Zerayohannes
66. Chapter 2- Strip Method for SlabsY direction middle stripsCantilever my = wb2/8= 1552/8=46.88 kNm/mnegative mys = 46.882/3=31.25 kNm/mpositive myf = 46.881/3=15.63 kNm/mY direction edge strips are 1/8 middle strip values66Dr.-Ing. Girma Zerayohannes
67. Chapter 2- Strip Method for SlabsBecause of the hole, certain strips lack support at one end. To support them, 0.3m wide strong bands will be provided in the x direction at the long edges of the hole and 0.6 m wide strong bands in the y direction on each side of the hole.The y dir bands will provide for the reactions of the x dir bands.67Dr.-Ing. Girma Zerayohannes
68. Chapter 2- Strip Method for SlabsWith the distribution of loads shown in figure, strip reactions and moments are found as follows:Strip A-A: Assuming propped cantilever action with the restraint moment along the slab edge taken as 31.2531.25+w1(0.3)(1.75)-15(1.6)2/2 = 0 w1=-7.95 kN/m68Dr.-Ing. Girma Zerayohannes
69. Chapter 2- Strip Method for SlabsThe negative value indicates that the cantilever strips are serving as supports for strip DD, and in turn for the strong bands in the y-direction, which is hardly a reasonable assumption.Hillerborg notes that the restraint moment should stay as close to the basic value” w/o w1 being negativew1=0 (cantilever alone)69Dr.-Ing. Girma Zerayohannes
70. Chapter 2- Strip Method for SlabsNote: with w1 = 0 chosen -kw = 0 k=0 loading on the strong band = (1+k)w = w = 15 KN/m2Now mys = 15(1.6)2/2 = -19.2 kNm/mStrip B-B:mxs = 7.81 kNm/m (basic value) 7.81+w2(0.6)(2.5)-15(1.25)2/2 = 0 w2=2.61 kN/m70Dr.-Ing. Girma Zerayohannes
71. Chapter 2- Strip Method for SlabsBecause of the positive reaction by the strong band the load dispersed in the y direction must be greater than 15 kN/m2Determine k-kw = -15k = -2.61 kN/m2 k=0.174 load dispersed in strong band in y-dir in the middle: (1+k)w = (1.174)15=17.61 kN/m2 71Dr.-Ing. Girma Zerayohannes
72. Chapter 2- Strip Method for Slabs load dispersed in strong band in y-dir near the edge: (1+k/2)w = (1.087)15=16.31 kN/m2 Determine max span moment:Shear is zero at: 15x = 0.6(2.61) x = 0.1mmxf = 0.6(2.61)(0.3+0.95+0.1)-15(0.1)2/2 = 2.04 kNm/m72Dr.-Ing. Girma Zerayohannes
73. Chapter 2- Strip Method for SlabsStrip C-C:Negative and positive moments and the reaction to be provided by strip C-C, are al one-half the corresponding values for strip B-B.Strip D-D:The 0.3m wide strip carries 15kN/m2 in the x-direction with reactions provided by the strong bands E-E (loading 150.3 = 4.5 kN/m)73Dr.-Ing. Girma Zerayohannes
74. Chapter 2- Strip Method for Slabs(we0.6)2 = 4.52.4 we= 9.0 kN/m (over a 0.3 m wide strip)mxf = 9(0.6)(1.2+0.3)-4.5(1.2)/2 = 8.1 – 3.24 = 4.86 kNm/(0.3m width)Strip E-E:Direct load dispersed (1+k)w and (1+k/2)w are 17.61 and 16.31 kN/m2 respectively74Dr.-Ing. Girma Zerayohannes
75. Chapter 2- Strip Method for SlabsReactions from strong bands D-D is 9.0 kN/m over 0.3 m width or 9/0.3 = 30 kN/m2 300.6 = 18 kN/m over 0.6 m wide strip17.610.6 = 10.566 kN/m over 0.6 m wide strip16.310.6 = 9.786 kN/m over 0.6 m wide strip75Dr.-Ing. Girma Zerayohannes
76. Chapter 2- Strip Method for SlabsDetermine moments:Cantilever 9.786(2.5)2/2+(10.566-9.786)(1.25)(1.25+0.625)+18(0.3)(1.6+0.15) = 41.86 kNm (per 0.6 m width)Negative: mys = 41.86(2/3) = 27.911 kNm (per 0.6 m width)positive: myf = 41.86(1/3) = 13.95 kNm (per 0.6 m width)76Dr.-Ing. Girma Zerayohannes
77. Chapter 2- Strip Method for SlabsReferenceNilson and Winter, 14th edition or newer77Dr.-Ing. Girma Zerayohannes