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strip method for g5 civil engineering student

1. Simple strip method for slab analysis Tutorial for GC March 27,2024 1 By: Tekalign Behailu ( Msc )

2. Introduction Was developed by Hillerborg and published for the first time in swedish in 1956. Circumstances motivating Hillerborg are; It is a lower bound theory based on the satisfaction of equilibrium requirements everywhere in the slab. Sometimes called an equilibrium theory and the first coming task in this method is determination of moment field and then after, the reinforcement at every point in the slab will determined for this moment field. March 27,2024 2

3. Continue… 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 the given external loading will represent a lower bound of a true carrying capacity. It gives a result always in a safe side and which is certainly preferable in practice. It is a design method by which the need reinforcement can be calculated. It encourages the designer to vary the reinforcement in a logical way. March 27,2024 3

4. Continue… This leading an economical arrangement of steel and as well as safe design. It is generally simple even to use for slabs with holes and irregular boundary conditions. The work was tested or validated by an experiment of Armer and conclude that the method produces safe and satisfactory design. In contrast to the yield line method(provides no inducement to vary bar spacing) ,it encourages the use of strong bands of steel where needed, such as around openings or over columns , and hence it improves economy and reducing excessive cracking or large deflection under service load. March 27,2024 4

5. BASIC PRINCIPLES The governing equilibrium equation for a small slab having side of dx and dy is Where: q is external load per unit area m x , m y is bending moment per unit width in the x and y direction respectively m xy is a twisting moment March 27,2024 5

6. Continue… According to the lower bound theorem, any combination of mx, my,and mxy that satisfies the equilibrium equation 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. Considering the torsional effect assumed to be zero , and therefore the reinforcement is arranged in parallel to the axes in recti leaner coordinate system. m xy =0 March 27,2024 6

7. Continue… Therefore it can be reduces to : This can be split to twistless beam strip action In many regions in a slabs, the value of K will be either zero or one. In other regions, it may be reasonable to be the load is divided equally in two directions. K =0.5 March 27,2024 7

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9. CHOICE OF LOAD DISTRIBUTION The desired goal is to arrive at an arrangement of steel that is safe and economical and that will avoid problems at the service load level associated with excessive cracking and deflection. For an economical decision its best to based the elastic moment distribution. To see and illustrate the choices open to the designer for about which one way of stripping makes feasible as per the economy and even simplicity in practice specially during reinforcement placement, consider the following three scenarios having simply supported slab with a square dimension of “a” length. March 27,2024 9

10. Continue… Scenario1 : The load distribution setting with K = 0.5 over the entire slab. March 27,2024 10

11. Continue… The load on all strip in each direction is then equal to q/2. and this gives a maximum moment value of: over the whole slab with a uniform lateral distribution across the width of the critical section. this not represent an economical and serviceable solution because it is recognized that curvatures below, hence moments, must be greater in the strips near the middle of the slab than near the edges parallel to the direction of the edges. Extention of moment distribution accompanied by further cracking in highly stressed regions(middle strip). March 27,2024 11

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13. Continue… Scenario2: With the region of different load dispersion, separated by the dash-dotted “ discontinuity lines ” which follows the diagonals. March 27,2024 13

14. Continue… For a strip at A-A at a distance y< =a/2 from the x axis the moment is : Regarding to the reinforcement it develops a continuously varying bar spacing and this obviously impractical. its economical to the previous scenario that is because of the stress become higher in middle strip and this a logical distribution and makes an economical and it provides an appropriate reinforcement arrangement. March 27,2024 14

15. Scenario3: Division is made so that the load is carried to the nearest support , its as before but the load near the diagonals has been divided, with one half taken in each direction . Thus gives K values zero and one along the middle edges and a value half in the corner and center of the slab. As compared to the previous scenarios, it leads a practical arrangement , one with constant spacing through the strip of width a/2 & wider spacing in outer strip where the elastic moment and curvature is known to be small. March 27,2024 15

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17. Continue… For an x direction strip along section A-A max. moment is: For x direction strip along section B-B m ax. Moment is: Way of stripping the slab in this method is assumed to be in the form of scenario three since it leads an economical and practical arrangement of reinforcements. March 27,2024 17

18. RECTANGULAR SLABS It is reasonable to assume that In rectangular slab through out most of the area the load carried by the shorter direction. Hillerborg first scenario: For a rectangular section with simple supported BCs.the possible division is to be with discontinuity lines originating from the slab corners at an angle depending on the ratio of shorter to longer sides of the slab. But this scenario requires continuously varying reinforcement. March 27,2024 18

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20. Continue… Scenario 2: To avoid continuously varying reinforcements from the pervious scenario,Hillerborg suggests the following distribution with discontinuity line parallel to the side of the slab. The edge band width is equal to ¼ of the shorter side of the slab. The load in the corner is divided equally in the x & y directions while elsewhere all of the load is carried in the direction indicated by arrows. March 27,2024 20

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22. Continue… For the economical reinforcement arraingement,determine the moment distribution diagrams .(Exercise) March 27,2024 22

23. FIXED EDGES AND CONTINUITY CASE The discontinuity lines are shifted with a factor of alpha to account the greater stiffness of strips of fixed ends or the continuity, around the supports. The factor alpha for the slab shown below is taken less than 0.5. “alpha” related directly to the ratio of negative to positive moment in the strips . The sum of the absolute value of positive span moment and negative at the left or right end , ml , mr depends only on the respective end condition and numerically equals to the negative moment if the strip carries the load as a cantilever. March 27,2024 23

24. Continue… Hillerborg suggests as a general rule for fixed ends, the support moment equal to 1.5 to 2.5 times the span moment in the same strip. March 27,2024 24

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27. Specialty of the method… The real power of the strip method is becomes evident when dealing with non standard problems, such as slab with unsupported edges, holes or slabs with reentrant corners like L-shaped slabs. Slab with unsupported edge The special case in this analysis is that the strip along unsupported edge takes a greater load per unit area than the actual. According to Wood and Armer such a strips referred as a strong band . March 27,2024 27

28. Continue… This strips usually have the same depth to the remaining slab region but with high concentration of reinforcement. and sometimes may be made deeper than other regions but this will not be always necessary. Lets take a rectangular slab caring a uniformly distributed load per unit area with fixed edge in three sides and no support in one shorter side. And the discontinuity flow as shown from figure below. March 27,2024 28

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31. Continue… How to decide K? March 27,2024 31

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33. Continue… If the unsupported edge is in the long span direction, then a significant fraction of the load in the slab central region will be carried in the direction perpendicular to the long edges, and the simple distribution shown below is more suitable. Strong band along the free edge serves as an integral edge beam , with width normally chosen as low as possible considering limitations on tensile reinforcement ratio in the strong band. March 27,2024 33

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35. Continue… How to decide K1 & K2 ? March 27,2024 35

36. Slab with Holes ( Reading? ) March 27,2024 36

37. EXAMPLES March 27,2024 37

38. Revision on TOS II Helpful for the analysis of maximum moments in the span or supports of each strips once the load distribution is prepared in correct ways presented before. Moment distribution method/ Kani method Its an iterative method through the following main points. 1.Expressing external applied load with form of fixed end moments. 2. Distributing the unbalanced moments from the free joints to adjacent members based on their flexural stiffness. March 27,2024 38

39. Continue… 3.As second iteration, if the first iteration is not converged rebalance the moments from the free joints and distributing it to the adjacent members and proceeding this step until the unbalanced moment from the free joint becomes decline. Distribution factor : Once the moment is distributed to the members, depending on the support type found at end of the member , some part of the member moment is transferred to the support with factor called carry over factor. March 27,2024 39

40. FEM computation tables March 27,2024 40

41. Example-1 Determine strip moments for the following slab shown below. March 27,2024 41

42. Continue… Example -2 For the previous simply supported rectangular slab loaded with live load of 110KN/m2 and dead load of 150KN/m2,detemine the design moment distribution in the shorter span direction. March 27,2024 42

43. THANK YOU! March 27,2024 43