The topic of reinforced concrete slabs is somewhat related to that of beams but also needs to be considered as a separate unit. Types of slab. 1. Single span solid slab. 2. Single span ribbed slab. 3. Two way “waffle” slab. 4. Flat slab – No column drops Analysis of slabs. Theoretically all slabs span too greater or lesser extent in both directions. In practice many slabs are designed to only span in one direction and although there will be a small moment in the orthogonal direction in practice it is neglected.
According to classical plate theory, a two way spanning slab is subject to the following forces; Mix, My, Mix, Mix, Vs. and W. In practice we only design for Mix and My and provide restraint against twisting at corners (applies to external slabs). Slab analysis can be undertaken using grilles or finite element analysis. In this course we will only examine simple techniques including the yield line method of analysis. One way spanning slabs. Analyses as a continuous “beam” strip spanning between beams and subject to the most unfavorable set of loads.
Moment re-distribution (as with beams) may be undertaken) . The main reinforcement is positioned with the least cover and as such provides the maximum resisting moment. Transverse reinforcement is used to form a mesh. Sometimes reinforcement mesh is purchased as a manufactured item but often the reinforcement “mesh” is built before being placed. Reinforcing mesh. Stronger in one direction than the other Building a reinforcing mesh. Probably for two-way spanning slab Slab reinforcement at a column head. Bending moments and shear forces in singly spanning slabs.
Two-way spanning slabs. Assuming the slabs are simply supported, then With one way spanning slabs the bending moment for a simply supported slab is :- WALL/8 or 0. LOWLY. For a square two way simply supported slab the bending moment at the centre is :- WALL/16 or 0. LOWELL The general expression for bending moment is M = ;wall Where :- w is the uniformly distributed load on the slab, L the span in question M the moment and a coefficient With simply supported one way spanning slabs the load is distributed evenly to the supports either side.
Two way spanning slabs shed load to the supports as indicated :- Most slabs have a combination of simply supported (discontinuous) and continuous edges. One practical problem with simply supported slabs is that the corners tend to curl upwards, behavior which has been commonly observed in practice. To overcome this the corners of slabs are usually restrained by including additional reinforcement in the corners of slabs. The reinforcement required is usually set at 0. 5 of the maximum reinforcement required to overcome maximum mid span moments. This additional reinforcement is provided as an orthogonal mesh both at he top and bottom of the slab, and it must extend one fifth of the span length from the corner. It should be noted that discontinuous edges of slabs which are otherwise continuous are considered simply supported. The tables below indicate the coefficients p, which when substituted into the following equations enable the bending moments and shear forces per meter width of slab to be obtained.
Med = ;slowly Med = ;slowly Veda = ;vex Veda = ;vowel Med design value of the applied internal bending moment about the x axis. Med design value of the applied internal bending moment about the y axis Veda – design alee of the applied internal shear force perpendicular to x axis. Veda – design value of the applied internal shear force perpendicular to y axis ;xx – moment coefficient about x axis sys – moment coefficient about y axis ;vs. – shear coefficient perpendicular to x axis ;ivy – shear coefficient perpendicular to y axis Yield line method of analysis. Summary.
The object of a yield line analysis is to postulate a yield pattern from which the ultimate moment of resistance can be determined by: a) Considering the equilibrium of the slab elements or by b) Using the work equation. Theoretically, the Yield-line method is “unsafe” because an upper bound solution is produced which is an over-estimate of the slab strength. Physically, it is “Safe” – because the analysis ignores two important factors, Biz, a) Strain Hardening Moment of resistance of slabs is calculated ignoring strain hardening. It increases the moment capacity and therefore strength of the slab.
Actual and assumed bi-linear moment/curvature relationship: b) Tensile Membrane Action “This phenomenon occurs where the slab edges provide minimum restraint (e. G. Simply supported. Edges). The effect of: educing the bending load in the tension zone and it. Increasing the load in the compression zone. This preserves vertical equilibrium and reduces the maximum bending moment. C) Compressive Membrane Action This phenomenon occurs in slabs with restrained edges or continuous panels It is characterized by the development of very large membrane forces such that arching or Jamming occurs.
The collapse load may be several times greater than that predictable by yield-line theory. Thus, for the designer, the advantages of using the Yield-Line method of analysis may be summarized as follows: I. A good knowledge of yield line analysis will give the designer a greater understanding of slab behavior at the ultimate limit states. It. Yield line patterns can be approximately predicted for various loading, boundary conditions, and slab shapes. Iii.
An understanding of yield line patterns allows the designer to identify highly stressed areas requiring additional reinforcement. Lb. Yield line analysis is an upper bound solution, which renders economy of design. V. Once cracking begins within the slab, moments are redistributed. This redistribution renders an elastic moment diagram invalid. On the other hand, the yield line analysis gives a truer representation of internal moments. The disadvantages may be summarized as follows: I. An upper bound solution inherently reduces the safety factor against collapse. T. To simplify the analysis, corner levers and fan mechanism are commonly ignored which may introduce significant errors; therefore, the designer should have a good knowledge of the limitations of yield line analysis to have a “feel” for the magnitude of error. Iii. A series of yield line patterns are assumed to represent collapse mechanisms. Not have been identified. V. A greater probability of error is also present when formulating yield line patterns in irregular shaped slabs or if the slab is subject to a complex loading.
This method of analysis is an upper bound method of determining the load carrying capacity of slabs. This means that the predictions will be high when compared to experimental findings, not necessarily exceeding them but possibly lowering factors of safety. If the correct yield line is selected, then an accurate solution is determined. If the wrong yield line is selected, then unsafe predictions will result. The method is eased on assumed patterns of yield lines along which the reinforcement is assumed to have or be yielding.
Reinforced concrete segments between yield lines are assumed to be rigid and as such not to deflect. The location of the yield lines is dependent on the loading and boundary conditions. To analyses the slab a virtual work approach is adopted. In this, the work done by the actions on the slab are equated with the energy dissipated along the yield lines. The figure below indicates the yield lines in a single span slab built in either end. The notation is evident from the figure. Notation. Yield Lines. The selection of reasonably correct yield lines is important because this method gives an upper bound solution.
The aim is to find the pattern which gives the lowest load carrying capacity. However, when the yield pattern is adjusted to its critical dimensions, the ratio of the ultimate resistance to the ultimate load reaches its maximum. This means that when analyzing a slab algebraically, if we differentiate the ratio and set this to zero, we find the critical dimensions. These can then be substituted back into the expression for capacity to find the optimum solution. However it is usually acceptable to select a few obvious patterns if an algebraic solution is not possible and undertake the design using these.
Assumptions used in Yield Line Analysis 1) The slab is assumed to be governed by flexure alone: Other effects such as shear and deflections are to be separately considered. At failure. 3) The slab deforms plastically at failure and is separated into slab segments by yield lines. 4) The bending and twisting moments are assumed to be uniformly distributed along yield lines and they are the maximum values provided by moment strengths in the two orthogonal directions (for two way slabs). 5) The elastic deformations are negligible compared with the plastic deformations: thus, the slab parts rotate as plane segments in the collapse condition. ) Yield lines are assumed concentrated at points of maximum principle moments. 7) Membrane effects are neglected. The 8) Slabs are assumed to be lightly reinforced: thus allowing a ductile failure. 9) The rotation capacity off plastic hinge is assumed to be adequate to permit the development of all other plastic hinges necessary to form a collapse mechanism. The following rules apply to yield lines. 1 . All yield lines must be straight and end at a slab boundary. 2. A yield line can only change direction at an intersection with another yield line. . The yield line separating two slab elements must pass through the intersection of their axis of rotation (which may be at infinity). 4. Axes of rotation lie along supported edges, pass over columns or cut unsupported edges. 5. All reinforcement intersected by a yield line is assumed to yield at the line. 6. Yield lines divide slabs into rigid regions which are assumed to remain plane, so all rotation takes place along the lines. . If the slab possesses any degree of symmetry, this is reflected in a corresponding symmetry of the yield pattern.
Typical yield line patterns. Experience and knowledge of the axis of rotation enable yield lines to be developed. Preliminary information. A band of reinforcement with a yield moment of resistance of m per unit width is represented by a line drawn normally to the direction of the reinforcement. By convention, a solid line represents a positive moment, a dashed line a negative moment. Moment vectors are drawn with a double headed arrow whose direction is that of the advance of a right handed screw turned in same sense as the moment.
Stepped Yield Criterion Enables equations to be set up parallel and perpendicular to the reinforcement mesh by resolving in orthogonal directions. Energy dissipated by rotation of yield lines bounding any rigid area is given by Unknown is mix (my) External work done. Mexico + multiply The work done by the external loading is the product of load x displacement at the centered of each region. Consider a square slab of side L, subject to a uniformly distributed load w which results in a deflection at the centre of the slab of unity.