Design and Analysis of Flat Slab [PDF]

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DESIGN & ANALYSIS OF FLAT SLAB

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INTRODUCTION

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Flat slabs are two-way spanning reinforced concrete slabs supported on an orthogonal (ie at 90°) grid of columns. The soffit is at the same level throughout, which makes them easy to construct and avoids downstand (or upstand) beams conflicting with services in the ceiling below.

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An important variant has a thickened area at each column formed by lowering the soffit, called a drop panel or simply a drop.

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The flexibility of flat slab construction can lead to high economy and yet allow the architect great freedom of form.

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It is important to recognize that the behavior of flat slabs is very different from other structural forms, both for strength and for serviceability.

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Typical forms of flat slabs are shown in the next couple of slides; proprietary systems include other acceptable forms (e.g. voided slabs).

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TYPICAL BEHAVIOR OF A FLAT SLAB •

A flat slab spans between column supports without the need for beams. For a regular layout of columns, failure can occur by the formation of hinge lines along the lines of maximum hogging and sagging moments. This can be most easily presented using the folded plate theory, as shown in next figure. A complementary set of yield lines can form in the orthogonal direction.

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NOTE: One misconception is to consider a reduced loading when analyzing in a particular direction. The moments applied in each orthogonal direction must each sustain the total loading to maintain equilibrium. There is no sharing of the load by partial resistance in each orthogonal direction. It is also quite wrong to treat the load as split 50-50 between each of the two directions.

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The deflected shape of an interior panel of a flat slab on a regular grid of columns under typical in-service conditions is a function of the sum of the deflections in each orthogonal direction as shown:

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The use of finite element methods shows that the distribution of bending moments per unit width is characterized by hogging moments that are sharply peaked in the immediate vicinity of the columns. The magnitude of the hogging moments locally to the column face can be several times that of the sagging moments in the mid-span zones. These moments do occur in practice and the design should take them into account. Redistribution allows a more uniform spread of reinforcement but increases the likelihood of cracking.

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A typical distribution of bending stresses for a uniformly distributed load on a flat slab with a regular layout of columns is illustrated in the next figure.

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MOMENTS IN THE SLAB •

Flat slabs must be designed to have adequate bending/flexural strength in each of the two principal directions separately. The range of available methods of analysis is discussed below, but all manual methods are based on carrying out two separate designs, one in each of the two principal directions, each for 100% of the load. BS 8110-1 permits the analysis to be based on the single load case of maximum design load on all spans, in which case the support moments should be redistributed downwards by 20%.

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There are several different ways of analyzing flat slabs in bending. The method which has traditionally been used and is covered in detail in BS 8110-1 is the equivalent frame method. A width of slab between the centerlines of adjacent bays and centered on a line of columns is treated as a single beam element and with the columns is analyzed as a multi-bay frame. The slab is generally analyzed as a beam on point supports, i.e. the stiffness of the columns is ignored. BS 8110-1 allows for the finite width of the supports by permitting a reduction in support moments used to determine the reinforcement of 0.15 F hc (cl 3.7.2.7) [hc is effective diameter of the column and F is the total design ultimate load].

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The slab element is marked out into column strips and middle strips (see BS 8110-1 figure 3.12), and the total moment in each panel is then divided between the column and middle strips in specified proportions, generally 55:45 in the span and 75:25 over the supports, see table 3.18 in BS 8110-1. Reinforcement is provided for these moments overall. In addition, two-thirds of the column strip support reinforcement is bunched into the central 50% of the width.

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Note that the peak moments over a column head can be as much as 2.5 times the average for the whole width of the column strip. The effect of bunching two-thirds of the reinforcement into half the width of the column strip is an increase of 1.33, which means that the peak moment can still be nearly twice as much as the reinforcement is designed for.

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Moments into the columns: Earlier we mentioned that, the slab is generally analyzed as a beam on point supports, i.e. the stiffness of the columns is ignored. The equivalent frame method treats the width of slab between two panel centerlines as a beam firmly connected to the columns, whereas in practice only a narrow strip is directly connected, with the connection from the rest being laterally via torsion; thin concrete slabs are notoriously weak in torsion. The method therefore overestimates moments transferred to the columns – in particular edge columns. Allowance should be made for this inaccuracy in the modelling of the edge slab/column joint. A reasonable approximation is to reduce the support moment by a factor equal to 0.7 of the elastic moment found from the equivalent frame analysis. This should be treated as a redistribution of the support moment and the moment in the span increased appropriately. Further redistribution of moments is permissible in according to the normal rules.

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What ACI 318 & IS 456 says about the strips? Column strip is a design strip with a width on each side of a column centerline equal to 0.25L2 or 0.25L1, whichever is less. Middle strip is a design strip bounded by two column strips.

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What are the different analysis methods: 1.

Elastic Plane Frame – Equivalent Frame method

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Grillage

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Yield Line – plastic method

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Finite Element analysis – elastic method

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Grillage programs are covered in CIRIA report 110, but are not thought to be entirely appropriate. There are little or no experience within the Structural Engineer’s fraternity.

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Johansen’s yield-line analysis is simple to analyze and gives very straightforward reinforcement arrangements; Kennedy and Goodchild have written a helpful guide Practical yield line design. Hilleborg’s strip method is also available. These are both theoretically correct at ultimate load, but give no guidance as to the behavior at service load and therefore how to distribute the reinforcement, risking higher deflections and crack widths than elastic methods. Some adjustments of reinforcement layout to correlate with elastic behavior should therefore be made.

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Finite element programs can give good results if the mesh is defined realistically. Although modelling the section as uncracked and unreinforced is a major oversimplification, it generally gives a reasonable pattern of bending moments although the 20% redistribution may need to be introduced by hand. A not-too-complex modification is to change the E-value (not the thickness) 3 so that when it is used with I = 𝑏ℎ 12 it matches the EI of the cracked section, allowing for sustained loading as necessary. Check also that the program adds the Wood-Armer moments (which convert torsional to orthogonal moments so that they can be reinforced for).

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STRUCTURAL ENGINEER’S HEADACHE •

There are two distinct challenges of flat slab design:

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1.

Punching shear or 2-way shear (failure path shown below)

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Deflection

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PUNCHING SHEAR •

Shear in flat slabs is called punching shear, in which the failure surface is the frustum of an upturned cone. The BS 8110-1 method checks the punching shear on a rectangular perimeter 1.5d from the face of the column or wall. Note that a circular perimeter should be used instead of a rectangular one for circular columns.

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EC2, Punching shear does not use the Variable Strut inclination method and is similar to BS 8110 methods though the basic control perimeter is set at 2d from the loaded area/column face.

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ACI 318 & IS 456 considers critical perimeter at d/2 from the column face.

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The shape of control perimeters have rounded corners in BS & EC whereas ACI & IS considers sharp corners.

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It is normal practice to deduct the load arising from inside the perimeter. Conservatively one can ignore.

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The calculated ultimate shear force is then increased by factors of 1.15, 1.5 and 1.4 for interior, corner and perimeter columns respectively (care must be taken to account any openings).

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It should be noted that, for flat slabs 200mm thick and over, the need to include punching shear reinforcement is common. For slabs less than 200mm, thick shear reinforcement is ineffective.

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If the punching shear resistance found (VRdc) to be lesser than the design shear then (VEd), punching shear reinforcement should be designed provided effective applied shear stress for the given critical perimeter is lesser than the maximum allowable shear resistance VRd,max; otherwise a redesign would be necessary.

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In other words, when the maximum shear resistance is greater than the effective applied shear stress at the column perimeter but the shear resistance at the basic control perimeter is less than the effective applied shear stress, shear reinforcement is required.

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The outermost perimeter of shear reinforcement should be placed at a distance not greater than 1.5d within the outer control perimeter, see next figure.

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The unreinforced shear capacity of slabs thin enough to be economic is often insufficient. Conventional shear reinforcement is difficult to detail and fix, and proprietary systems have been developed to provide the necessary additional strength more efficiently.

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General arrangement of placing shear reinforcement as per EC

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General arrangement of placing shear reinforcement as per ACI

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PROPRIETARY PUNCHING SHEAR REINFORCEMENT SYSTEMS

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Stud rails : This general term describes four discrete systems each produced by different manufacturers. The systems comprise two structural elements, the stud and the spacer rail onto which the stud is fixed. The stud is fabricated from standard straight reinforcement bar (plain or deformed) onto which an enlarged head is welded to either one or both ends. Figure 4 highlights the main manufactured variations; each system claims superiority backed up with laboratory tests, but all have their own inherent advantages and disadvantages.

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Figure a, b & c - Double headed studs require non-structural spacer bars. Note that a uses rod with 45° returns which provide extra stability from rotation during pouring of the slab.

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Figure d - Single headed studs require the bar to provide stud anchorage as well as spacing.

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The number of studs per rail combined with the diameter (range 10 to 25 mm) can be varied to provide the correct area of shear reinforcement Asv, while the stud height (120 mm upwards*) and c/c spacing are governed by the overall depth of the slab and the required cover.

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Shear ladders:

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Shear band strips:

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Stirrup mats:

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Shearhoops:

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ACI shear stirrups:

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ACI shear stirrups consist of conventional beam reinforcement cages that are sandwiched between the main longitudinal reinforcement. The shear stirrups are usually prefabricated off the critical path to save installation time and are assembled in a variety of shapes (X, L or T on plan) to deal with punching failure at internal, corner and edge columns respectively.

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The ACI has pioneered this system with well-documented design guidance and commentary in ACI 318. The system relies on the shear capacity of the slab to distribute the internal stresses to the closest arm. In doing so, the effective shear perimeter is transformed from a square to a much longer kite shape.

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ACI’s improvised arrangements of headed shear stud reinforcement:

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Structural steel shearheads:

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Structural steel shearheads rely on the same design principles as ACI shear stirrups except that the reinforcement cages are replaced with structural steel sections. The column capitals can be adapted from the cruciform element into a closed grid; the holes within the grid can then be used to accommodate risers adjacent to the column.

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Dome shearhead system is contained within the depth of the slab and relies on membrane action rather than bending stresses. Extra holes can be punched into the dome every 90º to assist the concrete pouring process (figure d, next page).

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DEFLECTION •

The combination of thin sections and two-way spanning means that flat slabs are prone to quite large deflections. The deflection at mid-span can be visualized by going from a column along the grid line to the halfway point, and then turning 90° to the mid-point of the bay. This means that the mid-bay deflection will be about twice the one-way deflection. Don’t forget that the reinforcement is in two layers, so either use an average effective depth or calculate the two directions separately. Remember also that key sections will probably be cracked, which means that conventional elastic methods will usually seriously underestimate deflection.

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NOTE: With their longer spans and more deflection-prone behavior, flat slabs are thought to be more at risk of uncomfortable vibrations. Checks should certainly be carried out before using flat slabs for dancing or aerobics.

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Deflection limits as per EC2 (7.4.1):

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Deflection limits as per ACI 318:

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It is important to note that ACI [ACI 318, 2008] does not impose a limit to deflection under self-weight. ACI’s recommendations address the amount of deflection subsequent to the installation of nonstructural elements likely to be damaged. The following table lists the ACI’s stipulation on deflections (TABLE 9.5(b)).

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LONG-TERM DEFLECTIONS

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A concrete member’s deformation changes with time due to shrinkage and creep. Shrinkage of concrete is due to loss of moisture. Creep is increase in displacement under stress. Under constant loading, such as selfweight, the effect of creep diminishes with time. Likewise, under normal conditions, with loss of moisture, the effect of deformation due to shrinkage diminishes. Restraint of supports to free shortening of a slab due to shrinkage or creep can lead to cracking of slabs and thereby an increase in deflection due to gravity loads. While it is practical to determine the increase in instantaneous deflection of a floor system due to creep and shrinkage at different time intervals, the common practice for residential and commercial buildings is to estimate the long-term deflection due to ultimate effects of creep and shrinkage.

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Shrinkage: It is the long-term shrinkage due to loss of moisture through the entire volume of concrete that impacts a slab’s deformation. Plastic shrinkage that takes place within the first few hours of placing of concrete does not play a significant role in slab’s deflection and its impact in longterm deflection is not considered. Long-term shrinkage results in shortening of a member. On its own, long-term shrinkage does not result in vertical displacement of a floor system. It is the presence of non-symmetrical reinforcement within the depth of a slab that curls it (warping) toward the face with less or no reinforcement. The slab curling is affine to its deflection due to selfweight, and hence results in a magnification of slab’s natural deflection. It is important to note that, deflection due to shrinkage alone is independent of the natural deflection of slab. It neither depends on the direction of deflection due to applied loads, nor the magnitude. The shrinkage deflection depends primarily on the amount and position of reinforcement in slab. A corollary impact of shrinkage is crack formation due to restraint of the supports. This is further discussed in connection with the restraint of supports. It is the crack formation due to shrinkage that increases deflection under gravity loads. Shrinkage takes place over a time period extending beyond a year. While the amount of shrinkage and its impact on deflection can be calculated at shorter intervals, the common practice is to estimate the long-term deflection due to the ultimate shrinkage value. Shrinkage values can vary from zero, when concrete is fully immersed in water to 800 micro strain. Typical ultimate shrinkage values are between 400 to 500 micro strain.

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Creep: Creep is stress related. It is a continued magnification of the spontaneous displacement of a member with reduced rate of creep with time. Values of creep vary from 1.5 to 4. Typical ultimate creep values for commercial and building structures are between 2 to 3.

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Restraint of Supports: Restraint of supports, such as walls and columns to free movement of a slab due to shrinkage can lead to tensile stresses in the slab and early cracking under applied loads. Early cracking will reduce the stiffness of the slab and increase its deflection.

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DESIGN CHECK LIST •

This list has been adapted from the list in CIRIA report 110, and only applies to non-prestressed flat slabs with the same solid thickness throughout.

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1. Assess required depth of slab from span/effective depth limitations factored down by at least 0.9. Decide which direction of reinforcement will go in which layer (or use the average for equal spans both ways), and allow adequate cover.

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2. Check whether slab at a typical internal column needs shear reinforcement. Consider effect of a 200 mm square hole close to the column.

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3. Calculate typical top reinforcement at an internal column and check possible congestion of reinforcement.

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4. Calculate typical mid-span reinforcement, check span/depth assumptions.

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5. Check moments and shears at typical edge and corner columns. Ensure that the junction is capable of transferring the required moment.

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6. Check effects of large columns and rigid corners such as core walls.

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7. Check likely position and effect of holes and openings. Decide whether they comply with the BS 8110 rules, and how to design them if they don’t. Re-check punching shear if necessary.

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