Designing Reinforced Concrete Slabs Engineering Essay

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Work done in yield lines rotating = work done in loads moving

Two of the most popular methods of application are the 'work method' and the use of standard formulae.

Thus, having postulated a valid mechanism by which the slab can fail, it is given a hypo-thetical displacement and the consequent expenditure of energy by the external loads due to this displacement is equated to the internal dissipation of energy in the yield lines in talking up compatible rotations. The equation formed in this way is called the Work Equation and gives the relationship between the slab strength and the applied loads for a particular layout of yield lines.

It is, however, the task of the analyst to seek the particular layout of the failure pattern which is the most critical. The term 'pattern' or 'mode' is used to indicate arrangement of yield lines having a particular general shape, the term layout indicates a particular configuration of a pattern. Whilst it is frequently possible to predict the correct failure pattern it is not immediately apparent which layout of that pattern is the most critical.

Advantages of yield line design

Yield line design has the advantages of

Economy

Simplicity and

Versatility

Yield line design leads to slabs that are quick and easy to design, and are and easy to construct. There is no need to resort to computer for analysis or design. The resulting slabs are thin and have very low amounts of reinforcement in very regular arrangements. The reinforcement is therefore easy to detail and easy to fix and the slabs are very quick to construct. Above all, yield line design generates very economic concrete slabs because it considers features at the ultimate limit state.

Yield design is a robust and proves design technique. It is a versatile tool that challenges designers to use judgment. Once grasped yield line design is an exceedingly powerful design tool.

The catch

Yield line design demands the familiarity patterns, eg. Knowledge of how slabs might fail. This calls for a certain amount of experience, engineering judgment and confidence, none of which is easily gained.

Yield line design tends to be a hand method. This may be seen as both an advantage and disadvantage. Each slabs has to be judged on its merits and individually assessed. The method allows complex slabs to be looked at in a simple way. And in an age of computers it gives an independent method of analysis and verification. This is especially important for those who are becoming disillusioned with the reliance placed on finite element analysis. They see a need to impart greater understanding and remind designers that reinforced concrete does not necessarily behave in an elastic manner. Nonetheless it is hoped that the option of suitable and accessible software for yield line design will become available in the near future.

3.1 One way spanning slabs

3.1.1 General

The general form of the formulae for one way spanning slabs uniformly distributed loads is as follows

m = nL2

2 (√1+ i1 + √1+ i2) 2

Where

m ultimate moment along the yield line [ kNm/m]

n ultimate load [kN/m2]

L span [m]

i1, i2 ratios of support to mid span moments, the values of which are chosen by the designer : i1= m'1/m, i2= m'2/m

Figure 3.1 one way spanning slab

L

m

i2= m'2/m

i1= m'1/m

m ( kNm/m)

m'2 ( kNm/m)

m'1 ( kNm/m)

n ( kNm/m)

Figure 3.2 axonometric view of a simply supported one way spanning slab.

Figure 3.3 loading, hinge, locations and bending moment diagram for simply supported slab

Plastic (ultimate) moment

m along yield line

SECTION A-A

L

m

i1

i2

m

m

m'1 - 0

m'2 - 0

U.D.L n (kN/m2)

m

Elastic bending moment diagram with maximum

Ultimate moment in location where yield line will form

Using the general formula and i1 = i2 = 0: (i1 = m'1/m = 0 and i2 = m'2/m = 0)

nL2

nL2

2 (√1+ 0 + √1+ 0) 2

2 (√1+ i1 + √1+ i2) 2m = =

nL2

nL2

8

2(2)2

m = = kNm/m

Therefore the plastic ultimate moment along the yield line is nL2/ 8, which is correct!

3.1.2. Design formulae

The formulae to establish the value of the maximum midspan moment 'm' for span within a continuous slab are given in table 3.1. cases 2, 3 and 4 are variations on the base formula for case 1.

The formula in table 3.1 enable the designer to choose a set of values for the negative and positive ultimate moments in each day of a continuous slab for the maximum design ultimate load 'n'. this is carried out on the assumption that all spans are loaded with the ultimate load 'n'. this is also the recommendation of clause 3.5.2.3 of BS 8110, but is subject to the restrictions that

pk/gk < 1.25,

pk ( excluding partitions) < 5 kN/m2 and

the bay areas exceed 30 m2.

With yield line theory these restrictions do not apply there are ways of investigating the effect that pattern loading has on the design ultimate moments chosen for any span. Tables 3.3 and 3.4 describe this procedure.

Please note that in table 3.1 opposite,

In case 1 and 2 the ratio of support moments to midspan moments have been fixed

In case 3 and 4 the magnitude of the support moments have been fixed

In case 5 and 6 the ratio of one support moment to midspan moment and the magnitude of the other support moment have been fixed.

Table 3.1 formulae for one way slabs to establish mid span yield line moment 'm' in a span of a continuous slab.

L

m

i2= m'2/m

i1= m'1/m

m ( kNm/m)

m'2 ( kNm/m)

m'1 ( kNm/m)

n ( kNm/m)

Case Diagram Formula

i2

i1

nL2

m1 m =

2(√1+i1 + √1+ i2)2

nL2

2(1 + √1+i2)2

m =

i2

i1 = 0

m2

nL2 - 4 m'1 + m'2 -

(m'1 - m'2)2

nL2

8

m'2

m'1

m =

m3

nL2 - 4 m'2 -

nL2

8

m =

( m'2)2

m

m'1= 0

m'2

4

nL2 - 4 m'2 -

( m'2)2

nL2

4 (1+0.5i1 + √1+i1)

m =

Approx

These formulae can err 3-10% on the high side, especially when there is a large difference between the end moments

m'2

i1

m5

nL2 - 4 m'1 -

( m'1)2

nL2

4 (1+0.5i2 + √1+i2)

m =

i2

m'1

m6

Where

m is the ultimate moment along the yield line [kNm/m]

n is the ultimate load per unit area [kN/m2]

L is the span, either centerline to centerline, or with integral supports, clear span [m]

i1- i2 are the ratios of support to midspan moments, whose values are chosen by the designer

m1- m2 are the support moments - the values of which are chosen by the designer.

The values could be established from analysis carried out on an adjacent bay [kNm/m]

3.1.3 Location of maximum midspan moments and points of contra flexure

Table 3.2 presents expressions for the location of maximum midspan moments and points of contra flexure. The parameters s1 and s2, the location of the points of contra flexure, are needed when checking the extent of top steel in accordance with BS 8110 clause 3.2.2.1 condition 3 which states : " resistance moment at any section should be at least 70% moment at that section obtained from an elastic maximum moments diagram covering all appropriate combinations of design ultimate load."

Table 3.2 location of maximum midspan moments and points of contra flexure

L

X1

X2

n ( kNm/m)

m'1 = i1 m

S1

S2

i1

i2

m'1 = i1 m

X1 =L

√1+i1

√1+i1 + √1+i2

X2 =L

√1+i2

√1+i1 + √1+i2

S2 =L

√1+i2 - 1

√1+i1 + √1+i2

S1 =L

√1+i1 - 1

√1+i1 + √1+i2

Where

X1' X2 are the distance to maximum span moment [m]

S1' S2 are distance to points of contra flexure, i.e. points of zero moment [m]

L span [m]

i1'i2 are ratios of support to midspan moments

m is the maximum midspan moment [kNm/m]

m'1 m'2 are the support moments [kNm/m]

3.6.1 Modes of failure

The collapse modes associated with flat slabs on a rectangular grid of columns are shown in figures 3.9, 3.10 and 3.11

Figure 3.9 flat slabs : folded plate collapse mode

A similar collapse mode, at right angles to the one shown should also be considered.

In figure 3.9 the fracture line pattern consists of parallel positive and negative moment lines with the negative yield line forming along the axis of rotation passing over a line of columns. This forms a folded plate type of collapse mode with maximum deflection taken as unity occurring along the positive yield line. A corresponding pattern could take place at right angles.

Figure 3.10 flat slabs : combined folded plate collapse mode

The assumed deflection at the column supports is 0, at midway between columns, the assume deflection is ½ and in the middle of the bay, 1. This mechanism is rarely investigated as there is no change in the collapse load compared to the mechanism shown in figure 3.9.

Figure 3.10 shows how these folded plate collapse mode could develop simultaneously both directions. However, as there is no change in the collapse load, this mechanism is rarely investigated. The axis of rotation for this combine mode passes over the columns. The maximum deflection (of unity) occurs at the center point in the bay and one half of the maximum deflection occurs at mid - point between columns along the negative yield lines. The fracture lines have been shown schematically on column center lines but in reality these will form along column faces (because the yield lines must be straight).

Figure 3.11 flat slabs : conical collapse modes (with isotropic reinforcement)

Figure 3.11 shows the other possible collapse mode consisting of inverted conical failure patterns occurring over each column. Around each column negative radial yield lines emanate from the center and a positive circumferential yield line forms at the bottom of the corn shaped surface. This collapse mode requires that the remaining slab, the peculiarly shaped central rigid portion of the slab, drops down vertically. This displacement is given the value of unity. The positive circumferential yield line is circular for isotropic reinforcement and elliptical for orthotropic reinforcement with the larger dimension parallel with the direction of the stronger reinforcement.

The formulae for local future patterns are shown in tables 3.10 and 3.11. with concentration of top reinforcement at supports, this mode of failure will generally not occur.

A separate check for punching shear is required.

Table 3.10 flat slabs : formulae for local failure pattern at internal column support (in slabs with isotropic reinforcement)

Where

m is the positive ultimate moment [kNm/m]

m' is the negative ultimate moment [kNm/m]

NB m' ≤3m otherwise formulae invalid

n is the ultimate uniformly distributed load [kNm/m2]

A is the area of column cross-section [m2]

S is the ultimate load transferred to column from the slab tributary area [kN]

Note : s may be equated to Vt, the design shear transferred to column as defined in BS 8110, clauses 3.7 notwithstanding BS 8110, CI 3.8.2.3, it is customary to allow for elastic reactions in calculating this load.

Table 3.11 flat slabs : formulae for local failure pattern at perimeter columns (in slabs with isotropic reinforcement)

Where

m is the positive ultimate moment [kNm/m]

m' is the negative ultimate moment [kNm/m]

NB m' ≤3m otherwise formulae invalid

---- is the angel described by edges of slab [rads]

NB 2π > ---- ≥ π/3 (or 360˚ > ----- ≥ 60˚ ) - otherwise formulae is invalid.

(Ï€ = 3.142 [rads])

n is the ultimate uniformly distributed load [kNm/m2]

A is the area of column cross-section [m2]

S is the ultimate load transferred to column from the slab tributary area [kN]

The extent of local failure patterns

For the patterns depicted in tables 3.10 and 3.11, the radius of the positive circumferential yield line (from the center of the column) may be calculated from :

r = c Ã- 3 √S/nÃ-A

Where

S, n and A are as above and

c is the radius of an equivalent circular column.

For a rectangular column of dimensions a and b the equivalent value of c = √aÃ-b /Ï€

Generally, except where columns are very large, r works out to be < 0.25L for internal columns and < 0.2L for perimeter columns.

The circular yield line is positive and requires bottom reinforcement. This reinforcement needs to be adequately anchored each side of the circular yield line - hence curtailment of bottom steel near supports is not advised. Top steel reinforces the top of the slab against radial negative yield lines within the area bounded by the circular yield line. To be effective this therefore needs only nominal anchorage, say 12 diameters, beyond the circular yield line. Limits of 0.25L for internal columns and 0.2L ( at right angles to the edge) for perimeter columns are advocated for curtailment of top reinforcement.

Designers may wish to check curtailment using tables 3.3 and 3.4 to ensure adequate anchorage in all situations. However, with the distribution of top steel advocated and usually employed, this local mode of failure is very rarely an issue that needs considering.

Curtailment of reinforcement

Similar to section 3.1.4, full ultimate loads are considered on each panel to determine the design moments in that panel. For alternate bay loading the same rule for extending top steel by 0.25L into adjoining panels applies. If the designer has any doubt about curtailment then tables 3.3 and 3.4 can be used if a one way failure is being investigated. If a two way failure mode is being considered then these tables no longer apply and a different approach might be required. However, applying tables 3.3 and 3.4 in such cases will err on the safe side.

4.1.1 Perimeter loads

A simple and conservative way of allowing for perimeter cladding loads is to incorporate this line load in to an equvalant uniformity distributed load over a chosen width of slab and consider the overall straight line failure patterns. Based upon elastic principles, BS 8110 clause 3.5.2.2 spreads the line load over a width of 0.3 Ã- span. However with yield line design, we should consider the global straight yield line failure mechanism for the combined load extending over the whole day. As a matter of engineering judgment and bearing serviceability requirements in mind, the designer can err on the safe side by dissipating the additional uniformly distributed load over a reduced length of yield line rather than over its whole length. A reasonable compromise would be to spread the total load over a length of yield lines equal to 0.6 of the span of the slab and assess the width of slab independently of the rest of the slab.

This method is conservative as it ignores the two way action that a local failure pattern, instigated by a line load, would induce. A comprehensive treatise of this type of local failure is given in chapter 1.4 of johansen's book.

4.1.6 Perimeter details

Local flexure modes are rarely critical for interior columns. However, reinforcement over internal columns is usually concentrated over a certain distance either side of the columns support to enhance the punching shear resistance.this concentration of reinforcement is also used in the layout of reinforcement at perimeter columns where printed supports are assumed in the design. The local failure modes used for establishing the minimum amount of this reinforcement, as given in table 3.11, is best carried out for m=m'. it is then imperative to ensure that there is at least this amount of reinforcement top and bottom each way for a distance of 0.2 * span from the center line of support of right angels to the edge of slab and 0.25 * span from the center line of support in each direction parallel with the edge of slab.

At corner columns too, it is best to proceed with the assumption that m = m and to provide 'U' bars at right angles to cater for this moment projecting as before for a distance of 0.2 *span from the center of the column at right angles to the edge of slab, corners with an angel of less than 90 o should be avoided as they are difficult to reinforce efficiently and design against punching shear failure is then likely to become the governing factor.

Around the perimeter between the concentrations of reinforcement at the columns it is recommended to provide a concentration of reinforcement equivalent to a minimum of 50% of end span bottom reinforcement in the form of 'U' bars with the top leg extending from the edge a distance of 0.2 * span from the centerline of support and the bottom leg having a tension lap with the bottom reinforcement.

Local to the column, the slab should also be capable of accommodating transfer moments, Mt, subject to Mtmax , derived from considering column design.

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