TWO-WAY EGE SUPPORTED SLABS

Foreword to slab support: In the observed set of cases we have 2 basic situations.1. The slab has one dominant load bearing (LB) direction so, it is - one way slab2. The load action is dispersed into 2 basic areal directions x,y. This case is called

two-way slab and is the main topic of this lecture.About the case 1 or 2 decides: The aspect ratio lx /ly, and condition of support: fixedhinge, non.(and designer‘s will. (In some limits.) This lecture deals step by step thesementioned conditions and their impact to load distribution. There are someconditions which helps us decide about structural behaviour of solved case. Thecalculation itself is a mix of simplified assumptions and indisputable facts. The mostimportant presumption and the fact is about deflection equality – se later. This is anexample of a fact which you have to memorise together with the knowledge how touse it in design of two-way labs. Therefore are these parts of theory papers marked asE-Essential. The rests is EP -Essential in principles. E.g. you have to understand theprinciples and be able to formulate them with your own words, or with the originalones. Eventual formulas is not necessary to memorise.Due to the limits of summer term 2019/20 , especially Covid19 pandemic , isdiscussion of more complex two-way slabs omitted.

2020 edition

Slabs are horizontally oriented 2D (plane x, y) members. (Notice: Are to walls the same with vertical orientation).According to the technical theory of thin (elastic) slabs (Kirchhoff) we identify following internal forces:

factorized

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We can differ: a) slabs with linear (continuous, edge) supportsb) slabs with point (small area supports)

ad a) support by walls or lintels (beam)TASK OF BL 005!

ad b) support by columns or

columns with drop panels

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About one/two way case will decide arrangement of edge support subfig 1 and 2

IN GRAY=NOT TASK OF BL 005!

Clean one-way One-way special

If there are 2 opposite supports (case a)) or one support only (case b)), we call the support one-way slabs. It was case of BL001 subject and in this subject was touched by footing strip SUW. WE can meet clear one way cases – see next two figures

Console – cantilever - corbel

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Even if 3 or 4 sides of a slab are supported, the case can be solved as one-way slab, if the ratio between its sides is

Commonly: by slabs the areal load p(x,y)divides into:

p(x,y) = px + pxy + py

approximately: (1 : 1,7) or greater – back slide.

By one-way slab remains only one LB direction:p(x,y) = px + pxy + py

The loads pxy and py are presumed to be= 0Notice: we can presume the loads pxy and py

are zero, but not the stresses sy. Why? (Help –see back-slide (once more).

The reason is relation between transversal and axial stress of a material = Poisson's ratio. For concrete is m=0,2 and therefore is for transversal direction min of 20% As used.

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Rev1

Snímek 5

Rev1 revisor; 09.04.2018

Slab supported by its 3 or 4 margins with side/side ratio 2:1 must be designed as so called

2(two)-way slab!In such case there are two sub-case here:

I. p(x,y) = px + pxy + py

orII. p(x,y) = px + pxy + py

Figures – Subcase I.

Subcase I.Slabs with yielding/deflecting (not stiff) support There are no torsional moments in the slab (mxy=0) and bending moments are directly equal principal moments (for design)

m1 = mx and m2 = my

The same effect is reached if the support is stiff and edges of the slab can lift–up.

Lifting corners

Deflecting supports in direction „x“

Deflecting (not stiff edge support)

In our role of an edge support, it can be

Walls – In a case of support walls

is usually presumed to be stiff enough. (is stiff in its verticalplane )

Beams = girders - lintelsIn a case of lintel support is usually compared stiffness of supporting beam (T-beam) with stiffness of supported slab – see figure

Older rule:

h≥2hs

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When the edges can lift-up

Edges of a slab can lift-up if there are not sufficient load above and the slab is not effectively anchored into support wall. It is relatively seldom a case of slabs under light roof, free RC covers etc.

Lifting of edges usually not welcome –structural failure, so the edges have to be hold in down position.

Gd ≥ 1/16 Lx. Ly . p

Where p is the total areal load (permanent + variable)The anchoring should be provided by the distance of 0,2 Lx and 0,2 Ly from the edge.The idenification of Ns should be based on the same principle.

When the edges cannot lift-up and the supports are stiff

This is relatively frequent case, the calculation of internal forces are based on the same principles the differences are mostly solved by detailing of the reinforcement.

Internal forces

For design of a reinforcement we need to find proper internal forces. They are:mx and my are the factorsed bending moments (for 1m of breath) in direction x or ymxy is the specific torsional moment for 1 m elementvx and vy are the specific shear forces for directions x or y

The principle of finding internal forces is (in all cases mentioned before) the same, and is based on division of the load into directionsx and y. For mx and my is the technique of calculation based on the fact, that the deflection calculated for the same point of a slab is equal for both directions.

for bendind moments

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E.G. For simply supported slab (around the whole perimeter) hold true:

5

384.�� . ��

�

�. � =

5

384.�� . ��

�

�. �

More overpx + py = p substitution py = p - px

E.I = constant for one slabAfter simple transformation we obtain equation with one unknown px

�� . ��� = (� − �� ). ��

�

Which we can solve and calculate the px and the py then

�� = �

� . �� . ��

� �� = �

� . �� . ��

�

For practical reasons (using of charts) the quantity Cx is defined, only. Other marking is sometime used (e.g. ax). Usage: px = p . Cx and py = p . (1- Cx ). For different support different Cx should be used!:

single line = simple support , doubled line = fixed support (or continuity)

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Another set of helping figures – deflections (marked wm) and extreme values of bending moments for different position and support conditions

Pay attention to the signs in figures!

With help of this and previous slide figures we can easy calculate load px and py

and equal values of factorised banding moment mx and my. (By single slabs).

• The values of m are used as calculated for cases with deflecting supports or lift-up edges,

• The positive (in spans) values of m may be reduced with coeff. k for cases with stiff supports. (Approx. calculation acc. to Marcus.) next slide

According to Marcus:

� = 5

6 .

��� . ��

�

��� + ��

�

For the field on both sides simple supported: mr = m . (1 - k)

For the field with one side simple supported and one fixed: mr = m . (1 – 2/3.k)

For the field with both sides fixed: mr = m . (1 – 1/3.k)

mr = reduced moment

Cracks in the bottom face by overload

EBy finding of loading areas for vertical supports give us very good idea the theory of yielding lines = strain (yielding) of rfcmt and cracks of similar real slab.

Shapes of flats from which is the load transferred to marginal edge supports.. Arrows show flow direction from loaded area into supports. How to make proper envelop is explained on the next slide.

Yielding lines letter-envelope technique of finding the load of verticall support

Rules of letter-envelope technique = finding of the Yielding lines by variable support conditions.

The principal can be expressed much more simple with words, only:• The angle of diagonal is 45˚ if the support condition of both crossing edges in the corner are

equal .• If the kind of condition of crossing supports is not equal, then the diagonal builds the angle

of 60˚ with fixed support. By free end no load flows into this not supported edge.

Detailing of 2 -way slabs reinforcement (RF)

Detailing is relatively complicated due to technical principles and due to some Eurocode rules. (e.g. Top RF by free support for ¼ of Med span) The second reason is a tendency of some lectures/books to apply all possible savings (of RF amount) which perhaps finally are more complications, not savings. Let us summarize basic principles of 2-way slab RF detailing:1. RF has to be positioned both at the bottom and by top surface.2. To each LB bar have to be present transversal bar (distributive or

load bearing), there.3. The only part/area theoretically free of RF is the top middle section

of 2-way slab, but it is recom. to place welded wire meshes for shrinkage cracks protection, here. By small slabs can be omitted.

4. All known common detailing rules relevant for slabs are valid here (cover, max and min distance of bars e.g.).

5. Some new rules applicable by free edges or by openings of slabs at the end of the lecture are presented.

possible mesh

Notice: On the figures above you can see typical “book detailing’’ which is created under the slogan “save as much reinforcement as you can, regardless of the too complex result”.

On the other hand lecturer prefers slogan: “Make it so simple as possible to prevent human mistakes.”(With full respect to code’s detailing.) Possible correction of fig1 is to use asx as constant with ¢ inconstant distance on the full bottom surface.

1.

Detailing: bottom fixed support

Alternative – both directions As,x≥As,y

Alternative – both directions As,x≥As,y

Top Reinforcement stiff supports

EUROCODE rules implementation: Top reinforcement is not reduced anywhere. Slab has to be reinforced by top

surface in fixed or partial fixed supports (length of reinforcement is depicted above) - green and blue colour.

In the corners where simple supports meet to each other must be reinforcement transferring torsion moments. The minimum area of this reinforcement (in each way) is a maximum from values As,x and As,y in mid-span - pink colour.

In corners where simple support meets fixed support is necessary one half of max (As,x; As,y) placed parallel with fixed support- pink colour.

In corners where fixed support meets fixed support is not needed reinforcement for torsion moments (this case is not in scheme above)

All reinforcement must be accompanied by distribution reinforcement.

No surfaces can be without any reinforcement due to volume changes of concrete (shrinkage). Thereby the surfaces without reinforcement should be accompanied by welded wire mesh.

Special reinforcement detailing applicable by one/two way slabs

Possible alternatives

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