Assignment. Design of a RC slab, Advanced Structural Design

ADVANCED STRUCTURAL DESIGN

ACADEMIC YEAR 2021/22

Prof. FABIO BIONDINI

Assis. Luca Capacci, Mattia Anghileri

Structural Design of a Reinforced Concrete Slab

Students:

POLITECNICO

MILAN01863

Academic Year 2021/2022

STRUCTURAL DESIGN

ADVANCED

Prof. Fabio Biondini

Strnctural Design of a Reinforced Concrete Slab

The Reinforced Concrete (RC) slab shown in Figure 1.a is to be designed at the ultimate limit

state under uniformly distributed loading using reinforcing steel bars B450C and appropriate

concrete class on

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Figure 1. RC slab: (a) geometry and boundary conditions; (b) reference grid.

ADVANCED STRUCTURAL DESIGN a.y. 2021 2022

TABLE OF CONTENT

Introduction ________________________________________________________________________1

a) Given data_________________________________________________________________2

b) Geometrical parameters and loads ______________________________________________2

Question 1 (Comparison between sections parallel to the edges and beam models _________________3

a) Comments and conclusion of the question 1______________________________________13

Question 2 (Design of steel reinforcement according to the Wood-Armer method)________________13

a) Wood-Armer______________________________________________________________13

b) Corner reactions ___________________________________________________________24

c) Comments and conclusion of the question 2______________________________________26

Question 3 (Principal bending moments)_________________________________________________27

a) Comments and conclusion of the question 3______________________________________34

Question 4 (Yield line theory)_________________________________________________________ 35

a) Global collapse ____________________________________________________________35

b) Local collapse _____________________________________________________________42

c) Comments and conclusion of the question 4______________________________________53

Question 5 (Strip method) ____________________________________________________________53

a) Strip method without strong bands _____________________________________________53

b) Strip method with strong bands________________________________________________58

c) Comments and conclusion of the question 5______________________________________66

Conclusion ________________________________________________________________________68

ANNEX A ________________________________________________________________________69

ANNEX B________________________________________________________________________ 70

ANNEX C________________________________________________________________________ 71

ANNEX D ________________________________________________________________________72

DRAWINGS

a) Technical drawings for the Wood-Armer method

b) Drawings for the Strip Method without strong bands

c) Drawings for the Strip Method with strong bands

INTRODUCTION / ABSTRACT

This project regards the structural design of a Reinforced Concrete Slab with a specific

geometry. Some assumptions are made on the plate and on the concrete class. It is

supposed to be submitted to an uniform distributed loading. The concrete belongs to the

class C25/30 and the reinforcing steel is B450C. After a brief comparison between the slab

behaviour and a equivalent beam model, a steel reinforcement will be designed according to

Wood-Armer method. Then we will suggest alternative layouts at four specific points. Using

the yield line theory, we will check the slab safety. Finally we will discuss about possible

improvements on our design, especially discuss the benefits of adding strong bands. 1

Given data

Steel bars: B450C

Our choice - Concrete: C25/30 (Table E.1N - EC2)

Geometrical parameters and loads:

Slab side:

a = 6.00m + Δa = 6.00m + 1.80m = 7.80m

Slab depth: h ≤ a/25 so in our case:

h ≤ a /25 = 7.80/25 ≤ 0.312m . We ch oose h = 0.32m

Permanent load:

Self-weight (γ = 25 kN/m3):

Weight of non-structural components:

The live load qk depends on the category of use as recommended by Eurocode. 2

That’s why in our case:

The bending moments Mxu, Myu, and torsional moment Mxyu, computed under uniform load

pu by assuming a Poisson’s ratio ν = 0.18, are listed in dimensionless form in Tables 1, 2,

and 3,respectively, for the grid of points shown in Figure 1.b (grid spacing Δx=Δy= a/

10=0.78m).

Question 1

The load is uniform along the plate’s surface. It means that in every direction considered it

acts with the same value per unit of squared meter. The Eurocode2 states that the external

load must be multiplied for a certain coefficient correlated to the nature of the load itself. It

also suggest the next values of the multiplier coefficients :

For permanent load:

For variable load:

Permanent load:

Design load:

We can compute Mxu,Myu,Mxyu based on initial data: 3

4

In this chapter, the bending moments along the sections parallel to the edges are compared

to beam models for the same "strips". (Sections are chosen with the highest moments in X

and Y direction). 5

In all directions, the load is considered acting along the entire span, so the load per unit

length is equal to:

In a beam model, for sections 2-2’, 4-4’ and 5-5’, the system would be unstable if we

consider the free edges unsupported. For each of these strips, we have to modify the model

and to make another modelization. Further details can be found below in the dedicated

sections.

Section 1-1: 6

Interpretation :

We can observe on the plot that the bending moments for the section 1-1’ are larger in the

beam model. For the boundary ξ=0, the bending moments are logically null given that we are

on the simply supported edge. We can notice the further from the boundary ξ=0 we are and

the higher the values of the bending moments are and as a consequence the closer they are

from the beam model.

The distribution of bending moments are quite similar between the beam model and the

different strips except for ξ=0.6 where the distribution is a little bit different. We can interpret

that as a consequence of the proximity of this strip to free edges (in ξ=0,65).

Section 2-2: 7

Interpretation:

Both ends of strip 2-2’ are free and a loaded beam with two free edges is obviously unstable.

In order to elaborate model, we consider an edge as a simple support and the other end is

set as a roller. This model is coherent considering the fact that at both extremities the

bending moment is null and besides we can see it on the plot. According to the plot, the

further we are from the supported edge in ξ=1 and the higher the bending moments are. 8

Section 3-3:

Interpretation:

The model for the strip 3-3’ is a simply supported / roller support. This is consistent with the

null bending moments at the ends. We can observe on the plot that the beam’s moments are

much higher the the slab ones. 9

2

p × l

u

+ +

M = 54,037 k Nm M = = 149,438 k Nm

ma x slab ma x beam 8

+

M ma x slab ≃ 0,36

M +

ma x beam [ ]

ξ ∈ 0 ; 0,6

Plus, we can easily notice on the graph that for approximately, the bending

moments tend to be “attenuated”. We can explain this by the fact that the plate is here larger

[ ]

ξ ∈ 0,65 ; 1

than for and furthermore clamped on one edge and then logically stiffer. Our

beam model does not take this phenomenon into account which explains the 64% reduction

of the maximum value of bending moment.

Section 4-4: 10

Interpretation:

Sections 4-4’ and 5-5’ have the same ends so the beam model is the same : simple support /

roller support. However the results are very different for these two sections. We can easily

notice on the plot for section 4-4’ that the results are far from the theoretical results of the

beam theory. The case η=0 has to be excluded because it corresponds to the clamped

boundary. For η=0,1 (still close to the clamped section), we have negative bending

moments. The influence of the clamped section is strong. For η=0,2, the distribution of

bending moments seems more “acceptable” but values stay lower compared to the beam

theory. The slab tends to be stiffer than a equivalent model of a beam.

Section 5-5: 11

Interpretation:

The distribution of bending moments for section 5-5’ is closer to the expected results of the

beam theory (compared to section 4-4’). On this edge of the plate, we don’t have the

influence of a clamped section but from a simple support. The bending moments are by the

way null on this section (η=1). The results for η=0,8 are closer to the beam theory than for

η=0,9. It seems that, similarly, the further we are for the boundary and the more we stick to

the beam model. However the influence of a simple support is weaker than the influence of

clamped section. 12

Comments and conclusions:

In this part we tried to compare the ability to bear a loading bi-directionally with the slab or

mono-directionally with the beam model. However, when elaborating our beam model we

fully neglect the presence of a perpendicular strip element carrying the load. The bi-

directionnal behaviour of the slab clearly improves its capacity to bear the loading since the

generated bending moments of the slab are of smaller absolute values compared to the

ones of equivalent beams. Plus, Poisson’s effect and the consideration of torsional stiffness

in the slab theory explain the fact that beams tend to be more flexible and then why slabs

have a better load bearing capacity.

Question 2

Once we obtain the value of Mxu,Myu,Mxyu for every point of the plate, the next step

is to evaluate the quantity of reinforcement necessary to withstand them. The design is

conducted with the Wood-Armer Method in this phase: it is based on the normal bending

moment inequalities and Johansen’s yield criterion. The criterion states: “It is assumed that

reinforcing bars in both directions crossing a yield line reach the yield strength. The ultimate

moment of resistance about a yield line, which is at some general angle to the

reinforcement, is assumed to be due to the components of the ultimate resisting moments

mxu,myu, in the direction of the reinforcement”.

Reinforcement has to be designed in the most effective arrangement both for the

bottom and the top of the slab and can be computed in every point of the grid = by summing

up the quantities written in tables above as now shown: 13

BOTTOM REINFORCEMENT: 14

TOP REINFORCEMENT: 15

Geometrical features of the RC slab:

Steel: B450C:

Concrete: C25⁄30:

Geometry: 16

Therefore, the minimum amount of steel:

And the maximum amount of steel by EC2: 17

18

Of course, we should remember such things as anchorage length.

Meaning of anchorage length is the length required for development of stress in the rebars,

this is obtained by providing the required development length or hook/bends if sufficient

length cannot be achieved.

Using EC2 we can summarize all the equations for computing anchorage length:

By computing, considering all these factors↑ we get (in our assignment we consider bars in

tension, because we have no any situations with anchorage length in compression): 19

We should also remember about spacing of bars:

General situation:

Concentrated loads or maximum B.M.:

So, in our case:

General situation:

Concentrated loads or maximum B.M.

So, at this step we have all quantities, to make preliminary drawings. And after we do that,

we should also do a “checkout phase”, where we will compute MRD and compare it with

actual bending moments. The best way to do it is to draw cover diagrams which will clearly

show us the “situation” in slab.

It should also be mentioned that in order not to make lots of sections in one direction, we will

make only a few, but we will consider “the worst” situation in slab(We will take into account

20

the biggest Wood-Armer moments). While doing covering diagrams, we will consider a “shift”

rule → to include the effect of possible longitudinal cracks.

Checkout phase

To check the adequacy of provided reinforcement, the ultimate resisting moment is

calculated. 21

For computation we will use an Excel (Annex A) just to simplify our calculations. For better

visualization we will make “Covering diagrams for the Wood-Armer method”, please refer to

annex B to see covering diagrams in detail, which represent our MRD against actual

bending moments. 22

23

As we could see, all our resisting moments cover the acting bending moments, and we could

say now, that ULS check is done successfully.

We must also determine corner reactions and design steel reinforcement to prevent corner

uplifts.

Corners:

The physical model of effective shear force (that is given by a sum of integrals of shear

stress due to pure shear and torsion), indicates the existence of concentrated reactions at

the corners of the plates. These reactions R are directly correlated to torsional moments

since they appear only to balance it. The presence of R is essential to ensure the statical

equivalence of the assumed effective shear force distribution with the torsional moment

distribution, which has zero force resultant.

In the meantime, the external load Pu, (transversal to the midsurface) is entirely equilibrated

by the pure shear forces acting along the boundary that, in this case, are developing along

the slab's edges. So, the calculation of R is simply the sum of the torsional moment in the

corners of the slab; it is assumed that positive reactions point upward. The pictures below

show the convention used about torsional moment; moreover, it is explained how to

individuate the verse of the corner reaction R. 24

At a fixed edge there is no twisting moment because no rotation along the border is

possible; conversely, torque appears with a definite value when edges are simply supported

and exactly in the corner twisting moment gets its highest value.

As we see, we should consider only the bottom left corner for calculation.

Once assessed |R|, once estimated , it is easy to evaluate the area of reinforcement needed

to avoid uplifting as:

So, for the corner we can use 3∅10 with total reinforcement area: 235.6 mm2

The anchorage length in this case(for straight end): 403 mm

For bent end: 197 mm 25

SUMMING UP WOOD-ARMER METHOD:

So, we have done all the steps for the Wood-Armer Method. This method was developed by

considering the normal moment yield criterion (Johansen’s yield criterion) aiming to prevent

yielding in all directions. It is a local design method. In any orientation, we designed an

amount of steel reinforcement, which is sufficient to cover acting bending moments→

Resistance overcomes the demand. It is a good design method in the sense that torsional

effects are predominantly taken into account.

We also determined corner reactions and design steel reinforcement to prevent corners