RC Plate Design

ADVANCED STRUCTURAL DESIGN

Design of a Reinforced Concrete Slab

Student: Ferrari Daniele → 18

Parameter – Third letter of the Family name (S): → 14

Parameter – Third letter of the given name (N):

Assignment

The home assignment regards the analysis of several point about the design of a reinforced concrete

slab.

Let’s assume the following features of the slab at issue:

1. One fixed edge (x=0);

2. Three simply supported edges; g

3. Uniformly distributed loading (given by a sum of two different parts: that is the permanent

q

one and that is the variable one); C25/30;

4. Concrete class is assumed equal to

B450C.

5. Reinforcing steel type is

Moreover, the slab is characterized by these geometrical parameters and design loads depending on

literal parameters shown in the cover of this assignment:

After the evaluation of every quantity defined above, design data con be evaluated as shown now:

= 6,50 ∗ 1 + = 7,55

Slab side: ,

ℎ= = = 0,25

Slab depth: " = 25 ∗ 0,25 = 6,25 $ /

#

Self-weight: " = 1,00 $ /

#

Weight of non structural component: " = " + " = 7,25 $ /

# &# #

Permanent load: ' = 8,00 ∗ 1 − = 7,36 $ /

#

Variable Load: * * *

+, -, +-,

In this problem, it has been assumed that bending moments , and torsional moment

. / = 0, 12.

, . It is also given a Poisson ratio equal to

are computed under condition of uniform load 1

The following picture explains the spatial geometry of the problem considered:

x

It can be easily seen that the plate is symmetric and the axis is coincident with the symmetry axis.

(ξ ; η)

For the computation part, it is better to refer to a new reference system that is dimensionless.

It is necessary to conduct an analysis of various tasks listed next in the text from n°1 to n°8.

The aim of this report is to design a squared plate under uniform load from different points of view,

using different design methods and making comparison between them when it is requested.

Here below the 3D geometrical model of the plate is sketched. 2

.

,

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

multiply for a certain coefficient correlated to the nature of the load itself. It also suggest the next

3:

values of the multiplier coefficients

3 = 1,3 3 = 1,5

4 5

for permanent load for variable load

The difference between the value of the coefficients is due to the more uncertain of a variable load

that for its nature can’t be predict with an exact approximation. The use of the coefficient is justified

because of an increase of safety in the design.

Since multiplier coefficients has been defined, the external load needed in the design are:

" = " + " = 7,25 $ /

# &# #

Permanent load: ' = 7,36 $ /

#

Variable Load: = 6 7 + 6 9 = :0, ;0 <=/> :

7 8 9 8

Design Load: * * * ? ?

+, -, +-, +, -,

I compute , , (Table 4,5,6) from dimensionless moment , ,

Slab behaviour:

? +-, (Table 1,2,3) that were given as initial data.

TABLE 1

2

µ =M /(p a ) ξ= x/a

xu xu u

η=y/b 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00

0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000

-0,50 -0,0323 -0,0106 0,0050 0,0069 0,0106 0,0126 0,0133 0,0129 0,0112 0,0076 0,0000

-0,40 -0,0558 -0,0208 -0,0003 0,0119 0,0190 0,0228 0,0241 0,0233 0,0199 0,0130 0,0000

-0,30 -0,0717 -0,0288 -0,0017 0,0151 0,0250 0,0302 0,0321 0,0308 0,0260 0,0165 0,0000

-0,20 -0,0809 -0,0338 -0,0028 0,0168 0,0285 0,0348 0,0369 0,0353 0,0296 0,0185 0,0000

-0,10 -0,0839 -0,0354 -0,0032 0,0173 0,0297 0,0363 0,0385 0,0368 0,0308 0,0192 0,0000

0,00 -0,0809 -0,0338 -0,0028 0,0168 0,0285 0,0348 0,0369 0,0353 0,0296 0,0185 0,0000

0,10 -0,0717 -0,0288 -0,0017 0,0151 0,0250 0,0302 0,0321 0,0308 0,0260 0,0165 0,0000

0,20 -0,0558 -0,0208 -0,0003 0,0119 0,0190 0,0228 0,0241 0,0233 0,0199 0,0130 0,0000

0,30 -0,0323 -0,0106 0,0050 0,0069 0,0106 0,0126 0,0133 0,0129 0,0112 0,0076 0,0000

0,40 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000

0,50 TABLE 4

2

M =µ (p a ) ξ= x/a

xu xu u

η=y/b 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00

0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

-0,50 -37,74 -12,39 0,58 8,06 12,39 14,72 15,54 15,07 13,09 8,88 0,00

-0,40 -65,21 -24,31 -0,35 13,91 22,20 26,64 28,16 27,23 23,25 15,19 0,00

-0,30 -83,79 -33,65 -1,99 17,65 29,21 35,29 37,51 35,99 30,38 19,28 0,00

-0,20 -94,54 -39,50 -3,27 19,63 33,30 40,67 43,12 41,25 34,59 21,62 0,00

-0,10 -98,04 -41,37 -3,74 20,22 34,71 42,42 44,99 43,00 35,99 22,44 0,00

0,00 -94,54 -39,50 -3,27 19,63 33,30 40,67 43,12 41,25 34,59 21,62 0,00

0,10 -83,79 -33,65 -1,99 17,65 29,21 35,29 37,51 35,99 30,38 19,28 0,00

0,20 -65,21 -24,31 -0,35 13,91 22,20 26,64 28,16 27,23 23,25 15,19 0,00

0,30 -37,74 -12,39 0,58 8,06 12,39 14,72 15,54 15,07 13,09 8,88 0,00

0,40 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

0,50 3

TABLE 2

2

µ =M /(p a ) ξ= x/a

yu yu u

η=y/b 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00

0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000

-0,50 -0,0058 0,0006 0,0064 0,0107 0,0137 0,0153 0,0155 0,0143 0,0116 0,0069 0,0000

-0,40 -0,0100 -0,0010 0,0080 0,0155 0,0207 0,0237 0,0242 0,0221 0,0174 0,0100 0,0000

-0,30 -0,0129 -0,0025 0,0081 0,0172 0,0240 0,0278 0,0285 0,0260 0,0202 0,0114 0,0000

-0,20 -0,0146 -0,0036 0,0077 0,0177 0,0252 0,0296 0,0305 0,0277 0,0215 0,0120 0,0000

-0,10 -0,0151 -0,0039 0,0076 0,0178 0,0255 0,0301 0,0310 0,0282 0,0218 0,0122 0,0000

0,00 -0,0146 -0,0036 0,0077 0,0177 0,0252 0,0296 0,0305 0,0277 0,0215 0,0120 0,0000

0,10 -0,0129 -0,0025 0,0081 0,0172 0,0240 0,0278 0,0285 0,0260 0,0202 0,0114 0,0000

0,20 -0,0100 -0,0010 0,0080 0,0155 0,0207 0,0237 0,0242 0,0221 0,0174 0,0100 0,0000

0,30 -0,0058 0,0006 0,0064 0,0107 0,0137 0,0153 0,0155 0,0143 0,0116 0,0069 0,0000

0,40 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000

0,50 TABLE 5

2

M =µ (p a ) ξ= x/a

yu yu u

η=y/b 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00

0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

-0,50 -6,78 0,70 7,48 12,50 16,01 17,88 18,11 16,71 13,56 8,06 0,00

-0,40 -11,69 -1,17 9,35 18,11 24,19 27,69 28,28 25,82 20,33 11,69 0,00

-0,30 -15,07 -2,92 9,47 20,10 28,05 32,49 33,30 30,38 23,60 13,32 0,00

-0,20 -17,06 -4,21 9,00 20,68 29,45 34,59 35,64 32,37 25,12 14,02 0,00

-0,10 -17,65 -4,56 8,88 20,80 29,80 35,17 36,23 32,95 25,47 14,26 0,00

0,00 -17,06 -4,21 9,00 20,68 29,45 34,59 35,64 32,37 25,12 14,02 0,00

0,10 -15,07 -2,92 9,47 20,10 28,05 32,49 33,30 30,38 23,60 13,32 0,00

0,20 -11,69 -1,17 9,35 18,11 24,19 27,69 28,28 25,82 20,33 11,69 0,00

0,30 -6,78 0,70 7,48 12,50 16,01 17,88 18,11 16,71 13,56 8,06 0,00

0,40 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

0,50 TABLE 3

2

µ =M /(p a ) ξ= x/a

xyu xyu u

η=y/b 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00

0,0000 0,0173 0,0215 0,0193 0,0135 0,0056 0,0032 -0,0123 -0,0208 -0,0278 -0,0313

-0,50 0,0000 0,0155 0,0199 0,0181 0,0127 0,0053 -0,0030 -0,0115 -0,0194 -0,0256 -0,0281

-0,40 0,0000 0,0118 0,0160 0,0149 0,0106 0,0045 -0,0024 -0,0094 -0,0157 -0,0204 -0,0221

-0,30 0,0000 0,0078 0,0109 0,0104 0,0075 0,0032 -0,0017 -0,0066 -0,0109 -0,0139 -0,0150

-0,20 0,0000 0,0038 0,0055 0,0053 0,0038 0,0017 -0,0008 -0,0033 -0,0055 -0,0070 -0,0076

-0,10 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000

0,00 0,0000 0,0038 0,0055 0,0053 0,0038 0,0017 -0,0008 -0,0033 -0,0055 -0,0070 -0,0076

0,10 0,0000 0,0078 0,0109 0,0104 0,0075 0,0032 -0,0017 -0,0066 -0,0109 -0,0139 -0,0150

0,20 0,0000 0,0118 0,0160 0,0149 0,0106 0,0045 -0,0024 -0,0094 -0,0157 -0,0204 -0,0221

0,30 0,0000 0,0155 0,0199 0,0181 0,0127 0,0053 -0,0030 -0,0115 -0,0194 -0,0256 -0,0281

0,40 0,0000 0,0173 0,0215 0,0193 0,0135 0,0056 -0,0032 -0,0123 -0,0208 -0,0278 -0,0313

0,50 TABLE 6

2

M =µ (p a ) ξ= x/a

xyu xyu u

η=y/b 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00

0,00 -20,22 -25,12 -22,55 -15,78 -6,54 3,74 14,37 24,31 32,49 36,58

-0,50 0,00 -18,11 -23,25 -21,15 -14,84 -6,19 3,51 13,44 22,67 29,91 32,84

-0,40 0,00 -13,79 -18,70 -17,41 -12,39 -5,26 2,80 10,98 18,35 23,84 25,82

-0,30 0,00 -9,11 -12,74 -12,15 -8,76 -3,74 1,99 7,71 12,74 16,24 17,53

-0,20 0,00 -4,44 -6,43 -6,19 -4,44 -1,99 0,93 3,86 6,43 8,18 8,88

-0,10 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

0,00 0,00 4,44 6,43 6,19 4,44 1,99 -0,93 -3,86 -6,43 -8,18 -8,88

0,10 0,00 9,11 12,74 12,15 8,76 3,74 -1,99 -7,71 -12,74 -16,24 -17,53

0,20 0,00 13,79 18,70 17,41 12,39 5,26 -2,80 -10,98 -18,35 -23,84 -25,82

0,30 0,00 18,11 23,25 21,15 14,84 6,19 -3,51 -13,44 -22,67 -29,91 -32,84

0,40 0,00 20,22 25,12 22,55 15,78 6,54 -3,74 -14,37 -24,31 -32,49 -36,58

0,50 x

In green are marked the relevant values of max and min bending and torsional moments along and

y. And now the results are plotted in 3D to show the specific diagram of each quantities. 4

Graph 1

Graph 2

* *

+, -,

As expected, and are null where the plate is simply supported, negative in the clamped

section, positive almost in all span except for the ideal strip next to the fixed edge.

Graph 3 5

* * A

+-, +-, @ +

As expected, is null along the clamped edge, varies from 0 to along edges parallel to

* *

A

+-, @ + +-, @ +

x y

axis, is antisymmetric on simply supported edge parallel to varying from to .

* * * <=>.

+, -, +-,

The values of , , now evaluated for every point of the plate are given in

two different static schemes with the same span (due to squared geometry) and

Beam behaviour: x

different constraints at the edges can be extrapolated. One scheme develops along direction and the

y

other along direction. Internal actions (only bending moment is needed) are computed considering

only a strip of a unit width taken parallel to the plate’s edges. In both directions, the load is considered

acting along all the span, so the load per unit length is equal to:

<= <=

= :0, ;0 ∗ 1, 00 > = :0, ;0

> >

:

Since both the static schemes are well-known, the phase of calculation of internal action is omitted

and given directly from technical textbook that shows the solution of a literal problem in general.

x direction: redundant scheme with clamped and simply supported edges 6

It is necessary to compute the relevant bending moments which are the maximum negative one and

the maximum positive one using the formulas in the table above (no torsional moment is computed).

I obtain: 5 1 1

* = . B − . − . B [$ ]

+ , , ,

8 8 2

1 1

* (B = 0,625 ) = . = &lo