Experimental and Numerical Analysis on RC Structures Using Small Scale Models

8/12/2019 Experimental and Numerical Analysis on RC Structures Using Small Scale Models

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INSA de RENNES

Dpartement Gnie Civil et Urbain

Experimental and numerical analysis on

reinforced concrete structures using small

scale models

Adrian-Corneliu MIHALI

2014


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Table of contents

1. Introduction 1

2. Structural modeling 4

2.1. Generalities 4

2.2. Dimensional analysis 5

2.3. Classification of the structural models 7

3. The materials 8

4. Reinforced concrete beam 13

4.1. Ultimate bending moment 14

4.2. Cracking bending moment 19

4.3. Maximum mid-span deflection 21

4.4. Model beam three point bending. Experiment 26

4.5. Model beam three point bending. Finite Element simulation 29

5. Two dimensional frame 33

5.1. Vibration modes 33

5.2. Ductility test of the model 39

5.3. Earthquake response 41

6. Conclusions and future research 51

Appendix A 53

Appendix B 55

Appendix C 60

Appendix D 62

References


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1. Introduction

The current paper presents the results of 1:10 reinforced concrete scaled model tests.

The purpose was to find methods through which the behavior of full-sized structures (named

prototype) could be predicted using scaled structures (named models). The scaling of the

prototypes dimensions, materials characteristics, boundary conditions, loadings, and result

interpretation will be made based on the similitude theory.

The main advantages of small-scale experimenting are the ease of manipulating the

structure, fabrication costs and the small size of the equipment. The main disadvantages are

fabrication time and the challenges in simulating the complete behavior of the prototype. In

mots of the cases, some carefully chosen distortions are necessary that will lead to minimum

differences between the expected and the obtained result.

In the current study, the 1:10 scale has been used for experimenting on reinforced

concrete structures. The tests were made on a simple supported beam 3x6x50cm with the

support opening of 45cm and a concrete cover of 3mm. The similitude between the cracking

bending moment, the capable bending moment and the maximum deflection at breaking were

studied.

For the fabrication of the model the materials used are: a micro-concrete with a recipe

determined using the instructions from Manea ET. Al. [7] with 2mm aggregates due to the

concrete cover of the reinforcement. Following the results obtained by Kim ET. Al. [5], as

reinforcement threaded rod and steel wire for the stirrups were used. In figure 1.1 arepresented the reinforcement cages. Along with the beams, 10mm cubes and 4x4x20mm and

10x10x30cm prisms were cast in order to determine the characteristics of the concrete

(compression limit, tension limit and elasticity modulus). The model was subjected to a

physical test and the results compared to a finite element analysis while prototype was

analyzed only theoretically.

The second structure that has been studied is a two-dimensional frame with a single

span and two levels. The opening between the columns is 45cm and the levels are 60cm each.

The beams are 3x6cm and the columns are 5x5cm. Another micro-concrete with 2mm

aggregates whose recipe is presented in Appendix C was used. For the reinforcement

threaded rod and steel wire was used. Along with the three frames, 10mm cubes,

10x10x30cm prisms and hourglass shaped probes were cast to determine the concretes

characteristics. Both the frames and the probes were kept underwater for 28 days after

hardening.


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Figure 1.1 Reinforcement cages (left) and casted beams (right)

Figure 1.2 Reinforcement cages and framework (left) hardened frame (right)

The frames were subjected to dynamic tests: vibration modes, pushover analysis and a

time-history analysis in order to determine how the model simulates the behavior of the

prototype. The model and the prototype were both analyzed using the finite element method

in all three cases and an experimental modal analysis was performed on the model.

Structural similitude is known from the gothic masons period who used what isknown as proportionality laws through which a valid structural system remained so

independent of the size as long as the dimensions of the structure were proportional, as

observed by Huerta [8]. With the development of the PI Theorem by Buckingham [6],

structural modeling was made using a mathematical model (any physical system can be

expressed as a series of dimensionless products, called PI terms). Moncarz ET. AL. [1],


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Harris ET. AL. [2] and Dehousse ET. AL. [9] offer extensive information regarding the

challenges of structural modeling of reinforced concrete structures subjected to both static

and dynamic loading. Lu ET. AL. [3] report good results in modeling the vibration modes of

a structure using a 1:25 model and Wu ET. AL. [4] managed to predict the dynamic behavior

of a crane structure using a 1:10 model.

The present paper begins with an introduction in the similitude theory in chapter 2,

followed by the characteristics of the materials used in the experiments in chapter 3. The

experiments on the beams and frames of figures 1.1 and 1.2 are presented in chapters 4 and 5,

along with the results. In chapter 6 the conclusions and the future studies can be found.


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2. Structural modeling

2.1. Generalities

The structural model is any physical representation of a structure or a part of that

structure. In most of the cases, the model is built at a reduced scale. For the scaling of thestructures dimensions, characteristics, boundary conditions, loadings and result

interpretation, the laws of similitude must be used.

The model is the small-scaled structure for which the similitude laws are applied,

while the prototype is the full-scale structure. The advantages of physical modeling are:

- Experimental analysis of large structures- Numerous problems in structural engineering for which approximate solutions exist

are analytically resolved

- Useful in education, research and design in order to better understand the phenomenaand to validate analytical models

The limitations are the time and the impossibility of a complete simulation of the

prototypes behavior.

Structural models classify in many ways but the most relevant one is the models

designation. The following categories exist: elastic models, indirect models, direct models,

strength models, wind models, dynamic models, etc.

The choice of the models scale depends on the type of the experiment. A very small

model requires small loadings but fabrication and instrumentation difficulties appear, while a

large scale model is easier to build but requires larger testing facilities. Each model has an

optimal scale for testing, as seen in table 2.1.

Table 2.1 Scales for modeling different structures

Structure Elastic model Strength model

Usual column, beam, slab

structures

1/25 1/101/4

Roof structures 1/2001/50 1/301/10

Highway or railroad bridges 1/25 1/201/4

Wind models 1/3001/50 -Dams 1/2501/40 1/75

Nuclear reactors 1/1001/50 1/201/4


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A typical small scale study has the following stages:

- Defining the purpose of the experiment and choosing the proper model.- Specifying the similitude conditions for the models geometry, the materials, the loads

and the interpretation of the results.

- Establishing the size of the model and the accuracy level of the results.- Choosing the proper materials and building the model- Conducting the experiment and recording the results- Analyzing the results and writing the research paper

2.2. Dimensional analysis

Dimensional theory can be resumed in the following phrases:

1. Any equation which describes an aspect of nature must be dimensional homogenous:the equation must be valid independent of the chosen dimensions

2. As a consequence of dimensional homogeneity, it can be shown that any equation like(2.1)

F(X,X,,X)= 0 (2.1)Can be expressed as equation (2.2)

G(,,,)= 0 (2.2)where the PI terms are dimensionless products of the n physical quantities(X,X,,X),and m=n-r represents the number of fundamental dimensions used to describe the physical

process. This second observation was made by Buckingham [6].

As a consequence, physical systems differing only quantitatively will have the same

equation G(,,,)= 0, and the similitude conditions result imposing that the terms,j = 1,mto be equal in both the model and the prototype.

To form the PI terms, all variables must be used, the m terms must independent and

for any given problem there are multiple sets of P terms. The steps are as follows:

1. Choosing the n terms which significantly influence the physical phenomenon alongwith their dimensions. For example, in the case of a three point bending of a simplesupported beam as the one in chapter 4, variables like the concrete cover of the

reinforcements, the height of the beam and the effective depth can be all replaced with

the height of the compressed section. An equation like (2.1) is formed.

2. Choosing r main variables for the r dimensions which describe the physicalphenomenon and which are dimensionally independent. Dimensionless variables like


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Poissons coefficient or the damping ratio cannot be main variables. The main

variables are noted as Q,Q,,Q.3. The remaining m variables form the system of independent equations (2.3). The m PI

terms will be expressed as in (2.4):

[X][Q][Q] [Q] = 1[X][Q][Q] [Q]= 1

[X][Q][Q] [Q]= 1 (2.3)

where [X] represents the dimension of X. Any dimensionless parameter will automaticallybecome a PI term. = XQQ Q= 1 = XQQ Q = 1

= X

Q

Q

Q

= 1

(2.4)

4. For every PI term the values of a,= 1,will be determined and write an equationlike (2.2).

For example, consider a physical system which can be described with the following

variables: length, mass, concentrated force, stress, deformation, acceleration, displacement,

Poissons coefficient and the elastic modulus, n=9. Each of these variables can be expressed

with the following dimensions: force, length and time, r=3. In table 2.2 are all the parameters

along with their dimensions.

Table 2.2 The 9 variables with their dimensions

l Q M a E

F 0 1 1 1 0 0 0 0 1

L 1 0 -1 -2 0 1 1 0 -2T 0 0 2 0 0 -2 0 0 0

As main variables the mass, the elastic modulus and the length were chosen. The PI

terms will be determined solving each equation of the system at (2.5). The number of the PI

terms is m=n-r=9-3=6 and can be found in equation (2.6)

= [Q][M]

[E]

[l]

= 1

= [][M][E][l] = 1 = [a][M][E][l]= 1 = [][M][E][l]= 1 = [] = 1 = [] = 1 (2.5)


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The PI terms for this physical system are:

= ,= , = ,= , = , = (2.6)2.3. Classification of the structural models

The degree in which the PI terms are respected represents an important criterion in

classifying the structural models as follows:

1. Exact models which replicate entirely the studied phenomenon. It was shown that anyphenomenon can be reduced to a complete set of independent domensionless

products, called PI terms. Each of these terms can be expressed as in equation (2.7):

= (,,,) (2.7)Complete similitude is achieved imposing that the PI terms be equal both for the

model and the prototype as in equation (2.8):

= (,,,)(,,,)= 1 (2.8)where represents as defined for the model, and represents as defined for the

prototype. In complete similitude = , =, , =. Equation (2.8)will be named prediction equation.

Using the PI terms in previous example, equation (2.9) is obtained:

=

=>

=

=>

=

(2.9)

c = X/ Xdenotes the ratio between the values of the X parameter in the modeland the prototype. Equation (2.10) is obtained which describes the necessary force that must

applied to the model in order to achieve complete similitude.

c = cc =>Q= Qcc (2.10)The difficulties which appear in trying to obtain an exact model are very hard if not

impossible to overcome. In such cases, one must isolate the variables which dont influence

significantly the result and can be modified in order to minimize the errors in the results.

Such models are called adequate models.

In most of the cases, more often with the strength models (also used in the present

study), the alteration of a variable in inevitable and leads to important changes in the response

of the model. In this case the initial error must be considered when interpreting the final

result and correcting if necessary.


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3. The materials

The first step of the study was finding a proper micro-concrete recipe. Because of the

small size of the model having a 3mm cover of concrete, the aggregates used had to be no

larger than 2mm. Because of this restraint, the challenge was finding a micro-concrete with a

proper compression strength, while the elastic modulus and the tension strength will be scaled

accordingly.

Along with casting the models for the two experiments, probes were also cast (cubes,

prisms and hourglass shaped probes) in order to determine the properties of the concrete. The

recipes were determined following the indications in Manea ET. AL. [7]. Appendix C shows

the calculus made for establishing the recipe of the concrete used in making the second

model.

For building the beam model three recipes of micro-concrete have been tried: first,

with a water:cement:aggregates ratio of 0.507:1:3.196 with 9 ml of super plasticizer added.

The other recipes were 0.549:1:3.025 and 0.621:1:3.025 without super plasticizer. The

cement used was Portland 42.5 kept in a dry environment.

From these recipes 2 prisms with a cross section of 4x4cm were poured for each mix

along with two 10cm cubes. Each prism was subjected to a bending test to establish the

tension strength of the concrete. After the test each half of the prism was later tested to

determine the compression strength of the concrete. The results can be seen in table 2.1 and

the devices used ar in figure 2.1.

Table 2.1 Compression and tension strengths of the concrete

Recipe

(A:C:Ag)Prism 1 Cube 1 Prism 2 Cube 2

fct

(MPa)

fck 1

(MPa)

fck 2

(MPa)

fck 3

(MPa)

fct

(MPa)

fck 1

(MPa)

fck 2

(MPa)

fck 3

(MPa)

0.507:1:3.196 4.69 28.53 29.46 28.57 4.31 28.46 29.79 29.38

0.549:1:3.025 4.88 22.58 21.50 22.78 4.66 21.86 22.38 24.48

0.621:1:3.025 4.86 22.49 23.03 23.16 4.77 22.49 23.41 23.69

Although the best recipe is the one with the additive, the third one was used as not

enough super plasticizer was available. Thus obtaining a C16/20 class concrete.

For the determination of the elastic modulus Ecm , two 10x10x30cm prisms were cast

and kept in the same atmospheric conditions as the beams in chapter 4. The load stages were

8% and 40% of the compression strength. The value of 20.74 GPa was obtained.


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Figure 2.1 The devices used in obtaining the compression strength (left) and the tension

strength (right)

For the micro-concrete used in building the frames a better recipe was tried using

Portland cement 42.5 and 2mm aggregates and a W:C:A ratio of 0.42:1:2.57. For finding the

micro-concretes characteristics, five 10cm cubes were cast for the compression strength test,

two 10x10x30cm prisms for the elastic modulus test and 2 hourglass shaped probes for the

tension test. All these were kept underwater for 28 days before testing. The devices can be

seen in figure 2.2 and belong to the Central Laboratory for Concrete and for Structures of the

Technical University of Cluj-Napoca: one Lloyd LR10K press for tension tests and oneAvantest 9 for compression and elastic modulus tests. The results can be found in Appendix

D and are synthesized in table 2.2.

Table 2.2 Characteristics of the micro-concrete recipe

Probe 1 2 3 4 5 Average

Compression

[MPa]

40.70 45.15 47.81 43.27 44.59 44.30

Tension [MPa] 4.528 4.693 - - - 4.61

Elastic modulus

[MPa]

31130.98 29597.73 - - - 30364.36


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Figure 2.2 Probes (left) and devices used to determine the concretes compression strength

and elastic modulus (right) and tension tests (middle)

It can be seen from the table that the average compression strength obtained is

44.30MPa which corresponds to a concrete class C35/45. The elastic modulus reached

89.30% of the prescribed value by the Eurocode 2 and the tension strength is above with

44.10%. These differences are negligible when studying the breaking behavior and in the

elastic domain their effect can be controlled with the PI terms.

In order to simulate the reinforcement, the results of Kim ET. AL. [5] were consulted

and threaded rod was used as it properly simulates the bond between the reinforcement and

the concrete. Also, steel wire was used for the stir-ups. The diameters of the rod and the wire

were dictated by their availability on the market: 3mm and 4mm threaded-rod and

0.9mm and 1.2mm steel wire. Their characteristics were tested in the Strength of

Materials laboratory at the Faculty of Mechanics of Cluj-Napoca and can be found in

Appendix D. They are synthesized in table 2.3 and the stress-strain relationships are in

figures 2.3-2.6

Table 2.3 Steel wire and threaded rod characteristics

Reinforcements for the beams

0.9 3

Ultimate strength [MPa] 498.46 379.93

Elastic modulus [MPa] 105549.46 77485.08


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Reinforcements for the frames

1.2 3 4

Ultimate strength [MPa] 553.22 498.89 418.58

Elastic modulus [MPa] 147287.78 65505.92 67446.12

It can be seem that the characteristics above are close to that of the steel being used in

constructions. The differences between the values of the elastic modulus will be taken into

account while establishing the PI terms and correcting them if necessary.

Figure 2.3 Threaded-rods and steel wires used in fabricating the beams

Figure 2.4 Steel wire 1.2mm used in fabricating the frames


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Figure 2.5 3mm threaded-rods used in constructing the frames

Figure 2.6 4mm threaded-rods used in constructing the frames


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4. Reinforced concrete beamTo illustrate the theories presented in chapter 2 and to verify the behavior of the

materials in chapter 3, the following experiments were made: a simple supported reinforced

concrete beam submitted to three point bending and a two-dimensional frame with ope

opening and two levels submitted to dynamic tests. The purpose of these experiments was to

find proper scaling coefficients for 1:10 scaled models.

A beam like that in figure (4.1) was fabricated. Three identical beams with a cross-

section of 3x6cm, 50cm in length, reinforced with 23 mm threaded rod both superior and

inferior and 1 mm stir-ups with 10mm between them in the vicinity of the supports and

20mm in the middle. The concrete cover used was 3mm. The materials used are extensively

presented in the previous chapter.

Figure 4.1 Reinforcement cage and framework for the 1:10 scaled beam

Using known relationships, the capable bending moment, the cracking moment, and

the maximum deflection were determined both for the model and the prototype. The later was

considered as being built using a C16/20 concrete, with the characteristics in Eurocode 2,

reinforced using S500 steel. The area of the reinforcement varied from 214 to 240 both up

and down. This interval was chosen in order to obtain the minimum are as prescribed by the

national seismic norm, P100-1/2013 [26] and to ensure a ductile behavior at collapse.

The approach was as follows: every characteristic of the three studied was determined

both for the prototype and the model and the ratio between them; determine the scaling ratios

using the specifications of chapter 2 and the error between this value and the one previously

determined. The next step was to find a correction coefficient for the similitude determined


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ratio in order to be able to apply it at smaller scale models, from 1:15 to 1:100 with a

minimum error.

The calculus was compared to a Finite Element Analysis and a physical experiment

for the model structure.

4.1. Ultimate bending moment

It has been chosen to analyze to ultimate bending moment and not the applied force as

this information can be used in any static configuration and also, the influence of the support

opening is eliminated. For the calculation of the bending moment, equation (4.2) was used.

This was obtained from equation (4.1). Appendix A shows how this equation is obtained.

M = fxbd + A(d d) (4.1)M

=

f

xb

d

+ A

E

(d

d

) (4.2)

where: = 0.8, = 1 []fis the concretes compression strength [kN/ m]x is the height of the compressed area of the beam cross section [m]

b is the width of the beam [m]

d is the effective height [m]

Ais the reinforcement area at the upper part [m]

is the stress in the upper reinforcement in the moment of colapse [kN/ m]

dis the distance from the upper limit of the beam to the centroidof the upper reinforcement [m]

Eis the elastic modulus of the reinforcement [kN/ m]is the ultimate deformation of the concrete, 3.5. []The theoretical value of the capable bending moment for the model beam is

0.367kNm, resulting in the capable force of 3.253 for a simple supported beam.

Using the equation (4.2) again, the capable bending moment was determined for the

10:1 scale prototype reinforced in various ways at the upper and the lower part. The results

are in table 4.1. The missing values represent the combination of reinforcements which cause

brittle behavior of the beam


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Table 4.1 Capable bending moment [kNm] of the prototype reinforced in various ways

Sup

Inf

214 216 220 225 232 236 240

214 86.67 86.69 86.78 86.95 87.26 87.48 87.73216 112.28 112.26 112.25 112.28 112.42 112.54 112.69

220 173.23 173.18 173.02 172.79 172.53 172.41 172.33225 266.26 266.62 266.81 266.50 265.76 265.31 264.88

232 421.11 424.26 427.56 429.38 429.07 428.30 427.39236 - 524.40 531.17 536.61 538.56 538.02 537.01

240 - - - 651.19 658.34 658.94 658.27

All the variables in the (4.2) equation were chose to find the PI terms necessary to

express the perfect similitude between the model and the prototype. Awas replaced withA, the total area of reinforcement in the beam. As main variables, the height of thecompressed area of the beam section and the elastic modulus of the reinforcement were used.

The PI terms are in equation (4.3):

= , = , = , = , = (4.3)The similitude ratios obtained using this PI terms are in equation (4.4):

c = cc, = , = , = , = (4.4)It can be seen that many of this ratios cannot be fulfilled, making the model a

distorted one. The way each distortion in the model influences the final response is

impossible to establish. Therefore the ratio for the capable bending moment will be written as

in equation (4.5) in order to contain more relevant variables:

c = ccc (4.5)Table 4.2 contains the values of the capable bending moment of the prototype

reinforced in various ways using equation (4.5) and remembering that M . = cM..Table 4.3 contains the errors between these values and the ones in table 4.1

Table 4.2 Capable bending moment [kNm] of te prototypes cross-section determined using

the similitude theory:

Sup

Inf

214 216 220 225 232 236 240

214 262.20 302.40 398.70 549.20 816.10 998.10 1201.40

216 301.80 341.90 438.10 588.30 854.80 1036.40 1239.40

220 396.60 436.50 532.40 682.10 947.60 1128.60 1330.90

225 543.90 583.70 679.10 828.20 1092.50 1272.70 1474.10

232 803.30 842.80 937.60 1085.80 1348.50 1527.60 1727.80236 - 1018.30 1112.80 1260.40 1522.20 1700.60 1900.10

240 - - - 1454.70 1715.50 1893.30 2092.10


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Tabel 4.3 The error [%] between the value in table (4.1) and table (4.2):

SupInf

214 216 220 225 232 236 240

214 66.95 71.33 78.23 84.17 89.31 91.23 92.70

216 62.80 67.17 74.37 80.91 86.85 89.14 90.91

220 56.32 60.33 67.50 74.67 81.79 84.72 87.05225 51.04 54.32 60.71 67.82 75.67 79.15 82.03

232 47.45 49.66 54.40 60.45 68.18 71.96 75.26236 - 48.50 52.26 57.42 64.62 68.36 71.74

240 - - - 55.23 61.62 65.20 68.54

The error in table 4.3 must be as close to zero as possible. So, the scaling coefficient

in equation (4.5) will be corrected with a factor like in equation (4.6). This correction factor

will be determined like in equation (4.7) as a function of the ratio between the aria of

reinforcement at the upper part and the area at the lower part of the beam.

c = (4.5)= f (4.6)

Using the scales factors Cand C was preferred as equation (4.5) is expressed in scalingfactors.

will be determined as the error in table 4.3 approaches 0. To do so, in table 4.4 are

presented the values considered significant to determine the correction factor. The graph

generated by these values can pe seen in figure 4.2. The expression of was found as the one

in equation (4.7) whose graph can be also seen in figure 4.2 for comparison.

Figure 4.2 The points in table 4.4 compared to the graph of equation (4.7)


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= 1.47 + 1.68 (4.7)It can be seen the two graphs coincide. In table 4.5 the values for the capable bending

moment determined with equation 4.5 are synthesized, and in table 4.6 the error between

these values and those determined with equation (4.2) can be found.

Table 4.4 Significant values for determining the value of :

CAss/CAsi

0.2 1.88

1 3.18

2.07 4.80

3.23 6.54

4 7.608.16 13.73

Table 4.5 The capable bending moment [kNm] of the prototypes section determined with

equation 4.5:

SupInf

214 216 220 225 232 236 240

214 83.25 83.99 85.19 86.25 87.19 87.55 87.82216 107.59 108.54 110.15 111.66 113.06 113.62 114.05

220 165.23 166.57 169.00 171.51 174.09 175.17 176.05225 254.06 255.77 259.12 262.91 267.22 269.18 270.82

232 409.56 411.63 415.94 421.30 428.09 431.47 434.45236 - 516.80 521.52 527.60 535.70 539.89 543.69

240 - - - 645.30 654.57 659.54 664.15

Tabel 4.6 The error [%] between the value determined with equation (4.2) and equation 4.5:

SupInf

214 216 220 225 232 236 240

214 3.93 3.11 1.83 0.80 0.08 0.08 0.11

216 4.17 3.31 1.87 0.56 0.57 0.95 1.19

220 4.61 3.82 2.32 0.74 0.90 1.57 2.11

225 4.59 4.07 2.88 1.35 0.55 1.44 2.19

232 2.97 2.98 2.72 1.88 0.23 0.74 1.63

236 - 1.45 1.82 1.68 0.53 0.35 1.23

240 - - - 0.90 0.57 0.09 0.89

It can be seen that the error is now below 5%, while using the uncorrected similituderatio the maximum error was 92.70%.

Next, using equation 4.5, the capable bending moment for prototype beams scale 8:1,

20:1, 50:1 and 100:1 are calculated. A prototype beam scale 100:1 with a cross section of

600x1200cm is improbable but the purpose of the calculus was to illustrate the influence of

the correction factor on various scales, factor determined started from a 1:10 scale model.


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The reinforcements for the large scale beams were chosen in order to maintain a ductile

behavior of the beam and that the upper reinforcement does not yield before the lower

reinforcement.

Table 4.7 Capable bending moment [kNm] of the 8:1 scale prototype determined with

equation (4.5) and the error (%) to the value determined with equation (4.1)

SupInf

214 216 220 225 232 236 240

214 65.81(2.82) 66.40(1.94) 67.35(0.55) 68.19(1.49) 68.93(1.48) 69.22(1.73) 69.43(1.84)

216 85.02(2.99) 85.77(2.08) 87.04(0.53) 88.23(0.90) 89.34(2.18) 89.78(2.64) 90.12(2.96)220 130.45(3.22) 131.50(2.44) 133.42(0.90) 135.41(0.80) 137.44(2.57) 138.29(3.33) 138.99(3.94)

225 200.34(2.60) 201.68(2.26) 204.33(1.25) 207.32(0.29) 210.72(2.29) 212.26(3.26) 213.55(4.10)232 - - 327.44(0.21) 331.66(0.11) 337.01(1.66) 339.67(2.66) 342.02(3.62)

236 - - - 414.94(0.84) 421.31(1.50) 424.61(2.36) 427.60(3.28)240 - - - - 514.31(1.76) 518.21(2.25) 521.83(3.02)

Table 4.8 Capable bending moment [kNm] of the 20:1 scale prototype determined with


SupInf

240 825 732 640 840 1040 1240

240 540.9(4.26) 549.70(2.68) 557.20(1.39) - - - -825 869.50(5.25) 883.50(3.64) 896.60(3.10) 908.90(0.65) 917.60(0.37) 923.80(1.08) 928.30(1.55)732 1336.7(5.87) 1355.9(4.56) 1375.7(3.06) 1396.2(1.42) 1411.9(0.11) 1423.9(0.91) 1433.1(1.70)

640 2058.0(5.68) 2082.2(5.01) 2109.5(3.94) 2140.3(2.42) 2166.4(0.99) 2187.6(0.23) 2204.7(1.26)840 2931.1(4.40) 2958.9(4.45) 2992.0(4.11) 3032.3(3.11) 3068.7(1.83) 3100.4(0.56) 3127.1(0.59)

1040 3954.7(2.15) 3984.9(2.82) 4022.6(3.29) 4070.7(3.19) 4116.8(2.38) 4158.7(1.27) 4195.9(0.11)1240 - - - 5254.6(2.34) 5309.0(2.40) 5360.7(1.72) 5408.1(0.71)

Tabel 4.9 Capable bending moment [kNm] of the 50:1 scale prototype determined with


SupInf

825 732 640 840 1040 1240 1936

825 5476(4.79) 5543(3.61) 5597(2.65) 5641(1.89) 5676(1.30) - -

732 7778(5.72) 7572(4.54) 7953(3.51) 8022(2.63) 8079(1.89) 8127(1.28) 8167(0.78)640 10470(6.43) 10589(5.33) 10696(4.31) 10792(3.39) 10875(2.58) 10946(1.88) 11007(1.27)

840 13551(6.89) 13692(5.95) 13825(4.99) 13947(4.09) 14057(3.26) 14154(2.51) 14240(1.84)1040 17022(7.12) 17181(6.36) 17337(5.54) 17485(4.69) 17622(3.88) 17746(3.11) 17858(2.42)

1240 20527(7.13) 20698(6.57) 20872(5.91) 21405(5.17) 21567(4.41) 21718(3.67) 21857(2.97)

1936 25133(6.94) 25321(6.58) 25514(6.10) 25706(5.51) 25893(4.84) 26069(4.15) 26234(3.47)

Tabel 4.10 Capable bending moment [kNm] of the 100:1 scale prototype determined with

equation (4.5) and the error (%) to the value determined with equation (4.1)Sup

Inf

1936 1840 2240 3036 2840 3240 3640

1936 53460(5.08) 53770(4.53) 54050(4.02) 54310(3.56) 54550(3.14) 54760(2.76) 54960(2.42)

1840 63250(5.59) 63620(5.06) 63960(4.55) 64270(4.09) 64560(3.66) 64830(3.26) 65070(2.90)

2240 73850(6.04) 74270(5.53) 74660(5.04) 75030(4.58) 75380(4.14) 75690(3.74) 75990(3.37)

3036 85240(6.43) 85720(5.95) 86160(5.48) 86590(5.03) 86990(4.60) 87360(4.19) 87710(3.81)2840 97450(6.76) 97970(6.32) 98470(5.88) 94950(5.44) 99400(5.02) 99830(4.62) 100230(4.23)

3240 110460(7.02) 111020(6.63) 111570(6.22) 12100(5.81) 112610(5.41) 113100(5.01) 113550(4.63)3640 124280(7.21) 124890(6.87) 125480(6.51) 126060(6.14) 126620(5.75) 127160(5.37) 127670(5.01)


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It can be seen that the error is below 7.21%. This is obtained choosing the proper

scaling factors, which is always difficult.

4.2. Cracking bending moment

The cracking bending moment has been chosen in order to eliminate the influence of

the span of the beam which appears when determining the cracking force. The approach was

the same as in the case of the capable bending moment: determine the values for the

prototype, the 1:10 scale model by already known relationships and using similitude theory.

After that, applying the same relationships to beams of bigger scales in order to establish their

validity.

The well knows relationship of Navier for normal stresses for bent beams was used,

equation (4.8). The moment of inertia was determined for a composed section, considering

also the influence of the reinforcement. The cracking stress was considered equal to the

concretes tension limit, as defined by the Eurocode 2, equation (4.9).

M = = (4.8)f = max1.6 f,f (4.9)

where: fis the concretes tension limit [kN/ m]fis the concretes direct tension limit [kN/ m]z is the height of the tensed area of the beam section [m]

Iis the moment of inertia of the beams cross-section. [m]Using equation (4.8) the models crack bending moment was determined as 0.149

kNm, leading to a cracking force of 1.324 kN for a simple supported beam.

The PI term for scaling the cracking moment was determined using the variables in

equation (4.8) and can be found in equation (4.10); the main variables used were the

concretes tension limit fand the moment of inertia of the beams cross section, I. = (4.10)

The similitude ratio found based on the PI term in in equation (4.11). It was then used

to determine the cracking moment of a 10:1 scale prototype beam reinforced in various ways.

The results are found in table 4.11, along with the error induced by the similitude criterion.

M.= cM..c = cc (4.11)


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840 4569(0.45) 4592(0.52) 4619(0.61) 4649(0.72) 4682(0.84) 4719(0.97) 4758(1.13)1040 4602(0.16) 4625(0.09) 4652(0.01) 4682(0.11) 4716(0.23) 4783(0.37) 4793(0.52)

1240 4638(0.83) 4661(0.75) 4688(0.67) 4718(0.56) 4753(0.44) 4790(0.30) 4831(0.15)

1936 4676(1.54) 4699(1.47) 4727(1.38) 4758(1.28) 4793(1.16) 4831(1.02) 4872(0.87)

Tabel 4.15 Cracking moment [kNm] of the prototype scaled 100:1 determined using

similitude theory and the error [%] to the value determined with equation (4.8):Sup

Inf

1936 1840 2240 3036 2840 3240 3640

1936 49594(1.69) 49735(11.72) 49888(1.77) 50052(1.81) 50226(1.86) 50411(1.91) 50607(1.96)

1840 49735(1.47) 49877(1.50) 50031(1.54) 50195(1.59) 50371(1.63) 50556(1.69) 50753(1.74)2240 49888(1.22) 50031(1.26) 50185(1.30) 50350(1.35) 50526(1.39) 50713(1.45) 50910(1.50)

3036 50052(0.96) 50195(1.00) 50350(1.04) 50516(1.08) 50693(1.13) 50881(1.18) 51079(1.24)2840 50226(0.68) 50371(0.72) 50526(0.76) 50693(0.81) 50871(0.85) 51060(0.91) 51259(0.96)3240 50411(0.39) 50556(0.42) 50713(0.46) 50881(0.51) 51060(0.56) 51250(0.61) 51450(0.66)

3640 50607(0.07) 50753(0.11) 50910(0.15) 51079(0.19) 51259(0.24) 51450(0.29) 51652(0.35)

The error induced is below 10% and fades with the increase of the prototypes scale.

There was no need for a correction factor in order to maintain a minimum error because the

beam is in the elastic domain, where the similitude theory can yield acceptable results, as

observed by other authors like WU ET. Al. [4].

4.3. Maximum mid-span deflection

After studying the elastic and the breaking behavior, the mid-span deflection of a

simple supported beam submitted to three point bending was also studied. It was determined

using the similitude theory using a model as the one in figure (4.1) and then compared to the

value determined using Eurocode 2 relationships. The same similitude relationship was them

applied to larger scale prototype beams. The equation provided by the Eurocode is (4.12),

extended as in (4.13).

f = + (4.12)f = L + L (4.13)

= 1 (4.14)where f is the deflection at the middle of the beam [m]

is a distribution coefficient allowing for tension stiffening at a section0 for uncracked sections and determined with equation (4.14) if else []is a coefficient taking account of the influence of the duration of theloading or of repeated loading on average strain: =1 for single shortterm loading []


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is the stress in the tension reinforcement calculated on the basis of acracked section [kN/ m]is the stress in the tension reinforcement calculated on the basis of aCracked section under the loading conditions causing first cracking [kN/ m]

Mis the cracking moment [kNm]Eis the concretes modulus of elasticity [kN/ m]Iis the reinforced concrete cross-sections modulus of inertia [m]L is the span of the beam [m]

Eis the elastic modulus of the steel [kN/ m]z is the height of the tensed area of the cross-section [m]

is the deformation parameter considered (a strain, a curvature or a rotation); si are the values of the parameter calculated for the uncracked and fully cracked conditions

respectively.

Using equation (4.13), the mid-span deflection of the beam model in figure (4.1) was

determined as 2.879mm.

With the same relationship, the same characteristic was determined for a 10:1 scale

prototype reinforced in various ways. The values can be seen in table 4.16. The missing

values stand for the combination of reinforcements which lead to a brittle failure of the beam.

The PI terms which define this phenomenon where determined using the variables of

equation (4.13) and can be found in equation (4.15). As main variables L, the span of the

beam and E, the elastic modulus of concrete were used. = , = , = , = , = ,= (4.15)Table 4.16 The mid-span deflection [mm] for the prototype beam using equation (4.12):

SupInf

214 216 220 225 232 236 240

214 7.82 7.83 7.84 7.85 7.86 7.88 7.91

216 7.88 7.89 7.90 7.90 7.90 7.91 7.94

220 7.96 7.95 7.95 7.95 7.95 7.96 7.98

225 8.06 8.04 8.02 8.01 8.01 8.01 8.02232 8.18 8.14 8.11 8.09 8.07 8.06 8.06

236 8.41 8.35 8.29 8.25 8.21 8.18 8.14

240 9.26 9.13 8.99 8.86 8.75 8.61 8.42

The scaling ratios determined based on the PI terms are in equation (4.16). As in the

case of the capable moment, a similitude ratio which contains as many terms as possible was

used, equation (4.17)


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c = cc,c= c,c = c,c = c,c = c,c = c (4.16)c= (4.17)

Using equation (4.17) and knowing that f.= cf., the maximum deflection ofthe 10:1 prototype beam was determined and the values can be seen in table 4.17. The error

induced by the similitude theory as opposed to the values determined using equation (4.13) is

in table 4.18.

Table 4.17 The mid-span deflection [mm] of the 10:1 prototype determined using similitude

theory:

Sup

Inf

214 216 220 225 232 236 240

214 2.22 2.23 2.23 2.44 2.25 2.25 2.25

216 2.25 2.26 2.27 2.28 2.28 2.29 2.29

220 2.33 2.33 2.34 2.35 2.36 2.37 2.37225 2.44 2.45 2.46 2.47 2.48 2.49 2.49

232 2.64 2.65 2.66 2.68 2.69 2.70 2.70

236 2.78 2.78 2.79 2.81 2.83 2.84 2.84

240 2.92 2.93 2.94 2.96 2.98 2.99 3.00

Table 4.18 The error [%] induced by using similitude theory as opposed to equation (4.13)

SupInf

214 216 220 225 232 236 240

214 18.13 18.07 17.96 17.88 17.93 18.09 18.36

216 17.61 17.47 17.26 17.08 17.06 17.19 17.44

220 16.70 16.23 15.75 15.28 15.02 15.06 15.24225 16.25 15.44 14.07 12.90 12.07 11.90 11.94

232 17.76 16.34 13.48 10.43 7.81 7.04 6.64236 20.03 18.33 14.69 10.29 5.89 4.43 3.56

240 23.55 21.62 17.30 11.60 4.92 2.40 0.77

The error in the previous table must be minimal. So, equation (4.17) must be adjusted

using a correction factor, depending again on the ratio between the upper area of

reinforcement and the lower one, as they determine the capable moment of the cross-section.

The correction factor will be determined like in equation (4.19)

c= (4.18)= f (4.19)


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Table 4.19 Significant values for determining

CAss/CAsi

0.12 0.7640.25 0.820

0.31 0.853

0.48 0.8971 0.950

2.07 0.881

3.23 0.8494 0.848

8.16 0.816

will be determined in order for the error in table 4.18 to be as close to 0 as possible,

meaning that the values in tables 4.16 and 4.17 must be very close. Table 4.19 contains

significant values of the CAss/CAsi ratio. The graph generated by these points can be seen in

figure 4.3. was determined as in equation (4.20), whose graph can be seen in the same

figure 4.3 for comparison.

= 1.152. 0.19, < 10.15 .+ 0.954, 1 (4.20)

Figure 4.3 The graph generated by the points in table 4 .19 compared to equation (4.20)


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Next, the mid-span deflection for the prototype scaled 10:1 using similitude theory

was determined. The results can be found in table 4.20. The error between them and those

found with the equation given by the Eurocode is in table 4.21.

Table 4.19 Mid-span deflection [mm] of the 10:1 prototype beam using equation (4.18):

Sup

Inf

214 216 220 225 232 236 240

214 6.71 7.32 7.78 8.16 8.53 8.69 8.82

216 6.95 6.82 7.65 8.08 8.48 8.64 8.87220 7.53 7.33 7.08 7.94 7.43 8.62 7.77

225 8.28 8.07 7.72 7.47 8.44 8.68 8.87232 9.45 9.20 8.81 8.45 8.13 8.83 9.12

236 10.18 9.91 4.49 9.10 8.70 8.57 9.25240 10.96 10.67 10.22 9.80 9.37 9.17 9.06

Table 4.18 The error [%] between the value determined with equation (4.12) and using the

similitude theory after applying the correction factor:

Sup

Inf

214 216 220 225 232 236 240

214 14.19 6.53 0.85 3.49 7.24 8.69 9.75

216 11.90 13.50 3.10 2.02 6.26 7.56 9.04

220 6.60 8.81 11.69 0.79 4.81 6.82 8.28

225 1.50 3.36 6.35 8.81 3.48 6.24 8.17

232 2.00 0.83 061 1.90 3.38 5.20 8.54

236 1.77 0.97 0.51 0.75 0.29 0.16 8.21

240 0.41 0.84 0.33 1.52 3.48 3.60 3.85

Thus, it can be seen that the error reduced from 20.89% to 9.35%. Next, the mid-span

deflection of prototype beams scaled 8:1, 20:1, 50:1 and 100:1 will be determined using

equation (4.18). The error between these values and those determined using equation (4.12)

are also shown. The results can be found in tables 4.19 4.23.

Table 4.19 Mid-span deflection [mm] of 8:1 scale prototype beam determined using

similitude theory and the error [%] as opposed to Eurocod 2 equations:

Sup

Inf

214 216 220 225 232 236 240

214 6.17(4.81) 6.73(3.63) 7.16(9.23) 7.50(13.07) 7.82(16.09) 7.93(17.05) 8.01(17.59)216 6.43(2.04) 6.32(3.66) 7.09(7.51) 7.48(12.18) 7.82(15.67) 7.94(16.78) 8.03(17.43)

220 7.07(3.88) 6.89(2.93) 6.65(0.73) 7.47(10.56) 4.90(15.40) 8.05(16.91) 8.16(17.84)

225 7.93(8.04) 7.73(7.05) 7.41(5.35) 7.16(3.79) 8.07(15.46) 8.28(17.74) 8.43(19.17)

232 9.30(7.11) 9.07(7.46) 8.69(8.90) 8.34(10.34) 8.02(10.68) 8.68(18.43) 8.94(21.28)236 10.18(1.82) 9.92(3.09) 9.51(6.67) 9.13(11.07) 8.72(13.96) 8.59(14.61) 9.24(21.74)240 11.12(8.74) 10.83(6.39) 10.40(0.10) 9.99(8.20) 9.55(15.71) 9.33(17.40) 9.20(18.40)


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Table 4.20 Mid-span deflection [mm] of 20:1 scale prototype beam determined using

similitude theory and the error [%] as opposed to Eurocod 2 equations:

Sup

Inf

240 825 732 640 840 1040 1240

240 14.26(8.11) 16.04(3.82) 16.83(8.65) 17.41(11.97) 17.87(14.41) 18.23(16.25) 18.50(17.59)825 15.64(1.51) 15.15(3.190 16.84(7.92) 17.59(12.33) 18.15(15.32) 18.57(17.49) 18.50(19.07)732 17.30(4.26) 16.61(3.35) 16.23(2.93) 17.88(12.87) 18.61(16.84) 19.13(19.48) 19.52(21.37)

640 19.15(4.26) 18.39(3.35) 17.82(9.11) 17.50(9.50) 19.14(18.28) 19.84(21.79) 20.34(24.09)

840 21.23(8.61) 20.39(11.150 19.76(13.80) 19.26(15.27) 18.98(16.12) 20.61(23.85) 21.29(26.92)1040 23.55(6.80) 22.63(11.01) 21.94(15.74) 21.39(19.54) 20.91(21.52) 20.65(22.60) 22.29(29.36)

1240 26.14(1.03) 25.12(7.53) 24.37(14.61) 23.77(20.94) 23.25(25.43) 22.78(27.63) 22.52(28.77)

It can be seen from these two tables that the error in only increasing. This is because

equation 4.20 is not suitable as a correction factor. Also, from figure 4.3 it can be seen that

finding such an equation is very difficult.

Another purpose of this part of the study is to see if the materials used in building the

models are adequate for small scale experiments. The three beams in figure 4.1 weresubjected to three point bending and the results compared to those obtained in a Finite

Element Analysis. If the values obtained are close, a numeric simulation will be enough

without performing a physical experiment.

4.4. Model beam three point bending. Experiment

Three beams as in figure 4.1 were tested. The beams have a 3x6cm cross-section and

a 45cm span. The reinforcement cage is made of 43mm threaded rod and 0.9mm steel

wire for the stir-ups.

A 40tf press equipped with force and displacement transducers were used. The

displacements were also measured using a optical measuring device.

The cracks were observed on one side of the beam coated with white paint to observe

the contrast. The other side was left unmodified as the optical device works by measuring the

tones of grey on the element. The deflection was measured using a displacement transducer

found at the bottom of the beam as in figure 4.4.

Figure 4.6 shows the relationship applied force-time for all the three beams. For the

first beam the loading rate was slower as we didnt know what to expect, reading the

displacement transducer and using the optical device every 200daN. For the later two beams

the readings were made every 300daN. Force-displacement graphs are shown in figure 4.7.

All the three beams behaved in ductile manner as expected and successfully modeling the

behavior of prototype beams.


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Figure 4.4 Equipping the beams

The optical measurements were made on the second beam in order to verify the

results given by the displacement transducer. The device works by measuring using the

Digital Image Correlation. By comparing the grey tones of two successive images of the

analyzed element and using a special algorithm the software can track the movement of every

point a defined matrix on the surface on the element as seen in figure 4.5. The coordinates of

the points are known enabling to calculate the displacements and the strains of any section of

the beam.

Figure 4.5 Point matrix on the beam at initial stage (left) and the final stage (right)


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Figure 4.6 Time applied force graphs for the three beams

Figure 4.7 Force displacement graphs for beams 1 and 2

Figure 4.9 shows a comparison between the force-displacement relationship obtained

by mechanical means (press and transducer) and the one obtained with the optical device.


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Figure 4.9 Force-displacement graphs for beams 1 and 3

The three beams collapsed at 3.21kN, 3.96kN and 4.1kN, yielding an average

breaking force of 3.76kN. The mid-span deflections are 3.96mm and 5.67mm

4.5. Model beam three point bending. Finite Element simulation

In order to be able to reproduce effortlessly the behavior of a model made of the

materials used in this experiment, a finite element analysis using Abaqus 6.12 was performed.

Three materials were used for simulating the concrete, the threaded rod and the steel wire.

For the stir-ups an elastic, perfect plastic steel model was used with an elastic modulus of

105.56 GPa, a Poissons coefficient of 0.3 and a yield strength of 499MPa. The threaded rod

was simulated using a plastic steel model with an elastic modulus of 77.49 GPa and a

Poissons ratio of 0.3. The yield force was considered at 380MPa. The stree-strain

relationship for the rod can be found in Appendix D.

The concrete was simulated using the Concrete Damaged Plasticity model developed

by Lubliner [16] and modified by Lee [17]. The stress-strain relationship for the concrete was

determined using the equation developed by Saenz [19] in equation (4.21)

= ()() (4.21)where: si are the stress and the strain of concrete

E = 20740MPais the concretes modulus of elasticityis the stress in the concrete corresponding to the compression or tension limit,


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R =()

() ,R = ,E = , R = , R = . Usualy R = 4,R= 4 canbe employed with good results.

fis the concretes tension limit (1.815 MPa) or compression (23 MPa),

is the breaking stress of concrete

is the breaking strain of concrete.Using equation (4.21) the stress-strain relationships for concrete compression and

tension were obtained, as in figure 4.10.

The beam was modeled using solid elements (C3D8R) of 3x3x5mm, while the

reinforcement and the stir-ups were modeled using truss elements (T3D2) of 5mm and 3mm

respectively, in order for the nodes of the reinforcement and the concrete to coincide. Figure

4.11 shows the meshing of the elements.

Table 4.21 contains the values used for defining the concrete plasticity:

Table 4.21 Values used for defining the concrete plasticity

Dilation angle

()

Eccentricity fb0/fc0 K Viscosity

parameter

32 0.1 1.16 0.667 0.01

Figura 4.10 Relatia tensiune-deformatie pentru beton, folosind relatia (4.28)

For a greater accuracy, the supports and the metal piece through which the loads were

applied were also modeled. Theyre interaction with the concrete beam was modeled as a

General Contact with a Normal Behavior that allows the separation and a Penalty type

Tangential Behavior with a 0.45 friction coefficient. A Static-Riks analysis was conducted.

The force-displacement graph obtained using the finite element analysis is compared in figure


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4.12 with the ones obtained mechanically and optically. Figures 4.13 and 4.14 show the

concrete and reinforcement stresses and figures 4.15 and 4.16 show the cracks network.

Figure 4.11 The finite elements used to model the concrete beam and the reinforcement

Figure 4.12 Force-displacement graphs for beam 3

Figure 4.13 Concrete stresses

Figure 4.14 Equivalent plastic stress of reinforcement


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Figure 4.15 Cracks network

Figure 4.16 Cracks appeared during the experiment

In table 4.22 the values for the cracking moment, the capable moment and the mid-

span deflection are shown for the model in figure 4.1 determined using Eurocode 2, obtained

from the physical experiment and from the finite element analysis.

Table 4.22 Comparative values for the three results studied obtained in three separate ways

Cracking moment

[kNm]

Capable moment

[kNm]

Mid-span deflection

[mm]

Eurocode 2 equations 0.149 0.366 2.627Physical experiment 0.154 0.423 3.960Finite element analysis 0.159 0.450 4.024

The differences between the results obtained by computer analysis and the ones

obtained through a physical experiment are 3.1% for the cracking moment, 6% for the

capable moment and 1.6% for the mid-span deflection. Thus, the finite element model is

adequate in simulating the three points bending of a threaded rod reinforced micro-concrete

beam. Also, the cracks in the finite element model seem to agree with the behavior of the

actual beam.

Larger differences appeared between the values determined using Eurocode 2

equations and experimentally: 3.2% for the cracking moment, 13.5% for the capable moment

and 33.67% for the mid-span deflection. These differences are due to the fact that the speed

of loading was not taken into account when applying the fore mentioned equations.


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5. Two-dimensional frame

The second purpose of this study was to determine the response of a structure

subjected do dynamic actions by using a model scaled 1:10. The objectives are the modes of

vibration, the ductility of the model and the behavior during an earthquake

5.1. Vibration modes

The prototype frame is considered as part of a spatial structure with 4.5m openings in

both directions, a 12cm concrete slab and being an office building. Also, the columns are

considered fixed into the foundation. Thus, the loading acting on each level of the frame is

determined as in equation (5.1), according to Eurocode 0:

q = [1.35(g+ h) + 1.5q]L = 40.84kN/ m (5.1)where: g

= 1.5kN/ m

is the dead load

q = 2kN/ m is the live loadh = 12cm is the thickness of the concrete slab = 25kN/ m is the weight of the concreteL = 4.5m is the opening of the spatial on direction oneThe mass of each level was determined using the special combination of loads as

defined in Eurocode 0 like in equation (5.2):

q=[(g+ h) + 0.4q]T = 23.85kN/ m (5.2)

Figure 5.1 Prototype frame (left) and the model frame with the mass of each level (right)

As presented in chapter 3, for the fabrication of the model, a micro-concrete with a

compression strength of 44.30MPa and an elastic modulus of 30364.36MPa, corresponding to


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a C35/45 class. Thus, the prototype was considered as being built with a C35/45 concrete and

S500 steel. Reinforcing of the frame will be made in accordance to the prescriptions of P100-

1/2013 [26], the Romanian seismic design code for a structure in the class H (high) of

ductility and following an analysis using SAP2000. Also, the prototype was reinforced in

order to have the same reinforcement percentage for the columns as the model, because of the

availability of threaded-rod on the market.

Next, the similitude between the vibration modes of the prototype and of the model

will be studied. To establish the necessary parameters for the PI terms, equation (5.3) was

used.

= (5.3)where: is the pulsation of the fundamental mode [1/s]

k is the stiffness of the structure [kN/m]

m is the mass of the structure [kNs/ m]The fundamental period of a structure is determined using equation (5.4). So, will

be replaced by T. To this parameters, the added mass m [kNs/ m] used to simulate theloads acting on the structure and the gravitational acceleration g [m/s] are added.

T =

(5.4)

Thus, the parameters used in the similitude study are found in equation (5.5), the PI

terms are in equation (5.6) and the similitude ratios are in equation (5.7).

P = {T,k,m,m,g} (5.5) = , = (5.6)c =, c = c (5.7)

The model frame was reinforced using the same rules as stipulated in P100, with

some modifications: the middle steel bar was removed because of the difficulties in

fabricating the model, as in figure 5.2. Also, the threaded rod used gave a 2% reinforcement

percentage because of their availability on the market.

For the modal analysis of the two frames, a finite element analysis in Abaqus 6.12

was performed. The concrete and the reinforcement were defined as at paragraph 4.5. In

figure 5.3 the prototype frame with the added mass can be seen. This was simulated as a

50x50x50cm rigid body made of a material with a density of 132828.23kg/ m. Placing itwas made so that the frames stiffness will not be affected.


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For modeling the concrete elements, 100x100x100mm blocks (C3D8R) were used for

the prototype and 10x10x10mm blocks for the model. The reinforcement elements were

defined as truss (T3D2) having 100mm in the case of the prototype and 10mm in the case of

the model.

Figure 5.2 Reinforcement for the prototype (left) and the model (right)

The stiffness of each level was determined using the method presented in the work of

Barsan [10], like in Appendix B. The influence of the reinforcements were also taken intoaccount. The mass of each level was determined like in figure 5.1. These and the stiffness are

found in table 5.1

Figure 5.3 FE models for the frame with added mass (left) and reinforcement cage (right)


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Table 5.1 Masses and stiffness of each level of the frames

Mass [/] Stiffness [kN/m]Level 1 2 1 2

Prototype 16603.53 14691.56 141385.41 94256.94

Model 16.604 14.692 11795.44 7863.63

Ratio 0.001 0.001 0.083 0.083Table 5.2 contains the periods for the first 9 natural modes of the prototype and the

model. The ratio between them is compared to the ratio as obtained with similitude theory,

equation (5.7). It can be seen that the difference is below 2.47%.

Table 5.2 Vibration periods [s] for the model and prototype frames

Vibration

mode

Prototype Model Tm/Tp Ratio

obtained with

eq. 5.7

Error (%)

1 0.1387 0.0149 0.1072

0.1098

2.47

2 0.0447 0.0048 0.1086 1.15

3 0.0426 0.0046 0.1085 1.19

4 0.0384 0.0042 0.1081 1.55

5 0.0191 0.0013 0.1121 2.03

6 0.0191 0.0012 0.1121 2.077 0.0118 0.0008 0.1079 1.74

8 0.0107 0.0008 0.1101 0.309 0.0074 0.0007 0.1093 0.44

In order to validate the finite element analysis results, an experimental modal analysis

was performed using an impact hammer as that in figure 5.5 on the model in figure 5.4.

Figure 5.4 Reinforcement and formwork of the model frame (left) and the hardened model

(right)


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Figure 5.5 The devices used in the experimental modal analysis

Figure 5.6 shows the network of points used in the experiment. Figures 5.7 and 5.8

show a comparison between the mode shapes as obtained in the finite element analysis and in

the experiment.

Only the first two modes are presented because they were the only ones clearly visible

during the experiment . Table 5.3 contains the values of the mode periods obtained in each

case.

Figure 5.6 The network of points used in the experimental modal analysis


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Figure 5.7 Vibration mode 1 shape

Figure 5.8 Vibration mode 2 shape

Table 5.2 Vibration periods for the model frame

Vibration mode Period obtained throughFEM

Period obtained throughexperiment

1 0.008872 0.0869562 0.002619 0.003419

A significant difference between the two cases is observed: the period measured

experimentally is greater than the one obtained theoretically, which translates in lower


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stiffness of the model. This is to be expected as the supports for the columns are not perfectly

fixed and a level of fixation in impossible to determine. Also, the structural faults appeared

during fabrication are a significant factor, whose influence on the stiffness of the frame is

impossible to determine.

5.2. Ductility test of the model

Next, using a Pushover analysis, the ductility of the model frame will be studied,

comparing it to the ductility of the prototype. This is made by comparing the multiplication

factor /. The reinforcement of the two frames has been presented in figure 5.2. Thematerials used in the analysis have been presented in chapter 3 and their characteristics can

be found in Appendix D. A Static-Riks analysis was performed with forces applied at each

level.

Two types of analysis were made with two different distributions of the lateral forces,

as defined in P100-1/2013 and seen in figure 5.9:

- One where the lateral forces as proportional with the level masses without regard totheir position on the height of the structure, in order to evaluate the maximum shear

force

- One where the forces are according to the shape of the most influential vibrationmode; also, a simplified triangular distribution is accepted, with the tip of the triangle

at the base of the structure. This helps establish the maximum bending moments.

Figure 5.9 Lateral forces distribution


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From table 5.1, the mass ratio is R = m/ m = 1.11, so the force ratio will beR = 1.11in the first distribution. The predominant vibration mode is the first one, with aratio of the level displacement of R= d/ d = 0.5, leading to a force ratio of R = 0.5inthe second distribution.

After a finite element analysis, the graphs in figure 5.10 were obtained. Also, the

multiplication factor /was determined and the results are in table 5.3Table 5.3 The multiplication factor for the two frames in the two lateral forces distribution

Distribution 1 Distribution 2 /Model 1.756 1.867 1.756

Prototype 1.679 1.791 1.675

Figure 5.10 Base shear force displacement graphs for the model (left) and the prototype

(right) for each distribution of the lateral forces

The two frames appear to have the same behavior when submitted to lateral forces

with the same distribution and the multiplication factor is close, having a difference of below

4.83%. Figure 5.11 shows the plastic hinges appeared at breaking in the model and the

prototype. It can be notices that they seem to form at the same places: the base of the columns

and the extremities of the beams.


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Figure 5.11 Plastic hinges developed in the prototype (left) and the model (right) at

breaking

Next, the behavior of the frames will be studied during an earthquake. The analysis

will be made using a finite element model. Following the laws of similitude, the goal is to

predict the behavior of the prototype starting from that of a model scale 1:10.

5.3. Earthquake response

For establishing the PI terms, the equation (5.8) was used.

Mu+ Cu + Ku = p(t) (5.8)where: M is the mass matrix

K is the stiffness matrix

C is the damping factor matrix

u is the displacement vector

u

is the velocity vector

u is the acceleration vectorp(t) is the applied forces vectorBecause of the configuration of the frames, the masses and the stiffness of the two

levels are equal, as seen from table 5.1. Thus, in equation (5.9) the mass and the stiffness

matrix can be replaces with the mass of a level, m and the stiffness of a level, k. Also, the

mass has been replaced with the weight of the level, G. The damping ratio has not been taken


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Figure 5.13 Vrancea 1977 scaled time-history used in analyzing the model

A Dynamic Implicit finite element analysis has been performed using Abaqus 6.12,

with an acceleration applied at the base of the columns on the longitudinal direction. The

concrete and the reinforcement was defined as in the previous experiments.

Figures 5.14 and 5.15 show a comparison of the displacements obtained with a finite

element analysis and following the rules of similitude for each level of the prototype. Scaling

the displacements of the model has been done using equation (5.12). Figures 5.16 5.21

show the same graph on smaller intervals in order to be easier to analyze: 5s 15s, 15s 25s

si 25s 35s.

d = ,t = (5.12)In the process of finding the PI terms, only the period of the first vibration mode was

used. Thus, it can be seen in the following graphs that the displacement of each level

determined with similitude theory and by a finite element analysis are comparable. But, after

25s the differences increase due to the influence of the higher vibration modes. Regarding the

behavior of the reinforcement, nor the model neither the prototype arent developing plastic

hinges. They will be submitted to a stronger earthquake, Northridge 1944 like in figure 5.22

and scaled as in figure 5.23 for the analysis of the model.


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Figure 5.14 Level 1 displacement



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Figure 5.16 Level 1 displacement on the 5s 15s interval



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Figure 5.22 Northridge 1994 time-history used in the analysis of the prototype

Figure 5.23 Northridge 1994 time-history scaled and used in the analysis of the model

Figures 5.24 and 5.25 show a comparison of the displacements of each level of the

prototype obtained by a finite element analysis and by scaling the displacements of the model

using the similitude ratio in equation (5.12). The plastic hinges developed by the two

structures are found in figure 5.26.

It can be seen that the values of the displacements differ significantly. Also, the errors

start to appear after t=6s, when plastic hinges begin to form in the prototype, while the

reinforcements of the model remain in the elastic domain for the entire analysis.


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Figure 5.26 Plastic hinges developed in the reinforcements of the prototype (left) and the

model (right)


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6. Conclusions and future research

It has been shown in chapter 3 that is difficult for the materials used in the modeling

the reinforced concrete structures using a 1:10 to have the same characteristics as those used

in the construction industry but satisfactory values can be obtained, while the others are

subjected to similitude relationships.

The micro-concrete in the models is submitted mostly to compression stresses so it

needs to have a compression strength similar to that of the concrete used in the prototype.

Following the rules presented in Manea ET. AL. [7] regarding the concrete recipe, such

materials have been obtained. The tension strength and the elastic modulus had different

values due to the use of 2mm aggregates. These differences are mostly visible in the elastic

domain, where carefully chosen similitude parameters can minimize their influence. A micro-

concrete with the water:cement:aggregate ratio of 0.42:1:2.57 yielded the best results and

proved to be effective in 1:10 scale model testing.

Regarding the reinforcement, the experiments of Kim ET. AL. [5] proved to be

helpful and threaded-rod was used as it simulates properly the bond between it and the

concrete. Because of its availability on the market, 3mm and 4mm was used. Steel wire

for the stir-ups was also employed. After testing them (results available in Appendix D) it has

been seen that their breaking point and ultimate strain are close to those of the steel used in

the industry. The elastic modulus differs in a significant way but again, by carefully choosing

the similitude parameters satisfactory results can be obtained. Thus, the materials used forsimulating the reinforcement are considered to be adequate for 1:10 scale model testing.

In chapter 4 a simple supported reinforced concrete model beam has been tested at

three point bending. The studied results are: cracking moment capable moment and ultimate

mid-span deflection.

The cracking moment was simulated with no difficulty, as similitude theory is known

to yield good results in the elastic domain. For the capable moment some modifications of the

similitude ratios were necessary in order to obtain satisfactory results. The ultimate deflection

has proven to be hard to simulate because of its variation along with the variation of the ratiobetween the upper reinforcement sectional area and the lower one.

Chapter 5 presents the experiments made on a two dimensional frame with one

opening and two levels. The results studied were: vibration modes, ductility and earthquake

response.


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For the vibration modes, satisfactory results were obtained, as the frames remain in

the elastic domain. The ductility was studied by performing a pushover analysis following the

indications in P100-1/2013 [26], the Romanian code for seismic design and using a finite

element software, Abaqus 6.12. The obtained values of the multiplication factor proved to be

close for the model and the prototype. The harder to simulate proved to be the earthquake

response because of the phenomenons complexity which extends on the determination of the

similitude criteria. Aspects like the cyclic behavior of concrete and the influence of the

superior vibration modes have been taken into account in the present study.

The small scale experiments on reinforced concrete models imply a partial similitude,

as defined in chapter 2. But, after the two tests of the present study some indications have

been developed for minimizing both the fabrication difficulties and the errors in the results:

- The cross-section dimensions and the concrete cover should follow the similituderatios and the ratios between them

- A micro-concrete with a compression strength and an ultimate strain as close aspossible to the one of the prototypes concrete should be used

- The reinforcement used should have the ultimate strength and the ultimate strain asclose as possible to that of the reinforcement of the prototype

- In the case of a dynamic test the reinforcement ratio of the columns should be thesame in the model and the prototype

As original contributions the following can be named: the micro-concrete recipe used

in the experiments and the equations found by means of similitude theory used in the

determination of the cracking and capable moment of the reinforced concrete beam using a

1:10 scale model and in the determination of the vibration modes of a two dimensional frame

with one span and two levels using a 1:10 scale model.

As future research one can name:

- Studying the plastic behavior of concrete and micro-concrete and determination of theplasticity parameters for defining it in the Lubliner [16] model.

- Studying the behavior of concrete and micro-concrete to cyclic loading and theirinfluence in the final response of the structure along with a similitude study

- Studying the influence of the superior vibration modes and the damping ratio on thefinal response of the structure along with a similitude study between a prototype and

its model.


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Appendix A: The expression of the bending moment of a double reinforced

concrete cross section

Figure A.1 Strains and stresses in the double reinforced concrete cross section

h

bd1

d

As1

MRd

x Fc

Fs1

x

zc

cu

s1

fcd

As2 s2 Fs2

zs


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In calculating the value for the model beam in chapter 4, characteristic values for the

concrete and steel strengths have been used.


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Appendix B: Determination of the level stiffness and masses for the two

frames presented in chapter 5

The masses of each level was determined using the mass distribution in figure 5.1

For calculating the stiffness of each level, the method presented in the work of Barsan[10] was employed. The bending stiffness of the columns and beams was determined taking

into account the influence of the reinforcements.

Index p was used for the characteristics of the prototype and index m was used for the

characteristics of the model.

1. Loads:

2. Prototype frame:2.1.The geometry:


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2.2.Stiffness of the reinforced concrete elements:

2.2.1. The column:

2.2.2. The beam:

2.3.Stiffness of each level:

2.3.1. Level 1:


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2.3.2. Level 2:

2.4.Level masses:

3. Model frame:3.1.The geometry:


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3.2. Stiffness of the reinforced concrete elements:

3.2.1. The column:

3.2.2. The beam:

3.3. Stiffness of each level:

3.3.1. Level 1:


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3.3.2. Level 2:

3.4.Mass of each level:


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Appendix C: Micro-concrete class C35/45 recipe


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4 5 0 .0

0. 0

5 0 .0

1 0 0 .0

1 5 0 .0

2 0 0 .0

2 5 0 .0

3 0 0 .0

3 5 0 .0

4 0 0 .0

Time (s)

6 0 .00 .0 5 .0 1 0.0 1 5.0 20. 0 25 .0 30 .0 35 .0 4 0.0 45. 0 50 .0 55 .0

Cube

Advantes t 9

Compression test on concrete : EN 12 390-3

Certificate number Certificate date 1/1/1904:

Testing apparatus

Customer

Contact

Specimen type Cem.content kg/m

Cement type Preparation date 12/12/2013:

Specimen conditions:

On receipt At time of test

Sampling location Sampling date 1/1/1904

Specimen preparationSpecimen ID.

Dimensions 100.0a(mm) 100.0b(mm) 100.0c(mm) Mass (Kg) 2.450

Area (mm) 10000.0 Test date 1/1/1904:

Load of failure (kN) 407.0 Stress (MPa) 40.70

Failure appearance Satisfactory

Notes

Operator

Density (Kg/m) 2450.0

No parts of this certificate may be reproduced without permission

:

Test results

"L" distance between lower rollers 0.0 Upper bearers number 1

Load rate MPa 0.800

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6 5 .00 .0 5. 0 10. 0 15. 0 20. 0 2 5.0 3 0.0 3 5.0 4 0.0 4 5.0 5 0.0 5 5.0 6 0.0

Cube

Advantes t 9



Testing apparatus

Customer

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Area (mm) 10000.0 Test date 1/1/1904:



Notes

Operator



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Test results


Load rate MPa 0.800

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5 0 0 .0

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Time (s)

7 0 .00 .0 5.0 1 0.0 15 .0 20 .0 25. 0 3 0.0 35 .0 40 .0 4 5.0 5 0.0 55 .0 60. 0 6 5.0

Cube

Advantes t 9



Testing apparatus

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Area (mm) 10000.0 Test date 1/29/2014:



Notes

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Test results


Load rate MPa 0.800

Age 48 zile

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4 5 0 .0

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6 5 .00 .0 5. 0 10. 0 15. 0 20. 0 2 5.0 3 0.0 3 5.0 4 0.0 4 5.0 5 0.0 5 5.0 6 0.0

Cube

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Testing apparatus

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Area (mm) 10000.0 Test date 1/29/2014:



Notes

Operator



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Test results


Load rate MPa 0.800

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Static elastic modulus determination -

:Customer

:Identification number

: 1/25/2013Sampling date

: 3/11/2013Test date

: cond STANDARDDescriptions

TEST RESULTS

: prismaSpecimen type

: 100x100x300 mmDimensions (mm)

:Mass (Kg)

:Notes

:Position

: 31130.98Elastic Modulus (N/mm)

: 10000.0Area mm

:Standard N : 3/11/2013Standard date

:Testing apparatus

Name

:Notes

Page 1 of 3

:Ag e

:Base of measurement (mm)

:Extensometer no.:Type of extensometer

Specimen description

= 4.68 = 16.64

= 140.2 = 524.3

N/mmN/mm

e e

Reference values for elastic modulus elaboration :


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20.00

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

s

500.00.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0 400.0 450.0

STRESS / TIME diagram

:Posit ion

:Standard N : 3/11/2013Standard:Testing apparatus

Name

:Notes

Page 2 of 3

20.00

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

e

600.00.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0 400.0 450.0 500.0 550.0

Strain ch 5

Strain ch 6

Strain ch 7

Strain ch 8

Average Strain

Average Strain

STRESS / STRAIN diagram


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DEVIATIONS (%) / TIME diagram

:Posit ion

:Standard N : 3/11/2013Standard

:Test ing apparatus

Name

:Notes

Page 3 of 3

Step Max Delta ch5 (%) Max Delta ch6 (%) Max Delta ch7 (%) Max Delta ch8 (%) M (e)

1 19.9 7.0 - - 20.8

2 15.1 8.9 - -

495.93 12.7 23.0 - - 76.7

4 10.2 6.3 - - 516.7

5 23.6 32.3 - - 91.1

6 7.5 3.9 - - 528.6

7 29.1 37.6 - - 103.7

8 5.2 1.6 - - 537.7

9 36.4 43.1 - - 113.4

10 2.9 0.0 - - 545.9

11 38.6 45.5 - - 119.6

12 1.3 2.2 - -

550.7

13 41.2 46.2 - - 127.2

50.00

-50.00

-40.00

-30.00

-20.00

-10.00

0.00

10.00

20.00

30.00

40.00

s

500.00.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0 400.0 450.0

Strain ch 5

Strain ch 6

Strain ch 7

Strain ch 8

Average Strain

Stress

Scost +

Scost -

10.0Limits (%)

Cumulated deviations (%) for selected steps


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Static elastic modulus determination -

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: 1/25/2013Sampling date

: 3/11/2013Test date

: cond STANDARDDescriptions

TEST RESULTS

Documents

Experimental and Numerical Analysis on RC Structures Using Small Scale Models