MINISTRY OF EDUCATION MINISTRY
AND TRAINING OF CONSTRUCTION
HANOI ARCHITECTURAL UNIVERSITY
LAM THANH QUANG KHAI
STUDY THE DEFORMATION STRESS STATE
OF MULTI-LAYER REINFORCED CONCRETE
DOUBLY CURVED SHELL ROOF
FIELD OF STUDY: CIVIL ENGINEERING
(CIVIL AND INDUSTRIAL CONSTRUCTIONS)
CODE : 62.58.02.08
SUMMARY OF DOCTORAL THESIS IN ENGINEERING
HANOI – 2019
The thesis was completed at Hanoi Architectural University
Supervisors:
1. Assoc. Prof. PhD. Le Thanh Huan
2. Prof. PhD. Nguyen Tien Chuong
Reviewer 1:
Reviewer 2:
Reviewer 3:
This thesis was presented and defended at Doctorate Examination
Council at Hanoi Architectural University
At .... date .... month .... year 2019
The thesis is available at the National Library of Vietnam and
Library of Hanoi Architectural University
LIST OF PUBLISHED SCIENTIFIC ARTICLES
OF THE AUTHOR RELATED TO THE THESIS
1. Lam Thanh Quang Khai (2016), Some methods in calculating
stresses and deformations of reinforced concrete shell roof structures.
Vietnam Journal of Construction (ISSN 0866-0762), No. 6/2016, pp
(165-168).
2. Lam Thanh Quang Khai, Le Thanh Huan (2016), Surveying the
stress-deformation of the laminated shell by anisotropic shell theory and
equivalent thickness diagram. Vietnam Journal of Construction (ISSN
0866-0762), No. 8/2016, pp(190-194).
3. Lam Thanh Quang Khai, Le Thanh Huan, Nguyen Tien Chuong
(2016), Surveying the stress-deformation of the 5-layer shell roof by
reinforced concrete with different boundary conditions. Vietnam Journal
of Construction (ISSN 0866-0762), No. 10/2016, pp(136-140).
4. Lam Thanh Quang Khai (2018), Research the stress-
deformation of double-layer reinforced concrete shell by experiment.
Vietnam Journal of Construction (ISSN 0866-8762), No. 3/2018, pp (58-
61).
5. Lam Thanh Quang Khai, Do Thi My Dung (2018), Stress-strain
in multi-layer reinforced concrete doubly curved shell roof. 15th World
Conference On Applied Science, Engineering And Technology,
12/2018, India (ISBN: 978-81-939929-2-0).
1
INTRODUCTION
1. Reasons for choosing the topic
In calculating the reinforced concrete thin shell roof, with thin shell roof types such as: one
or two-dimensional curved shell, cylindrical shell, spherical shell... according to calculus,
numerical methods, experimental... With a curved two- dimensional shell roof, the shell is quite
special because of the change in curvature on the shell, because different types of boundary
structures will greatly affect the deformation stress of the shell and there are few studies for this
type of structure.
Some typical research on two-dimensional curved shells, including analytical studies were
introduced by Vlasov [63], Le Thanh Huan [12][13][15][16][65], Ngo The Phong [21]. Some
research by numerical methods: Ahmad and his colleagues [27], Nguyen Hiep Dong [9][11],
Harish and his colleagues [40], Stefano and his colleagues [60]. Some experimental studies by Le
Thanh Huan [65] and studies of Meleka and his colleagues [51], Sivakumar [59]…
However, in fact, using two-dimensional curved shell roofs in Vietnam, there are other
layers besides the main bearing concrete shell layer such as waterproof layer, heat-resistant layer
or reinforcement layer, reinforcing the shell... creating multi-layer shell structure. In it, analytic
studies were introduced by Ambarsumian [26][66], Le thanh Huan [68], An-dray-ep and Nhi-me-
rop-ski [69] With the assumption that the inner layers of the shell are tightly bound, it is possible
to put the multi-layer shell into an equivalent one-layer shell.
In addition to studying grade composite shells or in addition to shell oscillation or
stabilization studies, multilayer shell studies were introduced by authors Rao [56], Mohan [50],
Nguyen Dang Quy and his colleagues [52], Ferreira and his colleagues [34], Francesco and his
colleagues [35]... However, these studies are not really clear and complete in calculating the
deformation stress state, the ability to split and slip between layers in the shell and it is still quite
complicated in calculation.
However, in calculating the structure of reinforced concrete thin shell roof with single layer
or multiple layers, there are still many issues to be studied and solved such as: It is necessary to
solve the system of high-order differential equations, it is not easy to clearly know the stress state
of each shell layer, there are not many experimental studies on the type of reinforced concrete
roofs in a layer or many layers…
From reference to domestic and foreign sources, there are very few studies on the treatment
of multi-layer reinforced concrete shell roofs, the ability to split and slip between layers in a
comfortable multi-layer sloped shell roof and the use of metal fiber reinforced concrete layer
dispersed in the shells.
2
Therefore, the author sees the need to study the topic: "Study the deformation stress state of
multi-layer reinforced concrete doubly curved shell roof" to clarify the above problems of multi-
layer shell is practical in both scientific and practical meaning.
2. Objectives of the study
Study the deformation stress state of two-layer curved reinforced concrete cshell roof with
two positive dimensions.
Study the effect of parameters on shear stress in the two-layer sloped shell roof and consider
the ability to split and slip between layers.
3. Object and scope of the research
Object of the research: two-layer curved reinforced concrete sloped shell roof with two
positive dimensions.
Scope of the research: studying deformation stress state of two-layer curved reinforced
concrete sloped shell roof under the impact of uniformly distributed load in the period before
concrete appeared cracks, in case the shell has constant thickness.
4. Research Methods
Study the theory of combining analysis on Sap2000 software and ANSYS numerical
simulation. Experimental studies were also conducted with shells made of reinforced concrete
materials. Methods are synthesized, analyzed and compared to evaluate results.
5. Scientific and practical significance of the topic
Scientific significance: The thesis contributes to elucidating deformation stress and the
ability to split and slip between the shells of the structure of multi-layer curved reinforced concrete
sloped shell roof with two positive dimensions.
Practical significance: the problem of positive two-dimensional curved sloped shell roof
made of multi-layer reinforced concrete material under load, with experimental calculation and
numerical simulation, the thesis has drawn some technical comments, so it has practical
significance.
6. The thesis structure
In addition to the introduction, conclusions, recommendations and appendices. The thesis is
presented in 4 chapters, the content of each chapter is as follows:
Chapter 1: Overview of studies of two-dimensional curved reinforced concrete sloped shell
roof.
Chapter 2: Theoretical calculation of multi-layer curved reinforced concrete sloped shell
roof with two positive dimensions.
3
Chapter 3: Study the deformation stress state of two-layer reinforced concrete sloped shell
roof by experiment.
Chapter 4: Study the state of deformation stress of two-layer sloped shell roof by numerical
simulation and parameter survey.
7. New contributions of the thesis
1. The contribution of an experimental research result on the behavior of two-layer doubly
curved shell roof by concrete and steel fiber reinforcement concrete through the construction of
diagrams: deformation, stress, internal force, deflection, load - slip deformation. Evaluate the
degree of bonding of the shell to the stage before concrete appears cracks.
2. Based on experimental research, numerical simulation of ANSYS software, it is
concluded that the doubly curved shell roof is made of non-slip concrete materials, capable of
working together as a single-layer shell model equivalent to suitable boundary and load conditions.
3. Using the built model, studying the effect of shell parameters on the deformation stress
state of the comfortable sloped shell roof, including: layer thickness, fiber concrete layer position,
fiber content in concrete…
CHAPTER 1: OVERVIEW OF STUDIES OF TWO-DIMENSIONAL CURVED
REINFORCED CONCRETE SLOPED SHELL ROOF
1.1. Overview of theoretical and experimental studies on a signle-layer two-dimensional
curved reinforced concrete sloped shell roof
1.1.1. Theoretical studies
1.1.1.1. Analytical studies
To solve the problem of reinforced concrete sloped shell roof, Vlasov [63] has set up a
system of two differential equations with two stress and displacement functions to be found as
and w bear the vertical load of yxq , :
yxqy
w
yx
w
x
wD
yk
xk
y
wk
x
wkEh
yyxx
,2
02
4
4
22
4
4
4
2
2
22
2
1
2
2
22
2
14
4
22
4
4
4
(1.1)
On that basis, Le Thanh Huan [15][65], Bai cop V.N [67] used the point method (semi-
analytical) to solve the system of equations Vlasov to find the internal force values, the stress in the
positive two-dimensional curved sloped shell roof in different boundary conditions.
In addition to solve the system of equations Vlasov, Ngo The Phong [21] In addition to
solving the system of equations Vlasov, Ngo The Phong used Navier's double trigonometric series,
4
the single trigonometric series of Levi, the method of general torque theory to be distributed to
determine internal forces and bending moments for curved shells.
1.1.1.2. Studies according to numerical methods
a) Method of successive approximations
The essence of this method is to solve the generalized second-order differential equation of
the form:
pwwwwwwwwww n
i
ii
ii
ii
ii
ii
1
2
22
2
2
2
22
2
2
This method of successive approximations is also used by author Nguyen Hiep Dong
[9][10][11] in his doctoral dissertation and articles published in the country.
b) Finite Element Method
Method using flat plate type elements: Using flat triangular elements, flat quadrangular
elements have been presented quite well in the documents: Richard [55], Lee and his colleagues
Method using curved shell elements: In order to better approach the geometry of the shell
structure, in analysis using curved shell elements, there are also many documents that are quite
well presented.: [31][36][66]...
Thanks to the application of Finite Element Method, with the support of computer facilities,
many forms of thin shell structure have been studied and developed by many domestic and foreign
authors, such as:
Bandyopadhyay and his colleagues [29] analyzed the curvature of a two-dimensional
curved shell structure. The displacement fields are made of polynomial approximations.
Do Duc Duy [8], Dang Van Hoi [18], Tram Anh Tu [17]... have further clarified the
deformation stress of a two-dimensional curved sloped shell roof, solving complex problems that
have almost never been solved before, such as the impact of air temperature, the influence of
boundary structures…
Hyuk Chun Noh [39] I have studied the limited capacity of large-scale reinforced concrete
thin shell structure, taking into account both geometric nonlinearity and nonlinear shell material..
Harish and his colleagues [40] I have studied the stress deformation of two-dimensional
curved concrete shell with Sap2000 software under load evenly distributed to the shell.
In addition to studying deformation stresses of shells, Stefano [60] have also studied new
design methods to minimize the use of shell materials such as shell shape, boundary conditions,
and loads…
1.1.2. Experimental studies
5
Le Thanh Huan [65] studied the deformation stress in a positive two-dimensional curved
sloped shell with a square model of organic glass material.
Recently, Meleka and his colleagues [51] carried out to evaluate the repair and reinforcement
of reinforced concrete shell with openings with polymer reinforced fiberglass materials (GFRP).
Sivakumar and his colleagues [59] studied the stress and curvature of the curved shell with
the rectangular surface, the curvature at the top of the shell is 80mm, the edge beam is 40 × 50mm,
with the shell thickness of 20mm and 25mm.
Jeyashree and his colleagues [45] I studied the stress and displacement of the comfortable
sloped shell with two-dimensional curved squares with the size of 68 × 68cm under the
concentrated load at the top.
General comments on theoretical and experimental studies of single-layer shells: The study
of theory or experimentation of two-dimensional curved sloped shell roofs only stops at the
comfortable one-layer sloped shell roof type, not to mention the multi-layer shell structure.
Therefore, the thesis continues to focus on the study of multi-layer two-dimensional curved
sloped shell roofs.
1.2. Overview of theoretical and experimental studies on multi-layer two-dimensional curved
reinforced concrete sloped shell roof
From the equation system of Vlasov, Ambarsumian [26] From the equation system of
Vlasov, Ambarsumian has developed an anisotropic multi-layer shell theory for thin shell
problems and is considered a theoretical basis for multi-layer shell studies..
Ambarsumian has concluded that the layers work in the elastic phase, not sliding on each
other to allow us to no longer consider the strain stress of each individual layer.
Rao [56] has developed stiffness matrices for multi-layer anisotropic sloped shells in
rectangle, the deformation stress state of the shell is calculated based on the intermediate surface
of the shell.
After that, Le Thanh Huan [14][68] in his study was based on Ambarsumian's anisotropic
multi-layer shell theory, which continued to be developed for the multi-layer positive two-
dimensional reinforced concrete sloped shell roof problems with the assumption that the layers
stick together.
In 2001, An-dray-ep and Nhi-me-rop-ski [69] has published its work on plates and
anisotropic multi-layer shells, bending, stability and vibration with a different approach to the shell
theory of Ambarsumian. Equal and continuous equations are written in tense form.
In the study, Carrera [30] studied multilayered shells, but only general theory studies, not to
mention the possibility of sliding separation of shells.
6
Francesco and his colleagues [35] studied the positive two-dimensional curved sloped shell
on the Winkler-Pasternak elastic foundation by general differential method.
Currently, from many domestic and foreign sources, no empirical studies have been found
on the behavior of the sloped shell roof and considering the possibility of splitting and sliding of
multi-layer positive two-dimensional curved shell roofs with reinforced concrete materials in large
sizes.
To elucidate the deformed stress of the multi-layer positive two-dimensional curved
reinforced concrete sloped shell roof and consider the possibility of sliding separation of layers,
the thesis presents the following research contents.
1.3. The contents need to be studied by the thesis
Study the deformation stress state of multi-layer shell roof according to analytical solution
and solution of the solution method through Sap2000 software.
Study the state of deformation stress of two-layer sloped shell roofs by experiment
Study the state of deformation stress of two-layer sloped shell roof by numerical
simulation.
Study the effect of each layer thickness, fiber concrete layer position to the deformation
stress state of the sloped shell roof and consider the ability to split and slip between shells by
numerical simulation.
CHAPTER 2: THEORETICAL CALCULATION OF MULTI-LAYER POSITIVE TWO-
DIMENSIONAL CURVED REINFORCED CONCRETE SLOPED SHELL ROOFS
2.1. Concepts and applications of thin shell roofs
2.1.1. The concept of thin shell roof
Two-dimensional curved shell roof: The two-dimensional curved reinforced concrete shell
roofs is called slope when the slope of any point on the surface of the shell compared to the bottom
plane does not exceed 180 or the curvature ratio f is the largest (The height from the center of the
plane contains 4 corners to the top of the shell roof) on the short side 5
1
a
f [15].
2.1.2. Application scope and advantages of thin shell roof
Thin reinforced concrete roof: Widely used in construction works. Thin reinforced concrete
roof is a form of space structure with advantages [15]: Suitable for large aperture works, large
space without intermediate columns. Compared with plans for using flat structures with the same
aperture, the thin shell roof has a self weight reduction of 20-30%, creating architectural works
with rich and impressive shapes thanks to the curved surface and large scale of the roof.
2.1.3. Two-dimensional curved sloped shell roof has been built in and out of the country
Table 2.1: Construction of two-dimensional curved sloped shell roofs has been built
7
No. Name of works Construction
location Surface size
Thickness
of shell
Year of
completion
1 The works of Wiesbaden Germany 3030m 9cm 1931
2 Brynmawr rubber factory UK 18.925.9m 9cm 1951
3 Smithfield Poultry market UK 38.168.5m 7.6cm 1963
4 Hall of Hanoi National
University Viet Nam 1818m 7cm 1996
2.2. Basic calculation theory of a 1-layer positive two-dimensional curved sloped shell roof
2.2.1. Vlasov's equation system [63]
yxqy
w
yx
w
x
wD
yk
xk
y
wk
x
wkEh
yyxx
,2
02
4
4
22
4
4
4
2
2
22
2
1
2
2
22
2
14
4
22
4
4
4
There are many methods to solve the system of differential equations of level 4 (2.5), but it is
not very simple. The complexity is that for a sloped reinforced concrete roof, two functions and
w must be found so that they both satisfy the system of equations (2.5) and satisfy different
boundary conditions.
2.2.2. The calculation of the shell according to the non-torque state
2.2.2.1. Use Navier's double trigonometric series [21]
The non-torque internal force of the shell is determined by the formula (2.7):
b
yn
a
xm
nkmkm
nqN
m n xy
sinsin
162222
2
1
b
yn
a
xm
nkmkn
mqN
m n xy
sinsin
1622222
(2.7)
.2.2.2. Application of Lévi single trigonometric series [21]
Non-moment internal force of the shell is determined by the formula (2.9):
xbnCh
yChqRN n
n
n
sin
41
nn
n
n xbCh
yCh
n
qRN
sin1
142 (2.9)
2.2.2.3. Application of point method (semi-analytical method)
Depending on the requirement of works use, marginal structures have different forms, such
as structure of flat trusses, beam, wall or rows of column or pillars at 4 corners…Bai cop [67] and
L. T. Huan [15][65] have presented 3 circumstances upon applying the non-torque theory.
8
2.2.3. Shell calculation based on moment states
2.2.3.1. Shell calculation based on marginal effect theory[15][21]
a). In case of shell having pinned connection with marginal strucrure:
Figure 2.9: Bending moment diagram in case of shell having pinned connection with margine
b). In case of shell having clamped connection with marginal structure:
Hình 2.10: Bending moment diagram in case of shell having clamped connection with margine
2.2.3.2. Shell calculation based on general theory of moment [21]
Expressions of internal force and moment as follows:
yxnkmkm
nKqN
m n xy
mn212222
2
1 sinsin16
yxnkmkn
mKqN
m n xy
mn2122222 sinsin
16
yx
nmn
KmqaM mn
2122224
2
1 sinsin116
yx
nmm
KnqaM mn
2122224
22
2 sinsin116
2.3. Calculating theory of doubly curved shell roof with multiple layers
2.3.1. System of equation for solution of reinforced-concrete doubly curved shell
roof with multiple layers by rectangle [26][68]
Figure 2.12. Quantity of roof layers
See the following equation:
9
2.3.2. Stress and deformation of doubly curved shell roof with multiple layers
Internal force of shell roof is determined as follows:
b
n
a
m
m
nqaN
m n
sinsin
16
...3.1 ...3,1 1
3
6
26
2
2
b
n
a
m
n
mqaN
m n
sinsin
16
...3.1 ...3,1 1
3
6
26
2
2
(2.32)
Normal stress and bidirectional moment:
b
n
a
m
mn
mAnAmaAnaAqa
m n
iiiii
sinsin
16
...3.1 ...3,1 1
2
24
7
22
2
2
6
2
3
4
5
2
3
24
4
6
2
b
n
a
m
mn
nAmAnaAmaAqa
m n
iiiii
sinsin
16
...3.1 ...3,1 1
2
222
11
22
210
22
3
4
9
2
3
4
8
6
2
b
n
a
m
mn
anAmAnAmA
qaM
m n
iiii
sinsin
16
. . .3.1 ...3,1 1
2
3
422
17
2
162
22
15
2
14
4
2
b
n
a
m
mn
amAnAmAnA
qaM
m n
iiii
sinsin
16
. . .3.1 ...3,1 1
2
3
42
20
22
192
2
15
22
18
4
2
2.4. Solution of curved shell roof with multiple layers based on equivalent single-layer shell
theory
2.4.1. Double-layer shell roof
2.4.1.1. Analytical solution
Figure 2.13. Orthotropic double-layer shell roof [68]
Internal force is:
8 8 8 8 8 6
1 3 5 4 2 18 6 2 4 4 2 6 8 6
6 6 6 4 4 42 2
3 4 2 2 1 2 14 2 2 4 6 4 2 2 42
P P P P P O
O O O k k k k Z
10
b
n
a
m
mn
mnmqRN
m n
sinsin
16
...3.1 ...3,1 0
2222
2
2
1
b
n
a
m
mn
nnmqRN
m n
sinsin
16
...3.1 ...3,1 0
2222
2
2
1
(2.39)
Vertical displacement (deflection):
b
n
a
m
mn
nm
EhC
qRw
m n
sinsin
16
...3.1 ...3,1 0
2222
2
2
1
(2.40)
Bending moment:
b
n
a
m
mn
Fnnm
ha
RPvnmnm
C
hqRM
m n
sinsin216
. . .3.1 ...3,1 0
22222
2
1
2
122222222
2
1
b
n
a
m
mn
Fmnm
ha
RPvnmnm
C
hqRM
m n
sinsin216
. . .3.1 ...3,1 0
22222
2
1
2
122222222
2
1
Example 2.1:
Positive two-way curved shell roof with the square surface dimension a=b=36m, curve
radius R1=R2=1.25a. Layer I: the concrete layer with thickness of hI=10cm, B25, modulus of
elasticity EI=315000kG/cm2. Layer II: the concrete layer with steel mesh B20, thickness of
hII=4cm, EII=265000kG/cm2. Similar poisson ratio v=0.2. Load, including dead load and live load
on the roof is 500kG/m2. Calculating internal force, stress and deflection of shell roof with
margine is pin-connected frame system.
Solution:
Figure 2.14. Internal force, stress, deflection diagram of double-layer shell [68]
11
2.4.1.2. Solution of finite element method by Sap2000 software
a). Construction of shell roof structure model
Figure 2.18. Internal force and deflection diagram of double-layer shell
Note: Red line: based on analytical solution [68]; Blue line: based on Sap2000
Remarks:
1. Variance of internal force is from 11.8% - 31%, variance of deflection is from 12% - 21%.
2. For reinforced concrete structure, various marginal structures have different hardness,
significantly influencing on deformation and stress of shell structures.
In order to clarify stress and deformation at marginal region as well as influence of shell
layers, the author carries out calculating of 5-layer shell roof with both marginal conditions: clamp
and pin, as follows:
2.4.2. 5-layer shell roof
2.4.2.1. Stress and deformation of 5-layer shell roof with pinned marginal condition
a). Analytical solution
Example 2.2: Positive two-way curved shell roof with the square surface a=b=36m,
R1=R2=45m, including 5 layers: layer 1 (bottom) by concrete B25 with thickness h1=3cm,
21 /315000 cmkGE ; layer 2 with thickness h2=19cm, 2
2 /141750 cmkGE ; layer 3 by concrete B25
with thickness h3=3cm, 23 /315000 cmkGE ; layer 4 by concrete B20 with thickness h4=5cm,
24 /264915 cmkGE ; layer 5 (top) by concrete B5 with thickness h5=2cm, 2
5 /10710 cmkGE .
12
Similar Poisson ratio v=0.2. Load, including dead load and live load on the roof if 500kG/m2.
Calculating internal force, stress and deflection of shell roof with margine is pin-connected frame
system.
Figure 2.20. Internal force and stress diagram of pinned marginal 5-layer shell
b). Solution of finite element method by Sap2000
Hình 2.22. Internal force diagram of pinned marginal 5-layer shell
Remark: “Stress distribution of multi-layer shell depends on the number of layer and
modulus of elasticity of each layer”, these are problems on reinforced concrete multi-layer shell
roof which are not clearly assessed.
2.4.2.2. Stress and deformation of 5-layer shell roof with clamped marginal condition
a). Analytical solution
Thus, in the analytical method, application of double trigonometric series sin , sin is not
still appropriate.
b). Solution of finite element method by Sap2000 software
13
Figure 2.25. Internal force, stress and deflection diagram of clamped marginal 5-layer shell
Remark: Internal force N at a location approaching the clamped marginal condition is less
than the pinned marginal condition, and internal force N is more than the pinned marginal
condition. The result shows influence of marginal conditions is very important.
2.5. Remark
Through the value of internal force and stress in shell, it is shown that “Stress distribution of
multi-layer shell depends on the number of layer and modulus of elasticity of each layer”
The results of internal force, stress and deflection based on analytical solution and Sap2000
are similar, thus, the theory of single-layer shell may be applied to determine stress and
deformation in shell with suitable load.
CHAPTER 3: RESEARCH ON STRESS AND DEFORMATION IF REINFORCED
CONCRETE DOUBLE-LAYER SHELL ROOF BASED ON EXPERIMENT
3.1. Objective and content of experimental research
3.1.1. Objective of experimental research
a) To studye working capacity of 2 concrete layers with different strength
b) To build diagrams : deformation, stress, internal force, deflection, relation between load -
creep deformation of shell.
3.1.2. Content of experimental research
Including: Making design of experiments, carrying out experiments and handling
experimental results
14
3.2. Basis for design of experimental models
3.2.1. Basis for design of experimental models
In theoretical researches [26][66][68], it is assumed that layers of shell roof are adhered, but
it is not specified which marginal structure is applied and how load is limited
3.2.2. Establishment of experimantal models for thesis
- Because the model of reinforced concrete shell roof is relatively large and the way to form
and make experiments on multi-layer shell roof is complicated with a lot of time, through the
ANSYS simulation, it is showned that if surfance dimension 33m, it is qualified to sensitively
response to the load.
- In application of shell roof in Vietnam, the underlying layer of shell roof is a main bearing
layer, waterproof concrete layers, heat-resistant concrete layers with low strength lies on shells. In
case of shell repairment, it will be researched by simulation with BTS layer lying on plain
concrete.
3.3. Design and manufacture of experimental models
3.3.1. Materials
- Concrete B20 (M250) for plain concrete and B30 (M400) for fiber reinforced concrete.
- steel fiber (0.5-L30mm): stell fiber meets ASTM A820-01 [23], fiber direction ratio is
from 50 to 100 meeting ACI 544.1R-1996 [22]
3.3.2. Experimental models
Figure 3.2. Design of shell roof 33m based on experiments
.3.3. Purpose, type and position of pasting strain gage
- Purpose of pasting strain gage (tenzomet resistor): measure deformation on concrete
surfaces and on reinforcement in each layer, thereby determining stresses and internal forces at the
pasting positions.
15
- Type of strain gage and strain gauge equipment: using strain gage type BX120-30AA, Leaf
form with 30mm long, 3mm wide, resistor Rgage=120, gage coefficient =2.081%. Using the
strain gage device of Data loger TDS-530 (30 channels), Data loger TDS-601 (10 channels) by
Vietnam Institute for Building Science and Technology IBST and Strain Indicator P-3500, set of
channel switch SB10 (10 channels).
- Position of pasting strain gage: from preliminary calculation results and simulation results
on ANSYS software
- Paste method: [48]
3.3.4. Manufacture laboratory samples
The steps are as follows:
- Step 1: processing the formwork in accordance with the shape of the positive two-direction
curved shell roof
- Step 2: pouring concrete for layer 1, which is steel fiber concrete with 2% content of steel
fiber in concrete,
- Step 3: continue to process reinforcement, strengthen edges, paste strain gage to the
middle steel, the strain gage is welded with anti-interference electric wires.
3.3.5. Sample maintenance: according to TCVN 8828-2011 [5]
3.4. Test for physical and mechanical properties of materials
3.4.1. Test for determining compressive strength of concrete: TCVN 3118-1993 [1]
3.4.2. Test for determining elastic modulus of concrete: TCVN 5726-1993 [2]
3.4.3. Steel tensile test
In the shell, there is no bearing steel bar in the shell, so it is not experimented for steel
tensile.
3.5. Test of 2-layer reinforced concrete shell roof
3.5.1. Layout diagram of test equipment
e) Position pasting strain gage on the underside f) Position pasting strain gage on the topside
16
g) Paste strain gage on steel at layer 1 h) Paste strain gage on steel at layer 2
Figure 3.11. Position pasting strain gage on the shell
3.5.2. Conduct the test: Implemented as follows: Step 1: Preparation work, Step 2: Install and
check measuring devices, Step 3: Start the test
3.5.3. Test result of 2-layer shell roof
Figure 3.15. Relationship between the load and the shear deformation of the shell
Comment: Shear deformation (Figure 3.15) at load level 611kG/m2 là 4.310-5, vis still small
compared to the extreme relative deformation of concrete which is cu=3.510-3. See as at the shell
edge of shear deformation, the layers are very small and can be ignored, meaning that the steel
pins linked between the two shells are not effective.
Figure 3.16. Deformation in the layers of the shell
SHEAR
DEFORMATION
17
Figure 3.17. Stress in layers of the shell
Figure 3.18. Internal force Nx, Ny of the shell
Figure 3.19. Compare stress and deflection results by EXP and SAP
Comment: We see that the position inside the shell has negligible stress difference between
the experiment and Sap2000.
3.6. Comment
The layers of shell roof do not slip on each other, working together as a multi-layer structure,
can use the equivalent single-layer shell model when the load is appropriate..
CHAPTER 4: STUDY ON THE DEFORMATION STRESS STATE OF TWO-LAYER
SHELL ROOF BY NUMERICAL SIMULATION AND PARAMETER SURVEY
18
4.1. Introduction about ANSYS software and study contents
4.1.1. Brief introduction about ANSYS software
The sequence of solving structural problems in works with ANSYS software includes the
following basic steps and is divided into 3 groups: data processing, calculation and processing of
calculation results.
4.1.2. Contents of numerical simulation study
- Build the FEM for the two-layer roof shell of the two layers to experiment
- Completing the FE model by adjusting the input parameters from the test results of normal
concrete materials, fiber concrete and steel fibers.
4.2. Select modeling of fiber reinforcement smeared in concrete
To modeling fiber reinforcement in concrete, three models are used: smeared model,
embeded model and discrete model [24] [32]. Thus, in this study, fiber reinforcement smeared in
concrete should use the smeared model to be reasonable.
4.3. Select modeling cracks in concrete
Currently, cracks in concrete are modeled in two basic forms: discrete model and smeared
model [38]. In this study, we select the smeared model for cracks in concrete.
4.4. Select the interface model between 2 layers of concrete
In calculation, it is possible to use the interface element or thin layer element to simulate the
shear interface surface between two different concrete layers. [19].
4.5. Building finite element model for the shell roof
4.5.1. Elements in the model
Concrete simulation element: SOLID65 element
Interface element: simulated by the type of TARGE170 element for 3D interface. The object
surface is modeled by the type of CONTA173 element.
4.5.2. Divide mesh for the model
The principle when dividing the mesh must ensure that elements shall share nodes together,
so we divide by shell thickness equal to layer 1 (ESIZE, ALL, H1) and divide the mesh freely
(MSHKEY, 0) with mesh shape divided into 3D tetrahedral blocks (MSHAPE, 1,3D).
4.5.3. Edge conditions and effective load
The shell is rigidly linked with edge curved beams. The effective load distributes on the top
side of the shell at the nodes of tetrahedral block mesh (NSLA, R, 1), by P compressive force
evenly distributed on the shell surface (SF, ALL, PRES, P).
4.6. Material model
4.6.1. Concrete material model
19
Figure 4.10. Strain stress curve of concrete when pulling and compressing an axis [28]
4.6.1.1. Strain stress model of concrete under compression
Through serveying strain stress models of concrete under compression is presented above
and the results of strain stress of tested concrete (Figure 3.9), we see that the test results are
consistent with Kachlakev's model..
4.6.1.2. Strain stress model of concrete when under the tensile
This model is already defined in ANSYS (Figure 4.15). [24]
4.6.2. Destruction standards of concrete
Willam and Warnke's destruction standards are used in this study and defined in ANSYS.
Concrete will be cracked or crushed if it satisfies the equation (4.10). [64]
4.7. Input parameters for the model
In ANSYS, to enter the input parameters for concrete element SOLID65, we must enter the
following 8 basic parameters: Cutting force transmission coefficient when crack is opened 0 , 2.
Cutting force transmission coefficient when crack is closed C , 3. Cracking stress when tensiling
an axis rf , 4. Crushing stress of an axis '
Cf , 5. Coefficient is reduced and weak due to cracking
when tensiling (default select as 0.6), 6. Elastic modulus CE ,7. Poisson's coefficient, 8. Curve of
stress strain relationship of concrete.
4.8. Research results between the test and numerical simulation
4.8.1. Deflection in the shell
Figure 4.17. Deflection of methods
20
4.8.2 Stress in the shell
Figure 4.18. Stress of methods
4.8.3. Deflection and stress of the shell roof at the load level starts appearing concrete cracks
Period starts appearing concrete cracks: Load level P=14kN/m2=1400 kG/m2, stress
13.38kG/cm2 then, the first crack appears in the shell, running along the lower edge of the BTS
layer, the maximum deflection at the top of the shell is 0.17mm.
4.8.4. Comment
The analytical results show that this FE model is suitable with the test and other software
(Sap2000). It is possible to use the ANSYS model to servey the effects of layer thickness, fiber
concrete layer position to strain stress in the shell and the ability to split and seperate among
layers.
4.9. Survey the parameters affecting the strain stress of the shell roof by numerical
simulation
4.9.1. Thickness parameters of each layer
a) The deflection of the shell of the survey cases
b) Stress x c) Stress y
Figure 4.22. Deflection and stress of survey cases
21
4.9.2. Position parameters of fiber concrete layers
a) Shell feflection of case 2 and case 4
b) Stress x c) Stress y
Figure 4.23. Deflection and stress of case 2 and case 4
4.9.3. Servey shear layers in the shell roof
Table 4.7: The result of the largest tangential stress calculation
Stress component BTS 2cm under the
BTT 3cm layer
BTS 3cm above the
BTT 2cm layer
Tangential Stress max 0.094MPa 0.069MPa
Corresponding normal stress 0.346MPa 0.276MPa
Comment: When the affected shell of the load is evenly distributed on the top of the shell
perpendicular to the shell surface, there is a phenomenon of shearing among layers in the roof.
After being affected by the load, at the interface position between the two shell layers, there is a
relative strain between the two layers equal to 110-3 and still much smaller than the relatively
limited strain of concrete cu = 3.510-3
4.10. Stress strain state of the roof shell 3636m
a). Stress and deflection of the shell in case of considering nonlinear of materials
22
a) When concrete starts to crack b) When concrete starts to sabotage
Figure 4.26. Stress of the shell when the content of steel fiber core changes
a) When concrete starts to crack b) When concrete starts to sabotage
Figure 4.27. Deflection of the shell when the content of steel fiber core changes
b). Comparing the stress and deflection of the shell when analyzing linear and nonlinear of
materials
a) When concrete starts to crack b) When concrete starts to sabotage
Figure 4.28. Stress of the shell when analyzing linear and nonlinear
23
a) When concrete starts to crack b) When concrete starts to sabotage
Figure 4.29. Deflection of the shell when analyzing linear and nonlinear
c). Shearing among shell layers
Table 4.12: Results of shearing calculation when steel fiber content changes
Value Content of 0% fiber Content of 8% fiber
Tangential Stress max 0.408MPa 0.389MPa
Corresponding Normal Stress 1.705MPa 1.774MPa
Load causes shearing 900 kG/m2 950 kG/m2
Comment: When this load causing shearing has not been passed, the layers inside the shell
do not have a shearing phenomenon, so in calculating, the roof shell can use the equivalent single-
layer shell theory
4.11. Comment
A survey among layers shows that shear strain is very small, can be ignored and able to
return to the equivalent single-layer shell theory when the load is appropriate..
In addition to the numerical simulation of the tested roof shell, the thesis also extends the
studied problem with layer span shell roof and nonlinear materials.
CONCLUSIONS AND RECOMMENDATIONS
I. Conclusion
1. The thesis has designed, fabricated and tested a shell model with quite large size of 3 ×
3m made of concrete and smeared steel fiber reinforced concrete, which are usually done on small
models and convertible materials. Can evaluate the linking level among the shell layers.
2. Has implemented the calcultation of numerical simulation of experimental shell roofs
using ANSYS software. Comparing numerical simulation results with results calculated with
Sap2000 software and experimental results to evaluate simulation parameters, from which there is
a basis for surveying the layer parameters and evaluating the the shearing level among layers
3. Through the experimentation and calculation of numerical simulation, it has determined
that the load causing shearing. When this load has not passed, the inner layers of the shell do not
have a shearing phenomenon, so in calculating the roof shell can use the equivalent single-layer
shell theory
II. Recommendations
- Recommendations:
▪ When the load applied to the shell is equal to the load itself and the live load on the roof, it
is possible to completely replace the reinforcement of the bar in the double-sided curved shell roof
24
with many layers using steel fiber reinforced concrete. When the load is overtaken, the shell roof
will be cracked, so it is necessary to arrange steel bars with the outer structure of the fiber
reinforcement.
▪ When calculating the shell design, in addition to the position near the edge with complex
strain stress, it is also necessary to consider at the position of the shell top and the shell angles..
▪ It is possible to use an equivalent single-layer shell theory with appropriate edge and load
conditions.
- Development direction of the topic:
▪ Subsequent studies can be developed for shell roof with openings and in different types of
thin shell roofs such as: Spherical shell roof, cylindrical roof, negative double-sided curved shell
roof, etc. for problems of heat and wind ..., different types of edge conditions.
▪ Studying to build general equations containing shell thickness parameters affecting the
strain stress state in the shell, or studying to build equations containing shell span parameters...
