Dissertation Summary.pdf

NhËn xÐt 3 §å thÞ h×nh 4 MINISTRY OF EDUCATION AND TRAINING THE UNIVERSITY OF COMMUNICATION AND TRANSPORT NGUYEN ANH TUAN RESEARCH FOR METHOD ANALYSING CONCRETE PAVEMENT AND CONSIDERING THE EFFECTS OF[.]

MINISTRY OF EDUCATION AND TRAINING

THE UNIVERSITY OF COMMUNICATION AND TRANSPORT

NGUYEN ANH TUAN

RESEARCH FOR METHOD ANALYSING CONCRETE PAVEMENT AND CONSIDERING THE EFFECTS OF TRANSVERSE SHEARING DEFORMATION

Specialization: Construction of Highways and City Streets

Code: 62.58.30.01

DISSERTATION SUMMARY

HA NOI-2013

Completion in:

THE UNIVERSITY OF COMMUNICATION AND TRANSPORT

Scientific Guider:

1 As-Prof Dr La Van Cham

2 Prof Dsc Ha Huy Cuong

Critic 1: Prof Dsc Nguyen Van Lien

Critic 2: Prof Dr Vu Dinh Phung

Critic 3: Prof Dr Nguyen Xuan Dao

Dissertation is protected in The Council in the University of Communication and Transport

………

In time ……….hour…….on date……month…….year 2013

Can find out about this dissertation in:

- University Library

- National Library

ISSUE

The current sheet method, based on Kirchhoff plate theory, without considering the effects of transverse shear strain induced by shear forces, while not directly allow satisfying 3 the boundary conditions on the edge of plate

Solve plate on elastic foundation, according to Kirchhoff plate theory, can not be determined accurately on the boundary and internal plate corner, is not the ground state of stress

Thesis research allows us to identify the state of stress and deformation of the plate and of the foundation simultaneously, directly satisfy the boundary conditions on the edges 3 plates

CHAPTER 1 ELASTIC FOUNDATION MODELS AND PLATE ANALYSIS 1.1 Elastic foundation and Structure-Soil Interaction

- “Plate on elastic foundation” is solved two basic problems: Plate and Elastic foundation

- Elastic foundation models have done Winker foundation and half-space foundation are used to the most popular

1.2 Plate theory of G.R.Kirchhoff

1.2.1 Basic assumptions

* Surface of the average plate in not distortion

* The section is plane and perpendicular with repect to the average section of the plate

* Separate layers are not prevention together

Based on this assumptions, they only study average section where has vertical displacement w x y and internal forces impaction  ,

1.2.2 Balance equations and boundary conditions

1.2.2.1 Balance equation between external forces and deflection:

1.2.2.2 Boundary conditions of rectangular plate:

a/ Edge plate is associated restraint:

Deflection and Rotation are zero

b/ Edge plate is associated joint :

Deflection and bending moment are zero

c/ Edge plate is free:

Bending moment and converted shear are zero

To do this work, Kirchhoff’s theory is not considering effect of shears, and considered equivalent to the slip modul of materials G  So, the problem is much sipler, and it is true when what is a thin plate, and when the impact load is not on edge plate

1.2.3 Effects of shear deformation

1.2.3.1 Basic content of E.Reissner’s plate theory

Eq (1.40) is satisfied for 4 boundary conditions on the two edges

of the plate similarly Kirchhoff’s theory

To find Q Q x, y functions, E.Reissner chosed any stress function

 , what is satisfy conditions :

2

100

However,  stress function is not general Do this work, will have difficulty to satisfy boundary conditions and is approximately for stress function  when use numerical method By side, conditionM xy  0 on the free edge is not mentioned

1.2.3.2 Plate theory has based on Timoshenko beam theory

E.Reissner’s theory and Timoshenko beam theory are only different a one constant

1.3 Basic on Gauss’s principle extreme to construct balance equations

1.4 Conclude chapter 1

Basic on Gauss’s principle extreme to construct balance equations

and boundary conditions of the plate

E.Reissner’s theory and Timoshenko beam theory are only different the constant and Post-Graduate will use Gauss’s principle extreme and Timoshenko beam theory to construct and solve “Plate on the elastic

foundation” in myself thesis

CHAPTER 2 PLATE ON ELASTIC FOUNDATION THEORY HAS

CONSIDERING INFLUENCE OF SHEAR DEFORMATION 2.2 Plate on Winkler foundation

- Solve the plate on Winkler foundation problem followwing Kirchhoff’s theory, only have to solve equation (2.10) to definite deflection functionw( x, y ), and then to definite internal forces in the plate:

- Solve the plate on Winkler foundation problem while have considering effects of shear deformation have to solve eq (2.58), (2.59) and (2.60) to definite functionsw( x, y ), Q x y x , and Q y x y : ,

Boundary conditions of rectangular plate:

On x = 0 and on x = a edge, and not impaction of load on that:

a/ Edge plate is associated joint :

y x xy

w

Q Q

Q Q w

y x xy

Q Q

Q Q w

On y = o and y = b edge, and not impaction of load on that: like over

2.3 Plate on elastic half-space foundation theory while have considering effects of shear deformation

And now, a V is soil volume, as Fig 2.4 That is impacted P straight load, as (1) volume, be showing dashed line The aim is be to definited      x, y, z, xy, xz, zy of V volume under impaction of P by the through      x o, o y, o z, o xy, o xz, zy o that’s knowns components based on R.D.Mindlin’s or J Boussinesq’s solutions

Fig 2.4 Using comparisons

based on Gauss’s principle

extremme for solve plate on

In this: u v w , , d are displacements to axis X Y Z , ,

Based on Gauss’s principle extreme, by through comparison, has functional Zd:

x y z xy xz zy

      what have defined

2.3.2 Plate on elastic half-space theory and when considering influence

Fig 2.5 Place on elastic half-space model, have considering effects of shear

deformation, based on Gauss’s princiole extreme

When z  0 and in scope    a b of the plate:

2

o o o

When z  0and out side of the plate, has (2.66)

Substitution (2.16) into (2.71), has:

2

5 1

o o o

of stress-deformation of the elastic foundation

 Satisfying three boundary conditions on edge plate by the way

considering effects of shear deformation

 Only have to solve equation (2.10) to define deflection function

  ,

 Solve equations (2.58), (2.59) and (2.60) could define deflection functionw x y   , and shear function Qx  x y , and shear function

  ,

y

Q x y while considering effects of shear deformation

 Solve plate on elastic half-space based on Gauss’s could

completion about bending plate theory

CHAPTER 3 SOLVE PLATE ON ELASTIC FOUNDATION, [5]

In this, Post-Graduate use 4 nodes rectangular element has 16 degrees of freedoms, called is BFS-16, interpolation about displacement and shear function of the plate by Hermite function Subgrade element is cubic finite-8 nodes and based on Mindlin’s solution The feature of Winkler foundation model is coefficient (k).And the feature elastic half-space model is elastic modulus (Eo) and coefficient poisson (o)

3.1 Establish programe by FEM

x = 1

( ) 1 3x 2x ( ) 4x 4x ( ) x 2x

Q Q Q

Fig 3.4 Shear element Qx

of 6 nodes plate element

Fig 3.5 Shear element Qy

of 6 nodes plate element

Like shear element Qx:

3.1.4 Stiffness matrix of plate element:

Displacement element w has 16 unknowns, displacement element

Qx has 6 unknowns and displacement element Qy has 6 unknowns Ttotal is

28 unknowns Called  U is vector having 28 unknowns:

  T

x y

U  w Q Q  (3.34) Expension vector  N w becoming   N what have 28 ingredients:   N    Nw, e z ro  1,12    (3.35) Deflection function w in any point of the plate element, eq (3.20), rewrite:

Element plate like plate and that has two-way link, so, extreme condition (2.29), becoming:

Have done this work with i = 1,2,…,28 , could have stiffness

matrix of plate element Ae 28 28  

Have connect stiffness matrixs of element, with external force function and boundary conditions and continuous conditions, could establish general stiffness matrix

3.1.5 Element of subgrade reaction  R based on Winkler model:

Subgrade reaction  R in any point of element:

2 3

(-1,1,-1)

(1,-1,-1) (-1,-1,-1)

(1,1,-1)

z

x o

z y

1 1

8 nodes element Load

Fig 3.9 Plate on elastic half-space foundation model, based on FEM

Deflection following x y z , , of any point in element:

1 8

1

, ,, ,, ,

i i i

i i i

      are vectors that have 8 compositions –

interpolation functions following (3.52)

3.1.7 General stiff matrix:

Functional ZZ Z min, when:

BLOCK DIAGRAM

To d e c l a r e

Plate Element Size Subgrade Model

Mo v in g l o a d

t o n o d e

WINKLER FOUNDATION / ELASTIC HALF-SPACE

Unnknown of Subgrade Stiffness matrix of Subgrade

Pl a t e

Unknown of Plate Parameters of Plate Interpolation function Stiffness matrix of Plate

Es t a b l is h

g e n e r a l s t if f n e s s ma t r ix

Load Boundary conditions Continuous conditions

s o l v e

r e s u l t s

Deflection Curly moment on edge Subsidence and stress of subgrade

c h e c k

Bending-tractive stress of plate < =

US kBending-tractive stress of material

0 5

3.4 Conclude chapter 3

 By the way considering effects of shear deformation and solve plate on elastic foundation program, good resuts:

- Satisfying three boundary conditrions on edge plate

- Not only define redistribution of internal in the plate, but also define contemporaneous stress and displacements of plate and foundation

 Change of stress and deformation status of plate:

- - Curly moment in free edge of the plate is not zero while ignore effects of shear deformation And else, curly moment is very small ( < 10-11) while considering effects of shear defomation, can be zero

- - Deflection maximum value is 5.5% difference-while ratio h/a  1/20 While 1/15  h/a  1/5 , this

Kich thuoc tam, 50cm x 10

Gia tri do lun, cm

- Having redistribution of internal in the plate Even, having reversal of moment-graph This change is very more clear when thickness plate is more increased, specially when the place be effected of load in corner

or in edge plate, so, 32.1% difference

- About moment value, when ratio (h/a) the more increasing, the more moment and deflection value is disparity, comparing that be in edge and corner to center plate

 Plate-Foundation model that Post-Graduate have choosed:

- Plate on elastic foundation-4 free edges

- Most adversary status: P – Converged load in midle length-edge of plate

 Have to reseach about effect of shear deformation, when:

CHAPTER 4

AN APPLICATION OF TC2BRP & TC32RP PROGRAME TO

DESIGN CONCRETE PLATE PAVAMENT

In this:

 Establish nomogram to define thickness of plate

 Calculate steel to reinforce for edge and corner of plate

 Calcuate foundation

 Comparison result of programe with some the other methods

4.1 Establish nomogram

Result 1:

Fig 4.1.Nomogram to define thickness of concrete pavament plate Based on model: plate on elastic half-space foundation

ch daN/cm

2

The nomogram is established for the aim:

- Based on parameters input,such as: External load, material, k, Eo,

, …put into the nomogram to define rudiment thickness plate (h)

- Put (h) and other parameters into TC2BRP or TC32RP programe, have calculated internal forces and stress and displacement of plate and of foundation under plate And then:

+ Calculate steel to reinforce for plate based on chart moment + Based on chart of stress and displacement of foundation to calculate and design foundation layer material

4.2 Calculate steel to reinforce for plate

4.2.1 Calculate steel to reinforce for edge of plate

0 2 4 6 8

- Seeing is 3 to 4 sheets of edge element has deflection most, Graduate proposals to strengthen reinforced this domain Based on the particle size, plate size and characteristics of elastic plates, Post-Graduate choose range enhance reinforcement from the back edge of the plate, by: Max (120cm, a / 3; 1.3L), (L is the elastic characteristics of the plate)

Post-Fig 4.3 Basic solution to calculate steel to reinforce for edge of plate

-1000 -500 0 500 1000 1500 2000 2500

Kich thuoc tam b = 1.3a, cm

4.3 Calculate and design foundation layer material

Based on the charts and graphs of stress subsidence of the ground, from which the decision of the foundation layer material

With each work load, plate material and plate thickness, we establish the relationship between the bending tensile stresses in the plate section is reviewed and elastic modulus of the ground / foundation

If the fixed plate material, thickness of nails beneath, to satisfy the conditions of bending tensile stresses in the plate, we completely determine the elastic modulus for subgrade requirements, and vice versa

Upright load: 55kN

Plate: h=24cm; E = 315000daN/cm b 2

Foundation: 18cm; E m

Ratio b/a: 1.3 Upright load: 65kN

Fig 4.8 Relationship between the stress of plate with module of foundation

According to calculations by graduate students, in the same condition as the other, then:

 When cement concrete sheet thickness increased / decreased

by 1 cm, modulus of elasticity of foundation under a reduced / increased by 40-50daN/cm2

 When the work load increase / decrease of about 10kN, thick concrete slabs increase / decrease of about 2cm

4.4 Some comparisons

Post-Graduate have compared the results of their calculations with analytical formulas of Westergaard and Shekter-Gorbunov-Pasadov, R805FAA software, KenSlabs-2003

The above comparison purposes only difference comes when and when not taking into consideration the effects of transverse shear strain in the plate, rather than specifying the value and stress torque plates to design, because not to mention other factors, such as considering the impact factor

of the load, for the same load, the effect of boundary links, the influence of temperature, humidity,

The biggest difference here is TC32RP program of Post-Graduate allows to define simultaneously determine the stresses and displacements of the plate and the subgrade

The authors above, who found the stress and displacement of foundation by other ways

CONCLUSIONS AND RECOMMENDATIONS

1.Conclusions:

Thesis research allows us to identify the state of stress and deformation of the plate and of the foundation simultaneously, directly satisfy the boundary conditions on the edges 3 plates

The new study results of the thesis, as follows:

a/ Theoretical:

 First:

By considering the effects of transverse shear strain and solving elastic plate-foundation system by the way of comparison method based on Gauss principle extreme theory, Post-Graduate have completed another step

computation theory " plate on elastic foundation ":

 The plates on Winkler elastic foundation, without considering the effect of transverse shear strain, just have to solve a eq (2.10), implicitly define a single deflection w x y   , of the plate, thereby determining the value of the internal force of plate