Research Collection Report Finite element-based elasto-plastic optimum reinforcement dimensioning of spatial concrete panel structures Author(s): Tabatabai, Seyed Mohammad Reza Publication Date: 1996 Permanent Link: https://doi.org/10.3929/ethz-a-001734882 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library
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Contents i
Contents ii
Contents iii
Abstract 1
Zusammenfassung 2
1.3 Assumptions and limitations 3
1. Problem Definition and Scope Formulation 4
1.1 Reinforcement dimensioning today 5
1. Problem Definition and Scope Formulation É ÉÉÉÉ ÉÉÉÉ ÉÉÉÉ ÉÉÉÉ ÉÉÉ ÉÉÉÉÉÉÉÉÉÉ ÉÉÉÉÉÉ ÉÉÉÉÉ ÉÉÉÉÉ ÉÉÉÉÉ ÉÉÉÉÉ ÉÉÉ ÉÉÉÉÉ ÉÉÉÉÉ ÉÉÉÉÉÉÉÉ ÉÉÉÉÉ ÉÉÉÉÉ ÉÉÉÉÉÉÉ ÉÉÉÉÉ (a) staircase with structural wall (b) foundation caissons of Storebaelt bridge (Denmark) Figure 1.1 Examples of spatial panel structures 6
1.2 Scope of this work dx dy 7
1. Problem Definition and Scope Formulation 8
1.3 Assumptions and limitations 9
1. Problem Definition and Scope Formulation 10
2.1 The finite element method 11
2. Basics of the Finite Element Method, Plasticity Theory and Linear Programming dy dy v y dy u v dx y u y dx u x dx x v x Figure 2.1 Linear strain-displacement relationship in a membrane panel 12
2.1 The finite element method y n x dy (n y n y dy) dx y x n yx dy f B y dxdy (n xy n xy dy) dx y h n xy dx f B x dxdy (n yx n yx dx) dy x dx n y dx (n x n x dx) dy x dy Figure 2.2 Stress resultants in an infinitesimal membrane element 1 n y n xy 1 y t n n yx n x 1 n x cos n yx cos n xy sin n tn n n 1 n t n nt n xy cos n x sin n yx sin x n y sin n y cos Figure 2.3 Normal stress transformation in a membrane element 13
2. Basics of the Finite Element Method, Plasticity Theory and Linear Programming z z y y y x w x z y Figure 2.4 Displacement components in plate bending x x x y 14
2.1 The finite element method z y q x dy m x dy (m xy m xy dy) dx y x m yx dy f B z dxdy (m y m y dy) dx y h dx m y dx m xy dx q y dx (m x m x dx) dy x (q y q y dy) dx y (m yx m yx dx) dy x (q x q x dx) dy x dy Figure 2.5 Stress resultants on an infinitesimal plate bending element 15
2. Basics of the Finite Element Method, Plasticity Theory and Linear Programming 16
2.1 The finite element method t y n x 1 1 q y q x t q x cos 1 q y sin q n n q y cos q t 1 t q x sin Figure 2.6 Shear stress transformation in a bending element n 17
2. Basics of the Finite Element Method, Plasticity Theory and Linear Programming f SZ f BZ f BY f SY f BX f SX Z y F iz z x X Y F ix F iy ÈÈ ÍÍ Figure 2.7 General three-dimensional body in equilibrium 18
2.1 The finite element method 19
2. Basics of the Finite Element Method, Plasticity Theory and Linear Programming 20
2.1 The finite element method 21
2. Basics of the Finite Element Method, Plasticity Theory and Linear Programming 22
2.2 Plasticity theory y y E tan E e p p (a) elastic-perfectly plastic (b) rigid-perfectly plastic Figure 2.8 Perfectly plastic materials in a uniaxial stress state 23
2. Basics of the Finite Element Method, Plasticity Theory and Linear Programming 24
2.2 Plasticity theory h A t N t M A b b Figure 2.9 Generalized strains and stresses for a beam section 25
2. Basics of the Finite Element Method, Plasticity Theory and Linear Programming S k S * S ds dq p S k N j S S ds j dq pj N j O S 1 1 O S 1 S n S n m (a) (b) Figure2.10 Yield surface in the generalized stress space 26
2.2 Plasticity theory 27
2. Basics of the Finite Element Method, Plasticity Theory and Linear Programming 28
2.3 Linear programming 29
2. Basics of the Finite Element Method, Plasticity Theory and Linear Programming 1 x 1 x j x n Z s 1. s i. s m c 0 b 1. b i. b m c 1 c j c n a 11 a 1j a 1n... a i1 a ij a in... a m1 a mj a mn O (m1)x(n1) Figure 2.11 Formulation of tableau of a general linear program x 1 x j x n x 1 s i x n c 0 c j c n c 0 b ic j a ij c j a ij c n a inc j a ij s 1 s i b i a i1 a 1j a ij pivot a in s 1 x j b i a ij a i1 a ij a 1j a ij 1 a ij a in a ij s m a mj b m a mn s m b m b ia mj a ij a mj a ij a mn a ina mj a ij (a) (b) Figure2.12 Tableau modification for one pivoting exchange step 30
2.3 Linear programming b i 0 exists? no optimum found yes pq exchange step for pivot a find min( b i 0) pivot row p i a pj 0 exists? no yes max b p find a pj pivot column q j no feasible solution Figure2.13 Exchange step from an unfeasible minimum towards a feasible optimum solution 31
2. Basics of the Finite Element Method, Plasticity Theory and Linear Programming c j 0 exists? no optimum found yes pq exchange step for pivot a find min(c j 0) pivot column q j a iq 0 exists? no optimum found yes min b i find a iq pivot row p i Figure2.14 Exchange step from a feasible non-optimum towards a feasible optimum solution 32
3.1 Yield conditions for a membrane infinitesimal element 33
3. Yield Conditions for an Infinitesimal Element h dy reinforced concrete = concrete + steel dx Figure 3.1 Reinforced concrete infinitesimal element in membrane state f c c tan E c c (a) c c c sliding failure c c c f c ÏÏ ÏÏ c ÏÏ Ï Ï Ï ÏÏ c c separation failure c f t 0 (b) c Figure 3.2 Concrete material model 34
3.1 Yield conditions for a membrane infinitesimal element y n c y n c 2 n c xy n c 1 x n c x (n c xy) 2 n c x n c y n c xy n c y n c 2 (a) O n c x r c m n c 1 r c m (n c xy) 2 (r c m n c x)(r c m n c y) (b) Figure 3.3 Yield conditions for an infinitesimal concrete element in a membrane state 35
3. Yield Conditions for an Infinitesimal Element n s xy n s y r s my y x Figure 3.4 Yield conditions for an orthogonal reinforcement net r s mx n s x n y 1 : n 2 xy (r s mx n x )(r s my n y ) 0 1 n xy r s my n x 2 : n 2 xy (r s mx n x )(r s my n y ) 0 3 : n 2 xy ( 1 2 rc m) 2 0 6 5 4 : n 2 xy (n x r s mx 1 2 rc m) 2 ( 1 2 rc m) 2 0 3 5 : n 2 xy (n y r s my 1 2 rc m) 2 ( 1 2 rc m) 2 0 2 4 7 r c m 6 : n 2 xy (n x 1 2 rc m) 2 ( 1 2 rc m) 2 0 7 : n 2 xy (n y 1 2 rc m) 2 ( 1 2 rc m) 2 0 8 : n x r s mx 0 9 : n y r s my 0 r c m rmx s 10 : n x r s mx 0 11 : n y r s my 0 Figure 3.5 Nonlinear yield conditions for an infinitesimal reinforced concrete membrane element 36
3.1 Yield conditions for a membrane infinitesimal element n xy n y 5 9 3 r c m 1 r s mx r s my 1,2 : n x n xy r s mx 0 7 r c m 2 n x 3,4 : n y n xy r s my 0 4 6 2 r c m 5,6 : n x n xy r c m 0 8 10 7,8 : n y n xy r c m 0 9,10 : n xy rc m 2 0 Figure 3.6 Linearized yield conditions of an infinitesimal reinforced concrete membrane element 37
3. Yield Conditions for an Infinitesimal Element dx s 2 dy = s 1 + + s i + y i x Figure 3.7 Superposition of reinforcement layers for an infinitesimal element 38
3.2 Yield conditions for a membrane-bending infinitesimal element Z Z f c h Y M p A s x A s h 2 h 2 d a X A s f y 1 Figure 3.8 Reinforced element subjected to pure bending 39
3. Yield Conditions for an Infinitesimal Element m x M b px m x m xy M b px m x m xy M b px m y M b py m y m xy M b py m x m xy M b px (M b px m x )(M b py m y ) m 2 xy m x m xy M t px m y m xy M b py m x M t px m y m xy M t py m y m xy M b py m y M t py m x m xy M t px (M t px m x )(M t py m y ) m 2 xy m x m xy M t px m y m xy M t py m y m xy M t py (a) (b) (c) Figure 3.9 Yield conditions for an orthogonally reinforced plate bending element (a) Nonlinear (b),(c) Linearized 40
3.2 Yield conditions for a membrane-bending infinitesimal element m xy m y y O m x x Figure3.10 Yield conditions for an infinitesimal concrete element in the bending state with M b px M b py M t px M t py 41
3. Yield Conditions for an Infinitesimal Element m x M b px m x m xy M b px M b pxy m y M b py m x m xy M b px M b pxy (M b px m x )(M b py m y ) (m xy M b pxy) 2 m y m xy M b py M b pxy m x M t px m y m xy M b py M b pxy m y M t py m x m xy M t px M t pxy (M t px m x )(M t py m y ) (m xy M t pxy) 2 m x m xy M t px M t pxy m y m xy M t py M t pxy m y m xy M t py M t pxy (a) (b) Figure 3.11 Yield conditions for an arbitrarily reinforced plate bending element (a) Nonlinear (b) Linearized Figure3.12 Stress components in a combined membrane-bending element 42
3.2 Yield conditions for a membrane-bending infinitesimal element top cover layer core bottom cover layer Figure3.13 Sandwich model for a membrane-bending element 43
3. Yield Conditions for an Infinitesimal Element z y (a) pure shear in uncracked core z x x y (c) membrane forces equilibrating q 0 cot y x y (b) diagonal compression field in cracked core x Figure3.14 Design for transverse shear 44
3.2 Yield conditions for a membrane-bending infinitesimal element Figure3.15 Sandwich model for a membrane-bending element including shear reinforcement 45
3. Yield Conditions for an Infinitesimal Element 46
4.1 Deficiencies of the usual approach 47
4. First Concept: Element-based Dimensioning and Yield Conditions for a Finite Element (a) (b) Figure 4.1 Evaluation of nodal average stresses in the finite element approach 48
4.2 Finite elements as dimensioning units 49
4. First Concept: Element-based Dimensioning and Yield Conditions for a Finite Element 50
4.2 Finite elements as dimensioning units 51
4. First Concept: Element-based Dimensioning and Yield Conditions for a Finite Element y y xy x x Figure 4.2 Transition from infinitesimal to finite element-based dimensioning 52
4.2 Finite elements as dimensioning units 53
4. First Concept: Element-based Dimensioning and Yield Conditions for a Finite Element 8 7 6 5 e 1 e 8 1 2 4 1 3 1 1 1 1 1 1 1 1 1 rigid body modes 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 constant strain modes 1 1 1 1 1 1 1 1 1 1 1 1 linear strain modes 1 1 1 1 Figure 4.3 Linearly independent nodal forces for an isoparametric membrane finite element 54
4.3 Generalized strains and stresses for a finite element 55
4. First Concept: Element-based Dimensioning and Yield Conditions for a Finite Element 56
4.3 Generalized strains and stresses for a finite element Figure 4.4 Strain modes for a rectangular membrane free formulation finite element 57
4. First Concept: Element-based Dimensioning and Yield Conditions for a Finite Element 58
4.3 Generalized strains and stresses for a finite element Figure 4.5 Strain modes for a rectangular bending finite element 59
4. First Concept: Element-based Dimensioning and Yield Conditions for a Finite Element 60
4.4 Yield conditions for a quadrilateral finite element b b s r D C a a (a) A B (b) Figure 4.6 Rectangular membrane finite element a m5 a m4 b 2 a 2 a m5 a m4 Figure 4.7 Admissible stress state for linear modes of membrane quadrilateral finite elements 61
4. First Concept: Element-based Dimensioning and Yield Conditions for a Finite Element 62
4.4 Yield conditions for a quadrilateral finite element s r j y x Figure 4.8 Reinforcement dimensioning direction for membrane finite element y s n y n xy n s n rs n r r n x x Figure 4.9 Stress transformation to a rotated coordinate system 63
4. First Concept: Element-based Dimensioning and Yield Conditions for a Finite Element 64
4.4 Yield conditions for a quadrilateral finite element r s r s A B C D D C r c r c A B r c 2 Figure4.10 Selected constraints in the four zones of a membrane finite element 65
4. First Concept: Element-based Dimensioning and Yield Conditions for a Finite Element s b D C r b a a (a) A B (b) Figure 4.11 Rectangular bending finite element a b5 a b4 a b4 Figure4.12 Admissible stress state for linear modes of bending quadrilateral finite elements a b5 66
4.4 Yield conditions for a quadrilateral finite element r s y t j s r t x b j y b x Figure4.13 Reinforcement dimensioning direction for bending finite element 67
4. First Concept: Element-based Dimensioning and Yield Conditions for a Finite Element A B C D r st D C r st A B r sb D C r sb A B Figure4.14 Selected constraints in the four zones of a bending finite element 68
4.4 Yield conditions for a quadrilateral finite element 69
4. First Concept: Element-based Dimensioning and Yield Conditions for a Finite Element 70
4.6 Transverse shear in bending finite elements Figure4.15 Strain modes for transverse shear (with no rotations) in bending 71
4. First Concept: Element-based Dimensioning and Yield Conditions for a Finite Element 72
4.7 Mesh parameter for error estimation 73
4. First Concept: Element-based Dimensioning and Yield Conditions for a Finite Element 74
4.7 Mesh parameter for error estimation 75
4. First Concept: Element-based Dimensioning and Yield Conditions for a Finite Element 3 8 3 1 1 3 3 8 3 3 3 1 1 1 3 1 1 1 1 1 1 1 1 1 8 1 1 1 1 3 8 3 3 3 1 3 3 3 1 3 Figure4.16 Membrane nodal force combinations in the unchecked subspace of a finite element 1 2 2 1 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 2 1 1 2 Figure4.17 Bending nodal load combinations in the unchecked subspace of a finite element 76
4.7 Mesh parameter for error estimation bounded full nonlinear yield conditions linearized yield conditions unbounded in some directions p or p or nonlinear yield conditions in subspace controlled linearized yield conditions in subspace Figure4.18 Projection of yield conditions in a subspace 77
4. First Concept: Element-based Dimensioning and Yield Conditions for a Finite Element Finite element model number of nodes number of degrees of freedom number of independent strain modes number of constant strain modes number of suggested higher order modes number of unchecked dimensions mesh parameter required triangular isoparametric membrane 3 6 3 3 0 0 no quadrilateral isoparametric membrane 4 8 5 3 2 0 no triangular free formulation membrane 3 9 6 3 0 3 yes quadrilateral free formulation membrane 4 12 9 3 2 4 yes triangular bending 3 9 6 3 0 3 yes quadrilateral bending 4 12 9 3 2 4 yes Figure4.19 Comparison between different element models in finite element-based dimensioning 78
5.1 Basic idea in a simple example l A E F p f c f y F 79
5. Second Concept: Optimization by Superposition of Self-Equilibrating Stress States 1 0 l 1 Ï 1 F 2 ÏÏ p e : F 2 F 2 p e : F 2 F 2 p dist : p 0 : AE 2 AE 2 AE p I : AE 2 AE 2 p : F 2 AE 2 F AE 2 2 Figure 5.1 Force redistribution in a two element bar system 80
5.1 Basic idea in a simple example 81
5. Second Concept: Optimization by Superposition of Self-Equilibrating Stress States (1 s ) A s 2 F f c A (2 c ) (1 s ) (2 c ) F f c A vector of the objective function 1 1 0 F f c A F 2f y A s 1 Figure 5.2 Graphical presentation of the optimization problem for the two element beam 82
5.2 Reinforcement evaluation for the linear elastic solution 83
5. Second Concept: Optimization by Superposition of Self-Equilibrating Stress States y p 1 As j ka s i 1 1 x p (a) A s i (b) Figure 5.3 Uniaxial (bar) and biaxial (net) reinforcement fields 84
5.2 Reinforcement evaluation for the linear elastic solution Ï Ï Ì Ì Ì Ì Ì membrane panel central layer (a) reinforcement fields (b) reinforcement zones bending panel Ì Ì top layer bottom layer Figure 5.4 Definition of a) user-defined reinforcement fields b) program-generated reinforcement zones in membrane and bending panels 85
5. Second Concept: Optimization by Superposition of Self-Equilibrating Stress States 86
5.2 Reinforcement evaluation for the linear elastic solution 87
5. Second Concept: Optimization by Superposition of Self-Equilibrating Stress States Figure 5.5 Flow diagram of reinforcement design for the linear elastic stress distribution Figure 5.6 Evaluation of the extreme state forces for the linear elastic solution 88
5.2 Reinforcement evaluation for the linear elastic solution Figure 5.7 Building the linear program for the linear elastic stress distribution X s 1 X s j X s NF Z 0. A z1 A zj A zn * * * * all zones of all panels all zones of a panel 1 zone s i s i (f i r s min i ) (f i r s min i ) 0. 0. or or (a s ij as ij ) (a s ij as ij ) X s min j 0. 0. 1. 0. 0. element with biggest (f i r s min i ) in r direction element with biggest (f i r s min i ) in s direction prescribed minimum reinforcement Figure 5.8 Setup of the tableau for reinforcement of the linear elastic solution 89
5. Second Concept: Optimization by Superposition of Self-Equilibrating Stress States no feasible solution slack fy ( e ) slack fy ( e dist 0 ) Figure 5.9 Checking all constraints 90
5.3 Force redistribution by introducing fictitious plastic strains 91
5. Second Concept: Optimization by Superposition of Self-Equilibrating Stress States fields / panel e.g. elements e.g. constraints / element full tableau linear elastic full tableau elasto-plastic 1 membrane panel 1 bending panel Figure5.10 Example of a full tableau size without progressive optimization Ç ÇÇÇÇ ÇÇÇÇÇ ÇÇ ÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇ linear ÇÇÇÇÇÇÇÇÇ elastic ÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇ ÇÇÇÇ ÇÇ new violated constraints new redistributions ÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇ linear ÇÇÇÇÇÇÇÇÇÇ elastic ÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇ ÇÇÇÇÇÇÇÇÇÇ Figure 5.11 Progressive optimization and expansion of the tableau 92
5.3 Force redistribution by introducing fictitious plastic strains stopping optimization due to: numerical instability or unacceptable mesh parameter or no new redistribution stress states possible or direct user interruption Figure5.12 Flow diagram of the optimization stage 93
5. Second Concept: Optimization by Superposition of Self-Equilibrating Stress States X s 1 X s j X s NF 1 m Z * A z1 A zj A zn 0. 0. * * * * * * s i (f i r s min i ) 0. or (a s ij as ij ) f I ij s ( min i 1 ) 0. 0. 0. 1. 0. 0. s ( max i 1 ) 0. 0. 0. 1. 0. 0. Figure5.13 Expansion of the tableau for new initial strains 94
5.3 Force redistribution by introducing fictitious plastic strains 95
5. Second Concept: Optimization by Superposition of Self-Equilibrating Stress States X s 1 X s j X s NF 1 m Z 0. A z1 A zj A zn * * * * * * * * s i (f i r s min i ) 0. or (a s ij as ij ) f I ij s i (f i r s min i ) 0. or (a s ij as ) ij f I ij s i (f i r c i) 0. 0. 0. f I ij Figure5.14 Expansion of the tableau for new violated constraints 96
5.3 Force redistribution by introducing fictitious plastic strains I 1m 1 0 1 I 2m 1 0 1 I 3m 0 1 1 I 4m 0 1 1 Figure5.15 Initial strain combinations shown on a square membrane finite element I 1b 1 0 1 I 2b 1 0 1 I 3b 0 1 1 I 4b 0 1 1 Figure5.16 Initial curvature combinations shown on a square bending finite element 97
5. Second Concept: Optimization by Superposition of Self-Equilibrating Stress States (a) (b) Figure5.17 Plastic curvature modes (a) with and (b) without concrete crushing 98
5.3 Force redistribution by introducing fictitious plastic strains Z 1 x T nk O 0 O k x 1 xnk T sk T k T Z s mk O 11 O 12 1+m k s mk O 11 O 12 O 1 22 O 21 O 12 O 1 22 s k O 21 O 22 k x k O 1 22 O 21 O 1 22 1+n k k 1+n k k Figure5.18 Regrouped tableau in the beginning O 0 and after k exchange steps O k 99
5. Second Concept: Optimization by Superposition of Self-Equilibrating Stress States Z 1 x T nk O k s T k x T n a x T n a s mk O 11 O 12 O 1 22 O 21 O 12 O 1 22 O 1a O 12 O 1 22 O 2a 1+m k O 1a x k O 1 22 O 21 O 1 22 O 1 22 O 2a O 2a k 1+n k k n a n a Figure5.19 Introduction of new columns (redistribution load cases) in the tableau at step k (O k ) Z 1 x T nk O k s T k 1+m k s mk O 11 O 12 O 1 22 O 21 O 12 O 1 22 k x k O 1 22 O 21 O 1 22 m a s ma O a1 O a2 O 1 22 O 21 O a2 O1 22 1+n k k m a s ma O a1 O a2 Figure5.20 Introduction of new rows (constraints) in the tableau at step k (O k ) 100
5.4 Ultimate load analysis (a) reinforcement minimization (b) ultimate load maximization Figure5.21 Comparison between reinforcement minimization and ultimate load maximization 101
5. Second Concept: Optimization by Superposition of Self-Equilibrating Stress States (a) (b) Figure5.22 Alternatives for ultimate load maximization transformed into a minimization problem Z 0. j 1. A zj X s j all zones of all panels all zones of a panel 1 zone s i s i * (f i r s min i (f i r s min i ) ) j j * (a s ij as ij ) Xs j (a s ij as ij ) Xs j Figure5.23 Adopted tableau for the ultimate load analysis 102
5.4 Ultimate load analysis 103
5. Second Concept: Optimization by Superposition of Self-Equilibrating Stress States 104
6.1 The structure of ORCHID FE FE LP FE LP PT Figure 6.1 Basic stages towards optimum reinforcement dimensioning 105
6. ORCHID: A Program for Optimum Reinforced Concrete Highly Interactive Dimensioning Figure 6.2 Flow diagram for Creating and solving the linear elastic finite element model 106
6.1 The structure of ORCHID reading geometrical input reading / writing FE model reading / writing load cases reading / writing reinforcement field input writing reinforcement output eventual data transfer to CAD programs for drafting drawing of whole structure and single panels drawing of displacements and stresses drawing of reinforcement and redistribution fields user-machine interaction interactive inputting, modifying and checking, zooming, scaling, rotating, animating, picking, selecting A: linear elastic FE B: linear optimization program FE meshing construction and expansion of tableau calculation of element stiffness types simplex algorithm and optimization assemblage of stiffness matrix and load vectors ultimate load analysis solution of equation system calculation of generalized stresses Figure 6.3 Modularity of ORCHID 107
6. ORCHID: A Program for Optimum Reinforced Concrete Highly Interactive Dimensioning 108
6.2 The graphical user interface of ORCHID commands of the active mode pull-down control menu optimization chart 2D view of active panel 3D view of spatial structure Figure 6.4 Graphical User Interface (GUI) of ORCHID 109
6. ORCHID: A Program for Optimum Reinforced Concrete Highly Interactive Dimensioning 110
6.3 Free Formulation element model 111
6. ORCHID: A Program for Optimum Reinforced Concrete Highly Interactive Dimensioning y v r z y xy x v x y u y x r z xy u x xy yx 1 2 ( x y ) 1 2 (v u x y ) r z 1 2 ( x y ) 1 2 (v u x y ) Figure 6.5 Definition of rotational degree of freedom in membrane free formulation elements 112
6.3 Free Formulation element model 113
6. ORCHID: A Program for Optimum Reinforced Concrete Highly Interactive Dimensioning w 1 3 w 1 ( 1 6, 2 3 ) 1 3 1 3 ( 1 6, 1 6 ) ( 2 3, 1 6 ) 1 3 1 3 Figure 6.6 Adopted integration points coordinates 114
6.3 Free Formulation element model y 3 3 (0,0,1) 1 x 1 C 1 2 (a) 1 (b) 2 0 A 2 A 3 2 (1,0,0) 3 0 (0,1,0) P A 1 1 0 Figure 6.7 Dimensionless coordinate systems for triangular elements for (a) membrane (b) bending 115
6. ORCHID: A Program for Optimum Reinforced Concrete Highly Interactive Dimensioning y x Figure 6.8 Local coordinates for a quadrilateral element 116
6.4 Specific features of the finite element model 117
6. ORCHID: A Program for Optimum Reinforced Concrete Highly Interactive Dimensioning y p panel node x p common node Z X Y Figure 6.9 Distinction between panel and common nodes 118
6.4 Specific features of the finite element model (a) (b) Figure6.10 Introduction of fictitious beams for adjoining membrane panels 119
6. ORCHID: A Program for Optimum Reinforced Concrete Highly Interactive Dimensioning m sl m y m1 x sl m2 y m1 sl x m2 (a) (b) (c) Figure 6.11 Master Slave specification 120
6.4 Specific features of the finite element model mi RX RY DZ DX DY RZ P sl P m1 P m2 Q sl Q m1 Q m2 b m bb mb b mm m RX RY DZ DX DY RZ sl (a) (b) Figure6.12 Eccentric panel connections 121
6. ORCHID: A Program for Optimum Reinforced Concrete Highly Interactive Dimensioning 122
6.4 Specific features of the finite element model Q sl Q mi (i 1, 2) k bb T PmiT bb k bb T Qmi bb T PmiT mb k mmt Qmi mb T PmiT mb k mmt Qmi mm k mm P sl T PmiT mm k mm T Qmi mb T PmiT mm k mm T Qmi mm P mi (i 1, 2) (a) (b) Figure6.13 Modification of the stiffness matrix for slave nodes 123
6. ORCHID: A Program for Optimum Reinforced Concrete Highly Interactive Dimensioning y z x Z Y X Figure6.14 Types of loads and boundary conditions implemented in ORCHID 124
6.6 Equation solver 125
6. ORCHID: A Program for Optimum Reinforced Concrete Highly Interactive Dimensioning 126
7.1 T-beam modelled by two panels 127
7. Examples Í Í Í Figure 7.1 T-beam modelled with continuum elements and two panels 128
7.1 T-beam modelled by two panels flange panel top view web panel side view Figure 7.2 Distribution of principal membrane stresses of the T-beam tension compression 129
7. Examples flange panel top view required reinforcement web panel side view Figure 7.3 Required reinforcement for the linear elastic stress distribution flange panel top view same layout for both layers web panel side view Figure 7.4 User-defined reinforcement fields for the T-beam 130
7.1 T-beam modelled by two panels flange panel top view required provided web panel side view Figure 7.5 Required and provided reinforcement for the linear elastic stress distribution flange panel top view required provided web panel side view Figure 7.6 Reinforcement for the optimized elasto-plastic stress distribution 131
7. Examples reinf. weight elastic before optimization elasto plastic after optimization field nr. elastic before optimization elasto plastic after optimization flange panel web panel total total Figure 7.7 Reinforcement comparisons for the T-beam 132
7.2 Three-panel plate with stiffening walls ÍÍ roller support ÍÄÄ ÉÉ ÉÉ ÉÉ ÉÉ ÍÍ É ÉÉÍ ÉÉÉ ÉÉ Í Í Í Figure 7.8 Structural geometry and deformed mesh ÍÍ simple support 133
7. Examples É É É É É É É É É É É É É É É É É É a) plate s principal moments b) plate s principal shear stresses positive moment negative moment intensity and direction c) load transfer mechanism tension compression d) plate s principal normal forces e) wall s principal normal forces Figure 7.9 Stress distribution for the linear elastic solution top layer required provided bottom layer É É É É É É É É É É user-defined reinforcement fields for both layers É É É É É É É É É É Figure7.10 Required reinforcement 134
7.2 Three-panel plate with stiffening walls ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ a) plate s principal shear stresses intensity and direction b) bottom layer reinforcement c) load transfer mechanism after optimization required provided Figure 7.11 Elasto-plastic load transfer no. 1 135
7. Examples required provided intensity and direction É É É É É É É É É a) plate s principal shear stresses É É É É É É É É É É É É É É É É É É b) bottom layer reinforcement É É É É É É É É É c) newly defined reinforcement field d) load transfer mechanism after optimization Figure7.12 Elasto-plastic load transfer no. 2 136
7.3 Effects of model assumptions in optimization required provided intensity and direction a) plate s principal shear stresses ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ b) bottom layer reinforcement ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ c) newly defined reinforcement d) load transfer mechanism after optimization Figure7.13 Elasto-plastic load transfer no. 3 137
7. Examples 3.0 ÉÉÉ ÉÉ ÉÉ ÉÉ ÉÉ 4 ÉÉ ÉÉ ÉÉ ÍÍ ÈÈ É ÉÉ ÉÉ ÈÈ ÈÈ 6 m ÉÉ ÉÉÉ 1.0 É ÉÍ ÈÈ ÈÈ ÈÈ Figure7.14 Simple one way slab with two vertical panels fully fixed support 138
7.3 Effects of model assumptions in optimization required reinforcement for linear elastic stress distribution reinforcement fields cases 1,2 required reinforcement for elasto-plastic stress distribution (a) top bottom top bottom top bottom (b) additional top reinforcement fields (c) bottom top Figure7.15 Required reinforcements for linear elastic and redistributed cases of a one way slab 139
7. Examples 140
8.1 On the proposed method 141
8. Closure 142
8.2 Suggestions for future work 143
8. Closure 144
Symbols 145
Symbols 146
Symbols 147
List of Figures 148
List of Figures 149
List of Figures 150
List of Figures 151
Literature 152
Literature 153
Literature 154
Literature 155