Draft

DRAFT

Lecture Notes in:

FINITE ELEMENT I Framed Structures

Victor E. Saouma Dept. of Civil Environmental and Architectural Engineering University of Colorado, Boulder, CO 80309-0428

Draft Contents 1 INTRODUCTION 1.1 Why Matrix Structural Analysis? 1.2 Overview of Structural Analysis . 1.3 Structural Idealization . . . . . . 1.3.1 Structural Discretization . 1.3.2 Coordinate Systems . . . 1.3.3 Sign Convention . . . . . 1.4 Degrees of Freedom . . . . . . . . 1.5 Course Organization . . . . . . .

I

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Matrix Structural Analysis of Framed Structures

2 ELEMENT STIFFNESS MATRIX 2.1 Introduction . . . . . . . . . . . . . . . . 2.2 Influence Coefficients . . . . . . . . . . . 2.3 Flexibility Matrix (Review) . . . . . . . 2.4 Stiffness Coefficients . . . . . . . . . . . 2.5 Force-Displacement Relations . . . . . . 2.5.1 Axial Deformations . . . . . . . . 2.5.2 Flexural Deformation . . . . . . 2.5.3 Torsional Deformations . . . . . 2.5.4 Shear Deformation . . . . . . . . 2.6 Putting it All Together, [k] . . . . . . . 2.6.1 Truss Element . . . . . . . . . . 2.6.2 Beam Element . . . . . . . . . . 2.6.2.1 Euler-Bernoulli . . . . . 2.6.2.2 Timoshenko Beam . . . 2.6.3 2D Frame Element . . . . . . . . 2.6.4 Grid Element . . . . . . . . . . . 2.6.5 3D Frame Element . . . . . . . . 2.7 Remarks on Element Stiffness Matrices . 2.8 Homework . . . . . . . . . . . . . . . . . 3 STIFFNESS METHOD; 3.1 Introduction . . . . . . 3.2 The Stiffness Method . 3.3 Examples . . . . . . . E 3-1 Beam . . . . . E 3-2 Frame . . . . . E 3-3 Grid . . . . . . 3.4 Observations . . . . .

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Part I: ORTHOGONAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 18 19 20 21 21 22 25

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29 29 29 30 31 32 32 32 35 36 39 39 40 40 40 42 42 43 43 45

STRUCTURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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47 47 47 49 49 51 54 56

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Draft CONTENTS

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111 111 112 113 115

6 EQUATIONS OF STATICS and KINEMATICS 6.1 Statics Matrix [B] . . . . . . . . . . . . . . . . . . . . . 6.1.1 Statically Determinate . . . . . . . . . . . . . . E 6-1 Statically Determinate Truss Statics Matrix . . E 6-2 Beam Statics Matrix . . . . . . . . . . . . . . . 6.1.2 Statically Indeterminate . . . . . . . . . . . . . E 6-3 Statically Indeterminate Truss Statics Matrix . E 6-4 Selection of Redundant Forces . . . . . . . . . 6.1.3 Kinematic Instability . . . . . . . . . . . . . . . 6.2 Kinematics Matrix [A] . . . . . . . . . . . . . . . . . . E 6-5 Kinematics Matrix of a Truss . . . . . . . . . . 6.3 Statics-Kinematics Matrix Relationship . . . . . . . . 6.3.1 Statically Determinate . . . . . . . . . . . . . . 6.3.2 ‡Statically Indeterminate . . . . . . . . . . . . 6.4 Kinematic Relations through Inverse of Statics Matrix 6.5 ‡ Congruent Transformation Approach to [K] . . . . . E 6-6 Congruent Transformation . . . . . . . . . . . . E 6-7 Congruent Transformation of a Frame . . . . .

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117 117 117 117 119 120 120 122 124 124 125 125 126 126 127 127 128 130

7 FLEXIBILITY METHOD 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Flexibility Matrix . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Solution of Redundant Forces . . . . . . . . . . . . 7.2.2 Solution of Internal Forces and Reactions . . . . . 7.2.3 Solution of Joint Displacements . . . . . . . . . . . E 7-1 Flexibility Method . . . . . . . . . . . . . . . . . . 7.3 Stiffness Flexibility Relations . . . . . . . . . . . . . . . . 7.3.1 From Stiffness to Flexibility . . . . . . . . . . . . . E 7-2 Flexibility Matrix . . . . . . . . . . . . . . . . . . 7.3.2 From Flexibility to Stiffness . . . . . . . . . . . . . E 7-3 Flexibility to Stiffness . . . . . . . . . . . . . . . . 7.4 Stiffness Matrix of a Curved Element . . . . . . . . . . . . 7.5 Duality between the Flexibility and the Stiffness Methods 7.6 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . .

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133 133 133 134 135 135 135 137 137 137 138 139 140 141 142

8 SPECIAL ANALYSIS PROCEDURES 8.1 Semi-Rigid Beams . . . . . . . . . . . . 8.2 Nonuniform Torsion . . . . . . . . . . . 8.3 Inclined Supports . . . . . . . . . . . . . 8.4 Condensation . . . . . . . . . . . . . . . 8.5 Substructuring . . . . . . . . . . . . . . 8.6 Reanalysis . . . . . . . . . . . . . . . . . 8.7 Constraints . . . . . . . . . . . . . . . .

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143 143 145 145 145 145 145 145

5.7

5.6.2.11 5.6.2.12 5.6.2.13 5.6.2.14 Homework . . .

5

Victor Saouma

Nodal Displacements Reactions . . . . . . . Internal Forces . . . . Sample Output File . . . . . . . . . . . . . .

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Finite Element I; Framed Structures

Draft CONTENTS

7

E 10-8 Tapered Beam; Fourrier Series . . . . . . . . . . . . . . . 10.4 † Complementary Potential Energy . . . . . . . . . . . . . . . . . 10.4.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Castigliano’s Second Theorem . . . . . . . . . . . . . . . . E 10-9 Cantilivered beam . . . . . . . . . . . . . . . . . . . . . . 10.4.2.1 Distributed Loads . . . . . . . . . . . . . . . . . E 10-10Deflection of a Uniformly loaded Beam using Castigliano’s 10.5 Comparison of Alternate Approximate Solutions . . . . . . . . . E 10-11Comparison of MPE Solutions . . . . . . . . . . . . . . . 10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Variational Calculus; Preliminaries . . . . . . . . . . . . . . . . . 10.7.1 Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . 10.7.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . E 10-12Extension of a Bar . . . . . . . . . . . . . . . . . . . . . . E 10-13Flexure of a Beam . . . . . . . . . . . . . . . . . . . . . . 10.8 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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186 188 188 188 189 189 189 190 190 191 191 194 197 198 199 201

11 INTERPOLATION FUNCTIONS 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Shape Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Axial/Torsional . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Generalization . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Flexural . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Constant Strain Triangle Element . . . . . . . . . . . . . 11.3 Interpolation Functions . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 C 0 : Lagrangian Interpolation Functions . . . . . . . . . . 11.3.1.1 Constant Strain Quadrilateral Element . . . . . 11.3.1.2 Solid Rectangular Trilinear Element . . . . . . . 11.3.2 C 1 : Hermitian Interpolation Functions . . . . . . . . . . . 11.4 Interpretation of Shape Functions in Terms of Polynomial Series

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203 203 203 204 205 205 207 209 209 210 211 211 212

12 FINITE ELEMENT FORMULATION 12.1 Strain Displacement Relations . . . . . 12.1.1 Axial Members . . . . . . . . . 12.1.2 Flexural Members . . . . . . . 12.2 Virtual Displacement and Strains . . . 12.3 Element Stiffness Matrix Formulation 12.3.1 Stress Recovery . . . . . . . . .

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213 213 213 214 214 214 216

13 SOME FINITE ELEMENTS 13.1 Introduction . . . . . . . . . . . . . . . 13.2 Truss Element . . . . . . . . . . . . . . 13.3 Flexural Element . . . . . . . . . . . . 13.4 Triangular Element . . . . . . . . . . . 13.4.1 Strain-Displacement Relations 13.4.2 Stiffness Matrix . . . . . . . . . 13.4.3 Internal Stresses . . . . . . . . 13.4.4 Observations . . . . . . . . . . 13.5 Quadrilateral Element . . . . . . . . . 13.6 Homework . . . . . . . . . . . . . . . .

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217 217 217 218 218 218 219 220 220 220 221

Victor Saouma

Finite Element I; Framed Structures

Draft CONTENTS

E B-3 Example . . . . . . . . . . . . . . . . B.2.3 Cholesky’s Decomposition . . . . . . E B-4 Cholesky’s Decomposition . . . . . . B.2.4 Pivoting . . . . . . . . . . . . . . . . B.3 Indirect Methods . . . . . . . . . . . . . . . B.3.1 Gauss Seidel . . . . . . . . . . . . . B.4 Ill Conditioning . . . . . . . . . . . . . . . . B.4.1 Condition Number . . . . . . . . . . B.4.2 Pre Conditioning . . . . . . . . . . . B.4.3 Residual and Iterative Improvements

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269 270 270 271 271 272 272 272 273 273

C TENSOR NOTATION 275 C.1 Engineering Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 C.2 Dyadic/Vector Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 C.3 Indicial/Tensorial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 D INTEGRAL THEOREMS 279 D.1 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 D.2 Green-Gradient Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 D.3 Gauss-Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

Victor Saouma

Finite Element I; Framed Structures

Draft List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Global Coordinate System . . . . . . . . . . . . . . . . . Local Coordinate Systems . . . . . . . . . . . . . . . . . Sign Convention, Design and Analysis . . . . . . . . . . Total Degrees of Freedom for various Type of Elements Independent Displacements . . . . . . . . . . . . . . . . Examples of Global Degrees of Freedom . . . . . . . . . Organization of the Course . . . . . . . . . . . . . . . .

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2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Example for Flexibility Method . . . . . . . . . . . . . . . . . . . . . Definition of Element Stiffness Coefficients . . . . . . . . . . . . . . . Stiffness Coefficients for One Dimensional Elements . . . . . . . . . . Flexural Problem Formulation . . . . . . . . . . . . . . . . . . . . . . Torsion Rotation Relations . . . . . . . . . . . . . . . . . . . . . . . Deformation of an Infinitesimal Element Due to Shear . . . . . . . . Effect of Flexure and Shear Deformation on Translation at One End Effect of Flexure and Shear Deformation on Rotation at One End . . Coordinate System for Element Stiffness Matrices . . . . . . . . . . .

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3.1 3.2 3.3

Problem with 2 Global d.o.f. θ1 and θ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Frame Example (correct K23 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Grid Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Arbitrary 3D Vector Transformation . . . . . . . . . . . 3D Vector Transformation . . . . . . . . . . . . . . . . . 2D Frame Element Rotation with respect to Z (or z) . . Grid Element Rotation . . . . . . . . . . . . . . . . . . . 2D Truss Rotation . . . . . . . . . . . . . . . . . . . . . Reduced 3D Rotation . . . . . . . . . . . . . . . . . . . Special Case of 3D Transformation for Vertical Members Complex 3D Rotation . . . . . . . . . . . . . . . . . . .

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60 61 62 63 64 65 66 67

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12

Frame Example . . . . . . . . . . . . . . . . . . . . Example for [ID] Matrix Determination . . . . . . Simple Frame Analyzed with the MATLAB Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simple Frame Analyzed with the MATLAB Code . Stiffness Analysis of one Element Structure . . . . Example of Bandwidth . . . . . . . . . . . . . . . . Numbering Schemes for Simple Structure . . . . . Program Flowchart . . . . . . . . . . . . . . . . . . Program’s Tree Structure . . . . . . . . . . . . . . Flowchart for the Skyline Height Determination . . Flowchart for the Global Stiffness Matrix Assembly

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69 72 73 75 80 82 86 87 91 92 94 96

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Draft List of Tables 1.1 1.2 1.3 1.4

Example of Nodal Definition . Example of Element Definition Example of Group Number . . Degrees of Freedom of Different

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure Types Systems

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2.1

Examples of Influence Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1

3D Transformations of Linear Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.1 6.2

Internal Element Force Definition for the Statics Matrix . . . . . . . . . . . . . . . . . . . 118 Conditions for Static Determinacy, and Kinematic Instability . . . . . . . . . . . . . . . . 124

10.1 10.2 10.3 10.4

Possible Combinations of Real and Hypothetical Formulations . . Comparison of 2 Alternative Approximate Solutions . . . . . . . Summary of Variational Terms Associated with One Dimensional Essential and Natural Boundary Conditions . . . . . . . . . . . .

. . . . . . . . . . . . Elements . . . . . .

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20 20 21 23

167 191 194 198

11.1 Characteristics of Beam Element Shape Functions . . . . . . . . . . . . . . . . . . . . . . 208 11.2 Interpretation of Shape Functions in Terms of Polynomial Series (1D & 2D) . . . . . . . . 212 11.3 Polynomial Terms in Various Element Formulations (1D & 2D) . . . . . . . . . . . . . . . 212

Draft

LIST OF TABLES

a A A b B [B0 ] [B] C [C1|C2] {d} {dc } [D] E [E] {F} {F0 } {Fx } {Fe } 0 {F } e {F } {F} FEA G I [L] [I] [ID] J [k] [p] [kg ] [kr ] [K] [Kg ] L L lij {LM } {N} {p} {P} P, V, M, T R S t b t u u ˜ b (x) u u, v, w U Victor Saouma

15

NOTATION Vector of coefficcients in assumed displacement field Area Kinematics Matrix Body force vector Statics Matrix, relating external nodal forces to internal forces Statics Matrix relating nodal load to internal forces p = [B0 ]P Matrix relating assumed displacement fields parameters to joint displacements Cosine Matrices derived from the statics matrix Element flexibility matrix (lc) Structure flexibility matrix (GC) Elastic Modulus Matrix of elastic constants (Constitutive Matrix) Unknown element forces and unknown support reactions Nonredundant element forces (lc) Redundant element forces (lc) Element forces (lc) Nodal initial forces Nodal energy equivalent forces Externally applied nodal forces Fixed end actions of a restrained member Shear modulus Moment of inertia Matrix relating the assumed displacement field parameters to joint displacements Idendity matrix Matrix relating nodal dof to structure dof St Venant’s torsional constant Element stiffness matrix (lc) Matrix of coefficients of a polynomial series Geometric element stiffness matrix (lc) Rotational stiffness matrix ( [d] inverse ) Structure stiffness matrix (GC) Structure’s geometric stiffness matrix (GC) Length Linear differential operator relating displacement to strains Direction cosine of rotated axis i with respect to original axis j structure dof of nodes connected to a given element Shape functions Element nodal forces = F (lc) Structure nodal forces (GC) Internal forces acting on a beam column (axial, shear, moment, torsion) Structure reactions (GC) Sine Traction vector Specified tractions along Γt Displacement vector Neighbour function to u(x) Specified displacements along Γu Translational displacements along the x, y, and z directions Strain energy Finite Element I; Framed Structures

Draft Chapter 1

INTRODUCTION 1.1

Why Matrix Structural Analysis?

In most Civil engineering curriculum, students are required to take courses in: Statics, Strength of Materials, Basic Structural Analysis. This last course is a fundamental one which introduces basic structural analysis (determination of reactions, deflections, and internal forces) of both statically determinate and indeterminate structures.

1

Also Energy methods are introduced, and most if not all examples are two dimensional. Since the emphasis is on hand solution, very seldom are three dimensional structures analyzed. The methods covered, for the most part lend themselves for “back of the envelope” solutions and not necessarily for computer implementation. 2

3 Those students who want to pursue a specialization in structural engineering/mechanics, do take more advanced courses such as Matrix Structural Analysis and/or Finite Element Analysis.

Matrix Structural Analysis, or Advanced Structural Analysis, or Introduction to Structural Engineering Finite Element, builds on the introductory analysis course to focus on those methods which lend themselves to computer implementation. In doing so, we will place equal emphasis on both two and three dimensional structures, and develop a thorough understanding of computer aided analysis of structures. 4

This is essential, as in practice most, if not all, structural analysis are done by the computer and it is imperative that as structural engineers you understand what is inside those “black boxes”, develop enough self assurance to be capable of opening them and modify them to perform certain specific tasks, and most importantly to understand their limitations.

5

6 With the recently placed emphasis on the finite element method in most graduate schools, many students have been tempted to skip a course such as this one and rush into a finite element one. Hence it is important that you understand the connection and role of those two courses. The Finite Element Method addresses the analysis of two or three dimensional continuum. As such, the primary unknowns is u the nodal displacements, and internal “forces” are usually restricted to stress σ. The only analogous one dimensional structure is the truss. 7 Whereas two and three dimensional continuum are essential in civil engineering to model structures such as dams, shells, and foundation, the majority of Civil engineering structures are constituted by “rod” one-dimensional elements such as beams, girders, or columns. For those elements, “displacements” and internal “forces” are somehow more complex than those encountered in continuum finite elements. 8 Hence, contrarily to continuum finite element where displacement is mostly synonymous with translation, in one dimensional elements, and depending on the type of structure, generalized displacements may include translation, and/or flexural and/or torsional rotation. Similarly, “internal forces” are not stresses, but rather axial and shear forces, and/or flexural or torsional moments. Those concepts are far

Draft

1.3 Structural Idealization

19

3. Type of solution: (a) Continuum, analytical, Partial Differential Equation (b) Discrete, numerical, Finite ELement, Finite Difference, Boundary Element

13

Structural design must satisfy: 1. Strength (σ < σf ) 2. Stiffness (“small” deformations) 3. Stability (buckling, cracking)

14

Structural analysis must satisfy 1. Statics (equilibrium) 2. Mechanics (stress-strain or force displacement relations) 3. Kinematics (compatibility of displacement)

1.3

Structural Idealization

Prior to analysis, a structure must be idealized for a suitable mathematical representation. Since it is practically impossible (and most often unnecessary) to model every single detail, assumptions must be made. Hence, structural idealization is as much an art as a science. Some of the questions confronting the analyst include: 15

1. Two dimensional versus three dimensional; Should we model a single bay of a building, or the entire structure? 2. Frame or truss, can we neglect flexural stiffness? 3. Rigid or semi-rigid connections (most important in steel structures) 4. Rigid supports or elastic foundations (are the foundations over solid rock, or over clay which may consolidate over time) 5. Include or not secondary members (such as diagonal braces in a three dimensional analysis). 6. Include or not axial deformation (can we neglect the axial stiffness of a beam in a building?) 7. Cross sectional properties (what is the moment of inertia of a reinforced concrete beam?) 8. Neglect or not haunches (those are usually present in zones of high negative moments) 9. Linear or nonlinear analysis (linear analysis can not predict the peak or failure load, and will underestimate the deformations). 10. Small or large deformations (In the analysis of a high rise building subjected to wind load, the moments should be amplified by the product of the axial load times the lateral deformation, P − ∆ effects). 11. Time dependent effects (such as creep, which is extremely important in prestressed concrete, or cable stayed concrete bridges). 12. Partial collapse or local yielding (would the failure of a single element trigger the failure of the entire structure?). Victor Saouma

Finite Element I; Framed Structures

Draft

1.3 Structural Idealization

21 Group No. 1 2 3

Element Type 1 2 1

Material Group 1 1 2

Table 1.3: Example of Group Number

Y

Y

X

X

X Z

2D TRUSS FRAME

BEAM

3D TRUSS GRID & FRAME

Figure 1.1: Global Coordinate System

1.3.2 22

Coordinate Systems

We should differentiate between 2 coordinate systems:

Global: to describe the structure nodal coordinates. This system can be arbitrarily selected provided it is a Right Hand Side (RHS) one, and we will associate with it upper case axis labels, X, Y, Z, Fig. 1.1 or 1,2,3 (running indeces within a computer program). Local: system is associated with each element and is used to describe the element internal forces. We will associate with it lower case axis labels, x, y, z (or 1,2,3), Fig. 1.2. 23 The x-axis is assumed to be along the member, and the direction is chosen such that it points from the 1st node to the 2nd node, Fig. 1.2.

24

Two dimensional structures will be defined in the X-Y plane.

1.3.3

Sign Convention

25 The sign convention in structural analysis is completely different than the one previously adopted in structural analysis/design, Fig. 1.3 (where we focused mostly on flexure and defined a positive moment as one causing “tension below”. This would be awkward to program!).

y, 2 x, 1

BEAM, TRUSS

x, 1 z, 3

GRID, FRAME

Figure 1.2: Local Coordinate Systems Victor Saouma

Finite Element I; Framed Structures

Draft

1.4 Degrees of Freedom

23

Figure 1.5: Independent Displacements various types of structures made up of one dimensional rod elements, Table 1.4. 34

This table shows the degree of freedoms and the corresponding generalized forces. Type

Node 1

Node 2

Fy1 , Mz2

{δ}

v1 , θ2

{p}

Fx1

Truss

u1 Fx1 , Fy2 , Mz3

u2 Fx4 , Fy5 , Mz6

Frame

{δ} {p}

u 1 , v2 , θ 3 Tx1 , Fy2 , Mz3

u 4 , v5 , θ 6 Tx4 , Fy5 , Mz6

Grid

{δ} {p} {δ}

θ 1 , v2 , θ 3

{p}

Fx1 ,

{δ} {p}

u1 , Fx1 , Fy2 , Fy3 , Tx4 My5 , Mz6

u2 Fx7 , Fy8 , Fy9 , Tx10 My11 , Mz12

{δ}

u 1 , v2 , w 3 , θ4 , θ 5 θ6

u 7 , v8 , w 9 , θ10 , θ11 θ12

Truss

[K] (Global)

4×4

4×4

2×2

4×4

6×6

6×6

6×6

6×6

2×2

6×6

12 × 12

12 × 12

1 Dimensional Fy3 , Mz4

{p}

Beam

[k] (Local)

v3 , θ 4 2 Dimensional Fx2

θ 4 , v5 , θ 6 3 Dimensional Fx2

Frame

Table 1.4: Degrees of Freedom of Different Structure Types Systems We should distinguish between local and global d.o.f.’s. The numbering scheme follows the following simple rules:

35

Local: d.o.f. for a given element: Start with the first node, number the local d.o.f. in the same order as the subscripts of the relevant local coordinate system, and repeat for the second node. Global: d.o.f. for the entire structure: Starting with the 1st node, number all the unrestrained global d.o.f.’s, and then move to the next one until all global d.o.f have been numbered, Fig. 1.6. Victor Saouma

Finite Element I; Framed Structures

Draft

1.5 Course Organization

1.5

25

Course Organization

Victor Saouma

Finite Element I; Framed Structures

Draft

Part I

Matrix Structural Analysis of Framed Structures

Draft Chapter 2

ELEMENT STIFFNESS MATRIX 2.1

Introduction

1 In this chapter, we shall derive the element stiffness matrix [k] of various one dimensional elements. Only after this important step is well understood, we could expand the theory and introduce the structure stiffness matrix [K] in its global coordinate system. 2 As will be seen later, there are two fundamentally different approaches to derive the stiffness matrix of one dimensional element. The first one, which will be used in this chapter, is based on classical methods of structural analysis (such as moment area or virtual force method). Thus, in deriving the element stiffness matrix, we will be reviewing concepts earlier seen.

The other approach, based on energy consideration through the use of assumed shape functions, will be examined in chapter 12. This second approach, exclusively used in the finite element method, will also be extended to two and three dimensional continuum elements. 3

2.2

Influence Coefficients

4 In structural analysis an influence coefficient C ij can be defined as the effect on d.o.f. i due to a unit action at d.o.f. j for an individual element or a whole structure. Examples of Influence Coefficients are shown in Table 2.1.

Influence Line Influence Line Influence Line Flexibility Coefficient Stiffness Coefficient

Unit Action Load Load Load Load Displacement

Effect on Shear Moment Deflection Displacement Load

Table 2.1: Examples of Influence Coefficients 5 It should be recalled that influence lines are associated with the analysis of structures subjected to moving loads (such as bridges), and that the flexibility and stiffness coefficients are components of matrices used in structural analysis.

Draft

2.4 Stiffness Coefficients

31

Virtual Force: Z

  δσ x εx dvol       M xy  Z l  δσ x = I M My σx dx δM δU = εx = E = EI Z  EI 0     y 2 dA = I     dvol = dAdx δW = δP ∆ δU = δW δU

=

Hence: EI |{z} 1 d11 = |{z} ∆

δM

Similarly, we would obtain: EId22 EId12

7

= =

Z

Z

L 0 L 0

L



1−

L 0

|

 x 2



            

l

δM 0

M dx = δP ∆ EI

L x 2 dx = 1− L 3 {z }

(2.3)

(2.4)

δM ·M

dx =

L 3

(2.5-a)

x x L dx = − L L 6

=

EId21

(2.5-b)

Those results can be summarized in a matrix form as: [d] =

8

Z

             Z

L 6EIz



2 −1

−1 2



(2.6)

The flexibility method will be covered in more detailed, in chapter 7.

2.4

Stiffness Coefficients

In the flexibility method, we have applied a unit force at a time and determined all the induced displacements in the statically determinate structure.

9

10

In the stiffness method, we 1. Constrain all the degrees of freedom 2. Apply a unit displacement at each d.o.f. (while restraining all others to be zero) 3. Determine the reactions associated with all the d.o.f.

{p} = [k]{δ}

(2.7)

Hence kij will correspond to the reaction at dof i due to a unit deformation (translation or rotation) at dof j, Fig. 2.2.

11

Victor Saouma

Finite Element I; Framed Structures

Draft

2.5 Force-Displacement Relations

33

Figure 2.2: Definition of Element Stiffness Coefficients

Victor Saouma

Finite Element I; Framed Structures

Draft

2.7 Remarks on Element Stiffness Matrices

43

Upon substitution, the grid element stiffness matrix is given by  α 1x x T1x GI  L V1y  0 M1z 0  Gi x T2x −  L V2y  0 M2z 0

[kg ] =

u1y 0

β1z 0

12EIz L3 6EIz L2

6EIz L2 4EIz L

z − 12EI L3

z − 6EI L2

0

6EIz L2

0

α2x x − GI L 0 0

z − 12EI L3 6EIz − L2 0

GIx L

12EIz L3 z − 6EI L2

0 0

2EIz L3

β2z  0  6EIz   L2 2EIz   L  0   z  − 6EI L2  4EIz 

u2y 0

(2.46)

L

Note that if shear deformations must be accounted for, the entries corresponding to shear and flexure must be modified in accordance with Eq. 2.42

52

2.6.5

3D Frame Element u1 t Px1 k11  Vy1 0 V z1  0 Tx1  0 My1  0 Mz1  0t Px2  k21 Vy2  0 V z2  0 Tx2  0 My2 0 Mz2 0

v1 0 b k11 0 0 0 b k21 0 b k13 0 0 0 b −k12



[k3df r ] =

w1 0 0 b k11 0 b k32 0 0 0 b k13 0 b k12 0

θx1 0 0 0 g k11 0 0 0 0 0 g k12 0 0

θy1 0 0 b −k12 0 b k22 0 0 0 b −k14 0 b k24 0

θz1 0 b k12 0 0 0 b k22 0 b k14 0 0 0 b k24

u2 t k21 0 0 0 0 0 t k22 0 0 0 0 0

v2 0 b k13 0 0 0 b −k12 0 b k33 0 0 0 b k43

w2 0 0 b k13 0 b k12 0 0 0 b k33 0 b −k43 0

θx2 0 0 0 g k12 0 0 0 0 0 g k22 0 0

θy2 0 0 b −k14 0 b k24 0 0 0 b −k34 0 b k44 0

θz2  0 k b 14   0   0   0   b  k24  0   b  k34  0   0   0  b k44

For [k3D 11 ] and with we obtain: 

Px1

  Vz1  0 Tx1 0  0 My1   0 Mz1  Px2 − EA L Vy2  0 Vz2  0 Tx2 0  0 My2  Vy1

[k

3df r

] =

u1 EA l 0

Mz2

0

v1

w1

θx1

θy1

θz1

u2

v2

w2

θx2

θy2

θz2

0 12EIz L3

0

0

0

0

0 6EIy − L2

0 12EIz − L3

0

0

− EA L 0

0

0 12EIy L3

0 6EIz L2

0 12EIy − L3

0

0 EIy −6 L2

0 6EIz L2

0 0 0 6EIz L2 0 12EIz − L3 0 0 0 6EIz L2

0 6EIy − L2

0 GIx L 0

0 4EIy L

0

0

0

0

0

0

0 12EIy − L3

0

0 EIy 6 L2

0 6EIy − L2 0

0 GIx − L 0 0

0 2EIy L 0

0

0

0

0

0

0

0 4EIz L 0 6EIz − L2

0 0 EA L 0

0 6EIz L2 0 12EIz L3

0

0

0

0

0

0 2EIz L

0 −

0 6EIz L2

−

0 GIx L 0

0 2EIy L

0

0

0

0

0

0

0 12EIy L3

0

0 6EIy L2

−

0

0

0 6EIy L2

0 6EIy L2 0

0 GIx L 0 0

0 4EIy L 0

0 0 0 2EIz L 0 6EIz − L2 0 0 0 4EIz L

(2.47)

                 

(2.48)

Note that if shear deformations must be accounted for, the entries corresponding to shear and flexure must be modified in accordance with Eq. 2.42

53

2.7

Remarks on Element Stiffness Matrices

Singularity: All the derived stiffness matrices are singular, that is there is at least one row and one column which is a linear combination of others. For example in the beam element, row 4 = −row Victor Saouma

Finite Element I; Framed Structures

Draft 2.8 Homework

2.8

45

Homework

Using the virtual force method, derive the flexibility matrix of a semi-circular box-girder of radius R and angle α in terms of shear, axial force, and moment. The arch is clamped at one end, and free at the other. Note: In a later assignment, you will combine the flexibility matrix with equilibrium relations to derive the element stiffness matrix.





O

Y

α Z

x z

R

Y

y

X

Victor Saouma

O

θ Z

X

Finite Element I; Framed Structures

Draft Chapter 3

STIFFNESS METHOD; Part I: ORTHOGONAL STRUCTURES 3.1

Introduction

In the previous chapter we have first derived displacement force relations for different types of rod elements, and then used those relations to define element stiffness matrices in local coordinates.

1

2 In this chapter, we seek to perform similar operations, but for an orthogonal structure in global coordinates. 3 In the previous chapter our starting point was basic displacement-force relations resulting in element stiffness matrices [k].

In this chapter, our starting point are those same element stiffness matrices [k], and our objective is to determine the structure stiffness matrix [K], which when inverted, would yield the nodal displacements.

4

5

The element stiffness matrices were derived for fully restrained elements.

This chapter will be restricted to orthogonal structures, and generalization will be discussed later. The stiffness matrices will be restricted to the unrestrained degrees of freedom.

6

From these examples, the interrelationships between structure stiffness matrix, nodal displacements, and fixed end actions will become apparent. Then the method will be generalized in chapter 5 to describe an algorithm which can automate the assembly of the structure global stiffness matrix in terms of the one of its individual elements.

7

3.2

The Stiffness Method

8 As a “vehicle” for the introduction to the stiffness method let us consider the problem in Fig 3.1-a, and recognize that there are only two unknown displacements, or more precisely, two global d.o.f: θ 1 and θ2 . 9

If we were to analyse this problem by the force (or flexibility) method, then 1. We make the structure statically determinate by removing arbitrarily two reactions (as long as the structure remains stable), and the beam is now statically determinate. 2. Assuming that we remove the two roller supports, then we determine the corresponding deflections due to the actual laod (∆B and ∆C ).

Draft Chapter 4

TRANSFORMATION MATRICES 4.1 4.1.1

Preliminaries [ke ] [Ke ] Relation

1 In the previous chapter, in which we focused on orthogonal structures, the assembly of the structure’s stiffness matrix [Ke ] in terms of the element stiffness matrices was relatively straight-forward. 2 The determination of the element stiffness matrix in global coordinates, from the element stiffness matrix in local coordinates requires the introduction of a transformation. 3 This chapter will examine the 2D and 3D transformations required to obtain an element stiffness matrix in global coordinate system prior to assembly (as discussed in the next chapter).

4

Recalling that {p} {P}

5

(4.1) (4.2)

Let us define a transformation matrix [Γ(e) ] such that: {δ}

Note that we use the same matrix Γ one). 6

= [k(e) ]{δ} = [K(e) ]{∆}

(e)

{p}

= =

[Γ(e) ]{∆} [Γ

(e)

]{P}

(4.3) (4.4)

since both {δ} and {p} are vector quantities (or tensors of order

Substituting Eqn. 4.3 and Eqn. 4.4 into Eqn. 4.1 we obtain [Γ(e) ]{P} = [k(e) ][Γ(e) ]{∆}

(4.5)

{P} = [Γ(e) ]−1 [k(e) ][Γ(e) ]{∆}

(4.6)

premultiplying by [Γ(e) ]−1

7

But since the rotation matrix is orthogonal, we have [Γ(e) ]−1 = [Γ(e) ]T and {P} = [Γ(e) ]T [k(e) ][Γ(e) ]{∆} {z } |

(4.7)

[K(e) ] = [Γ(e) ]T [k(e) ][Γ(e) ]

(4.8)

[K(e) ]

which is the general relationship between element stiffness matrix in local and global coordinates.

Draft

4.1 Preliminaries

61

where lij is the direction cosine of axis i with respect to axis j, and thus the rows of the matrix correspond to the rotated vectors with respect to the original ones corresponding to the columns. 11

With respect to Fig. 4.2, lxX = cos α; lxY = cos β, and lxZ = cos γ or

Y

y

β

VY

x

V α VX

VZ Z

X

γ

z

Figure 4.2: 3D Vector Transformation Vx

12

=

VX lxX + VY lxY + VZ lxZ

(4.14-a)

=

VX cos α + VY cos β + VZ cos γ

(4.14-b)

Direction cosines are unit orthogonal vectors satisfying the following relations: 3 X

lij lij = 1

i = 1, 2, 3

(4.15)

j=1

i.e: 2 2 2 = 1 or cos2 α + cos2 β + cos2 γ = 1 = δ11 + l13 + l12 l11

and 3 X

lij lkj

j=1

l11 l21 + l12 l22 + l13 l23

(4.16)

  i = 1, 2, 3 k = 1, 2, 3 = 0  i 6= k

(4.17-a)

= 0 = δ12

T

T

(4.17-b)

T

By direct multiplication of [γ] and [γ] it can be shown that: [γ] [γ] = [I] ⇒ [γ] = [γ] an orthogonal matrix. 13

14

−1

⇒ [γ] is

The reverse transformation (from local to global) would be T

{V} = [γ] {v} or

Victor Saouma

(4.18)

     lxX lyX lzX  Vx   VX  Vy VY =  lxY lyY lxY      Vz lxZ lyZ lzZ VZ {z } | [γ ]−1 =[γ ]T

(4.19)

Finite Element I; Framed Structures

Draft

4.2 Transformation Matrices For Framework Elements

Y Y

Y Y 1

γ

Y

β

X

α

2

γ

X

67

1

Y

2

γ

γ

Y

β

β

Z Z β

Z

γ

1

γ β

X

β

X

1

Z

β

β

α Z

Z

X

β

X

γ

1

γ

1

X

β

2

γ

Z

β

β

Z

γ

X

Y

Yβ

Y

Yα

β

γ

2

α Xγ Xα

α α

Zβ

Z

γ

α

α

Z

Zα

2

Figure 4.8: Complex 3D Rotation

Victor Saouma

Finite Element I; Framed Structures

Draft Chapter 5

STIFFNESS METHOD; Part II 5.1

Direct Stiffness Method

5.1.1

Global Stiffness Matrix

1 The physical interpretation of the global stiffness matrix K is analogous to the one of the element, i.e. If all degrees of freedom are restrained, then Kij corresponds to the force along global degree of freedom i due to a unit positive displacement (or rotation) along global degree of freedom j.

2

For instance, with reference to Fig. 5.1, we have three global degrees of freedom, ∆1 , ∆2 , and θ3 . and

P

0/0 /

A

EI

M B

P/2

w

?

@?@

L/2

A

)#)#))**


=

∆1

&%&

P

%

A B

J#I JI

C

$#" C$#" $"

C

B#B#B 1 D#C D#C DC A#A#A

K21

77

K22 K31

,#+,#+ ,+,+

775#6#5 6#5 65

A

K32

K23

43

4343

K12 B

K11

F#EC FE

θ3

w

L/2

B