v, saouma fem i.pdf

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Lecture Notes in:

FINITE ELEMENT IFramed Structures

CVEN4525/5525

cVICTOR E. SAOUMA,

Fall 2002

Dept. of Civil Environmental and Architectural Engineering

University of Colorado, Boulder, CO 80309-0428


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Victor Saouma Finite Element I; Framed Structures


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Contents

1 INTRODUCTION 111.1 Why Matrix Structural Analysis? . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Overview of Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Structural Idealization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.1 Structural Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.2 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.3 Sign Convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5 Course Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

I Matrix Structural Analysis of Framed Structures 111

2 ELEMENT STIFFNESS MATRIX 212.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Influence Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Flexibility Matrix (Review) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Stiffness Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5 Force-Displacement Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5.1 Axial Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5.2 Flexural Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5.3 Torsional Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5.4 Shear Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6 Putting it All Together, [k] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2112.6.1 Truss Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2122.6.2 Beam Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

2.6.2.1 Euler-Bernoulli . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2132.6.2.2 Timoshenko Beam . . . . . . . . . . . . . . . . . . . . . . . . . . 213

2.6.3 2D Frame Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

2.6.4 Grid Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2152.6.5 3D Frame Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

2.7 Remarks on Element Stiffness Matrices . . . . . . . . . . . . . . . . . . . . . . . . 2172.8 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

3 STIFFNESS METHOD; Part I: ORTHOGONAL STRUCTURES 313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 The Stiffness Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31


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3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33E 3-1 Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33E 3-2 Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35E 3-3 Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

3.5 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

4 TRANSFORMATION MATRICES 414.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.1 [ke] [Ke] Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1.2 Direction Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Transformation Matrices For Framework Elements . . . . . . . . . . . . . . . . . 444.2.1 2 D cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.1.1 2D Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.1.2 Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.1.3 2D Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.2 3D Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2.3 3D Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

5 STIFFNESS METHOD; Part II 515.1 Direct Stiffness Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1.1 Global Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.1.1.1 Structural Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . 525.1.1.2 Augmented Stiffness Matrix . . . . . . . . . . . . . . . . . . . . 52

5.2 Logistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2.1 Boundary Conditions, [ID] Matrix . . . . . . . . . . . . . . . . . . . . . . 535.2.2 LM Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2.3 Assembly of Global Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . 54

E 5-1 Assembly of the Global Stiffness Matrix . . . . . . . . . . . . . . . . . . . 555.2.4 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57E 5-2 Direct Stiffness Analysis of a Truss . . . . . . . . . . . . . . . . . . . . . . 57E 5-3 Analysis of a Frame with MATLAB . . . . . . . . . . . . . . . . . . . . . 512E 5-4 Analysis of a simple Beam with Initial Displacements . . . . . . . . . . . 515

5.3 Skyline Storage of Global Stiffness Matrix, MAXA Vector . . . . . . . . . . . . . 5195.4 Computer Program Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 5225.5 Computer Program Flow Charts (FORTRAN) . . . . . . . . . . . . . . . . . . 524

5.5.1 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5255.5.2 Element Stiffness Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 5285.5.3 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

5.5.4 Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5315.5.5 Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5315.5.6 Backsubstitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5315.5.7 Internal Forces and Reactions . . . . . . . . . . . . . . . . . . . . . . . . . 531

5.6 Computer Implementation with MATLAB . . . . . . . . . . . . . . . . . . . . . . 5355.6.1 Program Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

5.6.1.1 Input Variable Descriptions . . . . . . . . . . . . . . . . . . . . . 5365.6.1.2 Sample Input Data File . . . . . . . . . . . . . . . . . . . . . . . 5375.6.1.3 Program Implementation . . . . . . . . . . . . . . . . . . . . . . 538



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5.6.2 Program Listing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5385.6.2.1 Main Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5385.6.2.2 Assembly ofID Matrix . . . . . . . . . . . . . . . . . . . . . . . 5405.6.2.3 Element Nodal Coordinates . . . . . . . . . . . . . . . . . . . . . 5415.6.2.4 Element Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

5.6.2.5 Element Stiffness Matrices . . . . . . . . . . . . . . . . . . . . . 5425.6.2.6 Transformation Matrices . . . . . . . . . . . . . . . . . . . . . . 5435.6.2.7 Assembly of the Augmented Stiffness Matrix . . . . . . . . . . . 5445.6.2.8 Print General Information . . . . . . . . . . . . . . . . . . . . . 5455.6.2.9 Print Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5455.6.2.10 Load Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5465.6.2.11 Nodal Displacements . . . . . . . . . . . . . . . . . . . . . . . . 5475.6.2.12 Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5485.6.2.13 Internal Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5495.6.2.14 Sample Output File . . . . . . . . . . . . . . . . . . . . . . . . . 550

5.7 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

6 EQUATIONS OF STATICS and KINEMATICS 616.1 Statics Matrix [B] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.1.1 Statically Determinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61E 6-1 Statically Determinate Truss Statics Matrix . . . . . . . . . . . . . . . . . 62E 6-2 Beam Statics Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.1.2 Statically Indeterminate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64E 6-3 Statically Indeterminate Truss Statics Matrix . . . . . . . . . . . . . . . . 65E 6-4 Selection of Redundant Forces . . . . . . . . . . . . . . . . . . . . . . . . 676.1.3 Kinematic Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.2 Kinematics Matrix [A] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69E 6-5 Kinematics Matrix of a Truss . . . . . . . . . . . . . . . . . . . . . . . . . 610

6.3 Statics-Kinematics Matrix Relationship . . . . . . . . . . . . . . . . . . . . . . . 6106.3.1 Statically Determinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6116.3.2 Statically Indeterminate . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

6.4 Kinematic Relations through Inverse of Statics Matrix . . . . . . . . . . . . . . . 6126.5 Congruent Transformation Approach to [K] . . . . . . . . . . . . . . . . . . . . 612

E 6-6 Congruent Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 613E 6-7 Congruent Transformation of a Frame . . . . . . . . . . . . . . . . . . . . 615

7 FLEXIBILITY METHOD 717.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.2 Flexibility Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.2.1 Solution of Redundant Forces . . . . . . . . . . . . . . . . . . . . . . . . . 727.2.2 Solution of Internal Forces and Reactions . . . . . . . . . . . . . . . . . . 737.2.3 Solution of Joint Displacements . . . . . . . . . . . . . . . . . . . . . . . . 73E 7-1 Flexibility Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7.3 Stiffness Flexibility Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.3.1 From Stiffness to Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . 75E 7-2 Flexibility Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767.3.2 From Flexibility to Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . 76E 7-3 Flexibility to Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77



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7.4 Stiffness Matrix of a Curved Element . . . . . . . . . . . . . . . . . . . . . . . . . 797.5 Duality between the Flexibility and the Stiffness Methods . . . . . . . . . . . . . 7107.6 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711

8 SPECIAL ANALYSIS PROCEDURES 81

8.1 Semi-Rigid Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.2 Nonuniform Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838.3 Inclined Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838.4 Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838.5 Substructuring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838.6 Reanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838.7 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

II Introduction to Finite Elements 85

9 REVIEW OF ELASTICITY 91

9.1 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919.1.1 Stress Traction Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92E 9-1 Stress Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

9.2 Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949.3 Fundamental Relations in Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . 95

9.3.1 Equation of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95E 9-2 Equilibrium Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969.3.2 Compatibility Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

9.4 Stress-Strain Relations in Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 979.5 Strain Energy Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

10 VARIATIONAL AND ENERGY METHODS 10110.1 Work, Energy & Potentials; Definitions . . . . . . . . . . . . . . . . . . . . . . . . 101

10.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10110.1.2 Internal Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

10.1.2.1 Internal Work versus Strain Energy . . . . . . . . . . . . . . . . 10310.1.3 External Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

10.1.3.1 Path Independence of External Work . . . . . . . . . . . . . . 10510.1.4 Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10610.1.5 Complementary Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . 106

10.1.5.1 Internal Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . 10610.1.5.1.1 Elastic Systems . . . . . . . . . . . . . . . . . . . . . . 106

10.1.5.1.2 Linear Elastic Systems . . . . . . . . . . . . . . . . . . 10810.1.5.2 External Virtual Work W . . . . . . . . . . . . . . . . . . . . . 109

10.1.6 Complementary Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . 10910.1.6.1 Internal Complementary Virtual Strain Energy U . . . . . . . 109

10.1.6.1.1 Arbitrary System . . . . . . . . . . . . . . . . . . . . . 10910.1.6.1.2 Linear Elastic Systems . . . . . . . . . . . . . . . . . . 1010

10.1.6.2 External Complementary Virtual Work W . . . . . . . . . . . 101010.1.7 Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011



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10.1.7.1 Potential Functions . . . . . . . . . . . . . . . . . . . . . . . . . 101110.1.7.2 Potential of External Work . . . . . . . . . . . . . . . . . . . . . 101110.1.7.3 Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011

10.2 Principle of Virtual Work and Complementary Virtual Work . . . . . . . . . . . 101110.2.1 Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012

10.2.1.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012E 10-1 Tapered Cantiliver Beam, Virtual Displacement . . . . . . . . . . . . . . . 101510.2.2 Principle of Complementary Virtual Work . . . . . . . . . . . . . . . . . . 1017

10.2.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017E 10-2 Tapered Cantilivered Beam; Virtual Force . . . . . . . . . . . . . . . . . . 1018E 10-3 Three Hinged Semi-Circular Arch . . . . . . . . . . . . . . . . . . . . . . . 1020E 10-4 Cantilivered Semi-Circular Bow Girder . . . . . . . . . . . . . . . . . . . . 1021

10.3 Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102310.3.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102310.3.2 Euler Equations of the Potential Energy . . . . . . . . . . . . . . . . . . 102510.3.3 Castiglianos First Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1027E 10-5 Fixed End Beam, Variable I . . . . . . . . . . . . . . . . . . . . . . . . . . 102810.3.4 Rayleigh-Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029E 10-6 Uniformly Loaded Simply Supported Beam; Polynomial Approximation . 1031E 10-7 Uniformly Loaded Simply Supported Beam; Fourrier Series . . . . . . . . 1032E 10-8 Tapered Beam; Fourrier Series . . . . . . . . . . . . . . . . . . . . . . . . 1033

10.4 Complementary Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 103510.4.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103510.4.2 Castiglianos Second Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1035E 10-9 Cantilivered beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036

10.4.2.1 Distributed Loads . . . . . . . . . . . . . . . . . . . . . . . . . . 1036E 10-10Deflection of a Uniformly loaded Beam using Castiglianos second Theorem1036

10.5 Comparison of Alternate Approximate Solutions . . . . . . . . . . . . . . . . . . 1037

E 10-11Comparison of MPE Solutions . . . . . . . . . . . . . . . . . . . . . . . . 103710.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103810.7 Variational Calculus; Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 1041

10.7.1 Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104110.7.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045E 10-12Extension of a Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045E 10-13Flexure of a Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047

10.8 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048

11 INTERPOLATION FUNCTIONS 11111.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

11.2 Shape Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11111.2.1 Axial/Torsional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11211.2.2 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11311.2.3 Flexural . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11411.2.4 Constant Strain Triangle Element . . . . . . . . . . . . . . . . . . . . . . 115

11.3 Interpolation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11711.3.1 C0: Lagrangian Interpolation Functions . . . . . . . . . . . . . . . . . . . 118

11.3.1.1 Constant Strain Quadrilateral Element . . . . . . . . . . . . . . 11811.3.1.2 Solid Rectangular Trilinear Element . . . . . . . . . . . . . . . . 119



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11.3.2 C1: Hermitian Interpolation Functions . . . . . . . . . . . . . . . . . . . . 111011.4 Interpretation of Shape Functions in Terms of Polynomial Series . . . . . . . . . 1110

12 FINITE ELEMENT FORMULATION 12112.1 Strain Displacement Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

12.1.1 Axial Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12112.1.2 Flexural Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12212.2 Virtual Displacement and Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . 12212.3 Element Stiffness Matrix Formulation . . . . . . . . . . . . . . . . . . . . . . . . 122

12.3.1 Stress Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

13 SOME FINITE ELEMENTS 13113.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13113.2 Truss Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13113.3 Flexural Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13213.4 Triangular Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

13.4.1 Strain-Displacement Relations . . . . . . . . . . . . . . . . . . . . . . . . 133

13.4.2 Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13313.4.3 Internal Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13413.4.4 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

13.5 Quadrilateral Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13513.6 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

14 DYNAMIC ANALYSIS 14114.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14114.2 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

14.2.1 Explicit Time Integration; Central Difference . . . . . . . . . . . . . . . . 14314.2.2 Implicit Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

14.2.2.1 Linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14414.2.2.2 NonLinear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 14414.3 Time Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

14.3.1 Explicit Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 14514.3.1.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

E 14-1 MATLAB Code for Explicit Time Integration . . . . . . . . . . . . . . . . 14814.3.1.2 NonLinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . 149

14.3.2 Implicit Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14914.3.2.1 Newmark Method; Forward difference . . . . . . . . . . . . . . . 14914.3.2.2 Linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141114.3.2.3 NonLinear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 1411

14.4 Free Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141114.5 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413

15 GEOMETRIC NONLINEARITY 15115.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

15.1.1 Strong Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15215.1.1.1 Lower Order Differential Equation; Essential B.C. . . . . . . . . 15215.1.1.2 Higher Order Differential Equation; Essential and Natural B.C. 153

15.1.2 Weak Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156



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15.1.2.1 Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15615.1.2.2 Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

15.2 Finite Element Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15815.3 Elastic Instability; Bifurcation Analysis . . . . . . . . . . . . . . . . . . . . . . . 159

E 15-1 Column Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1510

E 15-2 Frame Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151315.4 Second-Order Elastic Analysis; Geometric Non-Linearity . . . . . . . . . . . . . . 1515

E 15-3 Effect of Axial Load on Flexural Deformation . . . . . . . . . . . . . . . . 1516E 15-4 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1519

15.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1522

16 REFERENCES 161

A REVIEW of MATRIX ALGEBRA A1A.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A1A.2 Elementary Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . A3A.3 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A3

A.4 Singularity and Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A4A.5 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A5A.6 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A5

B SOLUTIONS OF LINEAR EQUATIONS B1B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B1B.2 Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B1

B.2.1 Gauss, and Gaus-Jordan Elimination . . . . . . . . . . . . . . . . . . . . . B1E B-1 Gauss Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B2E B-2 Gauss-Jordan Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . B3

B.2.1.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B4

B.2.2 LU Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B4B.2.2.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B5E B-3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B5B.2.3 Choleskys Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . B6E B-4 Choleskys Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . B7B.2.4 Pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B8

B.3 Indirect Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B8B.3.1 Gauss Seidel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B8

B.4 Ill Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B9B.4.1 Condition Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B9B.4.2 Pre Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B9B.4.3 Residual and Iterative Improvements . . . . . . . . . . . . . . . . . . . . . B10

C TENSOR NOTATION C1C.1 Engineering Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C1C.2 Dyadic/Vector Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C1C.3 Indicial/Tensorial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C2



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D INTEGRAL THEOREMS D1D.1 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D1D.2 Green-Gradient Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D1D.3 Gauss-Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D1



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1.1 Global Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2 Local Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3 Sign Convention, Design and Analysis . . . . . . . . . . . . . . . . . . . . . . . . 161.4 Total Degrees of Freedom for various Type of Elements . . . . . . . . . . . . . . 171.5 Independent Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.6 Examples of Global Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . 191.7 Organization of the Course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

2.1 Example for Flexibility Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Definition of Element Stiffness Coefficients . . . . . . . . . . . . . . . . . . . . . . 252.3 Stiffness Coefficients for One Dimensional Elements . . . . . . . . . . . . . . . . . 262.4 Flexural Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 Torsion Rotation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.6 Deformation of an Infinitesimal Element Due to Shear . . . . . . . . . . . . . . . 292.7 Effect of Flexure and Shear Deformation on Translation at One End . . . . . . . 2112.8 Effect of Flexure and Shear Deformation on Rotation at One End . . . . . . . . . 2122.9 Coordinate System for Element Stiffness Matrices . . . . . . . . . . . . . . . . . . 212

3.1 Problem with 2 Global d.o.f. 1 and 2 . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Frame Example (correct K23) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Grid Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1 Arbitrary 3D Vector Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 3D Vector Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 2D Frame Element Rotation with respect to Z (or z) . . . . . . . . . . . . . . . . 454.4 Grid Element Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.5 2D Truss Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.6 Reduced 3D Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.7 Special Case of 3D Transformation for Vertical Members . . . . . . . . . . . . . . 494.8 Complex 3D Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

5.1 Frame Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Example for [ID] Matrix Determination . . . . . . . . . . . . . . . . . . . . . . . 545.3 Simple Frame Analyzed with the MATLAB Code . . . . . . . . . . . . . . . . . . 555.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.5 Simple Frame Analyzed with the MATLAB Code . . . . . . . . . . . . . . . . . . 5135.6 Stiffness Analysis of one Element Structure . . . . . . . . . . . . . . . . . . . . . 5165.7 Example of Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5205.8 Numbering Schemes for Simple Structure . . . . . . . . . . . . . . . . . . . . . . 521


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5.9 Program Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5265.10 Programs Tree Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5275.11 Flowchart for the Skyline Height Determination . . . . . . . . . . . . . . . . . . . 5295.12 Flowchart for the Global Stiffness Matrix Assembly . . . . . . . . . . . . . . . . . 5305.13 Flowchart for the Load Vector Assembly . . . . . . . . . . . . . . . . . . . . . . . 532

5.14 Flowchart for the Internal Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 5335.15 Flowchart for the Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5345.16 Structure Plotted with CASAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

6.1 Example of [B] Matrix for a Statically Determinate Truss . . . . . . . . . . . . . 626.2 Example of [B] Matrix for a Statically Determinate Beam . . . . . . . . . . . . . 646.3 Example of [B] Matrix for a Statically Indeterminate Truss . . . . . . . . . . . . 656.4 Example 1, Congruent Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6146.5 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616

7.1 Stable and Statically Determinate Element . . . . . . . . . . . . . . . . . . . . . 75

8.1 Flexible, Rigid, and Semi-Rigid Beams . . . . . . . . . . . . . . . . . . . . . . . . 818.2 Deformations of a Semi-Rigid Beams . . . . . . . . . . . . . . . . . . . . . . . . . 828.3 Fixed End Forces in a Semi-Rigid Beams . . . . . . . . . . . . . . . . . . . . . . 82

9.1 Stress Components on an Infinitesimal Element . . . . . . . . . . . . . . . . . . . 919.2 Stresses as Tensor Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929.3 Stress Traction Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939.4 Equilibrium of Stresses, Cartesian Coordinates . . . . . . . . . . . . . . . . . . . 959.5 Fundamental Equations in Solid Mechanics . . . . . . . . . . . . . . . . . . . . . 99

10.1 Strain Energy and Complementary Strain Energy . . . . . . . . . . . . . . . . . . 10210.2 Effects of Load Histories on U and Wi . . . . . . . . . . . . . . . . . . . . . . . . 104

10.3 Torsion Rotation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10710.4 Flexural Member . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10710.5 Tapered Cantilivered Beam Analysed by the Vitual Displacement Method . . . . 101510.6 Tapered Cantilevered Beam Analysed by the Virtual Force Method . . . . . . . . 101910.7 Three Hinge Semi-Circular Arch . . . . . . . . . . . . . . . . . . . . . . . . . . . 102110.8 Semi-Circular Cantilevered Box Girder . . . . . . . . . . . . . . . . . . . . . . . . 102210.9 Single DOF Example for Potential Energy . . . . . . . . . . . . . . . . . . . . . . 102410.10Graphical Representation of the Potential Energy . . . . . . . . . . . . . . . . . . 102510.11Variable Cross Section Fixed Beam . . . . . . . . . . . . . . . . . . . . . . . . . . 102810.12Uniformly Loaded Simply Supported Beam Analysed by the Rayleigh-Ritz Method103110.13Example xx: External Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . 1034

10.14Summary of Variational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 103910.15Duality of Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 104010.16Variational and Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . 1042

11.1 Axial Finite Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11211.2 Flexural Finite Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11411.3 Shape Functions for Flexure of Uniform Beam Element. . . . . . . . . . . . . . . 11611.4 Constant Strain Triangle Element . . . . . . . . . . . . . . . . . . . . . . . . . . . 11611.5 Constant Strain Quadrilateral Element . . . . . . . . . . . . . . . . . . . . . . . . 119



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11.6 Solid Trilinear Rectangular Element . . . . . . . . . . . . . . . . . . . . . . . . . 1110

15.1 Level of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15115.2 Euler Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15215.3 Simply Supported Beam Column; Differential Segment; Effect of Axial Force P . 154

15.4 Solution of the Tanscendental Equation for the Buckling Load of a Fixed-HingedColumn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15515.5 Summary of Stability Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523



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1.1 Example of Nodal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2 Example of Element Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3 Example of Group Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Degrees of Freedom of Different Structure Types Systems . . . . . . . . . . . . . 18

2.1 Examples of Influence Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.1 3D Transformations of Linear Elements . . . . . . . . . . . . . . . . . . . . . . . 47

6.1 Internal Element Force Definition for the Statics Matrix . . . . . . . . . . . . . . 626.2 Conditions for Static Determinacy, and Kinematic Instability . . . . . . . . . . . 69

10.1 Possible Combinations of Real and Hypothetical Formulations . . . . . . . . . . . 101210.2 Comparison of 2 Alternative Approximate Solutions . . . . . . . . . . . . . . . . 103810.3 Summary of Variational Terms Associated with One Dimensional Elements . . . 104110.4 Essential and Natural Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 1046

11.1 Characteristics of Beam Element Shape Functions . . . . . . . . . . . . . . . . . 11511.2 Interpretation of Shape Functions in Terms of Polynomial Series (1D & 2D) . . . 1111

11.3 Polynomial Terms in Various Element Formulations (1D & 2D) . . . . . . . . . . 1111


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NOTATION

a Vector of coefficcients in assumed displacement fieldA AreaA Kinematics Matrixb Body force vectorB Statics Matrix, relating external nodal forces to internal forces[B] Statics Matrix relating nodal load to internal forces p = [B]P[B] Matrix relating assumed displacement fields parameters to joint displacementsC Cosine[C1|C2] Matrices derived from the statics matrix{d} Element flexibility matrix (lc){dc}[D] Structure flexibility matrix (GC)E Elastic Modulus[E] Matrix of elastic constants (Constitutive Matrix)

{F

}Unknown element forces and unknown support reactions

{F0} Nonredundant element forces (lc){Fx} Redundant element forces (lc){Fe} Element forces (lc){F0} Nodal initial forces{Fe} Nodal energy equivalent forces{F} Externally applied nodal forcesFEA Fixed end actions of a restrained memberG Shear modulusI Moment of inertia[L] Matrix relating the assumed displacement field parameters

to joint displacements

[I] Idendity matrix[ID] Matrix relating nodal dof to structure dof J St Venants torsional constant[k] Element stiffness matrix (lc)[p] Matrix of coefficients of a polynomial series[kg] Geometric element stiffness matrix (lc)[kr] Rotational stiffness matrix ( [d] inverse )[K] Structure stiffness matrix (GC)[Kg] Structures geometric stiffness matrix (GC)L LengthL Linear differential operator relating displacement to strains

lij Direction cosine of rotated axis i with respect to original axis j{LM} structure dof of nodes connected to a given element{N} Shape functions{p} Element nodal forces = F (lc){P} Structure nodal forces (GC)P , V , M , T Internal forces acting on a beam column (axial, shear, moment, torsion)R Structure reactions (GC)S Sinet Traction vector



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t Specified tractions along tu Displacement vectoru Neighbour function to u(x)

u(x) Specified displacements along uu,v,w Translational displacements along the x, y, and z directions

U Strain energyU Complementary strain energyx, y loacal coordinate system (lc)X, Y Global coordinate system (GC)W Work Coefficient of thermal expansion[] Transformation matrix{} Element nodal displacements (lc){} Nodal displacements in a continuous system{} Structure nodal displacements (GC) Strain vector

0Initial strain vector

{} Element relative displacement (lc){0} Nonredundant element relative displacement (lc){x} Redundant element relative displacement (lc) rotational displacement with respect to z direction (for 2D structures) Variational operatorM Virtual momentP Virtual force Virtual rotationu Virtual displacement Virtual curvatureU Virtual internal strain energy

W Virtual external work Virtual strain vector Virtual stress vector Surfacet Surface subjected to surface tractionsu Surface associated with known displacements Stress vector0 Initial stress vector Volume of body

lc: Local Coordinate systemGC: Global Coordinate System



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Chapter 1

INTRODUCTION

1.1 Why Matrix Structural Analysis?

1 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 whichintroduces basic structural analysis (determination of reactions, deflections, and internal forces)of both statically determinate and indeterminate structures.

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

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

4 Matrix Structural Analysis, or Advanced Structural Analysis, or Introduction to StructuralEngineering Finite Element, builds on the introductory analysis course to focus on those meth-ods which lend themselves to computer implementation. In doing so, we will place equalemphasis on both two and three dimensional structures, and develop a thorough understandingof computer aided analysis of structures.

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

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 elementone. 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. Assuch, the primary unknowns is u the nodal displacements, and internal forces are usuallyrestricted to stress . The only analogous one dimensional structure is the truss.

7 Whereas two and three dimensional continuum are essential in civil engineering to modelstructures such as dams, shells, and foundation, the majority of Civil engineering structuresare constituted by rod one-dimensional elements such as beams, girders, or columns. Forthose elements, displacements and internal forces are somehow more complex than those


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Draft12 INTRODUCTION

encountered in continuum finite elements.

8 Hence, contrarily to continuum finite element where displacement is mostly synonymous withtranslation, in one dimensional elements, and depending on the type of structure, generalizeddisplacements 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 torsionalmoments. Those concepts are far more relevant in the analysis/design of most civil engineeringstructures.

9 Hence, Matrix Structural Analysis, is truly a bridge course between introductory analysisand finite element courses. The element stiffness matrix [k] will first be derived using methodsintroduced in basic structural analysis, and later using energy based concepts. This laterapproach is the one exclusively used in the finite element method.

10 An important component of this course is computer programing. Once the theory and thealgorithms are thoroughly explained, you will be expected to program them in either Fortran(preferably 90) or C (sorry, but no Basic) on the computer of your choice. The program(typically about 3,500 lines) will perform the analysis of 2 and 3 dimensional truss and frame

structures, and many students have subsequently used it in their professional activities.

11 There will be one computer assignment in which you will be expected to perform sim-ple symbolic manipulations using Mathematica. For those of you unfamiliar with the BechtelLaboratory, there will be a special session to introduce you to the operation of Unix on Sunworkstations.

1.2 Overview of Structural Analysis

12 To put things into perspective, it may be helpful to consider classes of Structural Analysiswhich are distinguished by:

1. Excitation model

(a) Static

(b) Dynamic

2. Structure model

(a) Global geometry

small deformation ( = ux ) large deformation (x = dudx +

12

dvdx

2

, P- effects), chapter 15

(b) Structural elements element types: 1D framework (truss, beam, columns) 2D finite element (plane stress, plane strain, axisymmetric, plate or shell ele-

ments), chapter 13

3D finite element (solid elements)(c) Material Properties:

Linear Nonlinear



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Draft1.3 Structural Idealization 13

(d) Sectional properties:

Constant Variable

(e) Structural connections:

Rigid Semi-flexible (linear and non-linear)(f) Structural supports:

Rigid Elastic

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 ( < f)

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

15 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. Someof the questions confronting the analyst include:

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 claywhich may consolidate over time)

5. Include or not secondary members (such as diagonal braces in a three dimensional anal-ysis).



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6. Include or not axial deformation (can we neglect the axial stiffness of a beam in a build-ing?)

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, andwill underestimate the deformations).

10. Small or large deformations (In the analysis of a high rise building subjected to windload, the moments should be amplified by the product of the axial load times the lateraldeformation, P effects).

11. Time dependent effects (such as creep, which is extremely important in prestressed con-crete, or cable stayed concrete bridges).

12. Partial collapse or local yielding (would the failure of a single element trigger the failureof the entire structure?).

13. Load static or dynamic (when should a dynamic analysis be performed?).

14. Wind load (the lateral drift of a high rise building subjected to wind load, is often themajor limitation to higher structures).

15. Thermal load (can induce large displacements, specially when a thermal gradient ispresent.).

16. Secondary stresses (caused by welding. Present in most statically indeterminate struc-tures).

1.3.1 Structural Discretization

16 Once a structure has been idealized, it must be discretized to lend itself for a mathematicalrepresentation which will be analyzed by a computer program. This discretization shoulduniquely define each node, and member.

17 The node is characterized by its nodal id (node number), coordinates, boundary conditions,and load (this one is often defined separately), Table 1.1. Note that in this case we have two

Node No. Coor. B. C.X Y X Y Z

1 0. 0. 1 1 0

2 5. 5. 0 0 03 20. 5. 0 0 04 25. 2.5 1 1 1

Table 1.1: Example of Nodal Definition

nodal coordinates, and three degrees of freedom (to be defined later) per node. Furthermore,



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Draft1.3 Structural Idealization 15

a 0 and a 1 indicate unknown or known displacement. Known displacements can be zero(restrained) or non-zero (as caused by foundation settlement).

18 The element is characterized by the nodes which it connects, and its group number, Table1.2.

Element From To GroupNo. Node Node Number

1 1 2 12 3 2 23 3 4 2

Table 1.2: Example of Element Definition

19 Group number will then define both element type, and elastic/geometric properties. Thelast one is a pointer to a separate array, Table 1.3. In this example element 1 has element code

1 (such as beam element), while element 2 has a code 2 (such as a truss element). Materialgroup 1 would have different elastic/geometric properties than material group 2.

Group Element Material

No. Type Group

1 1 12 2 13 1 2

Table 1.3: Example of Group Number

20 From the analysis, we first obtain the nodal displacements, and then the element internalforces. Those internal forces vary according to the element type. For a two dimensional frame,those are the axial and shear forces, and moment at each node.

21 Hence, the need to define two coordinate systems (one for the entire structure, and one foreach element), and a sign convention become apparent.

1.3.2 Coordinate Systems

22 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 axislabels, 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 internalforces. 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 pointsfrom the 1st node to the 2nd node, Fig. 1.2.

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



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X X

Y

BEAM2D TRUSS

FRAME

X

Y

3D TRUSSGRID & FRAME

Z

Figure 1.1: Global Coordinate System

BEAM, TRUSS

x, 1x, 1

y, 2

z, 3

GRID, FRAME

Figure 1.2: Local Coordinate Systems

1.3.3 Sign Convention

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

26

In matrix structural analysis the sign convention adopted is consistent with the prevailingcoordinate system. Hence, we define a positive moment as one which is counter-clockwise, Fig.1.3

27 Fig. 1.4 illustrates the sign convention associated with each type of element.

28 Fig. 1.4 also shows the geometric (upper left) and elastic material (upper right) propertiesassociated with each type of element.

1.4 Degrees of Freedom

29 A degree of freedom (d.o.f.) is an independent generalized nodal displacement of a node.

30 The displacements must be linearly independent and thus not related to each other. For

+-

+ve M

+ve M

Analysis Sign COnventionDesign Sign Convention (US)

Figure 1.3: Sign Convention, Design and Analysis



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Draft1.4 Degrees of Freedom 17

I , Lx

v1

v 2 3 4

A, I , Lx

v2

3

1 4

v5

u1

u1

I , I , Lx y A, L A, I , I , I , Lx y z

u1 u 2

v2

v8

u1

E A, L EBeam 2D Truss 2D Frame E

6

u2

v3

4

5

6u

7

v9

10

11

12

E E EGrid 3D Truss 3D Frame

v2

v

u4

5

3 6

Figure 1.4: Total Degrees of Freedom for various Type of Elements

Figure 1.5: Independent Displacements

example, a roller support on an inclined plane would have three displacements (rotation , andtwo translations u and v), however since the two displacements are kinematically constrained,we only have two independent displacements, Fig. 1.5.

31 We note that we have been referring to generalized displacements, because we want this termto include translations as well as rotations. Depending on the type of structure, there maybe none, one or more than one such displacement. It is unfortunate that in most introductorycourses in structural analysis, too much emphasis has been placed on two dimensional structures,and not enough on either three dimensional ones, or two dimensional ones with torsion.

32 In most cases, there is the same number of d.o.f in local coordinates as in the global coordinate

system. One notable exception is the truss element. In local coordinate we can only have oneaxial deformation, whereas in global coordinates there are two or three translations in 2D and3D respectively for each node.

33 Hence, it is essential that we understand the degrees of freedom which can be associatedwith the 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.

35 We should distinguish between local and global d.o.f.s. The numbering scheme follows the



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Type Node 1 Node 2 [k] [K]

(Local) (Global)

1 Dimensional

{p} Fy1, Mz2 Fy3, Mz4Beam 4 4 4 4

{} v1, 2 v3, 42 Dimensional{p} Fx1 Fx2

Truss 2 2 4 4{} u1 u2{p} Fx1, Fy2, Mz3 Fx4, Fy5, Mz6

Frame 6 6 6 6{} u1, v2, 3 u4, v5, 6{p} Tx1, Fy2, Mz3 Tx4, Fy5, Mz6

Grid 6 6 6 6{} 1, v2, 3 4, v5, 6

3 Dimensional

{p}

Fx1, Fx2Truss 2 2 6 6

{} u1, u2{p} Fx1, Fy2, Fy3, Fx7, Fy8, Fy9,

Tx4 My5, Mz6 Tx10 My11, Mz12Frame 12 12 12 12

{} u1, v2, w3, u7, v8, w9,4, 5 6 10, 11 12

Table 1.4: Degrees of Freedom of Different Structure Types Systems



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Draft1.5 Course Organization 19

Figure 1.6: Examples of Global Degrees of Freedom

following simple rules:

Local: d.o.f. for a given element: Start with the first node, number the local d.o.f. in the sameorder as the subscripts of the relevant local coordinate system, and repeat for the secondnode.

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.

1.5 Course Organization



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Figure 1.7: Organization of the Course



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Part I

Matrix Structural Analysis ofFramed Structures


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Draft


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Draft

Chapter 2

ELEMENT STIFFNESS MATRIX

2.1 Introduction

1 In this chapter, we shall derive the element stiffness matrix [k] of various one dimensionalelements. Only after this important step is well understood, we could expand the theory andintroduce 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 stiffnessmatrix of one dimensional element. The first one, which will be used in this chapter, is based onclassical 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.

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

2.2 Influence Coefficients

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

Unit Action Effect on

Influence Line Load ShearInfluence Line Load MomentInfluence Line Load Deflection

Flexibility Coefficient Load Displacement

Stiffness Coefficient Displacement Load

Table 2.1: Examples of Influence Coefficients

5 It should be recalled that influence lines are associated with the analysis of structures sub-jected to moving loads (such as bridges), and that the flexibility and stiffness coefficients are


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Draft22 ELEMENT STIFFNESS MATRIX

Figure 2.1: Example for Flexibility Method

components of matrices used in structural analysis.

2.3 Flexibility Matrix (Review)

6 Considering the simply supported beam shown in Fig. 2.1, and using the local coordinatesystem, we have

12

=

d11 d12d21 d22

p1p2

(2.1)

Using the virtual work, or more specifically, the virtual force method to analyze this problem,(more about energy methods in Chapter 10), we have:l

0M

M

EIzdx

Internal

= P + M External

(2.2)

where M, MEIz , P and are the virtual internal force, real internal displacement, virtualexternal load, and real external displacement respectively. Here, both the external virtual forceand moment are usualy taken as unity.



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Draft2.4 Stiffness Coefficients 23

Virtual Force:

U =

x x dvol

x =Mxy

I

x = xE = MyEIy2dA = I

dvol = dAdx

U = l0

MM

EIdx

W = PU = W

l0

MM

EIdx = P (2.3)

Hence:

EI 1M

d11

= L

0 1 xL2

MM

dx =L

3(2.4)

Similarly, we would obtain:

EI d22 =

L0

x

L

2dx =

L

3(2.5-a)

EI d12 =

L0

1 x

L

x

Ldx = L

6= EI d21 (2.5-b)

7 Those results can be summarized in a matrix form as:

[d] =L

6EIz

2 1

1 2

(2.6)

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

2.4 Stiffness Coefficients

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

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.



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{p} = [k]{} (2.7)

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

12 The actual stiffness coefficients are shown in Fig. 2.3 for truss, beam, and grid elements interms of elastic and geometric properties.

13 In the next sections, we shall derive those stiffness coefficients.

2.5 Force-Displacement Relations

2.5.1 Axial Deformations

14 From strength of materials, the force/displacement relation in axial members is

= E AP

=AE

L1

(2.8)

Hence, for a unit displacement, the applied force should be equal to AEL . From statics, the forceat the other end must be equal and opposite.

2.5.2 Flexural Deformation

15 Our objective is to seek a relation for the shear and moments at each end of a beam, in termsof known displacements and rotations at each end.

V1 = V1(v1, 1, v2, 2) (2.9-a)

M1 = M1(v1, 1, v2, 2) (2.9-b)

V2 = V2(v1, 1, v2, 2) (2.9-c)

M2 = M2(v1, 1, v2, 2) (2.9-d)

16 We start from the differential equation of a beam, Fig. 2.4 in which we have all positiveknown displacements, we have from strength of materials

M =

EI

d2v

dx2

= M1

V1x (2.10)

17 Integrating twice

EI v = M1x 12

V1x2 + C1 (2.11-a)

EI v = 12

M1x2 1

6V1x

3 + C1x + C2 (2.11-b)



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Draft2.5 Force-Displacement Relations 25

Figure 2.2: Definition of Element Stiffness Coefficients



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Figure 2.3: Stiffness Coefficients for One Dimensional Elements

V

M

M

Vvv

1

1

1

12

2 2

2

Figure 2.4: Flexural Problem Formulation



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18 Applying the boundary conditions at x = 0

v = 1v = v1

C1 = EI 1C2 = EI v1 (2.12)

19 Applying the boundary conditions at x = L and combining with the expressions for C1 andC2

v = 2v = v2

EI 2 = M1L 12 V1L2 EI 1EI v2 = 12 M1L2 16 V1L3 EI 1L EI v1

(2.13)

20 Since equilibrium of forces and moments must be satisfied, we have:

V1 + V2 = 0 M1 V1L + M2 = 0 (2.14)

or

V1 =(M1 + M2)

LV2 = V1 (2.15)

21 Substituting V1 into the expressions for 2 and v2 in Eq. 2.13 and rearrangingM1 M2 = 2EIzL 1 2EIzL 2

2M1 M2 = 6EIzL 1 + 6EIzL2 v1 6EIzL2 v2(2.16)

22 Solving those two equations, we obtain:

M1 =2EIz

L(21 + 2) +

6EIzL2

(v1 v2) (2.17)

M2 =2EIz

L(1 + 22) +

6EIzL2

(v1 v2) (2.18)

23 Finally, we can substitute those expressions in Eq. 2.15

V1 =6EIz

L2(1 + 2) +

12EIzL3

(v1 v2) (2.19)

V2 = 6EIzL2

(1 + 2) 12EIzL3

(v1 v2) (2.20)

2.5.3 Torsional Deformations

24 From Fig. 2.2-d. Since torsional effects are seldom covered in basic structural analysis, andstudents may have forgotten the derivation of the basic equations from the Strength of Materialcourse, we shall briefly review them.

25 Assuming a linear elastic material, and a linear strain (and thus stress) distribution alongthe radius of a circular cross section subjected to torsional load, Fig. 2.5 we have:

T =

A

cmax

stress

dAarea

Force

arm

torque

(2.21-a)



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Figure 2.5: Torsion Rotation Relations

=max

c

A

2dA J

(2.21-b)

max =T c

J(2.21-c)

Note the analogy of this last equation with = McIz .

26

A

2dA is the polar moment of inertia J. It is also referred to as the St. Venants torsion

constant. For circular cross sections

J =

A

2dA =

c0

2 (2d)

=c4

2=

d4

32(2.22-a)

For rectangular sections b d, and b < d, an approximate expression is given by

J = kb3

d (2.23-a)

k =0.3

1 +

bd

2 (2.23-b)For other sections, J is often tabulated.

27 Note that J corresponds to Ixx where x is the axis along the element.

28 Having developed a relation between torsion and shear stress, we now seek a relation betweentorsion and torsional rotation. In Fig. 2.5, we consider the arc length BD

maxdx = dc

d

dx= maxc

max = maxG d

dx= maxGc

max = T CJ d

dx

=T

GJ

(2.24)

29 Finally, we can rewrite this last equation as

Tdx =

Gjd and obtain:

T =GJ

L (2.25)

Note the similarity between this equation and Equation 2.8.



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dvs

y

xdx

dy

Figure 2.6: Deformation of an Infinitesimal Element Due to Shear

2.5.4 Shear Deformation

30 In general, shear deformations are quite small. However, for beams with low span to depthratio, those deformations can not be neglected.

31 Considering an infinitesimal element subjected to shear, Fig. 2.6 and for linear elastic ma-terial, the shear strain (assuming small displacement, i.e. tan ) is given by

tan = dvsdx

Kinematics

=

GMaterial

(2.26)

where dvs

dxis the slope of the beam neutral axis from the horizontal while the vertical sections

remain undeformed, G is the shear modulus, the shear stress, and vs the shear induceddisplacement.

32 Along a beam cross section, the shear stress is not constant. For example for rectangularsections, it varies parabolically, and in I sections, the flange shear components can be neglected.

=V Q

Ib(2.27)

where V is the shear force, Q is the first moment (or static moment) about the neutral axis ofthe portion of the cross-sectional area which is outside of the section where the shear stress isto be determined, I is the moment of inertia of the cross sectional area about the neutral axis,

and b is the width of the rectangular beam.33 The preceding equation can be simplified as

=V

As(2.28)

where As is the effective cross section for shear (which is the ratio of the cross sectional areato the area shear factor)



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34 Let us derive the expression of As for rectangular sections. The exact expression for theshear stress is

=V Q

Ib(2.29)

where Q is the moment of the area from the external fibers to y with respect to the neutral

axis; For a rectangular section, this yields

=V Q

Ib(2.30-a)

=V

Ib

h/2y

bydy =V

2I

h2

4 y2

(2.30-b)

=6V

bh3

h2

4 y2

(2.30-c)

and we observe that the shear stress is zero for y = h/2 and maximum at the neutral axiswhere it is equal to 1.5 Vbh .

35 To determine the form factor of a rectangular section such that As = A (clarify)

= V QIb= k VA

Q =

h/2y

bydy =b

2

h2

4 y2

k = QAIb =A2I

h2

4 y2

bhA

=

A

k2dydz

= 1.2 (2.31)Thus, the form factor may be taken as 1.2 for rectangular beams of ordinary proportions,

and As = 1.2AFor I beams, k can be also approximated by 1.2, provided A is the area of the web.

36 Combining Eq. 2.26 and 2.28 we obtain

dvsdx

= VGAs

(2.32)

Assuming V to be constant, we integrate

vs =V

GAsx + C1 (2.33)

37 If the displacement vs is zero at the opposite end of the beam, then we solve for C1 andobtain

vs = VGAs

(x L) (2.34)

38 We define

def=

12EI

GAsL2(2.35)

= 24(1 + )A

As

r

L

2(2.36)

Hence for small slenderness ratio rL compared to unity, we can neglect .

39 Next, we shall consider the effect of shear deformations on both translations and rotations



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Draft2.6 Putting it All Together, [k] 211

1

6EI

6EI

12EI

12EI

L2

L2

L3

L3

Figure 2.7: Effect of Flexure and Shear Deformation on Translation at One End

Effect on Translation Due to a unit vertical translation, the end shear force is obtained fromEq. 2.19 and setting v1 = 1 and 1 = 2 = v2 = 0, or V =

12EIzL3 . At x = 0 we have, Fig.

2.7, and Eq. 2.34vs =

V LGAs

= V

V = 12EIzL3 = 12EIGAsL2

vs = (2.37)Hence, the shear deformation has increased the total translation from 1 to 1+. Similar

arguments apply to the translation at the other end.

Effect on Rotation Considering the beam shown in Fig. 2.8, even when a rotation 1 isapplied, an internal shear force is induced, and this in turn is going to give rise to shear

deformations (translation) which must be accounted for. The shear force is obtained fromEq. 2.19 and setting 1 = 1 and 2 = v1 = v2 = 0, or V =

6EIzL2 . At x = 0,

vs =V L

GAsV = 6EIzL2 = 12EIGAsL2

vs = 0.5L (2.38)in other words, the shear deformation has moved the end of the beam (which was supposedto have zero translation) by 0.5L.

2.6 Putting it All Together, [k]

40 Using basic structural analysis methods we have derived various force displacement relationsfor axial, flexural, torsional and shear imposed displacements. At this point, and keeping inmind the definition of degrees of freedom, we seek to assemble the individual element stiffnessmatrices [k]. We shall start with the simplest one, the truss element, then consider the beam,2D frame, grid, and finally the 3D frame element.

41 In each case, a table will cross-reference the force displacement relations, and then the elementstiffness matrix will be accordingly defined.



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4EI2EI

6EI

6EI

LL

L2

L2

0.5L1

0.5L1

1

1=1

Figure 2.8: Effect of Flexure and Shear Deformation on Rotation at One End

X

Y

Beam Element

X

Y

Z

Frame Element

X

Z

Y

Grid Element

Figure 2.9: Coordinate System for Element Stiffness Matrices

42 Fig. 2.9 illustrates the coordinate syatems used for the element stiffness matrix definitionsin this section.

2.6.1 Truss Element

43 The truss element (whether in 2D or 3D) has only one degree of freedom associated witheach node. Hence, from Eq. 2.8, we have

[kt] =AE

L

u1 u2p1 1 1p2 1 1 (2.39)

2.6.2 Beam Element

44 There are two major beam theories:



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Euler-Bernoulli which is the classical formulation for beams.

Timoshenko which accounts for transverse shear deformation effects.

2.6.2.1 Euler-Bernoulli

45 Using Equations 2.17, 2.18, 2.19 and 2.20 we can determine the forces associated with eachunit displacement.

[kb] =

v1 1 v2 2

V1 Eq. 2.19(v1 = 1) Eq. 2.19(1 = 1) Eq. 2.19(v2 = 1) Eq. 2.19(2 = 1)M1 Eq. 2.17(v1 = 1) Eq. 2.17(1 = 1) Eq. 2.17(v2 = 1) Eq. 2.17(2 = 1)V2 Eq. 2.20(v1 = 1) Eq. 2.20(1 = 1) Eq. 2.20(v2 = 1) Eq. 2.20(2 = 1)M2 Eq. 2.18(v1 = 1) Eq. 2.18(1 = 1) Eq. 2.18(v2 = 1) Eq. 2.18(2 = 1)

(2.40)

46 The stiffness matrix of the beam element (neglecting shear and axial deformation) will thus

be

[kb] =

v1 1 v2 2V1

12EIzL3

6EIzL2 12EIzL3 6EIzL2

M16EIz

L24EIz

L 6EIzL2 2EIzLV2 12EIzL3 6EIzL2 12EIzL3 6EIzL2M2

6EIzL2

2EIzL 6EIzL2 4EIzL

(2.41)

2.6.2.2 Timoshenko Beam

47 If shear deformations are present, we need to alter the stiffness matrix given in Eq. 2.41 in

the following manner

1. Due to translation, we must divide (or normalize) the coefficients of the first and thirdcolumns of the stiffness matrix by 1 + so that the net translation at both ends is unity.

2. Due to rotation and the effect of shear deformation

(a) The forces induced at the ends due to a unit rotation at end 1 (second column)neglecting shear deformations are

V1 = V2 = 6EIL2

(2.42-a)

M1 = 4EIL (2.42-b)

M2 =2EI

L(2.42-c)

(b) There is a net positive translation of 0.5L at end 1 when we applied a unit rotation(this parasitic translation is caused by the shear deformation) but no additionalforces are induced.



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(c) When we apply a unit rotation, all other displacements should be zero. Hence,we should counteract this parasitic shear deformation by an equal and oppositeone. Hence, we apply an additional vertical displacement of 0.5L and the forcesinduced at the ends (first column) are given by

V1 = V2 =12EI

L31

1 + kbt11

(0.5L) vs

(2.43-a)

M1 = M2 =6EI

L21

1 + kbt21

(0.5L) vs

(2.43-b)

Note that the denominators have already been divided by 1 + in kbt.

(d) Summing up all the forces, we have the forces induced as a result of a unit rotationonly when the effects of both bending and shear deformations are included.

V1 = V2 = 6EIL2

Due to Unit Rotation

+ 12EIL3

11 +

kbt11

(0.5L) vs

Due to Parasitic Shear

(2.44-a)

= 6EIL2

1

1 + (2.44-b)

M1 =4EI

L Due to Unit Rotation

+6EI

L21

1 + kbt21

(0.5L) vs


(2.44-c)

=4 +

1 +

EI

L (2.44-d)

M2 =2EI

L Due to Unit Rotation

+6EI

L21

1 + kbt21

(0.5L) vs


(2.44-e)

=2 1 +

EI

L(2.44-f)

48 Thus, the element stiffness matrix given in Eq. 2.41 becomes

[kbV] =

v1 1 v2 2V1

12EIzL3(1+y)

6EIzL2(1+y)

12EIzL3(1+y) 6EIzL2(1+y)M1

6EIzL2(1+y)

(4+y)EIz(1+y)L

6EIzL2(1+y)(2y)EIz

L(1+y)

V2 12EIzL3(1+y) 6EIzL2(1+y) 12EIzL3(1+y) 6EIzL2(1+y)M2

6EIzL2(1+y)

(2y)EIzL(1+y)

6EIzL2(1+y)(4+y)EIz

L(1+y)

(2.45)



45/304


2.6.3 2D Frame Element

49 The stiffness matrix of the two dimensional frame element is composed of terms from thetruss and beam elements where kb and kt refer to the beam and truss element stiffness matricesrespectively.

[k2df r] =

u1x v1y 1z u2x v2y 2zP1x k

t11 0 0 k

t12 0 0

V1y 0 kb11 k

b12 0 k

b13 k

b14

M1z 0 kb21 k

b22 0 k

b23 k

b24

P2x kt21 0 0 k

t22 0 0

V2y 0 kb31 k

b32 0 k

b33 k

b34

M2z 0 kb41 k

b42 0 k

b43 k

b44

(2.46)

Thus, we have:

[k2df r] =

u1x v1y 1z u2x v2y 2z

P1xEA

L 0 0 EA

L 0 0V1y 0

12EIzL3

6EIzL2 0 12EIzL3 6EIzL2

M1z 06EIz

L24EIz

L 0 6EIzL2 2EIzLP2x EAL 0 0 EAL 0 0V2y 0 12EIzL3 6EIzL2 0 12EIzL3 6EIL2M2z 0

6EIzL2

2EIzL 0 6EIzL2 4EIzL

(2.47)

50 Note that if shear deformations must be accounted for, the entries corresponding to shearand flexure must be modified in accordance with Eq. 2.45

2.6.4 Grid Element

51 The stiffness matrix of the grid element is very analogous to the one of the 2D frame element,except that the axial component is replaced by the torsional one. Hence, the stiffness matrix is

[kg] =

1x u1y 1z 2x u2y 2z

T1x Eq. 2.25 0 0 Eq. 2.25 0 0V1y 0 k

b11 k

b12 0 k

b13 k

b14

M1z 0 kb21 k

b22 0 k

b23 k

b24

T2x Eq. 2.25 0 0 Eq. 2.25 0 0V2y 0 k

b31 k

b32 0 k

b33 k

b34

M2z 0 kb

41 kb

42 0 kb

43 kb

44

(2.48)



46/304


Upon substitution, the grid element stiffness matrix is given by

[kg] =

1x u1y 1z 2x u2y 2zT1x

GIxL 0 0 GIxL 0 0

V1y

0 12EIzL3

6EIz

L20

12EIz

L3

6EIz

L2M1z 06EIz

L24EIz

L 0 6EIzL2 2EIzLT2x GixL 0 0 GIxL 0 0V2y 0 12EIzL3 6EIzL2 0 12EIzL3 6EIzL2M2z 0

6EIzL2

2EIzL3 0 6EIzL2 4EIzL

(2.49)

52 Note that if shear deformations must be accounted for, the entries corresponding to shearand flexure must be modified in accordance with Eq. 2.45

2.6.5 3D Frame Element

[k3df r] =

u1 v1 w1 x1 y1 z1 u2 v2 w2 x2 y2 z2

Px1 kt11 0 0 0 0 0 k

t21 0 0 0 0 0

Vy1 0 kb11 0 0 0 k

b12 0 k

b13 0 0 0 k

b14V z1 0 0 kb11 0 kb12 0 0 0 kb13 0 kb14 0Tx1 0 0 0 k

g11 0 0 0 0 0 k

g12 0 0

My1 0 0 kb32 0 k

b22 0 0 0 k

b12 0 k

b24 0

Mz1 0 kb21 0 0 0 k

b22 0 kb12 0 0 0 kb24

Px2 kt21 0 0 0 0 0 k

t22 0 0 0

v, saouma fem i.pdf

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