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www.icivil.ir نارمع هیسدىهم ي نایًجشواد عماج لاترپ نارمع یسدىهم ناگیار تايسج ي اهباتک هئارا نارمع زير تلااقم هیرترب ي هیرتهب نارمع یسدىهم یصصخت یاه همجوا نارمع یسدىهم یصصخت هاگشيرف
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Draft DRAFT Lecture Notes in: MATRIX STRUCTURAL ANALYSIS with an Introduction to Finite Elements CVEN4525/5525 c VICTOR E. SAOUMA, Fall 1999 Dept. of Civil Environmental and Architectural Engineering University of Colorado, Boulder, CO 80309-0428
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Draft 0–2 Blank page Victor Saouma Matrix Structural Analysis
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Draft Contents 1 INTRODUCTION 1–1 1.1 Why Matrix Structural Analysis? . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–1 1.2 Overview of Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–2 1.3 Structural Idealization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–4 1.3.1 Structural Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–5 1.3.2 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–6 1.3.3 Sign Convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–6 1.4 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–9 1.5 Course Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–11 I Matrix Structural Analysis of Framed Structures 1–15 2 ELEMENT STIFFNESS MATRIX 2–1 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–1 2.2 Influence Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–1 2.3 Flexibility Matrix (Review) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–2 2.4 Stiffness Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–4 2.5 Force-Displacement Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–7 2.5.1 Axial Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–7 2.5.2 Flexural Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–7 2.5.3 Torsional Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–9 2.5.4 Shear Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–10 2.6 Putting it All Together, [ k ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–13 2.6.1 Truss Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–14 2.6.2 Beam Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–14 2.6.2.1 Euler-Bernoulli . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–15 2.6.2.2 Timoshenko Beam . . . . . . . . . . . . . . . . . . . . . . . . . . 2–15 2.6.3 2D Frame Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–17 2.6.4 Grid Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–18 2.6.5 3D Frame Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2–18 2.7 Remarks on Element Stiffness Matrices . . . . . . . . . . . . . . . . . . . . . . . . 2–19
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Draft 0–2 CONTENTS 3 STIFFNESS METHOD; Part I: ORTHOGONAL STRUCTURES 3–1 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–1 3.2 The Stiffness Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–2 3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–4 E 3-1 Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–4 E 3-2 Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–6 E 3-3 Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–9 3.4 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–13 4 TRANSFORMATION MATRICES 4–1 4.1 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–1 4.1.1 [ k e ] [ K e ] Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–1 4.1.2 Direction Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–2 4.2 Transformation Matrices For Framework Elements . . . . . . . . . . . . . . . . . 4–6 4.2.1 2 D cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–6 4.2.1.1 2D Frame, and Grid Element . . . . . . . . . . . . . . . . . . . . 4–6 4.2.1.2 2D Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–8 4.2.2 3D Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–8 4.2.2.1 Simple 3D Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–9 4.2.2.2 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–12 4.2.3 3D Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–15 5 STIFFNESS METHOD; Part II 5–1 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–1 5.2 [ ID ] Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–2 5.3 LM Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–3 5.4 Assembly of Global Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 5–3 E 5-1 Global Stiffness Matrix Assembly . . . . . . . . . . . . . . . . . . . . . . . 5–4 5.5 Skyline Storage of Global Stiffness Matrix, MAXA Vector . . . . . . . . . . . . . . 5–6 5.6 Augmented Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–9 E 5-2 Direct Stiffness Analysis of a Truss . . . . . . . . . . . . . . . . . . . . . . 5–14 E 5-3 Assembly of the Global Stiffness Matrix . . . . . . . . . . . . . . . . . . . 5–19 E 5-4 Analysis of a Frame with MATLAB . . . . . . . . . . . . . . . . . . . . . 5–21 5.7 Computer Program Flow Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–25 5.7.1 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–26 5.7.2 Element Stiffness Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–26 5.7.3 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–29 5.7.4 Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–32 5.7.5 Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–32 5.7.6 Backsubstitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–32 5.7.7 Internal Forces and Reactions . . . . . . . . . . . . . . . . . . . . . . . . . 5–32 5.8 Computer Implementation with MATLAB . . . . . . . . . . . . . . . . . . . . . . 5–36 5.8.1 Program Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–36 Victor Saouma Matrix Structural Analysis
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Draft CONTENTS 0–3 5.8.1.1 Input Variable Descriptions . . . . . . . . . . . . . . . . . . . . . 5–37 5.8.1.2 Sample Input Data File . . . . . . . . . . . . . . . . . . . . . . . 5–38 5.8.1.3 Program Implementation . . . . . . . . . . . . . . . . . . . . . . 5–40 5.8.2 Program Listing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–40 5.8.2.1 Main Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–40 5.8.2.2 Assembly of ID
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