Chapter2 - Chapter 2 Deformations and Displacements in...

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© K. H. Ha - BCEE 343 Chapter 2 - v2.3 2.1 Chapter 2 Deformations and Displacements in Systems 1. Member Deformations and Displacements . .......................................................................... 2 1.1 Axial Deformations. ..................................................................................................... 2 Example 1 . .................................................................................................................. 3 1.2 Bending Deformations. ................................................................................................ 4 1.3 Member flexibility relation. ........................................................................................... 6 Example 2 . .................................................................................................................. 8 1.4 Matrix of members’ flexibilities . .................................................................................... 9 Example 3 . ................................................................................................................ 10 Problem 1................................................................................................................. 16 1.5 Deformations versus Displacements . ........................................................................... 18 2. Strain Energy in Members and Systems . ........................................................................... 20 2.1 Work done by External Forces . ................................................................................... 20 2.2 Work done by Internal Forces. .................................................................................... 20 2.3 The Principle of Energy Conservation. .......................................................................... 21 3. The Virtual Work Principle . .............................................................................................. 21 3.1 Nodal Displacements of Trusses, Beams, and Frames. .................................................... 22 Example 4 . ................................................................................................................ 24 3.2 Proof of the Virtual Work Principle. .............................................................................. 25 Problem 2................................................................................................................. 26 4. Nodal Displacements in Truss Systems . ............................................................................ 27 Example 5 . ................................................................................................................ 28 Example 6 . ................................................................................................................ 31 Problem 3................................................................................................................. 35 5. Nodal Displacements in Beam and Frame Systems . ............................................................ 36 5.1 Bending Moments and Deformed Shapes. ..................................................................... 36 5.2 Deformed Shapes of Beams and Frames . ..................................................................... 38 5.3 Factors Affecting Deformations . .................................................................................. 38 5.4 The Role of the Supports and Connections. ................................................................... 39 5.5 Nodal Displacements by Virtual Work. .......................................................................... 39 Example 7 . ................................................................................................................ 41 Example 8 . ................................................................................................................ 45 Example 9 . ................................................................................................................ 47 Example 10 . .............................................................................................................. 48 Problem 4................................................................................................................. 53 Problem 5................................................................................................................. 55 6. Generation of B and Q for Frames . ................................................................................... 56 Example 11 . .............................................................................................................. 59 Problem 6 .................................................................................................................... 60 7. Flexibility Coefficients. .................................................................................................... 62 7.1 Application . ............................................................................................................. 62 7.2 The System Flexibility Matrix F . .................................................................................. 63 8. Nodal Displacements due to Support Movements. ............................................................... 63 8.1 Virtual Work Principle for Rigid Bodies. ......................................................................... 64 8.2 Application to Displacements Caused by Support Movements . ......................................... 65 Example 12 . .............................................................................................................. 65 Problem 7................................................................................................................. 67
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© K. H. Ha - BCEE 343 Chapter 2 - v2.3 2.2 1. Member Deformations and Displacements Material deforms whenever it is subject to stresses and/or thermal expansion or shrinkage. The deformations change the shape of the body. Conversely, if the shape remains unchanged, then the deformations are zero even though the body might have displaced. In actual structural systems, the deformations and displacements 1 are usually so small that they may not be visible. We will consider the following two types of deformations: Axial deformations (found in trusses, cables, springs, beams and frames). Bending deformations (found in beams and frames). 1.1 Axial Deformations Axial deformations in a member are caused by Axial force in the member, and Uniform thermal expansion or shrinkage. By superposition of the two effects (Fig.2.1), the total axial deformation (elongation 2 or shortening) in a member is e F AE L e o + = ( 2 . 1 ) where F = axial force in the member. o e = axial deformation caused by free thermal expansion or shrinkage. In the case of thermal expansion due to a change of temperature T : T L e = α o where is the coefficient of thermal expansion. F F T F F T e Figure 2.1 Axial deformations 1 Displacements are movements from the original undeformed position. Rigid-body displacements involve no deformations. 2 Elongation is a positive axial deformation consistent with tension being a positive force.
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© K. H. Ha - BCEE 343 Chapter 2 - v2.3 2.3 Example 1 Members of the truss in Example 1.1 have constant A = 1.36 × 10 -3 m 2 , E = 200GPa and coefficient of thermal expansion α = 1.2 × 10 -5 / o C. Let node 2 of the truss be subject to a downward vertical force P = 20kN and a horizontal force P /2
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Chapter2 - Chapter 2 Deformations and Displacements in...

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