2 - Virtual Work - Compound Structures

# 2 - Virtual Work - Compound Structures - Structural...

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Structural Analysis IV Chapter 2 – Virtual Work: Compound Structures Dr. C. Caprani 1 Chapter 2 - Virtual Work: Compound Structures 2.1 Introduction . ........................................................................................................ 3 2.1.1 Purpose . ......................................................................................................... 3 2.2 Virtual Work Development . ............................................................................... 4 2.2.1 The Principle of Virtual Work . ...................................................................... 4 2.2.2 Virtual Work for Deflections . ........................................................................ 8 2.2.3 Virtual Work for Indeterminate Structures. ................................................... 9 2.2.4 Virtual Work for Compound Structures . ..................................................... 11 2.3 Basic Examples . ................................................................................................. 14 2.3.1 Example 1 . ................................................................................................... 14 2.3.2 Example 2 . ................................................................................................... 21 2.3.3 Example 3 . ................................................................................................... 32 2.3.4 Example 4 . ................................................................................................... 40 2.3.5 Problems . ..................................................................................................... 48 2.4 Past Exam Questions . ........................................................................................ 52 2.4.1 Sample Paper 2007 . ..................................................................................... 52 2.4.2 Semester 1 Exam 2007 . ............................................................................... 53 2.4.3 Semester 1 Exam 2008 . ............................................................................... 54 2.4.4 Semester 1 Exam 2009 . ............................................................................... 55 2.4.5 Semester 1 Exam 2010 . ............................................................................... 56 2.4.6 Semester 1 Exam 2011 . ............................................................................... 57 2.5 Appendix – Trigonometric Integrals . .............................................................. 58 2.5.1 Useful Identities . .......................................................................................... 58

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Structural Analysis IV Chapter 2 – Virtual Work: Compound Structures Dr. C. Caprani 2 2.5.2 Basic Results . ............................................................................................... 59 2.5.3 Common Integrals . ...................................................................................... 60 2.6 Appendix – Volume Integrals . .......................................................................... 67 Rev. 1
Structural Analysis IV Chapter 2 – Virtual Work: Compound Structures Dr. C. Caprani 3 2.1 Introduction 2.1.1 Purpose Previously we only used virtual work to analyse structures whose members primarily behaved in flexure or in axial forces. Many real structures are comprised of a mixture of such members. Cable-stay and suspension bridges area good examples: the deck- level carries load primarily through bending whilst the cable and pylon elements carry load through axial forces mainly. A simple example is a trussed beam: Other structures carry load through a mixture of bending, axial force, torsion, etc. Our knowledge of virtual work to-date is sufficient to analyse such structures.

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Structural Analysis IV Chapter 2 – Virtual Work: Compound Structures Dr. C. Caprani 4 2.2 Virtual Work Development 2.2.1 The Principle of Virtual Work This states that: A body is in equilibrium if, and only if, the virtual work of all forces acting on the body is zero. In this context, the word ‘virtual’ means ‘having the effect of, but not the actual form of, what is specified’. There are two ways to define virtual work, as follows. 1. Virtual Displacement: Virtual work is the work done by the actual forces acting on the body moving through a virtual displacement. 2. Virtual Force: Virtual work is the work done by a virtual force acting on the body moving through the actual displacements. Virtual Displacements A virtual displacement is a displacement that is only imagined to occur: virtual displacements must be small enough such that the force directions are maintained.
Structural Analysis IV Chapter 2 – Virtual Work: Compound Structures Dr. C. Caprani 5 virtual displacements within a body must be geometrically compatible with the original structure. That is, geometrical constraints (i.e. supports) and member continuity must be maintained. Virtual Forces A virtual force is a force imagined to be applied and is then moved through the actual deformations of the body, thus causing virtual work.

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2 - Virtual Work - Compound Structures - Structural...

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