Combined Structures 2009-10 - Structural Analysis IV...

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Structural Analysis IV Virtual Work – Combined Structures 4th Year Structural Engineering 2009/10 Dr. Colin Caprani Dr. C. Caprani 1
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Structural Analysis IV Contents 1. Introduction . ........................................................................................................ 4 1.1 Purpose . ........................................................................................................... 4 2. Virtual Work Development. ............................................................................... 5 2.1 The Principle of Virtual Work. ........................................................................ 5 2.2 Virtual Work for Deflections. .......................................................................... 9 2.3 Virtual Work for Indeterminate Structures. ................................................... 10 2.4 Virtual Work for Combined Structures. ........................................................ 12 3. Examples . ........................................................................................................... 14 3.1 Example 1 . ..................................................................................................... 14 3.2 Example 2 . ..................................................................................................... 15 3.3 Example 3 . ..................................................................................................... 16 3.4 Example 4 . ..................................................................................................... 22 3.5 Further Examples. .......................................................................................... 29 4. Exercises. ............................................................................................................ 30 4.1 Problems . ....................................................................................................... 30 4.2 Past Exam Questions . .................................................................................... 32 5. Appendix – Trigonometric Integrals. .............................................................. 36 5.1 Useful Identities. ............................................................................................ 36 5.2 Basic Results. ................................................................................................. 37 5.3 Common Integrals . ........................................................................................ 38 6. Appendix – Volume Integrals. .......................................................................... 45 7. Ring Beam Examples (Advanced) . .................................................................. 46 7.1 Example 1 . ..................................................................................................... 46 7.2 Example 2 . ..................................................................................................... 51 7.3 Example 3 . ..................................................................................................... 58 7.4 Example 4 . ..................................................................................................... 66 7.5 Example 5 . ..................................................................................................... 75 Dr. C. Caprani 2
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Structural Analysis IV 7.6 Review of Examples 1 – 5. ............................................................................ 96 Dr. C. Caprani 3
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Structural Analysis IV 1. Introduction 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: Our knowledge of virtual work to-date is sufficient to analyse such structures. Dr. C. Caprani 4
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Structural Analysis IV 2. Virtual Work Development 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. 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. Dr. C. Caprani 5
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Structural Analysis IV 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. Virtual forces must form an equilibrium set of their own. Internal and External Virtual Work When a structures deforms, work is done both by the applied loads moving through a displacement, as well as by the increase in strain energy in the structure. Thus when virtual displacements or forces are causing virtual work, we have: 0 0 IE EI W WW δ δδ = −= = where Virtual work is denoted W and is zero for a body in equilibrium;
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Combined Structures 2009-10 - Structural Analysis IV...

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