Note that for mode n of a uniform simply-supported beam of span L and mass/unit length m with f = sin(npx/L), the modal mass M* is: which is 0.5mL (half the total mass of the beam). The application of these concepts to the problems of floor vibration and wind- induced vibration is described in References 6 and 7. 12.3.4 Approximate methods to determine natural frequency Approximate methods are useful for estimating the natural frequencies of struc- tures not conforming with one of the special cases for which standard solutions exist, and for checking the predictions of computer analyses when these are used. One of the most useful approximate methods relates the natural frequency of a system to its static deflection under gravity load, d. With reference to section 12.2.2 the natural frequency of a single lumped mass system is: This may be rewritten, replacing the mass term M by the corresponding weight Mg, as: Since Mg/K is the static deflection under gravity load (d): (12.14) This formula is exact for any single lumped mass system. For distributed parameter systems a similar correspondence is found, although the numerical factor in Equation (12.14) varies from case to case, generally between 16 and 20. For practical purposes a value of 18 will give results of sufficient accuracy. When applying these formulae the following points should be noted. (1) The static deflection should be calculated assuming a weight corresponding to the loading for which the frequency is required.This is usually a dead load with an allowance for expected imposed load. (2) For horizontal modes of vibration (e.g. lateral vibration of an entire structure) the gravity force must be applied laterally to obtain the appropriate lateral deflection, as shown in Fig. 12.6(a). f g n = 15.76/ when is measured in mm = 2pd dd f gK Mg n = Ê Ë ˆ ¯ 2p f K M n = Ê Ë ˆ ¯ 1 2p Mm nxL x L *= dsin /p () [] Ú 2 0 Distributed parameter systems 365 Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ — — — / (0) ft t tt fit fit t I tilt t (b) UUUU (c) 366 Applicable dynamics (3) The mode shape required must be carefully considered in multi-span structures. In the two-span beam of Fig. 12.6(b) the lowest frequency will correspond to an asymmetrical mode as illustrated; the corresponding d must be obtained by applying gravity in opposite directions on the two spans. The normal gravity deflection will correspond to the symmetrical mode with a higher natural fre- quency, Fig. 12.6(c). These concepts can be extended to estimating the natural frequencies of primary beam – secondary beam systems. If the static deflection of the primary beam is d p and the static deflection of a secondary beam is d s (relative to the primary) the com- bined natural frequency is approximately Fig. 12.6 Use of gravity deflections to estimate natural frequencies. In each case f n ϴ 18/ √d. (a) For a vertical structure the gravity load is applied horizontally. (b) Contin- uous structure gravity load applied in opposite directions on alternate spans. (c) Continuous structure – symmetric mode has higher frequency. Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ Damping 367 from which it can be shown that (12.15) where f p and f s are the natural frequencies of the primary and secondary beams alone. Care must always be taken in using these formulae so that a realistic mode shape is implied.There will generally be continuity between adjacent spans at small ampli- tudes even in simply-supported designs, and in many situations a combined mode where both the primaries and secondaries are vibrating together in a ‘simply- supported’ fashion is not possible. 12.4 Damping Damping arises from the dissipation of energy during vibration. A number of mechanisms contribute to the dissipation, including material damping, friction at interfaces between components and radiation of energy from the structure’s foundations. Material damping in steel provides a very small amount of dissipation and in most steel structures the majority of the damping arises from friction at bolted connections and frictional interaction with non-structural items, particularly par- titions and cladding. Damping is found to increase with increasing amplitude of vibration. The amount of damping that will occur in any particular structure cannot be calculated or predicted with a high degree of precision, and design values for damping are generally derived from dynamic measurements on structures of a cor- responding type. Damping can be measured by a number of methods, including: • rate of decay of free vibration following an impact (Fig. 12.3(a)) • forced excitation by mechanical vibrator at varying frequency to establish the shape of the steady-state resonance curve (Fig. 12.4) • spectral methods relying on analysis of response to ambient random vibration such as wind loading. All these methods can run into difficulty when several modes close in frequency are present. One result of this is that on floor structures (where there are often several closely spaced modes) the apparent damping seen in the initial rate of decay after impact can be substantially higher than the true modal damping. 111 222 fff nps =+ f n ps = + () 18 dd Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ 368 Applicable dynamics Damping is usually expressed as a fraction or percentage of critical (x), but the logarithmic decrement (d) is also used. The relationship between the two expres- sions is x = d/2p. Table 12.4 gives typical values of modal damping that are suggested for use in cal- culations when amplitudes are low (e.g. for occupant comfort). Somewhat higher values are appropriate at large amplitudes where local yielding may develop, e.g. in seismic analysis. 12.5 Finite element analysis Many simple dynamic problems can be solved quickly and adequately by the methods outlined in previous sections. However, there are situations where more detailed numerical analysis may be required and finite element analysis is a versa- tile technique widely available for this purpose. Numerical analysis is often neces- sary for problems such as: (1) determination of natural frequencies of complex structures (2) calculation of responses due to general time-varying loads or ground motions (3) non-linear dynamic analysis to determine seismic performance. 12.5.1 Basis of the method As explained in Chapter 9 the finite element method describes the state of a struc- ture by means of deflections at a finite number of node points. Nodes are connected by elements which represent the stiffness of the structural components. In static problems the equilibrium of every degree of freedom at the nodes of the idealization is described by the stiffness equation: F = KY where F is the vector of applied forces, Y is the vector of displacements for every degree of freedom, and K is the stiffness matrix.Solution of unknown displacements for a known force vector involves inversion of the stiffness matrix. Table 12.4 Typical modal damping values by structure type Structure type Structure damping (% critical) Unclad welded steel structures 0.3% (e.g. steel stacks) Unclad bolted steel structures 0.5% Composite footbridges 1% Floor (fitted out), composite and 1.5%–3% non-composite (may be higher when many partitions on floor) Clad buildings (lateral sway) 1% Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ (0) _-0 -2jY3 1% eamelements mosses at nodes connecting nodes (b) _____ / (c) Finite element analysis 369 The extension of the method to dynamic problems can be visualized in simple terms by considering the dynamic equilibrium of a vibrating structure. Figure 12.7(a) shows the instantaneous deflected shape of a vibrating uniform cantilever, and Fig. 12.7(b) shows a finite element idealization of this condition. The shape is described by the deflections of the nodes, Y, and a mass is associated with each degree of freedom of the idealization. At the instant considered when the deflection vector is Y the forces at the nodes provided by the stiffness elements must be KY. These forces may in part be resisting external instantaneous nodal forces P and may in part be causing the mass associated with each node to accelerate. The equa- tions of motions of all the nodes may therefore be written as P - MŸ = KY or KY + MŸ = P where M is the mass matrix, Ÿ is the acceleration vector, and P is the external force vector. The natural frequencies and mode shapes are obtained by solving the undamped free vibration equations: Fig. 12.7 Finite element idealization. (a) Uniform beam in free vibration, (b) finite element representation, (c) fourth mode not accurately represented Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ KY + MŸ = 0 Assuming a solution of the form y = Ycosw n t, it follows that Ÿ =-w 2 n Y and hence This is a standard eigenvalue problem of matrix algebra for which various numeri- cal solution techniques exist. The solution provides a set of mode shape vectors Y with corresponding natural frequencies w n . The number of modes possible will be equal to the number of degrees of freedom in the solution. 12.5.2 Modelling techniques Dynamic analysis is more complex than static analysis and care is required so that results of appropriate accuracy are obtained at reasonable cost when using finite element programs. It is often advisable to investigate simple idealizations initially before embarking upon detailed models. As problems and programs vary it is pos- sible to give only broad guidance; individual program manuals must be consulted and experience with the program being used is invaluable. More detailed back- ground is given in Reference 8. The first stage in any dynamic analysis will invariably be to obtain the natural fre- quencies and mode shapes of the structure. As can be seen from Fig. 12.7(c) a given finite element model will represent higher modes with decreasing accuracy. If it is only necessary to obtain a first mode frequency accurately then a relatively coarse model, such as that illustrated in Fig. 12.7(b), will be perfectly adequate. In order to obtain an accurate estimate of the fourth mode a greater subdivision of the struc- ture would be necessary, since the distribution of inertia load along the uniform beam in this mode is not well represented by just four masses. Probably the most widely used approach for eigenvalue problems is subspace iteration.This is a robust solution method which maps problems with a large number of degrees of freedom on to a ‘subspace’, with a much smaller number of degrees of freedom, to reduce the problem size. The structure’s eigenvalues (frequencies) are the same as those of the subspace, and the eigenvectors (modeshapes) of the structure are calculated from the eigenvectors of the subspace.This method resolves the smallest eigenvalues with the highest accuracy. Most eigensolvers will also include Sturm sequence checks to ensure that the eigenvalues found are the ones required, and that there are none missing from the sequence. Two approaches exist for calculating the element mass matrix. The consistent mass matrix is the most accurate way of representing the mass and inertia of the element, but when the aim is to calculate the dynamic response of the structure the simpler lumped mass approach can be more effective as it avoids single element modes of vibration. It is important to ensure that all the relevant mass is accounted for in a modal dynamic analysis. In many cases the engineer starts from a model, built for static KMY- [] =w n 2 0 370 Applicable dynamics Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ analysis, where ‘mass’ is applied as loading. It is important to ensure that all the mass present is correctly included as mass (not loading) in the dynamic analysis. Clearly, dynamically consistent units must be used throughout, and these units may not be the same as those used for a static analysis using the same model. A hand check of the first mode frequency using an approximate or empirical method is strongly advisable to ensure that the results are realistic. In addition, there is even more need than with static analysis to view computer analysis results as approxi- mate. It is very difficult to predict natural frequencies of real structures with a high degree of precision unless the real boundary conditions and structural stiffness can be defined with confidence. This is rarely the case and these are uncertainties that finite element analysis cannot resolve. 12.6 Dynamic testing Calculation of the dynamic properties and dynamic responses of structures still pres- ents some difficulties, and testing and monitoring of structures has a significant role in structural dynamics. Testing is the only way by which the damping of structures can be obtained, by which analytical methods can be calibrated and many forms of dynamic loading can be estimated. It is often an essential part of the assessment and improvement of structures where dynamic response is found to be excessive in prac- tice. Further details are contained in references 9 and 10. References to Chapter 12 1. Clough R.W. & Penzien J. (1993) Dynamics of Structures, 2nd edn. McGraw- Hill. 2. Dowrick D.J. (1987) Earthquake Resistant Design for Engineers and Architects, 2nd edn. John Wiley & Sons. 3. Warburton G.B. (1976) The Dynamical Behaviour of Structures, 2nd edn. Pergamon Press, Oxford. 4. Harris C.M. & Crede C.E. (1976) Shock and Vibration Handbook. McGraw- Hill. 5. Roark R.J. & Young W.C. (1989) Formulas for Stress and Strain, 6th edn. McGraw-Hill. 6. Construction Industry Research & Information Association (CIRIA)/The Steel Construction Institute (SCI) (1989) Design Guide on the Vibration of Floors. SCI Publication 076, SCI, Ascot, Berks. 7. Bathe K.J. (1996) Finite Element Procedures. Prentice Hall, Englewood Cliffs, NJ. 8. Bachmann H. (1995) Vibration Problems in Structures. Birkhauser Verlag AG. 9. Ewins D.J. & Inman D.J. (2001) Structural dynamics @2000: Current status and future directions. Research Studies Press, Baldock. Dynamic testing 371 Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ 10. Ewins D.J. (2001) Modal Testing: Theory, practice and application, 2nd edn. Research Studies Press, Baldock. Further reading for Chapter 12 Blevins R.D. (1995) Formulas for natural frequency and mode shape (corrected edition). Krieger, Malubar, FL. Chopra A.K. (2001) Dynamics of Structures – Theory and Applications to Earth- quake Engineering, 2nd edn. Prentice Hall, Upper Saddle River, NJ. National Building Code of Canada (1995) Commentary A – Serviceability Criteria for Deflections and Vibrations. 372 Applicable dynamics Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ [...]... http://shop.steelbiz.org/ Steel Designers' Manual - 6th Edition (2003) Design for axial tension rolled sections :i: Fig 14.1 Tension members JJ compound sections heavy rolled and built—up sections threaded bar flat round strand rope locked coil rope 385 Steel Designers' Manual - 6th Edition (2003) 3 86 Tension members This material is copyright - all rights reserved Reproduced under licence from The Steel. .. and Windus, London 2 The Steel Construction Institute (SCI) (2001) Steelwork Design Guide to BS 5950: Part 1: 2000, Vol 1: Section Properties Member Capacities, 6th edn SCI, Ascot, Berks Steel Designers' Manual - 6th Edition (2003) Chapter 14 Tension members by JOHN RIGHINIOTIS and ALAN KWAN This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute.. .Steel Designers' Manual - 6th Edition (2003) Chapter 13 Local buckling and cross-section classification by DAVID NETHERCOT This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ 13.1 Introduction The efficient use of material within a steel. .. is suitable for plastic design – plastic cross-section (Class 1) Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ 3 76 Fig 13.3 Local buckling and cross-section classification Rectangular... under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ where Ae = effective cross-sectional area = k1 k2 At At = net cross-sectional area A = gross area sy = nominal yield stress for steel given in BS 4 360 gm = partial safety factor.a Values are given in Table 2 of BS 5400: Part 3 This takes account... (1973) Canadian Structural Steel Design Canadian Institute of Steel Construction, Ontario Bresler B., Lin T.Y & Scalzi J.B (1 968 ) Design of Steel Structures Wiley, Chichester Dowling P.J., Knowles P & Owens G.W (1988) Structural Steel Design Butterworths, London Horne M.R (1971) Plastic Theory of Structures, 1st edn Nelson, Walton-on-Thames Owens G.W & Cheal B.D (1989) Structural Steelwork Connections Butterworths,... function of d/t for different web stress patterns 379 Steel Designers' Manual - 6th Edition (2003) 380 Local buckling and cross-section classification This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ In the introduction to this chapter... would render design calculations prohibitively difficult and it is Steel Designers' Manual - 6th Edition (2003) Economic factors 381 This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ therefore usual to make only a very approximate allowance... lies between -1 and +1 and is a measure of the stress ratio within the web Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ 382 Local buckling and cross-section classification only plastic... df1r In' d[lHt b _T b I HT T Fig 13.1 Structural cross-sections I 1j b T Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ Cross-sectional dimensions and moment–rotation behaviour Fig . 12 .6( a). f g n = 15. 76/ when is measured in mm = 2pd dd f gK Mg n = Ê Ë ˆ ¯ 2p f K M n = Ê Ë ˆ ¯ 1 2p Mm nxL x L *= dsin /p () [] Ú 2 0 Distributed parameter systems 365 Steel Designers' Manual. symmetric mode has higher frequency. Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute. damping. 111 222 fff nps =+ f n ps = + () 18 dd Steel Designers' Manual - 6th Edition (2003) This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute