paper: gordon/may
Observations on the grillage analysis of slabs Synopsis The results of the analyses of three slabs using the grillage analogy (with various methods for approximating the equivalent grillage members properties) and finite element analysis are compared with theoretical solutions, where available. The examples demonstrate some of the disadvantages of using the grillage analogy for analysing slabs which can lead to erroneous results. It is concluded that the use of finite element analysis is preferred and that the use of the grillage method should be avoided.
Introduction The authors’ were surprised that from discussions with practising engineers that some engineers still resort to carrying out a grillage analysis of slabs when better tools are now available, particularly when the grillage method is not only approximate but also can lead to the wrong results. Only recently a Past President1 of the Institution mentioned the approximate nature of the method. In order to show the problems associated with the grillage method the results of the analyses for three slabs are described. The results are compared with exact solutions and with finite element analyses. The examples are restricted to isotropic slabs but the analysis of other forms of construction is discussed. The design and analysis of any slab requires that initially the engineer will need to create simple conceptual models in order to understand the way in which the loads are carried and the way the slab deflects. From this simple conceptual model the sections will be sized. This stage may involve simple ‘rules of thumb’ or engineering judgement. At this stage a more detailed analysis will often be carried out to both validate and refine the initial simplified analysis. For the design of concrete slabs an elastic analysis is usually used. The advantage of elastic analysis is that it gives an equilibrium stress field and thus the lower bound theorem of limit analysis can be invoked. Reinforcement for the ultimate limit state is designed using ultimate load methods, for example the Wood and Armer equations2. The results from the analysis can also be suitably factored and used for serviceability checks. The authors appreciate that with the uncertainties in other aspects of the design, the enforcement of the requirement of high accuracy from the analysis may not be justifiable. However, the contention is that the use of finite element analysis gives increased accuracy with either no extra cost or even savings in cost.There is, of course, also the possibility of applying the grillage method to a structure of,for example, unusual geometry and obtaining completely erroneous results. One of the possible reasons for the continued use of grillages is the widespread use in Industry of the text by Hambly3 which states ‘The (finite element) method is much more complicated and expensive than the grillage, and since it does not produce significantly different results….’. Also Hambly states that ‘the method is cumbersome to use’, and ‘the choice of element type can be extremely critical and, if incorrect, the results can be far more inaccurate than those predicted by simpler models such as grillage or space frame’. The book was first published in 1976 and significant strides have been made in finite element technology. The data preparation for the finite element analyses carried out as part of this study proved to be quicker than for the grillage analyses and there was no significant difference in run times.The results presented will show that there can be differences. Hambly explains that the grillage is an approximation and identifies the shortcomings of grillage in his text as being that grillages have ‘no physical or mathematical principle that makes torques or twists automatically identical in orthogonal directions’. Additionally, he states that ‘the moment in any (grillage) beam is solely proportional to the curvature in it’. However, it is well known that plate behaviour is such that the moment in any given direction is dependent on the curvature in that direction and in the direction orthogonal to it, as influenced by Poisson’s ratio. Hence, the gril-
Stuart R. Gordon BSc, MSc, CEng, MICE Heriot-Watt University, Edinburgh
Prof. Ian M. May BEng, MSc, PhD, CEng, FIStructE, FICE Professor of Civil Engineering, Heriot-Watt University, Edinburgh Received: 08/02 Modified: 03/03 Accepted: 08/03 Keywords: Grillage analysis, Slabs, Plates, Simply supported © Stuart R. Gordon, Ian M. May
Fig 1. (top) Slab notation for moments and reactions Fig 2. (left) Square slab model (6 x 6 arrangement) Fig 3. (below) Square slab nondimensionalised Mxy moments along support section A-A
3 February 2004 – The Structural Engineer|35
paper: gordon/may
lage will lead to approximate results.
Analyses Three typical slabs have been used to highlight the shortfalls in grillage analysis.The analyses were undertaken using the LUSAS finite element package4. Four-noded quadrilateral thin plate bending elements were used throughout, except in the analysis of the skew orthogonal slab, where 3-noded triangular elements were mixed with the quadrilaterals. The notation for moments and reactions is shown in fig 1.
Fig 4. Rectangular slab model (6 x 6 and 8 x 8 arrangements)
Skew plate simply supported on two sides
Square plate simply supported on four sides The square plate studied was of edge length a,thickness a/20,with simple supports along the edges and rotational restraints parallel to the supports and Poisson’s ratio equal to 0.2. Various mesh configurations were adopted (4 × 4, 6 × 6, 10 × 10 and 20 × 20) to test convergence in the models. Grillages with the same nodal arrangements were used with grillage members connecting each node. Hence, for a 6 × 6 finite element mesh model the corresponding grillage will have 42 longitudinal and 42 transverse members.The section properties were calculated for the grillages using the properties proposed by West5. In addition, section properties were calculated using the principles of thin plate theory as proposed by Hambly3.A typical slab grillage layout is shown in fig 2. Section properties were calculated assuming a width of grillage member which comprised of half the distance between the grillage members on either side.The calculated values of the torsional stiffnesses of the grillage members were halved to account for the slab ‘element’ twisting in both orthogonal directions. The output from the grillage analyses were manipulated to determine the moment triad at any point using the method described by West5. Convergence for each analysis type was assessed for a uniformly distributed loading (UDL) w, over the entire slab.The series solution, which was taken as the exact solution, was proposed by Navier and is given by Timoshenko and Woinowsky-Krieger6. From fig 3, which shows the variation of Mxy along the supported edge, it can clearly be seen that as the mesh is refined the finite element models converge rapidly towards the series solution. Similar convergence occurs for the Mx and My moments in the finite element models throughout the slab. In the grillage analysis the Mx and My moments converged, however the twisting moment Mxy diverged for the West models. The Hambly models did not converge to the series solution but gave favourable agreement. In this slab, concentrated holding-down corner forces, R, occur, where R = 2Mxy. In the finite element models the corner forces converged towards the theoretical value,whereas in the grillages the corner forces show very slow convergence.
Rectangular plate simply supported on two sides The rectangular plate studied was of width b, span 1.5b, and 36|The Structural Engineer – 3 February 2004
thickness b/20, with simple supports along the two short edges and rotational restraints parallel to the supports.Various mesh/ grillage member configurations were adopted (6 × 6, 8 × 8, 10 × 10, and 20 × 20) to test convergence. The 8 × 8 mesh model was modified in accordance with Hambly’s recommendation to try to account for the transverse shear flow around the slab edges, by including a longitudinal member at 0.3 × slab depth from the deck edge. No edge stiffening was used. The section properties for the grillage analyses were calculated using both West’s and Hambly’s proposals.A typical slab grillage layout is shown in fig 4. Convergence for each analysis type was assessed for a uniformly distributed loading, UDL, of w over the entire slab. The series solution is that given by Timoshenko and WoinowskyKrieger. As with the square plate, the finite element analysis for the rectangular slab generally showed good convergence towards the series solution, fig 5. The grillages however, effectively model the slab as a series of beams.The longitudinal moment My at midspan predicted by the grillage analysis is wl2/8 as expected for a series of simply supported beams.However,the grillage analysis predicts zero transverse moment throughout the slab. The grillage analysis failed to give the correct twisting moments throughout the slab, fig 5, with the analyses giving virtually zero values throughout.The deflections from the finite element models gave a typical anti-clastic surface with increased longitudinal deflections at the edges. In contrast, the grillages gave a constant midspan deflection, fig 6. As in the square slab, concentrated corner forces, R = 2Mxy upwards also occurred in the rectangular slabs. The finite element analysis indicated good convergence to the corner forces, but no such forces were predicted in the grillage analyses.
Fig 5. (below) Rectangular slab model nondimensionalised Mxy moments at support section BB
A series of models were used to study a skew plate.The skew plate studied was of square span b, width 1.5b, thickness b/20, with simple supports along the long edges and rotational restraints parallel to the supports and a skew angle of 30°, fig 7 and fig 8. Skew and orthogonal grillages with corresponding finite element models with the same mesh arrangements, (6 × 6 and 20 × 20), were used to analyse the slab. The section properties for the grillage analyses were calculated using both West’s and Hambly’s proposals. An additional 40 × 40 finite element model was taken as the converged solution for comparisons with the grillages. A
paper: gordon/may
compared to the finite element converged solution, fig 9.The best results with West were with the 6 × 6 orthogonal grillage and the worst with the 20 × 20 skew grillage, that is refining the grillage gave worse results! The Hambly skew grillage models over-estimated Mv by about 35% at the centre compared to the finite element converged solution. Good results were found from Hambly’s orthogonal grillages. For comparisons with Muv a transverse section,B–B at midspan was considered, fig 10. Muv was under-estimated by up to 60% by West’s models at the slab centre. The best results with West’s models were with the 6 × 6 orthogonal grillage and the worst with the 20 × 20 skew grillage, again refining the grillage gave worse results.The Hambly skew grillage models under-estimated Muv by up to 40%, but the orthogonal grillages performed well.
Discussion
Fig 6. (above) Rectangular model non-dimensionalised deflections at midspan section AA Fig 7. (left) Skew slab model (6 x 6 arrangement)
Fig 8. (left) Skew orthogonal slab model (6 x 6 skew ortho arrangement) Fig 9. (below) Skew model nondimensionalised Mv moments for 30° skew slab at midspan section BB
The grillage and finite element analyses took similar times to run and the output for each was in a similar form, which was suitable for post-processing. Grillage analysis will give an equilibrium solution, which can be used in a lower bound solution for the reinforcement of slabs. However, grillage analysis does not necessarily give a set of moments that are accurate enough to be used to assess the serviceability behaviour of the slab. Since many slabs for buildings and bridge decks have been analysed and designed using the grillage analogy then it is reasonable to ask why such slabs appear to behave satisfactorily and do not exhibit unacceptable cracking. There can be a number of reasons for this. Slabs are generally designed for a range of load cases and hence an envelope of load effects is considered. From this,inadequacies in the modelling method may tend to be masked by the range of loadings applied. Load factors are required in bridge decks and buildings and generally designers will tend to provide additional steel to suit bar detailing. In addition, reinforcement requirements to cater for early thermal cracking and distribution are required in bridge decks. Hence, problems will not necessarily be observed at service loads, due to the additional amounts of steel present. However, if the loading were required to be increased, for example due to a change of use, then an analysis using the grillage method might fail to identify a slab that was not properly reinforced in areas of large twisting moments, for example. It is often considered that refining grillages locally can pick up local effects in bridge decks.However,for the slabs studied,the grillage analysis sometimes showed divergence on refinement and indicated that local refinement is not necessarily effective. None of the above problems would be encountered with a finite element analysis. In addition, by using finite element modelling the effects of edge stiffening, skew, curvature and orthotropy can readily be allowed for in slabs. The analyses of beam and slab bridge decks can also be carried out using a finite element analysis in which the slab is modelled using flat shell elements and the beams using a beam element which includes in-plane effects with
theoretical solution was not available for the 30° skew slab under uniformly distributed loading.However,a finite difference solution for a 45° skew rhombic plate under a central concentrated load analysed by Robinson and reported by Morley7 was used to demonstrate the accuracy of the chosen finite elements for a heavily skewed plate. Skew models using 20 × 20 meshes of quadrilaterals and triangular elements were used as a comparison and the results for the principal moments at the centre of the plates compared well with the finite difference results. The less distorted elements used in the analysis of the 30° skew slab should perform better than those used in the analysis of the 45° skewed slab. The finite element analysis converged for Mv (longitudinal moment on the skewed v axis), fig 9 and Muv (twisting moment relative to the u – v axes) fig 10. For comparisons with Mv a transverse section, B–B at midspan was considered. The West grillage models tended to over-estimate Mv by up to 45% at the centre, 3 February 2004 – The Structural Engineer|37
paper: gordon/may
grillage analogy to model slabs and the engineer should be aware of the limitations of the method. Finite element analyses will converge towards exact solutions with mesh refinement whereas this is not always the case with grillages. For the square slab and skew slab problems using the West rules, the coarse grillage often performed better than the finer models and suggests that refining the grillage is not always worthwhile. In the square slab, Hambly’s models performed well as did the orthogonal grillages in the skew slab. However, in the skew grillage analyses shortcomings were evident and the authors would advise against the use of such skew grillages. For the rectangular model the shortfalls in the modelling showed both West’s and Hambly’s models to be inappropriate in this case. The finite element analyses proved quicker to prepare compared with the grillage modelling. Also as previously stated there is, of course, the possibility of applying the grillage method to a structure of, for example, unusual geometry and obtaining completely erroneous results.
Acknowledgment The authors would like to thank Dr David Johnson of Nottingham Trent University for his advice during the preparation of this paper.
REFERENCES 1.
2.
both the beams and the slab positioned correctly. The advantage of such an analysis is that it gives a complete picture of the forces in the structure. Finite element models can easily be checked for convergence by refining the mesh either locally or globally.
Conclusions The slab examples presented demonstrate some of the disadvantages of using the grillage analogy for designing and assessing slabs. The authors would advise against using the West rules for
Fig 10. (above) Skew model nondimensionalised Muv moments for 30° skew slab at midspan section B-B
3. 4. 5.
6. 7.
Blockley, D. and Woodman, N.: ‘Civil/structural engineers and maths: the changing relationship’. The Structural Engineer, April 2002, 80/7 p 14-15 Wood, R. H.: ‘The reinforcement of slabs in accordance with a predetermined field of moments’, Concrete, 2/2, February 1968, p 69-76 Hambly, E. C.: Bridge Deck Behaviour, Chapman and Hall, 1976 LUSAS Version 13.4 Software, Finite Element Analysis Ltd, 2002 West, R.: ‘Recommendations on the use of grillage analysis for slab and pseudo-slab bridge decks’, C&CA/ CIRIA, London, 1974, p 24 Timoshenko, S. P. and Woinowsky-Krieger, S.: Theory of plates and shells, 2nd edition, McGraw Hill, 1959 Morley, L. S. D.: Skew plates and structures, Pergamon Press, 1963
EXAMINATIONS 1992-2003
IStructE has produced its latest examination CD, which contains examination question papers and examiners’ reports for the last 12 years. The CD also includes: • Personal points of view from a successful candidate and a chartered marking examiner’s position • An overview of the Institution’s examination process • Guidance advice to A-M and CM candidates preparing for the examinations • Seismic design information for the CM examination • Changes to the CM (2004) examination and the new format A-M examination introduced this year. The CD is priced £20 (including VAT), plus postage (UK: 60p; Europe: £1.32; Rest of the World: £1.61) Please remember to include postage when sending your order. For further information contact Alice Kingsley on 020 7235 4535 or email
[email protected] Candidates are reminded that the old format question papers are a useful tool for reference only
38|The Structural Engineer – 3 February 2004
