Why is there a step in slab 2D moment contours at changes in slab depth?
Not version-specific
Tekla Structural Designer
Environment
Not environment-specific
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Answer
Theory:
The answer lies in consideration of the moment-curvature relationship. From beam theory we have:
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Where:
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While this equation is from beam theory, the same principle applies to FE (shell) element mesh behaviour although, properly speaking, this is governed by plate theory.
Another fundamental equation to consider here is that for rectangular section inertia;Image
From the above we see that moment and curvature are directly related and that, for a given curvature, moment is proportional to the cube of the depth. Hence changes in depth have a very large effect on moment – other things being equal. The latter is not the main principle involved, but is worth considering at the same time since engineers are sometimes surprised by the degree of the effect on moment from changes in depth.
Explanation and Examples
We use the results of a number of simple models below to illustrate how consideration of this theory answers the question. You may wish to build similar models yourself. Full and exact dimensions/ properties are not given as this is a general effect which can be observed with many dimensions, properties and geometries. The model discussed is also attached as a zip file.
Consider the following model of two adjacent slab items, one double the depth of the other and with a (panel) line load applied along their junction line. Several closely spaced supports (approximating a continous support) are placed along two opposite edges such that the panels both span predominantly in one direction. From a study of the Solver model and displacements it will be confirmed that the slab panels are definitely meshed together and share the line load.
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Looking at the 2D deflection results it is observed that the contours are continuous and exhibit no step. Note especially that the deflections - and hence curvature - along the junction are the same for both panels, as they must be.
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Consider now the moment-curvature relationship discussed above; if the slabs have the same curvature, but different depths, then the moment in each panel cannot be the same. Tekla Structural Designer actually reports in the tool-tip the two separate panel moments at each node point along the junction. The panel moments are those used in slab design so this is an important principle for slab design.
Thus where panels of significantly different depth are adjacent and connected, a discontinuity in moment at their junction is to be expected. This can be misconstrued as indicating that the slabs are disconnected however this is not the case - the discontinuity is explicable by considering the moment-curvature relationship. Tekla Structural Designer displays this effect clearly as the 2D forces contouring is performed on an individual panel basis using the panel-specific nodal results.
Note that other analysis programs may ’blur’ this effect by simply averaging all of the nodal results when contouring and/or having no capability to differentiate these by association with specific (panel) areas of the FE mesh.
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For further illustration - and since some engineers may still have some uncertainty about FE results - consider a similar situation with beams of different depths.
Adjacent beams are connected at regular internals along their length by (stiff) analysis elements and so forced to deflect the same amount (ergo have the same curvature) and share load applied to them. Note that this is in effect a grillage of beam elements, which is actually how continous two-way spanning slabs were commonly modelled before the more widespread application of the finite element method (FEM) in commercial structural analysis programs. Load is applied only to the outer beams to reinforce the principle. Three beams are used to form a symmetrical arrangement and avoid twisting effects and asymmetric displacements. Hence this is not intended to be an exact equivalent of the two adjacent slabs discussed above, but to illustrate exactly the same principle.
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From the analysis results it is observed that, as intended, the beams deflect almost exactly the same amount (the diagram to the right shows their deflections superimposed by looking exactly from one side view)
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However the moments are radically different – the deeper beam is doing most of the ‘work’. In fact we can estimate the deeper beam moment from that of the smaller beams using Equation (2) since all other parameters and the curvature are constant and only the depth is changed;
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Deeper beam moment = 22.8x(2)3 = 22.8x8 = 182 kNm - very close to the observed value.
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We could also demonstrate the effect using Settlement Loads since, conveniently, these allow the application of a forced displacement. We used the deflection values of the beams discussed above.
Settlement loads applied along the length of the beams to force them into the same curvature:
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The expected difference in resulting moment is observed:
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