# Sway Effects Under Gravity Load & Meshed Wall and Core Forces Calculation

# Questions

For the overall 3D Building Analysis results we commonly receive the following kinds of questions:

*In my model there are sway and moments occurring in walls and or cores for gravity load cases with only vertical loads.**Is this correct?**Why is this happening?*

*How are the wall/ core moments calculated?*

Typical queried results are illustrated in the pictures below:

# Answers

Though lateral deflection under gravity loading may initially seem counter-intuitive, the explanation for it is quite simple and logical. Simply put; any asymmetry - in the structure or the vertical loading - can result in sway. The same applies to cores formed of walls. Since there is deflection, there will be force - i.e the sway will be accompanied by associated moments. The magnitude of such moments can be significant which seems to further surprise some engineers leading them to question their accuracy and derivation.

In the following article we will show that:

- This effect relates to differential axial compression within the lateral load resisting elements of the structure, which are commonly shear and core walls.
- The effect is real and not some artificial effect or error of computer analysis, despite the fact that it may have previously been routinely ignored in hand calculations.
- This effect is always exposed by any 3D analysis of a fully framed structure.
- The effect can be exposed in flat slab models which are stabilized by core walls.
- The calculation of both meshed wall and core moments derives from engineering first principles with which most engineers will be familiar and is generally quite straightforward and logical.

## Single Wall Example

Consider first a single wall with a point load applied to one end at the top, as shown in the pictures below. It can be further illuminating to consider the behavior under the same loading for both a **mid-pier** wall and **meshed **wall. For a full understanding of the differences between these see the Help topics __How mid-pier walls are represented in solver models__ and __How meshed walls are represented in solver models__. The key point here is that in general - certainly in the case of normally proportioned single separate walls - **the behavior and results of mid-pier and meshed walls are practically identical**, as explained in this article __Shear wall analysis - New modeling, same answers__.

First looking at the mid-pier wall, and considering the underlying analysis model (shown above right), we doubt any engineer would query the resulting sway and moment. It is clear the vertical load is eccentric to the analysis element at the center line of the wall and must therefore produce unresisted bending - i.e. curvature and sway - and associated moment in this. This is exactly what we find in the analysis results as shown below - the moment is as expected; Vertical load * eccentricity = 100 kN * 2m = 200 kNm.

Next, considering the meshed wall with the same geometry, section and loading, we could ask; ** why should the overall behavior and results of this be any different?** The answer is; they shouldn’t.

Though the underlying analysis model is quite different - being formed of a mesh of 2D elements of the actual wall dimensions and thickness - the overall ‘macro’ structural behavior is identical. The point load is eccentric to the centroid of the wall section, thus there is an intrinsic moment as for the mid-pier wall causing practically identical sway displacements and internal moment.

However the analytical ‘micro’ behavior of the meshed wall is quite different to that of the mid-pier wall (and more realistic it should be appreciated). There is differential axial compression and hence elastic compressive deformation in the elements under the point load. The 2D elements form an elastic continuum, thus the deformations propagate throughout the 2D element mesh, resulting in the observed sway and extension of the outer edge of elements on the opposite side to the load. There are no in-plane (major axis) moment results from the 2D elements as such. The in-plane elastic deformations produce in-plane vertical compression and tension stresses and forces which can be viewed via the Results View > 2D Results contours. For an understanding of these 2D results see the article __How do I interpret 2D Results (FE Contour Forces) and what are the Sign Conventions?__.

### Other Loading

The example above of a single concentrated point load at the end of a wall is more extreme and probably less common. However, similar sway and moment effects, of varying magnitude, will generally occur for any kind of asymmetric load e.g for a VDL (approximated with four UDLs as illustrated below). Again this is entirely logical and to be expected following the fundamental engineering principles discussed above.

## Core Wall Example

Next we will look at a simple core wall example. The sway behavior and its explanation for core walls is essentially identical to the single wall example discussed above - whenever asymmetric loads are applied to cores, some amount of sway and moment can result.

Consider the example below of a square core formed of 4m long walls with slabs supported by one core wall only. The slab decomposed gravity loads (circled) show that these are applied to only one wall and so are clearly eccentric to the center of rigidity of the core, which is at its centroid. Hence moment is applied to the overall core section and sway displacement results. From the above discussion, this should not be a surprise.

Just as for a single wall, the sway has associated core moments and reaction. From the decomposed loads, it is a simple matter to calculate the *applied *core moments from the total load applied at each level and its eccentricity ** e** from the core centroid which = 2m:

- Total load at each level; P = 24.5 kN/m * 4m =
**98 kN** - Top storey moment: M = 98 kN * 2m =
**196 kNm** - Middle storey moment M = 2*98 kN * 2m =
**392 kNm** - Bottom storey moment M = 3*98 kN * 2m =
**588 kNm**

For clearer understanding of this behavior, it is useful to think of the core walls as the ‘flanges’ or ‘webs’ of the core section, in the same manner that we commonly think about the side walls of rectangular hollow (RHS) steel sections. Mental comparison could then be made with a member with a similar hollow section shape; what would happen to a rectangular section column if you applied an axial compressive load to only one flange? Logically bending would result, since the load is eccentric to the major or minor axis. Exactly the same occurs with cores formed of concrete walls.

## More Practical Example

The above examples are deliberately simple to make the phenomenon and logical explanation of it hopefully very clear. However the effect can and will occur in ‘real world’ practical buildings - though the loading and behavior may be more complex, the explanation is just the same.

For illustration see the flat slab building model shown at the start of the article. Here the core is at one edge of the building so logically it will not receive uniform, symmetrical vertical loads from the gravity load cases. Looking at the decomposed loads (for the slab self weight load case) in detail, it can be seen that these are far from uniform/ symmetrical - the inside ‘flange’ wall receives the majority of load while the ‘web’ walls receive VDLs and the outer ‘flange’ wall receives no load.

Considering this it is possible to appreciate how the effective center of loading applied to the core is offset from its centroid (indicated by the center of rigidity arrow in this case). This eccentric load causes an unresisted curvature in the core and hence sway develops. The effect of wall openings in the ‘flange’ or ‘web’ walls can also exacerbate the effect, since they will be less stiff - both vertically and laterally. Openings would also complicate the behavior and distribution of wall forces involved - nevertheless the phenomenon and its explanation remain the same.

It is worth noting also that this example has an axis of symmetry of both geometry and loading aligned with the global axes. so there is sway in only one global direction. However where this is not the case - e.g. if there were openings in just one web wall or longer slab spans to one of them - there would be eccentricity of load and hence sway in two directions (i.e. at an angle to the global axes).

### Discrete Cores

It is worth noting that, where structures have discrete cores or numerous discrete walls providing stability, the sway effects will often counteract each other (provided the discrete cores/ walls are connected by diaphragms). Consider the example flat slab building above extended and with a similarly proportioned core on the opposite side as shown below. In this case the effect is counteracted as the two cores ‘lean against’ each other via the floor diaphragm connection. This stops any curvature from developing and no appreciable overall sway or moment is observed.

## Calculation of Wall and Core Forces

### Meshed Wall Section Moment

From __this article__, you will see that the **Fy** 2D results are (instantaneous) vertical axial forces which in this case are equivalent to the direct stresses acting on the section of a beam subject to bending. Just as we derive an overall beam section moment from the direct (bending) stresses, we can similarly derive an overall moment acting on a horizontal cross section of the wall by integrating the **Fy** force results along a horizontal line (note the instantaneous force **Fy** is obtained simply by multiplying the axial stress by the element thickness). TSD’s __Wall Lines__ and __Result Lines__ perform this integration automatically. Horizontal Result Lines have been added in the example shown above giving the expected constant moment of 200 kNm through the height of the meshed wall. Wall Lines also give the same result.

There are a number of ways the integration of nodal forces can be performed to derive the overall forces acting on a horizontal section. The picture below shows the Fy force diagram for the meshed wall and the individual nodal forces (which can be obtained from the cursor tooltip when the cursor is moved over a node) along a horizontal line towards the top of the wall.

Key points for deriving total section forces from 2D nodal results are:

- First note that the nodal forces are
*instantaneous*in terms of kN/m, not absolute. So deriving a total absolute section force is not simply a process of summation. - To yield an absolute force, the instantaneous nodal force must be multiplied by a finite length, with some average force considered over this length. Or it must be integrated over a finite length.
- The nodal values are discrete, while in reality the stress distribution would be continuous (it is linear in this case but will not always be). Hence some potential level of approximation should always be expected.
- The nodal values can also be a function of the mesh size depending on the nature of the stress distribution.

- We could calculate a quick approximate total axial force directly from the nodal forces, by multiplying each nodal value by a ‘tributary length’ based on the node spacing (i.e. the element horizontal size) which is a uniform 1.0m in the example. Thus internal nodes have a tributary length of 1m and the edge nodes 0.5m. Working from left to right and taking compression as positive this gives:
- -(49.7)*0.5 + (-12.5)*1 + 24.8*1 + 62.5+1 + 100*0.5 = -24.9 - 12.5 + 24.8 + 62.5 + 50 =
**99.9**

- -(49.7)*0.5 + (-12.5)*1 + 24.8*1 + 62.5+1 + 100*0.5 = -24.9 - 12.5 + 24.8 + 62.5 + 50 =
- As illustrated above, from the nodal forces we can also develop a function line for the continuous vertical force/ length distribution along the section line. The area under/ above this line (i.e. the integral of the force/ length function) is then the total absolute force.
- Calculating the areas of the force distribution triangles (see calculations below) to either side of the zero Fy point gives a similar value for total axial load;

133.9 - 33 =**100.9** - Using these areas and total forces we can also calculate the section
*moment*as shown below, giving the expected value of very close to 200 kNm. - It is worth bearing in mind that absolutely
*precise*agreement with ‘theoretical’ values (e.g. those actually deriving from beam theory) should generally not be expected from FEM (the finite element method).- It is also worth bearing in mind that beam theory - which engineers sometimes use either consciously or unconsciously to 'sense check' 2D (Finite) element results - also embodies assumptions and simplifications (such as plane sections remain plane and the neglecting of 2nd order terms) which do not apply to 2D (Finite) elements. Fundamentally the behaviour of 2D (Finite) elements follows Plate theory, not beam theory.

### Core Forces

The comparison with steel RHS sections also assists with understanding how core forces are actually derived from the wall (internal) forces (not the applied external loads). Just as for a rectangular hollow section, usually the majority of the bending moment is actually composed of axial forces in the ‘flange’ walls, which are perpendicular to the bending (sway) direction. There will also be some direct moments in the ‘web’ walls parallel to the bending direction. Thus, to derive an overall core moment we simply; calculate the moment produced by the ‘flange’ axial forces about the section centroid and add the web moments to give the total section moment. Note there may also be some small direct (minor axis) moments in the ‘flange’ walls, though these will usually be fairly insignificant.

The TSD 2D Integrated Results > **Wall Line** results automatically give the total sectional loads (derived from the 2D nodal forces as described above) acting on each wall of the core as shown below. Wall Line results are actually derived from automatically placed **Result Lines** at the top, bottom, middle and mid-fifth (0.4 and 0.6 h) points of each panel. We could use these for a quick calculation of core forces, however note that there may be some slight variation in the force through the height of the panel.

To check the moment and reaction at the base of the core more clearly, we can manually place Result Lines at the base of each wall as shown below.

The calculation of the total section moment - and reaction - at the base from these wall forces is then straightforward.

- 2m*(35.1 kN + 184.1 kN) + 2*74.8 kNm =
**588.6 kNm**

The calculated moment is, as expected, very close to the *applied *base moment calculated above - there may always be some slight variation from “theoretical” values due to the always approximate (but potentially highly accurate) nature of the FEM (finite element method) and averaging/ integration to develop absolute overall forces. The core forces throughout the height of the core are derived and can be checked in a similar manner.

## Closing Summary

In this article we have discussed *Sway Effects Under Gravity loads*, it is important to emphasize that this relates entirely to sway that is generated in the presence of purely vertical applied loads.

The possibility of sway developing from differential axial shortening is something that would have been routinely ignored in hand calculations by most engineers looking at most structures.

This phenomenon has nothing to do with notional loads (NL or NHL) or __equivalent horizontal forces__ (EHF), which are applied separately and will always be designed for provided you have included them in your design combinations

For framed models in Tekla Structural Designer, the effect is always included/ exposed by the 3D Building Analysis. For more about the various analyses performed by Tekla Structural Designer, and the rationale behind them, see the First Steps course __Background to the Analysis/Design procedures__.

For flat slab models the effect will not be exposed by an FE chase down analysis (which gives a more ‘traditional hand analysis’ distribution of loads and forces which ignores lateral sway effects). However Tekla Structural Designer automatically designs for the results of both 3D Building and FE chase down (by default) and so both sets of potential design forces are catered for. It is anticipated that the forces produced by sway effects under gravity loads are unlikely to be critical for design, other than potentially in very tall buildings and/ or quite extreme examples. However they fairly commonly raise questions, perhaps from engineers more familiar with using traditional hand calculations, hence it is worth appreciating their origin and logical nature.