Theoretically optimal bracing for preexisting building frames
Abstract
Bracing is commonly used to provide resistance to lateral forces in building structures. However, traditional bracing design approaches appear not to be underpinned by clear fundamental principles. Here, theoretically optimal arrangements of bracing members are sought for preexisting building frames, already designed to carry gravity loads. For sake of simplicity existing frame elements are assumed to be capable of carrying additional loads and three types of bracing are considered: tension only bracing, bracing intersecting only at the corners of the existing frame, and unconstrained optimal bracing, where bracing elements can intersect at any location. Layout optimization techniques are used to identify initial design solutions; these are then related to Michell trusses to obtain exact reference volumes, against which the efficiency of other bracing layouts can be judged. It is shown that from a theoretical standpoint tension only bracing is inefficient and that the optimal angle of intersection between a preexisting frame member and intersecting tension/compression bracing member pairs is 45^{∘}, something that can potentially be adopted as a basic principle when designing bracing for a preexisting frame.
Keywords
Buildings Bracing design Layout optimization1 Introduction
The lateral restraint system is an indispensable part of any building. This is particularly true in a tall building structure, where the requirements for lateral strength and stiffness can govern the layout of the whole building. Consequently, the design of efficient bracing layouts to resist lateral wind loading is an area of active interest to structural engineers.
Optimization of bracing systems has often focussed on geometry or size optimization. For example, Moon et al. (2007) analysed diagrid bracing systems, using the angles between bracing members as parameters which were varied to optimize the lateral stiffness of the structure. Bobby et al. (2013) and Lee and Tovar (2014) optimized an outrigger bracing system by using binary variables to represent the existence of belt trusses; Sala and Candiani (2014) used the size optimization technique proposed by Baker (1990) to identify the optimal section sizes of a fixed topology bracing system whilst Tangaramvong and TinLoi (2015) used a discrete approach to optimize the bracing topology using a sparsely populated ground structure (only members connecting neighbouring nodes were considered). Others have used continuum optimization techniques (e.g. Liang et al. 2000; Allahdadian et al. 2012; Stromberg et al. 2012; Kingman et al. 2015) or genetic algorithms (e.g. Baldock 2007; Yazdi and Sulong 2011; Richardson et al. 2013). However, to date classical layout optimization methods appear not to have been used to benchmark the efficiency of commonly used bracing systems.
The mathematical basis for the problem of finding the structural layout consuming the least volume of material was developed by Michell (1904). Whilst Michell’s analytical approach provides a strong foundation for the field, it is seldom applied to practical structural design applications, partly because of the difficulty of identifying the optimal structure for specified loading and support conditions. However, the approach readily lends itself to numerical implementation, where the optimal layouts of discrete members are found from a ‘ground structure’, comprising all possible interconnections of discrete node points within a design space. Using linear programming (LP) techniques numerical layout optimization results can be found which closely approximate classical Michell forms for any given problem (Dorn et al. 1964; Hemp 1973); more recently the computational efficiency of this method was improved by Gilbert and Tyas (2003). Here, numerical layout optimization will be used as a tool to investigate optimal bracing layouts for various different scenarios.
Stromberg et al. (2012) and Liang et al. (2000) have considered the optimization of bracing systems within a beam/column frame represented by discrete beam members. Stromberg et al. (2012) optimized the bracing topology with different type connections between continuum and discrete beam elements and the effect of column stiffness on the optimal bracing layout was also considered. Liang et al. (2000) assumed that the beam/column members sized in a gravity load analysis would also be adequate for lateral load cases, and optimized the bracing topology for several bay aspect ratios. This raises the question: if the beams and columns designed for gravity loads have sufficient reserves of strength, such that they do not need to be resized to act as part of the lateral stability system, can general rules governing the optimal layout of bracing members be found? This study focuses on optimizing the layout of bracing elements within a preexisting frame, where the members have already been designed to carry gravity loading, and are assumed to have the requisite additional reserves of strength to support lateral loads; other scenarios will be the subject of future research.
The paper is organized as follows: Section 2 outlines the assumptions made with respect to loading and preexisting frame performance; Section 3 describes the problem types considered and the numerical layout optimization method employed to obtain initial solutions; Sections 4 and 5 consider respectively single bay and multibay / storey building types, with exact analytical solutions derived for key problem types; finally conclusions from the study are drawn and the direction of future research is outlined in Section 6.
2 Assumptions
Here, for simplicity the concept of Notional Horizontal Load (NHL) (a small lateral load which is applied as part of the main gravity load case (1.a)) is ignored, and it is assumed that the bracing design is dominated by the wind load cases, (1.b) and (1.c).

Assumption 1: The loads generated in the columns in load case (1.b) and (1.c) are always less than those resulting from load case (1.a).

Assumption 2: Vertical gravity loads are only carried by the columns in all load cases.

Assumption 3: The sizes of preexisting horizontal members (i.e. floor beams and/or slabs) are dominated by the gravity load case (1.a), and the axial loads induced in them by lateral load cases (1.b) and (1.c) are small in comparison.
Assumption 1 means that the sizes of the columns are not determined by the lateral load cases, and thus that the columns may be assumed to be rigid with infinite strength when designing for lateral loads (i.e. accumulated reactions from bracing elements can be carried). Assumption 2 is widely adopted in design practice, where columns are often first designed to resist gravity loads, with the design of bracing members to resist lateral loads coming later. Together these two assumptions mean that the vertical loads in load combination (1.a) will not affect the optimal layout of the bracing system, and the wind load (i.e. 1.5W_{k}) is governing. Assumption 3 means that the structural members forming the floors effectively possess infinite reserves of strength, and can distribute applied lateral loads to either side of the frame. With these three assumptions, the design of a bracing system for a frame already designed to carry gravity loads is considered, where the bracing need only resist lateral loads. This effectively replicates one step in the traditional design procedure for buildings (e.g. Brown et al. 2009 suggest that the sizes of beams and columns are first determined by considering gravity loads, with the sizes of bracing members then determined by considering lateral loads). A similar approach has also been used in previous bracing topology optimization studies; for example Mijar et al. (1998), Liang et al. (2000), and Stromberg et al. (2012) take (or design) a columnbeam frame capable of carrying gravity loads and then, considering only lateral loads, carry out bracing topology optimization.
3 Problem specification
3.1 Cases considered
Two load cases involving horizontal loads of equal magnitude applied in opposite directions are used to replicate realworld conditions, and to also ensure the resulting layout is symmetric. According to the superposition theory proposed by Hemp (1973), a two load case solution can be obtained by superposing the results from two design problems involving single load cases. Therefore, all design problems considered herein can be treated as single load case problems.

Case 1: Tension only bracing. This is frequently employed in practice as it obviates the need to take action to ensure compression members do not buckle. Numerically, this is implemented by making the compression stress capacity of bracing members infinitesimally small.

Case 2: Bracing connected only at corners. Both tension and compression members are allowed to be present in the bracing system and the maximum tension stress is equal to the maximum compression stress. Numerically, bracing connected only at corners is implemented by removing the nodes along the perimeter of the design space, except at the corner points.

Case 3: Optimal reference bracing. Both tension and compression members can be present in the bracing system and the maximum tension stress is equal to the maximum compression stress. Connections between bracing members and the beams or columns can exist at any location. The resulting constraintfree bracing layout can be expected to be the most efficient of the three cases in terms of material consumption.
3.2 Single load case layout optimization formulation
The adaptive ‘member adding’ method proposed by Gilbert and Tyas (2003) is used to reduce the computational cost associated with solving problems; this involves solving a small initial problem involving sparse connectivity, and then iteratively adding further connections until a provably optimal solution is found. Additionally, to improve the quality of the optimized results, the geometry optimization rationalization procedure proposed by He and Gilbert (2015) is here used as a postprocessing tool.
Finally, although formulation (2) is strictly speaking a ‘plastic’ formulation, the optimal layouts from plastic and elastic (minimum compliance) design optimization procedures are identical when only a single load case is involved.
3.3 Practical considerations
Some of the solutions generated by layout optimization are complex in form, leading to justifiable questions over their practical usefulness. However, as Cox (1965) suggested, in the field of structural design there is a need for reference solutions, against which alternatives can be judged (Cox pointed out that just as there is a limit on the theoretical thermal efficiency of a heat engine, set by the Carnot cycle, so there is a lower limit on the volume of material necessary to form a structure). Thus even a complex solution provides a useful reference against which alternative designs can be judged, where the latter may include simplified variants of the optimal reference form.
Also, many of the layouts presented in this paper contain long compression members and it might reasonably be suggested that buckling will limit the capacity of such elements in practice. However, for the purposes of this study the effect of member buckling under compressive load will be ignored. Whilst incorporating buckling may, in principle, lead to different optimal layouts, there are two reasons for this omission. Firstly, as already stated, one goal of the study is to establish definite benchmarks for optimal layout and efficiency, against which realworld designs may be judged. The significance or otherwise of buckling effects in a given scenario depends to a great extent on specific design decisions, and this emphasizes the need for benchmarks to properly assess the effects of these decisions. Secondly, previous work by two of the present authors (Tyas et al. 2005) has indicated that the reduction in the strength of real steel members due to member buckling is rarely significant when suitable member crosssections are used. This latter study considered standard circular hollow structural steel sections, widely available in the UK, and showed that a near linear relationship held between member crosssectional area and the required compression capacity for a given unrestrained length. Euler buckling was significant only for very long, and/or very lightly loaded members, whereas, ‘...at most combinations of load and length, there exists a member which can effectively utilize the majority of its compressive crushing strength and minimise the effect of Euler buckling’. Also buckling can often be addressed by either selecting a suitable member crosssection in the detailed design stage, and/or by providing lateral restraint at intermediate locations (e.g. at floors), though this is likely to be more difficult in the case of small buildings, with lightly loaded members.
4 Optimized single bay bracing designs
4.1 Case 1: Tension only bracing
It is evident from Fig. 2a that tension only bracing leads to very simple optimized layouts. It is also evident from Fig. 3 that tension only bracing requires at least double the volume of material required by any other kind of bracing. This is because only one of the two inclined members carries a force when a horizontal load is applied. Thus, since the lateral load direction can be reversed, at least twice the volume of material must be consumed as when standard tension / compression bracing is used, assuming that the allowable stresses are equal in tension and compression. The issue of buckling must obviously be considered, but as mentioned earlier often this can be addressed by modifying the shape of the cross section and/or providing sufficient lateral restraint. The use of tension only bracing is therefore not a good choice when material savings are of paramount importance.
4.2 Case 2: Bracing connected only at corners
 Hypothesis 1:

The optimized result for the case with connections only at corners is a combination of two adjacent Michell cantilevers, contained within a laterally restricted design space (socalled Michell cantilevers in a half strip).
 Relation 1:

The volume of the optimal Case 2 layout for a problem with an aspect ratio h : b, where h : b ≥ 1, is twice the volume of a Michell cantilever of aspect ratio \({h\over 2}:b\) (though the latter is subjected to a loading at the centreline).
 Relation 2:

The volume of the optimal Case 2 layout for a problem with an aspect ratio h : b, where h : b ≤ 1, is \({2h}\over b\) times the volume of a Michell cantilever of aspect ratio \(h:{b\over 2}\) (though the latter is subjected to a loading at the centreline).
Therefore, using Relation 1 and 2, exact analytical solutions can be obtained for bracing problems, with numerical values for these solutions taken e.g. from Graczykowski and Lewiński (2010).
4.3 Case 3: Optimal reference bracing
In Case 3 bracing members are free to intersect preexisting members at any location around the perimeter of the design space, as shown in Fig. 2c.
 Property 1:

If a preexisting member with infinite reserves of strength actively carries load, then the optimal intersection angle between this and intersecting tension/compression member pairs will be 45^{∘}.
Comparison between optimal reference bracing (Case 3) and standard knee bracing volumes (where P is the magnitude of the horizontal load applied at the top left corner of the design domain, of breadth b and height h, and σ is the limiting tensile strength)
Aspect ratio (h:b)  1:2  1:3  1:4 

(Angle between knee bracing members)  (45^{∘})  (33.7^{∘})  (26.6^{∘}) 
Volume of Case 3 optimal bracing (× Pb/ σ)  1.00  0.71790  0.60691 
Volume of standard knee bracing (× Pb/ σ)  1.00  0.72222  0.62500 
Difference (%)  0  0.60175  2.9806 
 Relation 3:

The volume of the optimal Case 3 layout for a problem with an aspect ratio h : b, where h : b ≤ 1, is \(2h\over b\) times the volume of a Michell cantilever of aspect ratio \({b\over 2}:{2h}\) (though the latter is subjected to a loading at the centreline).
5 Multistorey / multibay buildings
Bracing members may cross several stories in multistorey buildings (Moon et al. 2007). In such a building each floorslab can be considered as a preexisting member. Therefore, in this section the focus is on considering the contribution of additional intermediate beams and columns. (Also, a pin support is added to the bottom of each intermediate column.)
Firstly, it is observed that for Cases 1, 2 and 3 all layouts revert to cross bracing once intermediate beams and columns are added. Therefore, for sake of clarity, only results for a 4:1 aspect ratio structure is considered further, as shown in Fig. 10a.
Since there are intermediate connections available across the height of the multiple storey building, Case 2 & 3 become identical, with the volume of the bracing structure shown in Fig. 10a being equal to 8.00Pb/ σ in both cases. However, the volume of the Case 1 (tension only bracing) structure will still be double the volume of the corresponding Case 2 & 3 (tension / compression bracing) structures, for the reason given in Section 4.1.
6 Conclusions

Case 1: Bracing systems comprising tension only members have been demonstrated to be inefficient from a theoretical standpoint. Replacing tension only bracing with tension/compression bracing will save at least 50% of the material needed if the limiting tensile and compressive material strengths are equal (i.e. assuming that buckling does not reduce the maximum compressive stress that can be sustained).

Case 2: When bracing is only allowed to connect at the corners of the preexisting frame it is shown that the optimal layout takes the form of two adjacent Michell cantilevers. This allows exact reference volumes to be obtained, against which the efficiency of other bracing layouts can be judged.

Case 3: When bracing is allowed to connect preexisting frame members at any location it is shown that intersecting tension/compression bracing member pairs meet each preexisting member at 45^{∘} in the optimal layouts. It is also shown that traditional cross bracing and knee bracing are very efficient from a theoretical standpoint.
In the case of multistorey frames, layouts involving diagonal crossbracing, with braces meeting preexisting members at 45^{∘}, are shown to be most efficient. Also it is evident that the theoretically optimal layouts identified herein quite closely resemble those which have been employed in braced tube building frames in practice. However, the exact reference solutions derived herein are only strictly applicable when bracing a preexisting frame; the more general holistic frame design case, where all framing elements are optimized simultaneously, under the action of multiple load cases, is the subject of future work.
Notes
Acknowledgements
The second and third authors acknowledge the support of EPSRC, under grant reference EP/N023471/1.
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