Similitude theory allows engineers, through a set of tools known as similitude methods, to establish the necessary conditions to design a scaled (up or down) model of a full-scale prototype structure. In recent years, to overcome the obstacles associated with full-scale testing, such as cost and setup, research on similitude methods has grown and their application has expanded into many branches of engineering. The aim of this paper is to provide as comprehensive a review as possible about similitude methods applied to structural engineering and their limitations due to size effects, rate sensitivity phenomena, etc. After a brief historical introduction and a more in-depth analysis of the main methods, the paper focuses on similitude applications classified, first, by test article, then by engineering fields.

# A Review of Similitude Methods for Structural Engineering OPEN ACCESS

**Alessandro Casaburo**

**Giuseppe Petrone**

**Francesco Franco**

**Sergio De Rosa**

Manuscript received April 6, 2018; final manuscript received April 9, 2019; published online June 3, 2019. Editor: Harry Dankowicz.

*Appl. Mech. Rev*71(3), 030802 (Jun 03, 2019) (32 pages) Paper No: AMR-18-1049; doi: 10.1115/1.4043787 History: Received April 06, 2018; Revised April 09, 2019

A fundamental step in the design of a product is experimental testing. While theoretical and numerical approaches are valuable tools, their predictions must be validated by extensive sets of experimental tests before going to production. This way, whether applied to the validation of a simple or complex system, one achieves the desired reliability, performance, and safety. An example given in Ref. [1] may help illustrate the necessary experimental effort: the final static tests of the Lockheed C-141A airlifter required 8 wing tests, 17 fuselage tests, and 7 empennage tests.

Crashworthiness evaluation requires both full-scale and drop tests. Moreover, it may be necessary to repeat some tests due to errors or to focus on unexpected phenomena. Full-scale experimental testing is expensive, in terms of both cost and time and, sometimes, hard to implement; in some cases, the usefulness of acquired data cannot justify the required effort. For these reasons, it is useful to be able to design a model of the original system, viz., a consistently scaled down replica of the full-scale prototype that can be tested at significantly lower cost and with less difficulty. Even if the model is a perfectly scaled-up or down variation of the prototype, it is however another structure, with its own static and dynamic responses that do not coincide with those of the original prototype. As a consequence, the recovery (or reconstruction) of the prototype response is not guaranteed.

Holmes and Sliter [2] provide an example of money and time saving when scaled models are used. For a single crash test, the authors estimate savings of between 1/3 and 1/4 the cost of the equivalent full-scale building and testing. The test times are reduced by 1/3 and more if also model fabrication is considered. An entire experimental program, with a mixture of subscale and full-scale models, would thus lead to greater economy in both financial and temporal terms.

Similitude theory provides the conditions to design a scaled (up or down) model of a full-scale prototype and to predict the prototype's structural response from the scaled results. The tools used are known as similitude methods.

In many modern applications, the increasing complexity of engineering systems makes theoretical and numerical analyses insufficient (and entirely unsuitable for very complex structures) for evaluating whether a system's performance meets design requirements. The extensive need for experimental tests, due to the disadvantages of full-scale testing, has resulted in a rapid growth in applications of similitude methods. In fact, such methods are used in a variety of engineering branches (aerospace, naval, civil, and automotive) and applications (free and forced vibrations, impact behavior, seismic response, etc.).

The aim of this paper is to provide a comprehensive review of similitude methods applied to structural engineering. A few related reviews have been already published: the first dates back to the early 2000s [3] and focuses, after an historical review, on the analysis of composite test articles by means of similitude theory applied to the governing equations of plates and shells. Besides being the first review on the topic, the importance of this work lies in the explanation of key terminology related to similitude theory that lacks precise definition, specifically *scaling* (or *scale effects*) and *size effects*. Generally, scale effects describe changes in the response to external causes due to changes in the geometric dimensions of a structure (or a structural component); size effects concern changes of strength and stiffness of the material as a consequence of the physical scaling process. According to Wissmann [4], when a size effect occurs, a physical phenomenon gains importance in a model due to the differences in size of the replica and the prototype. Notwithstanding these definitions, in many articles, the terminology scale effects are used also to refer to effects of size. To avoid any ambiguity, from now on, the terminology size effects will be used exclusively to describe the effects of physical scaling.

A comprehensive review is also provided by Ref. [5]. This combines historical, methodological, and application insights to trace the evolution of similitude theory. Recently, Zhu et al. [6] reviewed vibration problems of plates and shells using similitude theory based on the governing equations and sensitivity analysis. Rosen [7] surveyed the literature in the late 1980s and highlighted commonalities and relationships among all the scientific fields using similitudes. Other more limited reviews are present in the literature, including one on scaling models in marine structures by Vassalos [8] and on scaling techniques applied to the structural response of liquid metal fast breeder reactor vessels to hypothetical core disruptive accident in Cagliostro [9], and the work by Saito and Kuwana [10] on dimensional analysis applied to vibroacoustics.

The idea behind this paper is to give to the reader a new and up-to-date perspective by organizing the discussion around applications categorized in terms of test articles. In this way, all the contributions to a topic scattered across time and fields of study are organized in a new presentation.

Keeping this layout in mind, the paper is organized as follows. First, a short historical review is provided in which some information is given about the first publications and main manuals and books on the topic. Then, similitude theory is defined and a description of the most used methods is given, focusing on their relative advantages and disadvantages. Section 4 is the core of the paper. It is divided into several subsections, each one focused on the application of similitude methods to a particular test article (beams, plates, and cylinders). Section 5 is dedicated to the use of the theory in the study of complex structures across several engineering fields. In Sec. 6, the main points of the paper are summarized, with some final remarks of the authors about ways of employing similitude methods.

At the end of the paper, after the references, an appendix has been added. It contains three reference tables, introduced to give a useful synopsis to interested readers.

In this section, a brief historical review of similitude methods is provided, following the publication, in chronological order, of the first articles and books that introduced the main similitude methods, still in use today. A brief insight into the development of such methods according to test articles follows. Finally, some papers are discussed which do not fit clearly in this paper but are worthy of mention because of their historical and thematic relevance.

The first reference to similitude theory dates back to the 18th century, as reported in Ref. [6]. In fact, Galilei and Weston [11] stated that size and strength of an object do not decrease in the same ratio: if dimensions decrease, the strength increases. The curious aspect of this statement is that Galileo, in the 18th century, was already facing the problem of size effects. However, the first work in which scientific models based on dimensional analysis are discussed is due to Rayleigh [12]. Although Rayleigh's article aimed to underline the importance of similitude methods, especially in engineering, as discussed in Ref. [13], thirty years had to pass before the publication of another work in which the usefulness of similitude methods is highlighted: the NACA technical report by Goodier and Thomson [14] and the book by Goodier [15]. In these publications, dimensional analysis was applied for the first time with a systematic procedure to simple and complex problems. This resulted in a deep insight on the modeling of materials with nonlinear stress–strain characteristics or plastic behavior, buckling, and/or large deflections.

In the following years, many books were written on the topic. In their review, Simitses et al. [3] cite many works, e.g., Refs. [16–20], in which similitudes and modeling principles are based on dimensional analysis. Kline [21] gives a perspective on deriving similitude conditions with both dimensional analysis and direct use of governing equations. The latter method is accurately treated also in Ref. [22], while a whole chapter of Ref. [23] is dedicated to the former. In Ref. [5], good alternatives to Refs. [21] and [23] are given, such as Refs. [24] and [25]; furthermore, the authors underline that dimensional analysis has driven and justified the writing of several other manuals, such as Refs. [26–28], even though other methods were introduced and successfully applied. A wider modeling interpretation is given in Ref. [29], while Zohuri [30] provides first a perspective on classic dimensional analysis and, then, deepens the topic in Ref. [31], going beyond Buckingham's Π theorem and approaching self-similar solutions.

An overview of similitude methods is reported in Fig. 1. The methods on the vertical axis are dimensional analysis (DA), similitude theory applied to governing equations (STAGE), energy method based on the conservation of energy (EM), asymptotical scaled modal analysis (ASMA), similitude and asymptotic models for structural-acoustic research applications (SAMSARA), empirical similarity method (ESM), and sensitivity analysis (SA). More details on each of these are presented in Sec. 3. The horizontal bars represent the range of years in which each method has been used.

The development and application of similitude methods has not followed a linear path in terms of test articles. Contrary to expectations, the first methods were not applied to simple test articles first and more complicated ones later. For example, in the first relevant application of similitude theory to engineering [14], the authors first provide an extensive theoretical study on general structures (isotropic, composite, linear, and nonlinear). They then proceed to employ dimensional analysis to buckled thin square plates in shear, with and without holes. Many years had to pass to find the first instance in which dimensional analysis was applied to a stiffened panel [32]. Differential equations were used in Ref. [33] to investigate sandwich panels. The first application of similitude theory to a beam is reported in Morton [34] in which the author employs dimensional analysis. Some years later, the first analysis of unstiffened cylinder models is conducted by Hamada and Ramakrishna [35]; oddly, stiffened cylinders were already studied in a previous work by Sato [36] by means of dimensional analysis. An overview of progress in applying similitude methods to different test articles is shown in Fig. 2. The bars again illustrate the time range in which publications concerned with the corresponding test articles have been published.

A mention to two works with a great historical relevance is needed; both of them are dedicated to similitudes applied to shells.

The first is an analysis executed by Ezra [37]. This is motivated by a peculiar behavior of shells; under certain conditions, they can sustain pressures considerably larger than that which would produce static buckling. On the other hand, when applying a pressure rapidly with a long duration, the structure carries less than it would statically. The author uses dimensional analysis to investigate scale model determination of the buckling of a thin shell structure, with arbitrary shape and subject to an impulsive pressure load with duration not short enough to be considered as a pure impulse, nor long enough to be considered as static pressure. He shows that when the materials of prototype and model are similar, consistent predictions require that the magnitude of the applied pressure must be the same while the duration must be scaled proportionally. If magnitude and duration cannot be controlled, then a complete similitude can be achieved by a suitable choice of a different material for the model.

A second relevant work on the topic is that of Soedel [38]. In this paper, the author derives similitude conditions for free and forced vibrations of shells from Love's equations. Shells are characterized by both in-plane and transverse oscillations; as a consequence, when deriving the exact similitude conditions from the governing equations, the thickness is not independent from the surface geometry. By decoupling the membrane and the bending effects, the author derives two sets of approximate conditions in which the thickness is introduced as a parameter independent of the surface geometry. The choice of the set is dictated by the relative dominance of membrane and bending effects.

Finally, it is worth mentioning that some works by Sterrett consider the topic of similitude and, more generally, model in a wider manner; the epistemological setting of these articles may help to explain the concepts underlying similitude theory.

The application of fundamental laws to scale modeling is the main question in Ref. [39]; the author states that scale modeling must not mediate between an abstract/theoretical world and a phenomenological one, but rather to give insights into phenomena, so that it is possible to tell what happens in a situation that is not directly observable by means of another situation that can be observed.

A direct insight into dimensional analysis and Buckingham's Π theorem applied to both geometrical and physical similitudes is given in Ref. [40]. The usefulness of models is underlined in Ref. [41] in which the author shows how new areas of application and investigative research have been found. The topic is further developed in other articles [42–44].

Before proceeding to the core of this review, we present a brief introduction to similitude methods. After providing some useful definitions, we quickly describe the main characteristics of each method.

Fundamentally, similitude theory is a branch of engineering sciences which allows to determine the conditions of similitude between two or more systems. The full-scale system is known as *prototype*, while the scaled (up or down) one is the *model*. When a model satisfies the similitude conditions, it is expected to have the same response as the prototype (in a qualitative meaning). This is the reason why such tools are very useful. It is possible to overcome all the problems associated with full-scale testing by designing and investigating a scaled (usually down) model.

Some authors refer to *similitude*, others to *similarity*, thus a remark is necessary about the terminology. In fact, both terms are used interchangeably in the literature although with a slight difference: similarity is closer to the usage in fields of mathematics (for example, self-similarity solutions). An example of such an application is reported by Polsinelli and Levent Kavvas [45] in which Lie scaling methodology is introduced. Such a method performs symmetry analysis of the governing differential equations basing on Lie groups, special structures leading to invariant transformations. An extensive treatise on the topic is given in Ref. [46]. In this paper, we only use the word similitude.

In order to give a brief introduction to similitude methods, it is important to explain two concepts previously introduced, viz., similitude and similitude conditions.

Similitudes can be distinguished according to the parameters taken into account; it is possible to have

^{(1)}*Geometric similitude*, when geometric characteristics are equally scaled.^{(2)}*Kinematic similitude*, when homologous particles lie at homologous points at homologous times [24]. By recalling the ratio between space and time, it follows that kinematic similitude is achieved, simply, when homologous particles have homologous velocities.^{(3)}*Dynamic similitude*, when homologous parts of a system are subject to homologous net forces.

A formal definition of kinematic similitude, that also introduces the concept of scale factor, is given by Langhaar [17]:

The function *f′* is similar to function *f*, provided the ratio *f′/f* is a constant, when the functions are evaluated for homologous points and homologous times. The constant *λ = f′/f* is called the scale factor for the function *f.*

Baker et al. [24] explain the use of the term homologous in this definition as *corresponding but not necessarily equal values*.

Those listed above are the main types of similitude, but others can be defined. As an example, Baker et al. [24] add constitutive similitude to the list, achieved when there is similitude between the materials stress–strain curves of the prototype and the model, or between the constitutive properties of such materials. However, in general, only geometric, kinematic, and dynamic similitudes are considered, so that it is possible to say that two systems are similar if they share the aforementioned characteristics.

The sufficient and necessary condition of similitude between two systems is that the mathematical model of the one be related by a bi-unique transformation to that of the other.

So, if *X _{p}* and

*X*are two vectors of

_{m}*N*parameters, respectively, of the prototype and the model, then they are related one to each other in this way

where [Λ] is the following matrix:

The diagonal elements of matrix [Λ] are known as scale factors of the parameter *x _{i}* (

*i*=

*1, 2, …, N); herein they will be defined as $\lambda xi=xim/xip$ (where the subscript*

*m*is for model, while

*p*for the prototype), although the other authors use the inverse formulation.

According to the fulfillment of similitude conditions, different models can be defined (and another classification of similitudes can be done):

^{(1)}*True model*: All the conditions are fulfilled. This is referred to as complete similitude.^{(2)}*Adequate model*: First-order conditions, i.e., the conditions related to the main parameters are fulfilled. This is referred to as first-order similitude.^{(3)}*Distorted model*: At least one of the first order conditions is not satisfied. This is referred to as partial similitude.

The difference between true and adequate models is of relevance especially when using dimensional analysis. Here, special insight into a problem can be used to reason that some of the conditions are of “second-order” importance.

An interesting example is provided by Harris and Sabnis [26]: in rigid frame problems, axial and shearing forces are of second-order importance relative to bending moments as far as deformations are concerned. Thus, it may be adequate to model the moment of inertia but not the cross-sectional areas of members.

So, it is clear that the difference between true and adequate models relies on the choice of parameters that are accounted for when deriving the similitude conditions. For other methods, such as STAGE or SAMSARA (introduced below), that work according to other principles, such difference is absent and the concepts of true and adequate models can be joined.

When dealing with a scaled model, the main characteristics taken into account are the scaling effects, i.e., the change in response of the structure due to the geometric scaling procedure. In some applications, e.g., impact response, size effects must also be taken into account; they appear as a change in material properties, such as strength and stiffness, due to the scaling procedure. Simitses et al. [3] cite a workshop on the topic (Jackson [47]) in which, while all the presenters agree on the fact that size effect on stiffness is almost nonexistent, they disagree about its influence on strength.

Several similitude methods can be found in the literature. To give an exhaustive explanation of such tools is beyond the aim of this paper; however, it may be helpful to focus briefly on the working principles, advantages, and disadvantages of the main ones. The main aspects, advantages, and disadvantages of each method considered below are summarized in Table 1.

Dimensional analysis or, as Coutinho et al. [48] refer to, traditional similarity method, is based on the definition of a set of dimensionless parameters that govern the phenomenon of interest. It relies on the concept of dimensional homogeneity, i.e., an equation which describes a physical phenomenon must have sides with same dimensions. All these concepts are gathered in Buckingham's Π theorem [49]. According to this theorem, let *K* be the number of fundamental dimensions, required to describe the physical variables, and let $P1,P2,\u2026,PN$ be *N* physical variables. Suppose that it is possible to relate such variables through the functional relation

Equation (3) may be rewritten in terms of (*N* – *K*) dimensionless products, called Π products, as

Each Π product is a dimensionless product of *K *+* *1 physical variables so that, without loss of generality

In mechanics, usually *K *=* *3 and the fundamental dimensions are mass, length, and time: this is called the mass, length, and time (MLT) base. The choice of repeating variables *P*_{1}, *P*_{2}, …, *P _{K}* should be such that they include all

*K*fundamental dimensions, while each dependent variable of interest should appear in just one Π product.

Scale-modeling with dimensional analysis requires that all the dimensionless Π products are scaled in such a way that they are equal for both model and prototype, which means

When the condition (6) is fulfilled for each value of *j*, then complete similitude is achieved; if at least one condition is not satisfied, then the model is distorted. Generally, to simplify the calculations, only first-order conditions are considered, so that the difference between a true and an adequate model is neglected also when using dimensional analysis.

In some applications, dimensional analysis does not directly involve Buckingham's Π theorem. The scaling laws are determined by defining just one scale factor, then expressing the prototype/model ratio of parameters as a power law of such a scale factor. This step requires dimensional consistency, so that the method can be regarded as another version of dimensional analysis.

On the one hand, the described method is simple to use and useful for those systems without a set of governing equations, such as complex or new systems. On the other hand, it is clear that a phenomenologically meaningful choice of the parameters is needed: taking into account a parameter with low or no influence on the phenomenon would complicate the derivation of Π terms unnecessarily, while ignoring an important parameter would lead to an incomplete, and maybe wrong, conclusion. Hence, to use such a method, an experienced analyzer and a deep knowledge of the problem are needed.

Furthermore, Π terms may be not unique, which lead to a trial-and-error approach and to a significative calculation effort. Besides, not all Π terms have physical meaning and, in general, the equations characteristic of the phenomena under observation can be formulated only in an incomplete form. In conclusion, the procedure is not structured, so it cannot be easily implemented into an algorithm.