In Section 2, some basics concepts and background theories are first presented. Section 3 provides a brief overview of the methods and critical issues on the buckling failure of short columns, while slender columns are discussed in Section 4. Finally, Section 4.5 presents a subdiscussion on compressed members with variable stiffness due to thermal gradients and constructional materials that can be remarkably sensitive to degradation and hence to the premature column buckling collapse. It is important to mention that this work is primarily focused on the global buckling of the compression member.

#### **2. Basics**

The necessary preliminary analysis of the stability problem was proposed by Hooke in 1675, wherein it was shown that the displacement in any structural body is directly proportional to the load causing the displacement. This law can be applied to spring bodies, stone, wood, metal, etc., and it is commonly known as Hooke's law [1]. Further, Bernoulli studied the curvature and deflection of a cantilever beam using Hooke's law in 1705. It was Euler who was credited with the first systematic study of the stability problem in equilibrium. In his first publications, Euler investigated the stability of a hinged bar, having flexural rigidity (*EI*), in equilibrium, subjected to an axially compressive force (*p*) and uniformly distributed load (*q*) along the longitudinal axis (*z*) by two different approaches [5–8]. It is interesting to note that Euler has defined all his formulations in terms of *Ek*2 instead of *EI*, with *E* defined as strength property and *k*2 as a dimensional property of the column. Further, the transformation from *Ek*2 to *EI* requires the knowledge of Hooke's law, and it was Coulomb, who, for the first time, applied Hooke's law and equation of static equilibrium to develop the bending moment and normal stress due to the elastic bending in cantilever column as follows [9]:

$$EI\frac{d^3v}{dz^3} + qz\frac{dv}{dz} + p\frac{dv}{dz} = 0\tag{1}$$

As is evident from Equation (1), its solution will contain only three constants, and the equation has failed to satisfy four boundary conditions. Euler identified this error and presented a corrected differential equation in his third paper by including the presence of a horizontal force *N* [5]. However, it is interesting to note that Euler did a numerical mistake, and calculated the second eigenvalue instead of the first, which was later corrected by [10–12]. Thus, the equation of static equilibrium (Equation (2)) to develop bending moment and normal stress due to the elastic bending in a cantilever column is:

$$EI\frac{d^3v}{dz^3} + qz\frac{dv}{dz} + p\frac{dv}{dz} = N\tag{2}$$

Euler's analytical conclusion supported the experimental results obtained by Musschenbroek [13] for slender wooden columns. However, Coulomb discarded the result of Musschenbroek and concluded that the breaking strength was independent of length, based on experiments on masonry columns [9]. Duleau, Hodgkinson, Considère, and Engesser discussed Euler's formulation and its exclusive validity for "slender" columns [13–17]. Moreover, Hodgkinson proposed an empirical formula for the design of short columns based on the experimental investigations on cast-iron columns. In the year 1845, Lamarle proposed a critical load expression in terms of the critical stress and stated that Euler formulation is applicable when the critical stress (*σ*cr) is less than the elastic limit (*σ*0) for the constructional material in use. In other words, it is applicable for the struts whose slenderness ratio (*l*/*h*) is greater than the limit value given as follows i.e., Equation (3) [18]:

$$\left(\frac{l}{r}\right)^2 = \frac{\pi^2 E}{\sigma\_0} \tag{3}$$

where *r* is the radius of gyration about the weaker axis of the column. Although there is no record of whether Lamarle's suggestion was used anywhere practically, the formula suggested by Gordon provides the same result as Lamarle's model, and this is verified both for large and small slenderness ratios [19]. Figure 2 shows some typical design curves, as conventionally obtained in terms of stress and slenderness ratio, based on Lamarle and Gordon models.

**Figure 2.** Comparison of design curves for compression members.

The proportionality between the stress and the strain was proposed by Young [20]. Johnson et al. [21] suggested using Euler's formula by incorporating modifying constant, which is similar to the use of equivalent length coefficient, *k*.

It is to be noted that, despite considering all the assumptions to transform a real column into an ideal column, the existence of perfectly clamped or pinned boundary conditions at either end and no demand of flexural strength from compression members are hard to achieve. In real problems, these assumptions rarely meet since columns in framed structures are supposed to have sufficient flexural rigidity and restrain. Due to this gap, the use of interaction equations is favoured which is based on Ayrton-Perry's approach [22]. They first related the concept of the elastic critical stress to the failure stress, which was later simplified further in [23]. Herein, the average compressive stress (*f* c), the allowable compressive stress in an axially loaded strut (*p*c), the resultant compressive stress due to bending about the rectangular axis (*f*bc), and the allowable compressive stress for a member subjected to bending (*p*bc) are related as per Equation (4) using the well-known beam-column interaction:

$$\frac{f\_{\rm c}}{p\_{\rm c}} + \frac{f\_{\rm bc}}{p\_{\rm bc}} < 1\tag{4}$$

#### **3. Buckling Failures**

#### *3.1. Self-Buckling*

Self-buckling is a phenomenon wherein a column buckles under its own weight; these columns are commonly known as heavy columns. Generally, self-buckling is not considered since it is assumed that the weight of the column is small, compared to the applied axial loads. However, there may be cases in which self-buckling may govern and hence need attention. Self-buckling was first investigated in 1881 by Greenhill [24], and based on his analysis, he proposed that a vertical column may buckle under its own weight if its length exceeds, as given in the following (Equation (5)):

$$I \approx 7.8373 \left(\frac{EI}{\rho gA}\right)^{1/3} \tag{5}$$

where *ρ* is the density of column material, *E* is Young's modulus, *I* is the moment of inertia of column, *g* is gravitational constant, and *A* is the cross-sectional area of the column.

Duan and Wang [25] considered buckling of heavy columns and presented an analytical solution in terms of hypergeometric function. They highlighted the fact that buckling capacities were not only dependent on end support condition, shape, size, material but also on the weight. They suggested using a fourth-order differential equation instead of second order. Later on, Darbandi et al. [26] presented closed-form solutions for variable section columns subjected to distributed axial force. Herein, the column was modelled using the Euler's-Bernoulli theory, and solutions were presented using the singular perturbation method of Wentzel-Kramers-Berilloui (WKB), see [26].

In the year 2010, Wei et al. [27] outlined a procedure to compute the buckling load of prismatic and nonprismatic columns under self-weight and tip force. This method did not use Bessel's function as others [25], which strongly depends on the form of an ordinary differential equation with a variable coefficient [27]. Huang and Li [28] studied the column with a nonuniform section using Fredholm's integral equation and presented closed-form solutions. Fredholm's equation transformed the exercise of finding solutions of differential equations to simple algebraic expressions [28]. Later on, Riahi et al. computed the buckling capacity of columns with variable moment of inertia through the slope-deflection method, and dimensionless charts were proposed [29]. On the same line of study, columns with variable inertia (trigonometric-varied inertia column) iteration-perturbation method was applied and obtained results were compared with the result obtained by modelling the same column in ANSYS by Afsharfard and Farshidianfar [30]. Later on, detailed work was reported by Nikoli´c and Šalini´c [31], wherein they assumed that the column is doubly symmetric to apply the method of rigid elements in order to perform buckling analysis of columns with continuously varying cross section and multistepped columns under different boundary conditions.

The described method removes the limitation of the existing rigid body element approach. This method also serves an additional advantage that the boundary condition can be introduced without any extra calculation. However, the limitation of this method lies in the discretisation of elastic segments with rigid segments [31].

#### *3.2. Failure of Inelastic or Short Columns*

Duleau, Hodgkinson, Considère, and Engesser, while working independently, suggested that Euler's formula is valid only for slender columns. It is to be noted here that Hodgkinson had already suggested an empirical formula which was used for the design of short columns. However, there was a need to develop a theory which can govern the failure of short columns or columns with a smaller slenderness ratio [14–17]. Considering this, Engesser suggested tangent modulus theory, wherein he assumed that axial load was increasing during the transition from straight to the bent position and presented the value of critical stress in terms of tangent modulus (*E*t) as follows as given by Equation (6):

$$
\sigma\_{\rm cr} = \frac{\pi^2 E\_\rm tI}{\lambda^2} \tag{6}
$$

In the same year, Considère suggested that, if an ideal column is subjected to load greater than the proportional load, the column begins to bend, and stresses on the concave side increase according to tangent modulus theory, whereas, on the convex side, stress peaks decrease according to Hooke's law. He defined critical load by employing *E*, which is a function of average stress in the column. He also suggested that the *E* value should lie in between the modulus of elasticity and the tangent modulus. Later on, in 1995, the error in tangent modulus theory was put forward by Jasinski, and he pointed out that determination of function which describe *E* was impossible to find theoretically [32]. After this, a double modulus theory was developed by Karman and proposed the actual evaluation of *E* for rectangular cross section and idealised H-section consisting of infinitely thin flange and negligible web. The general expression for the critical stress *σcr* was thus defined in terms of reduced modulus, *E*r [33] by Equations (7) and (8), respectively as:

$$
\sigma\_{\rm cr} = \frac{\pi^2 E\_{\rm r} I}{\lambda^2} \tag{7}
$$

$$E\_{\mathbf{r}} = \frac{EI\_1 + EI\_2}{I} \tag{8}$$

where *I*1 and *I*2 are the moment of inertia of either side of the section about the neutral axis. Since then, the value of *E* has been evaluated by several authors.

In 1947, using an imaginary column, Shanley concluded that there will be bending once the tangent modulus load is exceeded following which axial load increases and reaches a maximum value which lies in between the tangent modulus load and reduced modulus load, and there will be stress reversal once the bending deformation becomes finite. In other words, Shanley's analysis clearly described that the first bifurcation will occur at tangent load and a sequence of equilibrium can be constructed in between two limiting loads, i.e., tangent load and double modulus load. Shanley thus proposed an interaction curve to link eccentricities to the tangent modulus theory in order to apply his theory for practical problem and design calculations [34,35]. In this regard, Figure 3 compares the average stress for different slenderness ratios, as collected from the experimental investigation of a specimen (aluminium solid round rod with 0.72 cm diameter having flat ends) discussed in [36].

Later on, a model similar to Shanley was analysed by Johnston by replacing the twoarea element with a solid rectangular segment and determined the magnitude of stress distribution for various loads above the tangent modulus load across the section [37]. With the advancement in computer technology, computer programs were written by Batterman [38] to find the maximum load for aluminium alloy H-section with finite web areas about weak as well as the strong axis of bending, in both initially straight position and with initial curvature [38]. In 1987, Groper and Kenig proposed the inelastic stability of stepped columns with the help of Newton's method or bisection method [39]. In general, the Engesser-Shanley definition for the critical load of a column in an inelastic range is

widely acceptable. The same concept is extended for structural steel columns having initial stress due to differential cooling, although the material is in an elastic range.

**Figure 3.** Comparison of experimental data and column theories.

#### *3.3. Failure of Imperfect Long Columns*

Long columns, more than short ones, are notoriously sensitive to initial imperfections, defects, etc. Hence, they necessitate careful investigations since a minor change in the loading and geometrical parameters may lead to their sudden failure. It is well accepted that perfect columns are theoretical identities, and in practice, their behaviour is altogether different. One of the important examples of such columns is a walking stick which is subjected to a large amount of eccentricity.

There exists a wide scatter of results for long columns, due to many reasons, and some of the motivations include nonideal supports, plastic behaviour, the interaction of buckling modes (wherein local buckling of columns is more important), along with possible residual stresses. Due to these imperfections, column behaviour is altogether different practically, in comparison with its theoretical treatment, and the reason for this may be attributed to the treatment of these imperfections. Thus, these columns primarily fail due to elastic instability. Section 4 reports further details about the failure of imperfect long columns.

#### **4. Imperfections in Long Columns**

#### *4.1. Imperfections Due to Large Deformations*

It is well understood that Euler's original formulation was based on some defined assumptions, and hence, he modelled the behaviour of ideal columns which hardly exist in the reality of the structures. In order to apply his theories to practical problems of engineering, it becomes important to understand the difference in the behaviour of a real and an ideal column. This result can be achieved by removing the various assumptions, one by one, and then analysing the column response. One of the prominent assumptions in Euler's theory, for example, is that all deformations are considered as "small". This results in curvature (1/*R*) of deflected shape of a column of length (*L*) with flexural rigidity (*EI*), subjected to axial load (*P*), with pinned boundary condition on either edge becomes equal to double differentiation of deflection (*y* ), thus neglecting (*y* ), as in the following [40] given by Equations (9) and (10) as:

$$\frac{\delta}{L} = \frac{2p}{\pi\sqrt{\frac{P}{P\_{\rm cr}}}}\tag{9}$$

$$p = \sin(\mathfrak{a}/2)\tag{10}$$

where *α* is the slope of deflected shape at support, and Equation (9) represents the solution in terms of mid-height deflection, δ, applied load, *P*, and Euler load, *P*cr.

According to [40], Figure 4 shows the variation of *P*/*P*cr with *δ/L* and highlights that the estimation of the expected critical load by linear theory is valid for a considerable range of deformations. The reason for such a behaviour is attributed to the fact that for most of the columns, a combination of bending and axial stresses reaches the proportionality limit long before the difference between linear and nonlinear theory becomes notable.

**Figure 4.** Normalised load-deflection curve.

#### *4.2. Imperfections Due to Initial Curvature and Eccentric Loading*

It was Young who, in 1807, tried to find out the effect of eccentricity (*e*) and initial curvature on the load-carrying capacity of a given column [41]. However, his original research results were not presented in usable form. Later on, during the year 1858, Scheffler [42] presented the complete solution for eccentrically loaded columns, by taking into account the effect of direct stress and bending stress. This solution is now commonly known as the "Secant Formula" (SF).

It is important to highlight, in this context, that the SF is accurate until the predicted stresses are within the elastic limit of the constructional material in use. The behaviour of a column with a given initial curvature or a column subjected to eccentric load is more or less the same, considering the fact that, in either case, the behaviour of the column is the same. Further, if the initial imperfections are small, the original Euler's formula results in a fairly accurate estimation of the total compressive load which a straight slender member can support.

$$\frac{P}{P\_{\rm cr}} = \left[\frac{2}{\pi} \cos^{-1} \left\{ \frac{1}{1 + \delta/e} \right\} \right]^2 \tag{11}$$

$$\frac{P}{P\_{\rm cr}} = 1 - \frac{a}{\delta} \tag{12}$$

Equations (11) and (12), in this regard, describe the correlation between the Euler's load for an ideal column (*P*cr) and the critical load (*P*) for a column either having a certain initial curvature (*a*) or subjected to eccentric loading (*e*). Figure 5 shows the graphical interpretation of Equations (11) and (12) for different values of eccentricity and initial curvature. The example calculations are carried out by taking into account Equations (11) and (12) by assuming the different values of δ, along with a and e to consider the initial curvature or eccentricity, respectively.

Along with that, the graph also shows that it does not matter how the initial imperfection is introduced in a perfect column, given that the critical load for an imperfect column will always be smaller than the critical load of the perfect one.

**Figure 5.** Normalised load versus deflection curves for a compressed member affected by various amplitudes of initial imperfection.

#### *4.3. Imperfections Due to Variable Stiffness*

Euler formula is derived for prismatic sections and in an attempt to increase the buckling capacity of columns, the researcher focused on the use of nonprismatic sections. This results in a variable moment of inertia (*I*) along the longitudinal axis. It is to be noted that columns with nonprismatic sections can be studied in two ways, as shown in Figure 6. Among these two major approaches, it can be noticed that several researchers employed a continuum approach using different functions and variables to report closed-form solutions. At the same time, other researchers employed numerical approaches/approximate methods to arrive at acceptable solutions.

**Figure 6.** Approaches for the solution of nonprismatic columns.

Finally, it is also well accepted that a given uniform section column is overdesigned everywhere, except the point at which the maximum bending moment occurs, and in need to optimise the buckling capacity of the column, some material can be taken out from the overdesigned section and placed at the point at which the maximum moment is occurring. A number of scientists developed this idea as reported in Table 1. The overall research efforts conducted by several researchers in understanding the behaviour of nonprismatic

#### columns using the two major approaches from Figure 6 are summarised in Table 1, with evidence of methods and outcomes.

**Table 1.** Summary of research on nonprismatic columns using continuum or numerical approaches.

#### **Table 1.** *Cont.*

**Table 1.** *Cont.*

\* Additionally, Weinberger (unpublished research).

#### *4.4. Imperfections Due to Functionally Graded Material*

From the basic equation of Euler's [5] one can directly infer that the buckling capacity of columns can also be varied by varying the modulus of elasticity. This method was not preferred until the technology advances to a level that variation of modulus of elasticity either in the axial or longitudinal direction was feasible. Based on the literature, it is observed that it is still a relatively unexplored area. In order to increase the buckling capacity of the column, it was suggested to vary the modulus of elasticity, but the solution of instability becomes difficult to compute. Signer [74] investigated the buckling solution for columns with continuous monotonic variations of flexural rigidity along the column. Fixing the origin at one end and *x* coordinate running along the centre line, variation of modulus of elasticity (*E*) and moment of inertia (*I*) were assumed as follows by Equation (13):

$$E(\mathbf{x})I(\mathbf{x})\eta(\mathbf{x}) = E(0)I(0) = E\_{\circ}I\_{\circ}, \eta(\mathbf{x}) = 1 + \beta(\mathbf{x}), \beta \in \mathbb{R} \tag{13}$$

It is to be noted that compressed members used in the civil structure are supported at the intermediate point (bracing). Considering the importance of the intermediate restraints, the study in [75] approximated a column with spatial variation of flexural stiffness due to material gradation or nonisoperimetric shape by an equivalent column with piecewise constant geometrical and material properties (Figure 7). This method uses a transcendental function that results in a closed-form solution of uniform columns. The suggested method was unique because the mathematical model preserves the properties of a continuous system by containing the infinite eigenvalues corresponding to all higher buckling modes [75]. The buckling analysis of axially graded columns was conducted by Huang and Li [28]. They transformed the governing differential equation with variable coefficients to Fredholm's integral equation which were further reduced to a system of algebraic equations. The accuracy of the suggested procedure was confirmed by comparing the obtained result with the available closed-form and numerical solutions. The significant role of their work was that, unlike other research, it was not restricted only to suitable buckling mode. Through this method, one can successfully solve the problem of buckling, if the variation of flexural rigidity was polynomial, trigonometric, or exponential function.

**Figure 7.** Examples of (**a**) nonuniform and nonhomogeneous axially graded columns, (**b**) piecewise continuous approximated model (of order *n*), and (**c**) axially graded column (order *n*) with multiple intermediate elastic restraints. Reproduced from [75] with permission from Elsevier®, license n.<PHONE_NUMBER>100.

Recently, Elishakoff [76] studied the buckling of columns made from functionally graded material in an axial direction. The study was limited to find the polynomial variation of modulus of elasticity, *E* such that the buckling value exceeds in case of cantilever column whose cross-sectional area was kept constant. In another recent investigation, Rychlewska [77] presented buckling solutions for a beam with clamped-clamped, hingedhinged, and hinged-clamped boundary conditions with an exponential variation of material properties in the axial direction and subjected to distributed load in exponential form.

#### *4.5. Imperfections Due to Elevated Temperature or Fire Exposure*

Specific attention can be paid to the buckling analysis of columns with variable stiffness, as in Section 4.3, but with a focus on stiffness variations due to the use of materials that are remarkably sensitive to temperature variations, as well as for resisting cross section that may suffer for long-term temperature exposure. This is the case of load-bearing members that are susceptible to elevated temperature exposure, and even fire, or any kind of phenomena that can be represented by a "thermal gradient" for the resisting cross section to analyse.

Most of the research studies, in this regard, are relatively recent and specifically focused on columns composed of steel [78–80], reinforced concrete [81–84], timber [85–87].

Developments in building technology and design strategies are even more frequently focused on innovative laminated glass solutions that are bonded by thermoplastic layers [88–91] or even composite-laminated insulated panels in which both mechanical and climatic loads can severely affect the overall column buckling performance. In this last case, thermal exposure effects do necessarily coincide with extreme accidents since fire loading can have marked effects on the overall mechanical performance, given the typically

small thickness that is of common use in structural glass applications. Besides different materials and characteristics that are used for these members, the common aspect of the above documents is represented by the progressive bending weakness deriving from the degradation of the constituent materials. Therefore, the total compressive load acts on a resisting section and member that prematurely collapses due to its lack of load-bearing capacity. From a practical point of view, the shared feature for the cited literature studies is the basic trend to define standardised design buckling curves for columns made of mostly different constructional materials and thus to collect, in a simplified and univocal formulation, all the possible uncertainties and effects due to material behaviours, eccentricities, imperfections [92].

Even more attention is indeed required for load-bearing members in general that can be subjected to scattered thermal patterns and are thus potentially characterised by a number of critical cross sections.

Additionally, in this latter case, the first efforts are certainly related to the classical material for buildings, thus steel members. Alpsten [93] showed, for example, that residual nonuniform thermal stresses can severely affect the column buckling performance of a given member and result in even more pronounced degradation than geometrical initial imperfections. Culver [94] also focused on the analysis of pinned columns with thermal exposure. The study proved that severe thermal gradients in the mid-span region of columns are typically associated with a remarkable loss of global buckling capacity. While such a concept can be intuitive—due to stiffness reduction—this is in contrast with the discussion by Hoffend [95]. The reason is in the idealisation of the thermal gradient profile. The generally recognised idea, finally, is that thermal gradient effects can be generally schematised in the form of an equivalent initial imperfection. Therefore, the overall buckling performance of an axially loaded member in compression can be severely compromised. On the other hand, this issue can be efficiently addressed for safe design by means of conventional calculation methods that include a given initial geometrical imperfection.

#### **5. Conclusions**

The problem investigated by Euler was much simpler since it did not involve finding the solution of differential equation with varying coefficient because neither the material properties nor the cross-sectional dimensions were changing. However, in an attempt to maximise the buckling capacity of the column, modifications were performed in the column due to which the differential equation governing the mathematical model is left with varying coefficients. This review paper provides a complete synopsis of the development of various theories related to column buckling. A significant number of methods were recalled to obtain close-form solutions, but providing evidence for each one of them had certain intrinsic restrictions—either the buckling shape was assumed to be governed by a specific function or the distribution of flexural stiffness was not random. Moreover, all the discussed methods in the literature showed rather good agreement with some experimental results available in the literature. However, which one of them is more suitable to find the solution for a given arbitrary variation of coefficients still remains an unanswered question. In the selfbuckling of columns, more research emphasis is required for the proper discretisation of elastic segments with rigid segments. For short column analysis, detailed experiments with various materials are one of the areas wherein research still needs to be carried out. In long columns analysis, more emphasis shall be put on the development of closed-form solution with variable moment of inertia with emphasis on varying material properties along the length of the column. Further, different functions can be developed to investigate the variation of modulus of elasticity and its effect on the buckling strength of columns. Furthermore, attention is indeed required for load-bearing members in general that can be subjected to scattered thermal patterns and are thus potentially characterised by a number of critical cross sections. This can be achieved by incorporating the effect of thermal gradient in terms of initial imperfections.

**Author Contributions:** The reported work is carried out authors as a team wherein Conceptualization, M.D.G.; methodology, M.D.G. and C.B.; software, A.S.; validation, A.S., A.P.K. and L.M.G.; formal analysis, M.D.G., C.B. and A.S..; investigation, M.D.G., C.B., A.S., A.P.K. and L.M.G.; resources, A.S. and M.D.G.; data curation, M.D.G., C.B. and A.S.; writing—original draft preparation, M.D.G., C.B. and A.S.; writing—review and editing, M.D.G., C.B., A.P.K. and L.M.G.; visualization, M.D.G. and C.B.; supervision, M.D.G.; project administration, L.M.G.; funding acquisition. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data will be available upon request.

**Acknowledgments:** Authors acknowledge the support and encouragement provided by Director, VNIT Nagpur, India. MDPI is also acknowledged for the invited free-of-charge submission of this manuscript in the special issue "Innovation in Structural Analysis and Dynamics for Constructions" (C.B. Guest Editor vouchers).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

## *Article* **Facial Expression-Based Experimental Analysis of Human Reactions and Psychological Comfort on Glass Structures in Buildings**

**Chiara Bedon \* and Silvana Mattei**

Department of Engineering and Architecture, University of Trieste, 34127 Trieste, Italy<EMAIL_ADDRESS>
**\*** Correspondence<EMAIL_ADDRESS>Tel.: +39-040-558-3837

**Abstract:** For engineering applications, human comfort in the built environment depends on several objective aspects that can be mathematically controlled and limited to reference performance indicators. Typical examples include structural, energy and thermal issues, and others. Human reactions, however, are also sensitive to a multitude of aspects that can be associated with design concepts of the so-called "emotional architecture", through which subjective feelings, nervous states and emotions of end-users are evoked by constructional details. The interactions of several objective and subjective parameters can make the "optimal" building design challenging, and this is especially the case for new technical concepts, constructional materials and techniques. In this paper, a remote experimental methodology is proposed to explore and quantify the prevailing human reactions and psychological comfort trends for building occupants, with a focus on end-users exposed to structural glass environments. Major advantages were taken from the use of virtual visual stimuli and facial expression automatic recognition analysis, and from the active support of 30 volunteers. As shown, while glass is often used in constructions, several intrinsic features (transparency, brittleness, etc.) are responsible for subjective feelings that can affect the overall psychological comfort of users. In this regard, the use of virtual built environments and facial expression analysis to quantify human reactions can represent an efficient system to support the building design process.

**Keywords:** structural glass; building design; human reactions; psychological comfort; experiments; virtual reality (VR)

#### **1. Introduction**

In building design, the end-user's comfort is a target for a multitude of applications. These include, for example, thermal comfort, indoor air quality, visual comfort, noise nuisance, ergonomics and vibrations. With around 90% of our lives spent in buildings, there is a strong link between comfort and the built environment [1,2]. For decades, various researchers have sought to understand how the characteristics of the built environment can impact the emotions, behaviors and physical well-being of end-users [2]. However, comfort itself depends on a great number of factors which can, if not addressed properly, lead to annoyance. Further, it is known that such analysis requires convergent teams from the humanities, arts, social sciences, technology, engineering and medicine. With a more explicit focus on engineering issues, comfort strictly relates to many topics of primary interest for building technology [3–5].

Structural issues and comfort can be, for example, considered in terms of vibration serviceability assessments. Usually, the evaluation of vibration issues in relation to possible annoyance risk is carried out in terms of acceleration peaks and recommended limit values. Similar methods can be used for the comfort analysis of floors under walking or standing conditions [6–8], and for tall buildings against wind [9–11]. As a matter of fact, human reactions and comfort levels strongly depend on engineering parameters, but also on the

**Citation:** Bedon, C.; Mattei, S. Facial Expression-Based Experimental Analysis of Human Reactions and Psychological Comfort on Glass Structures in Buildings. *Buildings* **2021**, *11*, 204. https://doi.org/ 10.3390/buildings11050204

Academic Editor: Geun Young Yun

Received: 9 April 2021 Accepted: 10 May 2021 Published: 14 May 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

physiological perception of the frequency and amplitudes of vibrations. The operational context, including location (street, gym, office, home, etc.), time of day (morning or evening) and stimuli duration (seconds or hours) also have severe impacts on the degree of human tolerance [8].

The structural use of materials and design concepts in buildings is an additional factor that can implicitly result in possible discomfort for the occupants. The well-known psychological effect of architectural solutions can have both positive and negative effects, and can evoke subjective feelings that reduce to a minimum the mechanical efforts of structural designers [12]. This is the major challenge of architects when evoking emotions in the design of so-called "emotional buildings" [13–18]. In the context of the built environment, experimental measurements can be carried out to quantify emotions [19]. Smart sensors or virtual walks are proven to represent useful tools in support of the analysis of human reactions to visual stimuli [20,21].

Among other things, structural glass is known to represent an attractive material for construction [22,23]. Key aspects are its transparency and abilities to adapt to various configurations and replace/interact with more traditional constructional materials. Typical examples can be found in facades, roofs, walkways, bridges and balustrades, as in Figure 1a–c. Besides, glass is also recognized as one of the most vulnerable components in constructions, and thus to need special structural design efforts in order to ensure appropriate safety levels. Glass facades and windows, for example, are the first physical barrier for building occupants, both in presence of ordinary operational conditions but also during accidental events that could result in potential shards and injuries (Figure 1d,e). The high aesthetic impact of glass components and structures is thus often in contrast with a basic discomfort and lack of safe feelings for end-users.

**Figure 1.** Examples of glass in buildings: (**a**,**b**) Facades and roofs for the Poly Plaza Cable-Net Wall (Beijing, CN) and the Kempinski Hotel (Munich, DE), (reprinted with permission from [24], Copyright 2021 Elsevier®, license number<PHONE_NUMBER>444); (**c**) a pedestrian system (walkway in Aquileia, IT; reprinted from [8] under the terms and conditions of CC-BY license); (**d**,**e**) cracks and shards due to a hazard (reprinted with permission from [25], Copyright 2021 Elsevier®, license number<PHONE_NUMBER>637).

Glass design can be highly demanding for pedestrian systems, due to the combination of dynamic mechanical parameters and complex human–structure interaction (HSI) phenomena that typically occur during a walk, but also due to subjective reactions [8].

In general terms, glass structures are often called "architectures of vertigo," given that transparent load-bearing systems are increasingly conceived as "spaces of visceral thrills with deep socio-spatial implications" [23]. Glass floors, in particular, can be recognized as intense psycho-physiological stimuli for pedestrians, as they induce unavoidable sensory experiences that go beyond the conventional HSI design procedures of analysis.

#### **2. Research Goals and Approach**

#### *2.1. Glazed Stimuli*

In this paper, a virtual experimental approach is presented in support of building design. The goal of the developed method is to explore and quantify the prevailing reactions and psychological comfort levels for end-users exposed to different visual built environments. At the time of analysis, careful attention was paid to the selection of visual stimuli characterized by a primary role of structural glass components in buildings and constructions (i.e., Figure 2a,b). The methodology can be adapted to different topics and design fields.

**Figure 2.** Examples of human reactions on glass structures: (**a**) Zhangjiajie bridge, Hunan, CN (adapted from [26], © Imagechina/REX/Shutterstock); (**b**) Cliffside skyway, CN (adapted from [27], © Visual China Group via Getty Images); and (**c**) a virtual experimental procedure.

While the use of glass in buildings has been increasing in the last few decades, with new challenges for structural designers [25], the impact on end-users is often severely affected by the material's transparency and brittleness, and also by the lack of technical knowledge regarding its mechanical properties [28,29]. In this regard, it is expected that the building design process could take advantage of subjective measurements and their possible combination with mathematical models and engineering calculations. As schematized in Figure 2c, the basic approach takes benefit from the quantitative measures of emotions and subjective feelings of individuals exposed to pre-selected visual stimuli. Based on the analysis of facial micro-expressions, comfort levels and trends can be analyzed by sub-groups of subjects of stimuli.

#### *2.2. Experimental Procedure*

The overall methodology was elaborated and designed at the University of Trieste (Department of Engineering and Architecture, Italy) in Winter 2020. The experimental study followed the procedural steps in Figure 3, and was carried out remotely.

**Figure 3.** Flowchart of the virtual experimental analysis. A detailed example: a glazed apartment floor in Paris, FR (adapted from [30], © Jerry Jacobs Design) and facial expression analysis (© C. Bedon).

More precisely:

• STEP 0: the participants were first recruited. The group of volunteers (30 in total) included mostly students and researchers, both females and males (58% and 42% respectively), with an age range of 20–56 (28.8 years-old was the average, ±8.4; 26 years-old the median value). Some volunteers were students or workers living in Trieste, or residing in the Friuli Venezia Giulia Region. The age demographics and geographic distribution of participants are shown in Figure 4. Their educational experience and background skills were characterized by different fields of study or activity. Most importantly, no preliminary technical knowledge on the use and features of structural glass in construction was required at the time of the experiment. Similarly, no direct experience on glass structures was needed to join the virtual investigation.

**Figure 4.** (**a**) Age demographics and (**b**) geographic distribution of participants.

#### *2.3. Strengths and Limitations of the Study*

The proposed experimental methodology took partial inspiration from past literature, which has proved that human emotions can be used as efficient guide for designers [32,33]. The study and measuring of human behaviors based on facial micro-expressions is in fact typical of many research and market fields, especially where understanding consumers feelings can help designers to establish an efficient emotional communication between consumers and products. It is in fact generally recognized that both verbal and non-verbal behaviors enable humans to communicate emotions [12,13]. Non-verbal communication includes physical behaviors that are commonly referred to as body language and gestures, including also facial expressions. Most importantly, facial expressions are essential for the quantitative measure of basic emotions, because they provide information about inner states of individuals.

For the present study, the use of a virtual experimental setup and its adaptation to the constructional field was suggested by several motivations. The first and most severe one was represented, at the time of the investigation, by severe movement limitations, remote teaching regulations and smart-working rules due to the COVID-19 pandemic

emergency (i.e., obligations for "red zone"/maximum risk measures in the Italian territory). As such, the experimental analysis was carried out with the active support of volunteers, but respectfully of social distance rules and general temporary prohibitions. Furthermore, technical trends and considerations suggested the design of the experimental study. The sanitary emergency revealed in fact that the use of transparent materials for constructions is expected to increase in the next future [29,34,35], so as to facilitate the definition of new design concepts for the organization of private or public spaces in buildings. Differently, this increasing use of glass in structures can be associated with a further magnification of "architecture of vertigo" concepts [23] earlier described. Balanced technical solutions should be thus necessarily developed. Both the cited aspects were merged to support the design of the experimental strategy herein presented.

Besides, the investigation was carried out with a clear preliminary analysis of intrinsic limitations that should be taken into account for future extensions. The number of participants (30), for example, was selected on the base of individuals availability (i.e., to join remotely the study, with own devices and high-speed internet connection). A possible extension of group size, and demographic/geographic distribution, could facilitate a more generalized analysis of reactions for specific sub-groups of participants. Another limitation of the study was represented by the fully remote experimental procedure. Under COVID-19 restrictions, the volunteers were invited to take part from homes/different regions of Italy. As such, the measure of facial expressions was carried out with shared screens only, without additional smart sensors, wearable devices or VR glasses [36,37].

#### **3. CAWI Survey**

The preliminary STEP 1 took the form of a CAWI survey that was designed to include 40 questions. The survey was designed with both the simple multiple choice (SMC) approach and the open question (OQ) method. The goal was to expose the participants to well-defined built conditions characterized by the presence of structural glass at different levels (residential buildings, open spaces, public offices, etc.), and to capture their feelings as active occupants. In this context, the survey revealed some important aspects, and suggested the development of the virtual experimental investigation herein discussed. For the majority of participants, it was proved that:

#### **4. Static and Dynamic VR Experiments**

#### *4.1. Methods*

The experimental investigation was based on a sequence of pre-selected input sources that were used to act as emotional visual stimuli for the involved participants. The so-called STEP 2 from the flowchart in Figure 3 was developed as schematized in Figure 5, and represented the core of remote analysis. STEP 2 as a whole consisted of three sub-steps for each participant. The process was repeated twice, first with a static stimulus (sequence of pictures described in Section 4.2) and later with a VR video clip of pre-recorded virtual

walks (Section 4.3). The remote experiments were carried out with the support of personal computers for all the volunteers, a shared screen for the visual stimuli and a webcam for recording the facial expressions of participants.

**Figure 5.** General procedure for the virtual experimental investigation (STEP 2).

Before watching the shared input source, the invited subjects were informed about the experimental procedure (STEP 2.1 in Figure 5). A special care was paid to verify the connection speed, and minimize possible delay in transmission and acquisition of shared data. Successively (STEP 2.2 in Figure 5), the visualization setup was also properly checked, so as to avoid disturbing effects for the quality level of signals, and thus for the post-processing analysis. More in detail, the attention was given to ensure an optimal and mostly uniform ambient illumination for the participants. The screen position was also assessed for each participant, in order to prevent severe facial distortions and to avoid introducing high head/eye inclinations through the experiments. Finally, the presence of participants with glasses or beard was separately noted, for a more refined post-processing analysis of measured data.

Once verified the basic operational conditions, the input source was shared separately for each volunteer (STEP 2.3 in Figure 5). No auditory stimuli were introduced. During the presentations, however, audio communication of volunteers with the project manager was allowed to provide technical feedback (when needed). During the presentations (STEP 2.3), the project manager webcam was used to record the facial expressions of participants, while looking at the shared screen. To this end, a preliminary analysis to capture the optimal webcam resolution and sampling rate was carried out. The final choice resulted in records with minimum 1280 × 720 (or 1920 × 1028) video resolution and around 4300 frames for each recorded signal (30–60 the range of frame rate; 12,432 kbits/s the average bitrate).

The analysis of recorded signals was carried out during STEP 3 of Figure 5. The interpretation of facial micro-expressions and emotions from the participants exposed to the shared input source was partly carried out with the use of the commercial software FaceReaderTM [31]. The automatic computational analysis was based on all the collected frames (≈4300 the average) for each one of the available records (30 volunteers × 2 stimuli), and required approximately 10 min of analysis/each. The overall elaboration of signals and data analysis was then carried out as in Section 4.4. For the current study, it is thus important to note that the automatic software analysis was used to provide row quantitative

data only of nervous states. The detection of human reactions and comfort trends for individuals exposed to glass environments was manually derived as in Section 4.5.

#### *4.2. Static Input Source*

The invited participants were first subjected to a shared input presentation that consisted in a selection of 27 pictures for various glass structures and scenarios. These pictures were equally spaced at time intervals of 5 s, so that the total duration of the visual stimulus could not exceed a maximum of 120 s, to avoid annoyance. Figure 6 shows an example of input pictures. A special care was spent for their selection (to capture emotions in the invited participants), but also for the definition of the optimal sequence. To that end, the pictures were selected from magazines, scientific journals, webpages of construction companies or newspapers. For all the participants, the sequence and duration of the static presentation was kept fix. Regarding the sequence, the choice was to alternate items possibly associated to comfort or negative feelings.

**Figure 6.** Selection of static input sources for the first stage of the static virtual experiment (examples from the set of 27 items): (**a**) private room (Off-grid itHouse, Pioneertown, USA; adapted from [38], © Airbnb); (**b**) Amsterdam RAI Hotel, NL (adapted from [39], © Egbert de Boer); (**c**) Hongyagu bridge, Hebei, CN (adapted from [40], © REUTERS/Stringer); (**d**) windows under a blast hazard (adapted from [29], © C. Bedon); (**e**) stairs of the Apple Cube, New York, USA (adapted from [41], © Sedak GmbH and Co KG); (**f**) facade and walls (adapted from [29], © C. Bedon); (**g**) window under vehicle impact (adapted from [42], © Times Colonist); (**h**) Space Needle tower (Seattle, WA, USA; adapted from [43], © Space Needle LLC and John Lok); (**i**) skydeck (Willis Tower, Chicago, USA; adapted from [44], © Ranvestel Photographic).

#### *4.3. Dynamic Input Source*

The setup schematized in Figure 5 was adapted and applied to the group of participants. The source consisted in a pre-recorded VR clip with total duration of 120 s, see Figure 7. The second stage of the investigation was in fact carried out to capture any kind of possible variation in the amplitude and trend of emotions of participants, when exposed to a virtual walk in a glazing environment. Major benefit was taken from the availability of a VR context representative of a case-study building located in Paris, and characterized by a large amount of glass components in facades, roofs, floors, balustrades. The VR context was used to pre-record a set of walks that could be used as stimulus. Based on preliminary studies, the clip was designed and divided onto three walks (40 s/each):

**Figure 7.** Selection of screenshot frames from the pre-recorded VR video clip inspired by the case-study palace in Paris, FR (adapted from [45], © Generali Real Estate French Branch): (**a**) general view of the building (from Rue Réaumur); (**b**) indoor open space; (**c**) top terrace; (**d**) glass conical envelope/facade; (**e**) glass floor (from the ground level); (**f**) glass roof (from the ground level).

Similarly to Section 5, no auditory stimuli were considered for the dynamic setup, and the participants were asked to watch the shared screen and join the virtual walks.

#### *4.4. Experimental Measurements*

The measurement of relevant data for comfort analysis was carried out in the postprocessing stage of records for each participant, when exposed to both static and dynamic VR glazed environments.

The FaceReaderTM software, in particular, was chosen because it allows to model accurately the face of a given individual, by taking advantage of 500 key points for the description of facial movements and micro-expressions. The software is based on Artificial Intelligence tools and can be used to detect subjective emotions and human reactions from input video records or even pictures. As a whole, the software output takes the form of a series of quantitative charts and data that can be used for the interpretation of basic emotional states. In addition to the reference "neutral" condition, the standard classification carried out by the software includes basic definitions for individual states (*N* = 6 in total) targeted as "happy", "sad", "scared", "disgusted", "angry" or "surprised". From a practical point of view, the software offers an instantaneous evaluation of Action Units (AUs, ranging from 0 to 1) for all these measured emotions. Additional useful feedback can be obtained by the quantitative analysis of output parameters such as valence, arousal, gaze direction, head orientation, heart rate (based on remote skin conductance analysis) and few additional personal characteristics of participants (such as gender, average age, presence of glasses, beard).

For the present study, AU records were collected for each participant/virtual stimulus/basic emotion, as a function of the time of experiment (with 0 ≤ *t* ≤ 120 s). AU data were exported from the software analysis with a sampling rate of 1 s. The typical example of AU records is shown in Figure 8 for one of the participants subjected to the dynamic VR setup. The chart, more in detail, gives evidence of specific AU values over time of virtual stimulus (i.e., STEP 2.3 in Figure 5), but omits the first instants of the experimental arrangement, when some preliminary discussion was carried out with the participant to provide instructions and create a comfortable condition.

**Figure 8.** Example of AU records of basic emotional states, as obtained for one of the invited participants exposed to the shared VR input source.

In the overall procedure, moreover, it has to be noted that the "neutral" state computed by the software was excluded from manual elaboration of measured data, as it is shown in Figure 8. In presence of VR input sources and visual stimuli that do not involve active movements of participants, it is in fact generally recognized that the expected reactions and emotions cannot be quantitatively and qualitatively compared to feelings evoked by real field experiments. Accordingly, combination of "neutral" data with other basic emotional signals for the present study would have alter most of the measured subjective reactions and comfort trends.

#### *4.5. Manual Post-Processing Strategy for AU of Basic Emotions*

The quantitative and qualitative analysis of human comfort was based on the detection and measure of the herein called "positive" (POS, in the following) or "negative" (NEG) feelings from all the participants, when visually exposed to specific glazed environments.

Given that several AU records were obtained (as for 30 participants, *N* = 6 basic emotions, 2 virtual scenarios), the first approach of manual data elaboration consisted in the individual analysis of collected AU records. This stage was developed to explore the subjective feelings of single volunteers when exposed to a specific building scenario (i.e., glass floor, roof, balustrade, etc.). In addition, AU calculations were used to correlate the subjective response of different participants when exposed to the same VR input, and thus to find a prevailing emotional reaction/comfort trend for groups of built environment conditions. This means that for each VR scenario, basic emotions were subdivided onto two major categories representative of the herein called POS or NEG feelings and comfort trends. In doing so, the approach schematized in Figure 9 was take into account. The automatic software analysis was first carried out at STEP 3.1.

**Figure 9.** Schematic procedure for the post-processing analysis of AU of basic emotions (STEP 3) and detection of comfort trends.

According to AU charts as in Figure 8, the preliminary requisite of analysis consisted in the normalization of AU records of basic emotions in the time of experiment, so as to facilitate the objective comparison of multiple AU signals (STEP 3.2 in Figure 9). Starting from the exported original AU charts (AUoriginal), more in detail, the average AU for each emotion at the beginning of the individual experiment/participant (AUinitial,avg) was taken as a reference for calculating the variation of subsequent AU measurements in time, that is:

$$\text{AU}\_{\text{norm}}(t) = \text{AU}\_{\text{original}}(t) - \text{AU}\_{\text{initial,avg}} \tag{1}$$

with 0 ≤ *t* ≤ 120 s and:

$$\text{pos}^\*(t) = \sum\_{i=1}^{N-1} \text{AU}\_{\text{norm}}(t) = f(happy) \tag{2}$$

Similarly, the neg\*(*t*) set included the input data from feelings marked as "sad", "angry", "scared", "disgusted":

$$\text{energy}^\*(t) = \sum\_{i=1}^{N-1} \text{AU}\_{\text{norm}}(t) = f(\text{sad}, \text{energy}, \text{scared}, \text{disgusted}) \tag{3}$$

Note in STEP 3.4 of Figure 9 that the "surprised" AU data ("SUR(*t*)") were omitted from the so-calculated pos\*(*t*) and neg\*(*t*) signals, and analyzed separately. This because (depending on the individual subject, context and stimulus) "surprised" states can be typically associated with both positive or negative feelings for the participants. The final marking of emotional data was thus based on the analysis of prevailing AUs at a given time instant *t*. The SUR(*t*) input was in fact superimposed as:

$$\text{POS}^\*(t) = \begin{cases} \text{pos}^\*(t) + \text{SUR}(t) & \text{if } \text{pos}^\*(t) > \text{neg}^\*(t) \\ \text{pos}^\*(t) & \text{if } \text{pos}^\*(t) < \text{neg}^\*(t) \end{cases} \tag{4}$$

or

$$\text{NEG}^\*(t) = \begin{cases} \text{neg}^\*(t) + \text{SUR}(t) & if \text{neg}^\*(t) > \text{pos}^\*(t) \\ \text{neg}^\*(t) & if \text{neg}^\*(t) < \text{pos}^\*(t) \end{cases} \tag{5}$$

An example of calculation results is shown in Figure 10, in terms of POS\*(*t*) and NEG\*(*t*) measured states for one selected participant.

**Figure 10.** Example of calculated (**a**) POS\* and (**b**) NEG\* measurements of AU signals, as obtained for one of the participants under the VR input source, as a function of the time of analysis.

In conclusion, the final detection of comfort trends for a given visual stimulus was carried out as in STEP 3.5 of Figure 9. To that end, the percentage variation of elaborated signals from STEP 3.4 was measured over the time of analysis. At each increment *t* of the experiments, the emotional variation was calculated for both POS\*(*t*) and NEG\*(*t*) data, and defined as:

$$
\Delta\% = 100 \frac{\Upsilon(t\_n) - \Upsilon(t\_{n-1})}{\Upsilon(t\_{n-1})} \tag{6}
$$

with Y = POS\*(*t*) or NEG\*(*t*) respectively, as shown in the example of Figure 11.

**Figure 11.** Examples of POS and NEG variation over the time of analysis, for a single participant exposed to (**a**) static and (**b**) dynamic VR stimuli.

The prevailing emotional state (and thus comfort trend) was established in terms of absolute maximum values of nervous feelings and max/min percentage variations in the time of analysis. Each time instant *t* was classified with a POS or NEG mark, uniquely detected from data elaboration. A "comfort weight" value C(*t*) = POS or NEG (based on the prevailing nervous state) was assigned to each participant/instant/scenario, to facilitate the comparative analysis of multiple signals and stimuli.

The global correlation of comfort trends with the VR source was in fact carried out based on the synchronized comparison of elaborated data for the 30 participants subjected to an identical stimulus (STEP 3.6). More precisely:

#### **5. Discussion of Results from Static Experiment**

#### *5.1. Prevailing Reactions*

Figure 12 shows the analysis of prevailing reactions in the group of participants, as obtained by the investigation of elaborated records in the time of analysis (Figure 12a), and by the comparison of group reactions for a given static stimulus (Figure 12b). POS and NEG values are reported in percentage values for the whole group of volunteers. As such, the comparative data can be used to address the prevailing emotional state (in terms of average or maximum/minimum reaction of participants), towards the imposed picture/context. Figure 12a, more in detail, shows a clear modification of human reactions for the assigned stimuli, with marked fluctuations of POS and NEG data. It can be easily perceived that the calculated POS signal prevails on the NEG one for some time intervals, while the opposite

condition can be noticed for other instants. In some other cases, rather balanced POS and NEG values can be also observed.

**Figure 12.** Detection of prevailing reactions (in percentage terms for the group of participants), as a function of (**a**) time of analysis or (**b**) static stimulus (27 pictures).

Most importantly, the POS and NEG trends were elaborated to derive the prevailing emotion for each input item, as shown in Figure 12b. For each time interval of 5 s, the average POS and NEG reactions were first calculated and associated with the 27 input pictures. Further, the maximum and minimum reaction peaks were also analyzed, as emphasized in the chart. This operation allowed us to derive a more precise analysis of human responses and nervous states, as a function of the imposed stimulus.

From the collected data, in particular, it was possible to assert that:

The analysis of average reactions in Figure 12b highlighted in fact that:

**Figure 13.** Static items with prevailing NEG reactions: (**a**) Glass Pavilion Rheinbach, DE (adapted from [46], © F. Wellershoff); (**b**) skywalk (Jasper National Park, CA; (adapted from [47], © Getty Images); (**c**) delamination (adapted from [48] under the terms and permissions of a CC-BY license and adapted from [49], Copyright 2021 Elsevier®, license number<PHONE_NUMBER>164); (**d**) Zhangjiajie bridge, Hunan, CN (adapted from [50], © VCG via Getty Images); (**e**) cliff concept boutique hotel (adapted from [51], © Hayri Atak).

The detailed analysis of any kind of discomfort revealed that:

The analysis of major comfortable states, finally, can be summarized as follows:

**Figure 14.** Static items with mostly balanced POS & NEG reactions: (**a**) private room (Off-grid itHouse, Pioneertown, USA; adapted from [38], © Airbnb); (**b**) glass bricked wall (Chanel Amsterdam Store, NL; adapted from [52], © MVRDV); (**c**) skywalk (Mahanakhon, Bangkok, THA; adapted from [53], © Tripadvisor); (**d**) roof (30 St Mary Axe Tower, London, UK; adapted from [54], © Nigel Young - Foster + Partners, Richard Bryant); (**e**) footbridge (Lisbon, PT; adapted from [55], © Schlaich Bergermann Partner); (**f**) Markthal, Rotterdam, NL (adapted from [56], © MVRDV) (**g**) walls for the 360 revolving restaurant Moon in Amsterdam, NL (adapted from [57], © A'DAM Lookout); (**h**) walkway (Coiling Dragon Cliff skywalk, CN; adapted from [58], © VCG via Getty Images); (**i**) swimming pool (Hubertus Hotel (Valdaora, IT; adapted from [59], © Design and Contract)); (**j**) underwater restaurant in Norway (adapted from [60], © Dezeen).

**Figure 15.** Static items with prevailing POS reactions: (**a**) mock-up of glass floor system (adapted from [61], © Vitroplena bvba, BE); (**b**); roof (Botanical Garden, Amsterdam, NL); (**c**) observation deck (360 Chicago Tilt, Chicago, USA; adapted from [62], © Tripster).

#### *5.2. Analysis of Reactions by Context of Pictures*

Based on the experimental outcomes as in Figure 12, a special care was paid for the analysis of POS or NEG reactions for pictures grouped by context/building configuration. Two major categories of items were considered, namely represented by:

In case of Group-A, a NEG reaction was typically found to prevail for the majority of pictures (8 of 11 items). For three pictures only (#10, #20, #23), the POS reaction minimally prevailed on the NEG measures. Comparative results are summarized in Figure 16a. Such an outcome suggests that pedestrian glass systems evoke the highest discomfort and emotional state for the selected stimuli, compared to other glazing solutions in buildings. The effect can be justified by the fact that, compared to other building conditions, the human interaction of pedestrians with the glass structure becomes predominant.

**Figure 16.** Detection of prevailing reactions (in percentage terms for the group of participants), as a function of static stimulus, for (**a**) Group-A or (**b**) Group-B.

Further, the few pictures from Group-A with a prevailing POS reaction were found associated with comfortable presence of individuals (i.e., item #10), and thus suggesting an appropriate safety level for the participants of the present experiment. Anyway, it is necessary to explore further this kind of scenario and extend the virtual experimental stage with field experiments, so as to capture the human reactions of active pedestrians and combine their emotional feelings with the structure vibrations and other mechanical parameters of primary interest for dynamic characterization.

Regarding the experimental results for Group-B of stimuli, NEG reactions were found to prevail on POS for most of the pictures (3 of 4 items), see Figure 16b. For the whole set, the presence of visual damage in glass proved to involve discomfort for the participants, thereby resulting in a rather good correlation of experimental predictions with the expectations of planned stimuli. The exception was represented by item #6 (Figure 6), characterized by a balanced POS/NEG average response.

#### *5.3. Analysis of Reactions by Sub-Groups of Participants*

The analysis of human reactions was also focused on the emotional state of sub-groups of participants, as, for example, obtained by gender or demographic sub-groups. As far as the POS comfort state is take into account, results according with Figures 17 and 18 can be observed. From Figure 17, more in detail, the total average POS response was found to prevail for the group of female volunteers, for most of items. This is the case for pictures #15 and #18 of Group-A and the whole Group-B. For some other items of Figure 17a, a more balanced reaction was indeed measured.

**Figure 17.** Detection of prevailing POS reactions (in percentage terms for participants), as a function of static stimulus, for (**a**) Group-A or (**b**) Group-B items. In evidence: the response for gender sub-groups of volunteers.

**Figure 18.** Detection of prevailing POS reactions (in percentage terms for participants), as a function of static stimulus, for (**a**) Group-A or (**b**) Group-B items. In evidence: the response for demographic sub-groups of volunteers.

Another aspect worth to discuss could be related to the emotional response for demographic sub-groups of volunteers, as in Figure 18. The distribution of volunteers onto age groups as in Figure 4 is not uniform and sufficiently extended to draw general conclusions. However, it is possible to notice that the trend of POS reactions modifies with age. Especially for Group-B with damage/hazard in Figure 18b, POS data decrease with age increase. Further investigations are hence needed in this direction.

#### **6. Discussion of Results from the Dynamic Experiment**

#### *6.1. Analysis of Reactions by Context of Walks*

For the dynamic setup, the post-processing stage of measured AUs was qualitatively carried out as in Section 4. However, major attention was focused on the analysis of absolute POS and NEG reaction peaks over the time of the experiment. This choice was suggested by the dynamic nature of the stimulus, with fast sequence of frames/scenarios and a different expected response compared to the static setup in Section 5.

Typical results can be seen in Figure 19, where the sequence of three different VR walks (40 s/each) is also emphasized. More precisely, the W*n* labels are used to denote:

**Figure 19.** Detection of prevailing reactions (in percentage values for the group of participants), as a function of the time of analysis for the dynamic VR scenario.

Key labels for selected frames in Figure 19 are defined as:

**Figure 20.** Selection of frames from the pre-recorded VR video clip with marked POS or NEG reactions (adapted from [45], © Generali Real Estate French Branch): (**a**)–(**b**) facade panels; (**c**) floor; (**d**) indoor balustrades; (**e**) indoor partition walls.

> Worth of interest in Figure 19 is that most of the POS or NEG peaks can be associated with well-defined s1–s10 scenarios in which glazing structural components have a primary role in the building context.

More precisely, for the W1 walk:

Finally, for the W3 walk, the experimental measurements were still found in line with the earlier scenarios, given that:

#### *6.2. Analysis of Reactions under Static or Dynamic Stimuli*

In conclusion, a comparative analysis was carried out in terms of subject/context of input stimuli for the group of volunteers, so as to capture any difference in the prevailing reactions due to a static or a dynamic setup. In Figure 21, results are shown in the form of average percentage values of measured POS and NEG reactions for selected categories of built environments and scenarios.

**Figure 21.** Analysis of prevailing POS or NEG reactions (average percentage results for the group of participants), as a function of context, as obtained from the static and dynamic VR experiments.

These are defined as:

For groups E1 and E2, the results are presented in Figure 21 for the dynamic input only, given that no direct correlation could be found with static items. Note, as also previously discussed, the remarkable NEG reaction of participants, both for outdoor or indoor balustrades. This reaction is typically observed when the end-user approaches the glass balustrades during virtual walks, and thus perceives a possible risk of falling.

Regarding the emotional states and reactions of participants towards glass facades and walls (E3 and E4 in Figure 21), the proposed experimental setup did not capture relevant modifications or prevailing nervous states. This effect can be quantified in mostly balanced POS/NEG reactions for both static and dynamic VR stimuli. Finally, it is possible to notice in Figure 21 that the configurations labelled as E5 (i.e., pedestrian systems) were mostly associated with prevailing NEG reactions. As shown, the discomfort and NEG reactions further increase with a dynamic stimulus, compared to the static one. This is not the case for glass roofs (E6) that do not involve direct contact with end-users and are characterized in Figure 21 by a slightly prevailing POS reaction, both under static and dynamic stimuli. In conclusion, it is worth noting that no items presenting a hazard or damage in the glass were taken into account for the comparative analysis of static and dynamic emotional states, due to the lack of appropriate comparative VR stimuli. Besides, data in Figure 21 still suggest a more concise emotional reaction of participants based on dynamic rather than static stimuli. As a further extension of the present analysis, it could be thus useful to take into account accidental scenarios and built environments in which glass walls are expected to act as physical barriers for the protection of occupants.

#### **7. Conclusions**

The optimization of human comfort in the built environment is a target for several design fields, and depends on a multitude of aspects. For engineering applications, some of these aspects can be mathematically controlled and limited to performance indicators, including structural, energy and thermal issues. Differently, human reactions are also sensitive to a various aspects that relate to subjective feelings, in the same way in which the so-called "emotional architecture" aims at evoking emotional responses from building users.

In this paper, an original experimental analysis was presented in support of the definition of new design strategies and tools. The goal was to assess the potential of virtual experimental methods and the possible combination of quantitative measures of subjective feelings with engineering targets, as well as architectural concepts and technology solutions for the constructional fields. Special attention was focused on glass structures and components in buildings, due to the material features, innovative use in constructions and emotional evocation for end-users. Major advantage was taken from the use of virtual experimental techniques and facial expression analyses able to capture the micro-reactions of the group of participants (30 volunteers).

By designing a visual stimulus composed of both static and dynamic virtual items for building configurations characterized by a prevailing role of structural glass elements at different context levels (roofs, floors, balustrades, etc.), the attention was focused on the quantification of "positive" or "negative" emotions and comfort feelings for the participants. As shown, the analysis of experimental measurements proved that the use of facial microexpression recognition tools in support of comfort analyses can represent and efficient approach for optimal design.

Moreover, the proposed experimental methodology highlighted that the design of structural glass elements for buildings can be severely affected by subjective feelings and nervous states of end-users. This is especially the case for structural components that involve direct interaction and contact from end-users (such as floor or balustrades which present a risk of falling), but also the condition of structural components with visual damage. Differently, mostly positive feelings were experimentally measured for the presence of structural glass in walls, facades and roofs. In this regard, the experimental investigation could be extended to different contexts, and to groups of volunteers characterized by different age demographics, geographic distributions or background skills.

**Author Contributions:** This study resulted from a joint collaboration of the authors. C.B.: conceptualization; software; data curation; writing—original draft preparation; writing – review and editing; project administration. S.M.: conceptualization; data curation; investigation; writing – original draft preparation; writing—review and editing. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Informed consent was obtained from all subjects involved in the study.

**Data Availability Statement:** Data will be shared upon request.

**Acknowledgments:** Seretti Vetroarchitetture S.r.l. (www.seretti.it (accessed on 12 May 2021)) is acknowledged for sharing the dynamic VR video clip for the case-study building in Paris (courtesy of © Generali Real Estate French Branch, adapted from www.helloworldparis.com (accessed on 12 May 2021)). All the participants that voluntarily and actively contributed to the experimental investigation are warmly acknowledged. A special thanks is for Corine Tetteroo (Noldus Information Technology bv, NL). Finally, Alice Iurlaro is acknowledged for supporting part of the recording stage.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

## *Article* **Calibrated Numerical Approach for the Dynamic Analysis of Glass Curtain Walls under Spheroconical Bag Impact**

**Alessia Bez 1, Chiara Bedon 1,\*, Giampiero Manara 2, Claudio Amadio 1 and Guido Lori 2**

**Abstract:** The structural design of glass curtain walls and facades is a challenging issue, considering that building envelopes can be subjected extreme design loads. Among others, the soft body impact (SBI) test protocol represents a key design step to protect the occupants. While in Europe the standardized protocol based on the pneumatic twin-tire (TT) impactor can be nowadays supported by Finite Element (FE) numerical simulations, cost-time consuming experimental procedures with the spheroconical bag (SB) impactor are still required for facade producers and manufacturers by several technical committees, for the impact assessment of novel systems. At the same time, validated numerical calibrations for SB are still missing in support of designers and manufacturers. In this paper, an enhanced numerical approach is proposed for curtain walls under SB, based on a coupled methodology inclusive of a computationally efficient two Degree of Freedom (2-DOF) and a more geometrically accurate Finite Element (FE) model. As shown, the SB impactor is characterized by stiffness and dissipation properties that hardly match with ideal rigid elastic assumptions, nor with the TT features. Based on a reliable set of experimental investigations and records, the proposed methodology acts on the time history of the imposed load, which is implicitly calibrated to account for the SB impactor features, once the facade features (flexibility and damping parameters) are known. The resulting calibration of the 2-DOF modelling parameters for the derivation of time histories of impact force is achieved with the support of experimental measurements and FE model of the examined facade. The potential and accuracy of the method is emphasized by the collected experimental and numerical comparisons. Successively, the same numerical approach is used to derive a series of iso-damage curves that could support practical design calculations.

 Introduction**

Glass curtain walls notoriously represent a challenging issue for designers. Given that they must satisfy specific performance requirements in terms of energy, light, acoustic insulation, etc., curtain walls represent in fact a physical barrier for the occupants. As such, careful consideration is required for their structural assessment, even under extreme events. In this regard, several literature studies have been dedicated to the analysis of glass facades under various loading conditions, including blast events, seismic actions, etc. [1–5]. Among others, the assessment of facade capacities under soft body impact (SBI) still represents one of the most frequent accidental design actions (i.e., due to impact of occupants) and also one of the most severe performance limitations, especially against the potential risk of fall [6], see Figure 1.

**Citation:** Bez, A.; Bedon, C.; Manara, G.; Amadio, C.; Lori, G. Calibrated Numerical Approach for the Dynamic Analysis of Glass Curtain Walls under Spheroconical Bag Impact. *Buildings* **2021**, *11*, 154. https://doi.org/ 10.3390/buildings11040154

Academic Editor: Giuseppina Uva

Received: 17 March 2021 Accepted: 5 April 2021 Published: 7 April 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

**Figure 1.** Example of curtain wall: (**a**) modular unit and (**b**) failure under impact.

So far, several research efforts have been spent on the analysis of the impact performance of glass systems. Relevant outcomes are available in the literature, but they are mostly related to specific design applications like simple glass panels [7], glass balustrades [8,9], load-bearing glass columns [10], existing traditional windows [11] or novel laminated glass windows [12]. In [13], a systemic experimental and numerical analysis has been dedicated to the analysis of glass constructional elements under the pendulum test, with a focus on the numerical description of the impactor features. This study follows the extended investigations reported in [14–16], in which the dynamic performance of glass panels under tire impact has been explored with the support of experimental and numerical tools. Often, within the research community, the SBI performance of glass elements is primarily explored for automotive applications [17], or in terms of danger for people [18,19], while a generalized methodology still lacks for the mechanical characterization of curtain walls.

In this paper, a careful consideration is paid for curtain walls under a spheroconical bag (SB) impactor, with the support of full-scale experiments, Finite Element (FE) numerical models (Code Aster [20]), two Degree of Freedom (2-DOF) modelling techniques and practical design tools that could be useful in support of daily practice (i.e., iso-damage curves). In more detail, Section 2 summarizes the key features of experimental methods in use for curtain walls under SBI, as well as the basic theoretical background of impact. Differing from "ideal" conditions, as shown, the SBI response of curtain walls is affected by the features of the facade (impacted body), by the type of impactor (SB or TT) and by the impact features (impact point, drop height, facade features and impactor properties). A novel approach is thus presented in Section 3, based on the coupled analysis of a simplified 2-DOF model and a more geometrically accurate FE model of the curtain wall unit object of study. The procedural steps that are used for the optimal calibration and minimization of the analysis efforts are hence discussed. Section 4, in more detail, presents an experimental campaign carried out on curtain wall unit specimens under a multitude of impact configurations (15 repetitions and arrangements), while the reliability of the proposed procedure is shown in Section 5. Given that the calibrated 2-DOF parameters are derived from a multitude of impact configurations, the general validity of the procedure is shown. Some further design outcomes are presented in Section 6, where comparative analyses are proposed for the optimal definition of the input impulse. For comparative analysis of the proposed methodology, additional numerical results are also derived from the commercial software SJ MEPLA [21] ("MEPLA", in the following). In conclusion, iso-damage curves are presented in support of design for glass curtain walls under SBI.

#### **2. State-of-Art and Background**

#### *2.1. Standardized Experimental Approach*

Glass curtain walls and facades are recognized to have a crucial role in buildings, given that they are expected to offer a physical barrier against falling of the occupants, and a physical protection from potential outdoor events. Accordingly, safety must be especially ensured against accidental impact events. The fall of possible debris, in the event of glass breakage, must be prevented with the protection of pedestrians that could be nearby the glass facade. Defenestration events must be also prevented. In this context, the use of laminated safety glass represents an efficient design solution. However, many other performance indicators must be properly assessed, at the component and assembly level.

For a long time, the SBI performance has been assessed through experiments, based on standardized testing procedures. Usually, these tests are carried out in-situ, or in laboratory conditions on full-scale mock-ups. The mostly used impactors, see Figure 2, are the spheroconical bag (SB) and the twin tire (TT). The former is filled with glass spheres and has a total weight *M* = 50 kg, while the latter consists of two pneumatic tires inflated with 3.5 bar air pressure. A steel mass (*M* = 50 kg) is included within the tires themselves. In this research paper, the tires consisted of Vredestein V47 pneumatics [22].

Major issues for manufacturers of glass curtain walls under SBI, in this regard, currently derive from the existence of a number of National norms and regulations that follow different procedures.

So far, the International EN 12600 standard [23] introduced the TT pendulum protocol at the European level, with the aim of replacing the SB impactor. The TT is in fact more easily describable in its basic components. Further, the German DIN 18008-4 (Annex A) regulations [24] confirmed the possibility of TT impact FE numerical simulations in place of full-scale experiments. In the years, several research studies highlighted the realistic description of TT features and effects with numerical tools [9,13–16,25]. The MEPLA commercial software, in this regard, also developed a lumped spring-mass model able to capture (with limits) the TT features, and thus offer a geometrically and mechanically simplified numerical model in support of design (see Section 2.2 and [26–28]).

On the other side, a set of standards exists that prescribe a pendulum setup still based on the SB impactor, and includes the French Cahier CSTB 3228 [29] and NF P 8-301 [30] for vertical or horizontal glazing, the CWCT TN 76 [31] (vertical glazing) and ACR[M]001:2005 [32] (horizontal elements) standards in the United Kingdom, but also the American ANSI Z97.1 document [33]. The performance evaluation after impact requires one to assess whether the glass system is able to pass a given setup, and to classify the specimen capacity depending on the lack of breakage, or on the size and features of cracks and fragments (if any). However, given that the cited references represent the primary regulation for various National technical committees, this condition forces facade designers and manufacturers to follow the original SB approach. In this regard, the use of efficient numerical tools could simplify the SB impact assessment for various practical applications. Besides, calibrated parameters for SB simulations are not available in the literature, and cannot be replaced by TT formulations. The Newman's hypotheses discussed in [34] asserts the greater severity of impact tests carried out with the TT impactor, rather than the SB. The latter is in fact recognized to have a behavior that is hardly comparable to an elastic body under impact. Moreover, the SB impactor generally offers a greater dissipative capacity, compared to the TT. Most of the dissipative phenomena are related to the glass spheres inside the bag of Figure 2, which are not bound to reciprocal movements. Accordingly, ideal theoretical assumptions for impact estimates are generally over-conservative for real experimental setup configurations with the SB, thus requiring more advanced characterization and calculation procedures.

In this paper, the proposed methodology aims at exploring the difference of SB and TT parameters, towards the definition and calibration of practical recommendations and practical approaches to equalize the SB effects to the TT, for a given envelope unit under SBI.

#### *2.2. Numerical Analysis of Glass Curtain Walls under Soft Body Impact*

#### 2.2.1. Ideal Impact

For a curtain wall under the impact of a pendulum in accordance with Figure 2, the dynamic features and response of the system can be notoriously sensitive to a multitude of parameters. As such, the reliable description of the input time history for the impact force can be challenging. The actual phenomenon still derives from the "ideal" impact configuration, but it should be further characterized. According to the classical theory, the limit condition is represented by an elastic impactor against a rigid body, and the expected force time history is expressed as:

$$F(t) = f(t) \, A \, p \tag{1}$$

where the key input parameters are represented by the impacted surface *A*; the peak pressure *p*, that depends on the drop height of the impactor; and the time factor *f*(*t*), whose typical trend is proposed in Figure 3.

**Figure 3.** Typical trend for the time factor *f*(*t*).

For a rigid impacted surface and elastic impactor, the incoming impulse *Iid* on a given surface *A* can be rationally calculated as:

$$I\_{\rm id} = \int \left[ F(t) \, dt = \int \left[ f(t) \, A\, p \right] \, dt = 2 \, M\_{\rm P} \, \upsilon\_{\rm P} \tag{2}$$

with *MP* denoting the mass of the impactor and *vP* its velocity, at the time of impact. Assuming that *MP* and *vP* are the assigned input data, the corresponding impact energy can be hence predicted in:

$$E\_{imp} = \frac{1}{2} M\_P \upsilon\_P^2 \tag{3}$$

and thus:

$$I\_{id} = \ 4 \frac{E\_{imp}}{\upsilon\_P} \tag{4}$$

The above assumptions, although theoretically accurate, involve a severely conservative description of input features, both on the side of the impactor and on the side of the impacted glass panel. Equations (2)–(4), more in detail, represent an upper limit (or "ideal") condition, since the real behavior is generally affected by the intrinsic flexibility of the impacted glass panel (i.e., stiffness of facade components, compared to a rigid surface), as well as on the impactor features (i.e., type, size, stiffness, impact energy), which manifest in a series of dissipative phenomena.

#### 2.2.2. Simplified Design Approach

The MEPLA computer software is largely used for structural glass design applications, given that it is specifically tailored to the needs of the civil engineering industry and offers some a useful support for the efficient analysis of standard configurations. For impact purposes, the software proposes a lumped spring-mass model that calculates a time history of impact forces for the so-called "default pendulum impactor" loading case based on EN12600 [23]. This "default pendulum impactor" option, compared to more refined modelling techniques (as for example in [9,13,28]), represents a simplified tool for the description of a test setup with TT impactor. At the same time, the approach assumes the hypothesis of elastic impact on a flexible, undamped facade unit. The final result, compared to an ideal impulse as in Section 2.2.1, is a partially reduced impulse *IMepla*:

$$I\_{\rm Merpla} = \,^\text{R}\_{\rm Merpla} \int \left[ f(t) \, \, A \, p \right] \, dt = \,^\text{R}\_{\rm Merpla} (2 \, M\_P \, \upsilon\_P) \, < \, I\_{\rm id} \tag{5}$$

in which the reduction factor *RMepla* < 1 accounts for some intrinsic aspects of the real dynamic phenomenon and bodies involved in the impact event. Research studies in [26,27] and others extensively verified the accuracy of such a schematic representation of the TT impactor, giving evidence of a good correlation with experimental records of various glazing systems. On the other side, see for example [13,28], the lumped mass-spring model cannot equal a geometrically refined characterization of TT impactor, in terms of component features and pressurized air volume in the tires. As a matter of fact, moreover, it lacks of an accurate geometrical and mechanical characterization of the impactor, as it would be required for the SB. Further effects, in the cited procedure, finally derive from the lack of dissipation features, both for the glazing system and for the impactor. These limitations end in a potential over-prediction of the expected accelerations, displacements and strains for the curtain wall, and thus in a consequent over-design of the load-bearing components against SBI.

#### **3. Design Approach for Glass Facades under Soft Body Impact**

#### *3.1. Concept, Assumptions and Goals*

In the current investigation, a new calibration procedure in support of design is proposed and validated by means of full-scale experimental tests. Further, the MEPLA software is used for comparative analyses on curtain wall units under variable SB impact. The goal is to predict the impact response of a given glass facade, by taking into account the actual impact force, with the support of analytical and numerical estimates in place of costand time-consuming experimental protocols. The flexibility and damping contributions are taken into account both on the side of the curtain wall and the impactor, thus allowing to obtain a generalized solution of the design issue. In this way, the amount of experiments can be reduced to a minimum, or even cancelled. From a practical point of view, see Figure 4, the magnitude of the impulse is expected to be:

$$I\_R = \,^r R\_R \, I \, \prec \, I\_{Mapla} \, \prec \, I\_{id} \, \tag{6}$$

with

$$R\_R = f\left(E\_{imp\prime}, K\_{imp\prime}, impactor\right) < R\_{Mapla} < 1\tag{7}$$

where *RR* is a refined reduction coefficient able to account for the boundary conditions and dynamic properties of the curtain wall under SBI. The advantage of Equation (7) is that the maximum theoretical impulse can be minimized to reliable estimates, while keeping mostly constant the total duration of the "ideal" impact event. To achieve this goal, however, some key procedural steps must be taken into account (Section 3.2).

**Figure 4.** Qualitative comparison of input impact forces for a given glass facade, as obtained under ideal conditions (rigid facade), from MEPLA (flexible, undamped facade) or by accounting for the proposed approach (impactor/facade properties and damping).

#### *3.2. Proposed Procedural Steps*

For a given glass curtain wall under SBI, a coupled approach based on a simplified 2-DOF model and a more accurate numerical model (Code Aster) is considered for the modular unit in Figure 5. For the 2-DOF model in Figure 5a, more in detail, components with subscript "F" refer to the facade systems, while the subscripts "P" and "C" are used for the impactor and the contact parameters. The design procedure herein proposed is achieved (and generalized) through the sequence of steps that are schematized in Figure 6. In the STEP 0, the features of the curtain wall and the desired impact energy *Eimp* are defined for the loading condition of interest. Given that *MP* = 50 kg, the corresponding impact velocity *vP* and the related ideal impulse *Iid* are first derived from Equations (3) and (4). At this stage, both the 2-DOF and the FE models of Figure 5 must be properly characterized.

**Figure 5.** Reference mathematical models for the proposed coupled analysis of glass curtain wall under soft body impact (SBI): (**a**) 2-DOF and (**b**) FE numerical model (Code Aster).

**Figure 6.** Reference procedural steps for the calibration of input parameters from the herein developed coupled analysis of glass curtain walls under SBI.

Considering the FE model in Figure 5b, the curtain wall module must be described with careful attention for materials, components, boundaries (Section 5). Then, a frequency analysis is carried out to predict the fundamental vibration period of the undamped system (*T*1, STEP 1). The equivalent stiffness *K*F of the curtain wall represents another key input parameter, given that it is affected by the facade features and also by the point of impact (STEP 2). Such a parameter can be derived from the same FE model, under a unit force that is statically assigned to the desired impact point, and thus based on the corresponding static displacement. The knowledge of *T*1 and *KF* is the starting point for the 2-DOF characterization, given that the equivalent mass *MF* for the modular unit can be calculated as (STEP 3a):

$$M\_F = \frac{K\_F}{\left(\frac{2\pi}{T\_1}\right)^2} \tag{8}$$

Finally, on the side of the curtain wall, the damping term *CF* must be also specified (STEP 3b), based on:

$$\mathbb{C}\_F = \mathfrak{F} \times \mathfrak{2} \sqrt{K\_F M\_F} \tag{9}$$

where *ξ* is a total damping term (including the effects of materials, possible damage propagation, and aero-elastic effects). A detailed calculation approach would require the experimental calculation of test records (i.e., logarithmic decrement, see Section 5). Given that the proposed approach is intended for efficient early-stage design calculations, however, the accurate total damping *ξ* may not be available. As a reference, accordingly, a minimum value should be set at least in:

$$
\mathfrak{F}\_{\text{min}} = \mathfrak{F}\_{\text{glass}} + \mathfrak{F}\_{\text{int}} + \mathfrak{F}\_{\text{sviliconc}} + \mathfrak{F}\_{\text{frame}} \tag{10}
$$

thus accounting for the material properties in use. Based on literature data, for example, the input values in Equation (10) can be expected in the order of 1% for glass, 2% for the soft layers and the metal frame, respectively [35–38].

Depending on the type of impactor in use (see Figure 6) the input parameters must be then properly specified to account for the TT or SB effects (STEP 4), as well as the effect that they transfer to the facade (STEP 5). Given that *MP* and *vP* are known, in accordance with the 2-DOF model in Figure 5a, such parameters are described in the form of a simple mechanical law for the contact itself (i.e., *KP* and *CP* are set to 0). The contact parameters *KC* and *CC* that are required for STEP 5, in other words, are responsible of the actual force (and time history features) that the impactor can transfer to the curtain wall, depending on the time of contact and on the impactor features. In this research study, all these parameters were iteratively calibrated, based on experimental results and numerical fitting (Section 6.2). Once the above parameters are known, the result of the 2-DOF model is a scaled impact force that implicitly accounts for the reduced impulse *IR* in Equation (6), and can be thus assigned to the FE model, to perform non-linear dynamic analyses of the system (STEP 6).

In presence of full-scale impact experiments (STEP 7), the FE dynamic analysis can be assessed towards the test records, or give a realistic prediction of the experimental outcomes, in terms of deflection in time (a), period of vibration of the facade (b), damping of the system (c), but also stress peaks, reaction forces, etc. In the present study, STEP 7 is specifically used for the validation. Otherwise, for curtain walls under SBI, the herein proposed calibrated parameters can be used for general applications.

Major efforts are in fact spent for an accurate characterization of the impactor and contact parameters for the 2-DOF model (STEPS 5 and 6), and the reliability of this characterization is supported by the extended experimental program inclusive of various impact configurations (Sections 4 and 5). The developed FE models, in particular, are able to capture the features of the impacted system and also the impactor/contact parameters, and this is proved by the close numerical match of the experimental displacement peaks for all the examined configurations. As far as a similar description of facade components

and features is taken into account for a given curtain wall module under variable SBI configurations, the proposed method results in a FE model that can be still used for SB setup investigations. In this way, based on the input facade/impact energy assumptions (STEP 0), the correction coefficient *RR* in Equation (6) is implicitly accounted in the 2-DOF procedure and the calculated impact force. For both the SB and TT impactors, simplified mechanical laws are thus presented in Section 6, while the final outcomes of such a calibration procedure are presented in Sections 6.1 and 6.2, giving evidence of the accuracy and potential of the overall procedure.

#### **4. Experimental Analysis of Curtain Walls under Soft Body Impact**

The numerical approach herein proposed was validated towards experimental measurements collected from SBI tests carried out on identical configurations of curtain wall panels under various impact conditions. The experiments were carried out in Vittorio Veneto (Italy), during February 2020, with an average ambient temperature of 8–10 ◦C.

#### *4.1. Materials and Methods*

The impact experiments were specifically focused on the dynamic analysis of a single curtain wall specimen configuration as in Figure 7a, with 15 test repetitions. The reference panel was characterized by the presence of:

**Figure 7.** Reference curtain wall specimen: (**a**) layout and (**b**) nominal cross-section of the visual section (dimensions in mm).

The specimen included a supporting aluminum frame (alloy type 6063-T6 [41]), with sectional properties reported in Table 1. The mechanical connection between the visual glass panel and the supporting frame was ensured by structural sealant joint (DowsilTM 993 silicone [42], with a joint thickness of 10 mm). The so-assembled prototype was thus clamped with four steel brackets to the test bench frame (two at the top and two at the bottom corners of the frame), see Figure 8a.

**Figure 8.** Impact experimental test with spheroconical bag: (**a**) tested curtain wall panel and (**b**) detail view of one of the laser sensors in use (bottom transom).

> To measure the out-of-plane deformations of the specimen under impact, three laser sensors were used. All the laser sensors were installed on the unexposed side of glass, in some key control points of the curtain wall unit that have been detected as:

To ensure the exhaustiveness and reliability of experimental measurements, the SBI tests were performed on glass specimens under different:

To this aim, as also in accordance with Figure 9, five different impact points were selected on the curtain wall specimen, namely represented by:

The experiments were first carried out for the specimen with spheroconical bag impactor. Successively, the more severe TT impactor was used for additional test repetitions. The final combination of testing configurations is summarized in Table 2.

**Table 2.** Selected experimental configurations. Key: TT = twin tire impactor; SB = spheroconical bag; x = not available.

#### *4.2. Test Results*

The analysis of test results was focused on the measured out-of-plane deformations of selected control points, as well as on the visual detection of possible cracks in glass, so as to evaluate the performance class of the tested specimens, according to the standards requirements. For all the loading configurations in Table 2, it is important to highlight that no glass damage was observed experimentally. In terms of FE numerical investigation herein discussed, accordingly, the lack of failure and crack propagation in the glass panel facilitated a simplified characterization of materials, especially glass, that was assumed to behave linear elastically (Section 5).

#### **5. Finite Element Numerical Analysis**

#### *5.1. Solving Approach*

The numerical study was carried out with the support of the Code Aster open-source computer software, is integrated into the Salome Meca platform [43]. A series of dynamic non-linear analyses was carried out under various impact configurations. The software choice was intentionally guided, at the time of the research study, by the need of a freely accessible FE software that could be accessible to designers, but at the same time highly refined to support non-linear dynamic simulations.

At the initial stage of the study, to make sure that the reference Code Aster FE model was properly calibrated in its key input parameters, a very high number of benchmarks has been developed under static and dynamic conditions. Based on sensitivity studies, the final FE assembly shown in Figure 4 resulted in brick elements (with 20 nodes and quadratic shape functions) for the glass panel and the sandwich spandrel. In the case of the supporting frame members, mono-dimensional beam elements were used. The mesh size was set in 10 mm. Besides such a series of basic assumptions, the attention was focused on the detailing of several influencing parameters that are summarized in the following sections.

#### *5.2. Reference FE Model for SBI Simulations*

The reference FE numerical model was realized to reproduce the actual geometry and dynamic behavior of the tested curtain wall specimens earlier described in Figure 7. Besides the reference mesh pattern that was chosen in Section 5.1, special attention was required for the calibration of some other important parameters.

#### 5.2.1. Geometry

The top spandrel, being composed of a sandwich section with insulation purposes only, was first simplified and numerically represented in the form of an equivalent monolithic section. The material properties were characterized in the form of elasticity and density able to reproduce the actual bending performance of the nominal spandrel section. An equivalent thickness concept was taken into account for the laminated glass section, given that PVB foils under short term and impact events are typically associated to a relatively stiff Young's modulus. Accordingly, a monolithic glass section of 10 mm in thickness was taken into account in place of the 5 + 5/0.76 composite panel.

Regarding the aluminum frame members, the attention was spent for the description of the inertial effects due to the presence of non-metallic thermal barrier components. Compared to inertial values reported in Table 1, the thermal barrier contribution was generally observed to greatly affect the overall period of vibration of the examined curtain wall system. Accordingly, the effective inertial properties for the frame members was herein calculated as prescribed in the Annex C of the EN 14024:2004 document [44]. The final input data are summarized in Table 3.

**Table 3.** Nominal properties of the frame members. Key: A = cross-sectional area, I = second moment of area; It = torsional moment.

#### 5.2.2. Gas Cavity

As known, the gas cavity represents a key parameter for insulated glass units, given that it is notoriously responsible of "load sharing" phenomena for the involved glass panels [45–47]. To simulate this interaction for the glass units, various numerical approaches can be found in the literature, and most of them are based on fluid interactions in the cavity, so as to include ideal gas formulations:

$$pV = nRT\tag{11}$$

In the current paper, given that the Code Aster software does not include any fluid interaction for the dynamic non-linear domain, a major simplification was assumed for the gas infill, see Figure 10.

In more detail, an equivalent, linear elastic material was taken into account, with an input modulus of elasticity that was calibrated under the assumption of an isothermal transformation for ideal gases (i.e., a transformation that can often occur under impact test conditions). For a single brick element of the gas cavity, the governing law:

$$pV = \text{cost} \tag{12}$$

was thus taken into account.

**Figure 10.** Reference FE numerical model: calculation of the input parameters for the equivalent cavity infill in (**a**) un-deformed and (**b**) deformed configurations.

According to Equation (12), the volume of a single brick element can be expressed as the product of its un-deformed surface area *Agas* (that was ideally kept constant under deformations) and the un-deformed thickness *h*0 (with *h*0 = 16 mm the cavity thickness, in this study):

$$p\_0 V\_{\text{gas}} = p\_0 A\_{\text{gas}} h\_0 = (p\_0 + dp) A\_{\text{gas}} (h\_0 + dh) \tag{13}$$

As far as small thickness variations *dh* are considered for the cavity subjected to external compressive loads agreeing with Figure 10, the pressure-deformation trend can be derived from measurements of the corresponding brick mesh nodes. Under the simplified hypothesis of a linear elastic material in the small-deformation regime, the so-derived constitutive law allows then to extrapolate an equivalent elastic modulus for the gas infill, that is:

$$E\_{cq, \text{gas}} = \frac{\Delta p}{dh/h\_0} \tag{14}$$

with Δ*p* the pressure variation calculated from Equation (11).

In the present study, the equivalent modulus was estimated in *Eeq*,*gas* = 0.08 MPa. Besides the intrinsic limits of such an assumption, compared to more accurate calculation approaches, the equivalent modulus *Eeq*,*gas* can be considered a reliable value, as far as the expected cavity deformations due to impact are small (i.e., in the same order of magnitude of *h*0 itself). On the other hand, *Eeq*,*gas* should be separately estimated under different geometrical configurations (and especially for variations in the un-deformed thickness *h*0).

#### 5.2.3. Materials

Given that no visible damage was detected in the reference full-scale experiments (see Section 4.2), linear elastic constitutive laws were used for all the materials in use. The corresponding Young's modulus *E*, Poisson' ratio *ν* and density *ρ* are summarized in Table 4. It is worth noting in Table 4 that no failure stress was explicitly considered for annealed glass (*ft* = 45 MPa the characteristic nominal value under quasi-static tensile loading [40]).

**Table 4.** Input mechanical properties for the materials in use.

\* = equivalent properties based on the monolithic thickness assumption; \*\* = equivalent properties based on the linear elastic behavior assumption.

The post-processing analysis of FE data was thus focused on the evolution of maximum deflections for the tested specimens, but also on a necessary analysis of stress peaks in the module components. Major attention was spent for glass, so as to ensure the reliability of a linear elastic material characterization for FE purposes. Among other constructional materials and literature studies, the discussion in [2] and Figure 11 about annealed glass confirms the well-known sensitivity of its compressive and tensile strength to strain rate. This effect typically results in the so called Dynamic Increase Factor (DIF) that can be quantified—for impact and impulsive configurations—in up to 3–4 times the nominal strength value *ft* under quasi-static loads. Simple stress estimates can be thus carried out towards the DIF value of strength [3,48,49].

**Figure 11.** Reference Dynamic Increase Factor (DIF) for glass strength in (**a**) compression or (**b**) tension, as a function of the imposed strain rate. Figure reproduced from [2] with permission from Elsevier®, copyright license agreement n.<PHONE_NUMBER>990 (April 2021).

5.2.4. Boundaries and Connections

A special attention was dedicated to the mechanical characterization of the actual supports for the examined curtain wall specimens. At the early design stage, or when no experimental feedback is available, the curtain wall module in Figure 7 should be reasonably described in the form of four point restraints at the corners of the panel. Such a choice would result in reliable calculations for the procedural steps in Figure 6.

A double check and calibration was also carried out for the examined facade modules, based also on the post-process of the available experimental records. On this regard, four axial "discrete" springs were introduced in the Code Aster model, in the region of top and bottom corners, so as to include a certain support flexibility for the restraints of the specimen. The axial stiffness of these springs was calculated with the support of the experimental displacements, given that they proved a certain out-of-plane deformation in the bottom transom. Accordingly, mostly rigid restraints were considered for the top corners (*Ktop* = 107 kN/m), while for the bottom corners the input stiffness was set in *Kbot* = 1700 kN/m. Finally, the connection of the glass panel with the supporting frame members was numerically reproduced in the form of a series of equivalent springs with stiffness given by:

$$K\_{\rm sil} = \frac{E\_{\rm sil} A\_{\rm sil}}{t\_{\rm sil}} = 1000 \text{ kN/m} \tag{15}$$

where *tsil* = 10 mm represents the nominal thickness of the silicone joint in use, *Esil* the modulus [42], while *Asil* is inclusive of the mesh size/spring influence area.

#### 5.2.5. Damping

In the reference FE model, damping was taken into account with the Rayleigh method, as a function of the mass of the curtain wall module. At first, as also in line with Equation (10), the input value was set in *ξ* = 8%. The so-calibrated model was thus preliminary taken into account for the 2-DOF characterization in the procedural steps of Figure 6.

Later on, the experimental records were also considered for a post-processing validation of the above assumption (i.e., STEP 7c in Figure 6). Based on the logarithmic decrease of the experimentally measured displacements *u*(*t*) for the glass panel:

$$\xi\_{test} = \frac{1}{\sqrt{1 + \left(\frac{2\pi}{\delta}\right)^2}}\tag{16}$$

and

$$\delta = \frac{1}{n} \ln \frac{u(t)}{u(t + nT\_1)} \tag{17}$$

and being *T*1 the fundamental period of vibration of the curtain wall, the damping was experimentally estimated in a mean value of *ξtest* = 12% that is in line with the approximate prediction of Equation (10), but also includes local effects that could derive from the impactor itself or possible local damage (i.e., frame, joints). In this context, it is worth mentioning that the effects of the so-derived damping term *ξ* have minimum effects on the overall dynamic estimates for the curtain wall analysis. On the side of the 2-DOF impact force, in particular, the curtain wall damping proved to have negligible effects on the corresponding time history of the force. On the side of the curtain wall, similarly, the maximum peak of impact effects (i.e., displacements, stresses, etc.) is notoriously not affected by any damping contribution, given that it can be perceived only in the vibrations that follow the instant of the impact. Accordingly, the overall damping estimation from Equation (10) was still found to be reliable.

#### 5.2.6. Impact Load and Impact Area

For each experimental configuration in Table 2, the impact load was properly considered in the FE model, based on a number of time histories that were derived from the 2-DOF model in Figures 5 and 6. Given that the input time history modifies with the impact features (impactor, impact point, etc.), the corresponding trends were separately collected for the corresponding FE analyses. Figure 12 shows a typical example of variation of the required input time history, as far as the examined curtain wall is subjected to different impact conditions.

**Figure 12.** Example of input time histories for the curtain wall under SB, at different impact points (*Eimp* = 900 J).

In order to facilitate the calibration of the key parameters for the 2-DOF and the corresponding FE models, the reference impact area was kept constant for all the Code Aster simulations, with a square shape that was set in *A* = 200 × 200 mm2. The latter, see Figure 13a, was experimentally measured in the case of the TT impactor. Similar experimental acquisitions, see Figure 13b, proved that the TT surface *A* is mostly half the spheroconical bag. For FE purposes, however, the initial *A* value was kept fix and the input force was scaled accordingly.

**Figure 13.** Experimental measurement of the reference impact area *A* corresponding to (**a**) TT or (**b**) SB impactors (acquisition for *Eimp* = 900 J).

#### **6. Experimental Validation of FE Results**

#### *6.1. Impact Response*

For the validation of the FE modelling assumptions and the coupled design approach herein proposed (i.e., impactor and contact parameters of the 2-DOF model), the outof-plane displacements for the three LS*n* control points were assessed for all the 15 test repetitions. Given that the coupled design approach schematized in Figures 5 and 6 is based on a first vibration mode of the curtain wall object of study, the period was first numerically calculated in *T*1 = 0.1 s, see Figure 14a.

For the curtain wall under various impact conditions, the typical deformed shape was still found to mostly agree with the global vibration mode of the system, see Figure 14b. The corresponding distribution of stress peaks in glass was calculated as in Figure 14c (exposed panel). For the same modular unit under different impact points, additional local effects were observed in major variations about the localization of stress peaks, and some modifications in the global deformation of the visual section (Figure 14d,e).

In Figures 15 and 16, selected data are shown for two testing configurations only (SB or TT, with P#1 and *Eimp* = 900 J). It is of interest that in both the cases, there is a rather good correlation for all the comparative plots, thanks to the reliable input parameters of the coupled approach. For the setup in Figure 15, for example, the maximum response peaks for the visual section under SB impact were quantified in an out-of-plane acceleration of around ≈155 m/s2, with a velocity peak of ≈1.7 m/s, after 0.03–0.05 s of analysis. The corresponding strain peak was measured in ≈0.0008.

**Figure 14.** Typical deformed shape of the examined curtain wall (Code Aster): (**a**) fundamental modal shape (frequency analysis) and (**b**–**e**) example of impact responses (legend values in mm and MPa). (**a**) *T*1 = 0.1 s. (**b**) Displacement (SB, P#1, *E*imp = 900 J). (**c**) Stress (SB, P#1, *Eimp* = 900 J). (**d**) Stress (SB, P#4, *Eimp* = 450 J). (**e**) Stress (SB, P#5, *Eimp* = 900 J).

**Figure 15.** Numerical and experimental out-of-plane displacements for the curtain wall under SB (P#1, *Eimp* = 900 J). In evidence, the control points (**a**) LS1 and (**b**) LS2, LS3.

**Figure 16.** Numerical and experimental out-of-plane displacements for the curtain wall under TT (*Eimp* = 900 J, impact point #1). In evidence, the control points (**a**) LS1 and (**b**) LS2, LS3.

It is worth noting that a general good accuracy was found also for the other testing conditions summarized in Figure 17a, where the numerical peak of out-of-plane displacements at the center of glass (LS1) is proposed under a multitude impact configurations, and compared to the experimental measurements. Given that the drop height, impact point and impactor changes for all the setup configurations, Figure 17a is a further confirmation of the proposal accuracy.

In this regard, it is important to remind that the P#*n* control points in Figure 17a are implicitly associated to different stiffness parameters and dissipation capacities for the impacted modular unit and for the SB or TT impactor in use.

For few impact configurations only, the FE predictions in Figure 17a tend to overestimate the corresponding experimental measurement. This limitation was justified by the sequence of test repetitions that were carried out on a single modular unit. Especially for few TT scenarios, the laser measurement control point gave evidence of minor damage.

Of interest is also the trend of numerically calculated stress peaks in glass as a function of the deflection amplitude, as schematized in Figure 17b for the panel exposed to impact. For the TT configurations (with P#1 impact point but different impact energy values), as expected, the stress peaks in glass reported in Figure 17b linearly increase with *Eimp* and the measured deflection peak. When the SB impactor is used, with different *Eimp* values, a clear correlation still exists between the calculated deflection-stress peaks data, as far as the impact point is kept fix. Moreover, as far as the impact point moves from P#1 towards the edges and the frame of the modular unit, the local stiffness variation results in even higher stress peaks in glass. Regarding the inner glass panel of the IGU visual component, the stress peak for the examined scenarios was numerically estimated in the range from ≈0.3 to ≈0.6 times the values in Figure 17b, depending on the impact configuration.

**Figure 17.** Numerical response analysis (Code Aster): (**a**) displacement peaks (LS1) for the curtain wall under various impact conditions, as a function of the experimental measurements, and (**b**) stress-displacement peaks (glass panel exposed to impact).

Finally, Figure 18 further confirms the general close correlation of FE estimates with the available experimental data, but also Newman's assertion in [34] about the higher severity of the TT impactor, for a given impacted body.

**Figure 18.** Comparison of numerical and experimental peaks of displacement (LS1) for the curtain wall under various impact conditions, as a function of the impact energy.

#### *6.2. Input Parameters for the Impactor*

The key aspect of all the comparisons in Section 6.1 is the use, for the Code Aster numerical model, of input forces that are derived from the 2-DOF model of the system, thus implicitly accounting for the *R*R coefficient in Equation (6).

In the current study, the parameters for contact (and thus for the impactor features) where calibrated with the support of experimental measurements, as well as the Code Aster estimates. The detailed procedure is shown in Figure 19, as a sub-section of the general approach in Figure 6. The final goal was obtained by matching the deflection peak of each specimen, for all the 15 test repetitions (STEP 7a in Figure 19), and thus minimizing the % scatter Δu for them (STEP 8a), so as to capture a schematic representation of the mechanical behavior for both the TT and the SB impactors (STEP 8b). For the first calibration of impact parameters, major advantage was represented by the available experimental-numerical feedback.

**Figure 19.** Developed calibration procedure for the 2-DOF input parameters of the impact contact, depending on the impactor in use (linear elastic for TT, elasto-plastic with hardening for SB).

Based on the selected impactors, more in detail, a linear elastic constitutive behavior was taken into account for the tire impactor (thus described in the form of an elastic stiffness *KC*). No damping was accounted for the TT (*CC* = 0), as also in line with past literature documents [9,12]. On the other side, to account for the different behavior of the bag, the latter was schematized in the form of an elasto-plastic law with hardening, with input parameters represented by the initial stiffness *KC*, a limit force *FC*,*limit* and a residual stiffness *K*C,u (with *KC*,*u*= 0.9 *KC*). With the support of benchmark parametric calculations, the limit force *FC*,*limit* was introduced in the contact description for the 2-DOF, so as to schematically reproduce the physical impact of the SB pendulum with the curtain wall, and the following separation of the two bodies, due to rebound. Such a plastic force *FC*,*limit* was in fact intended to act as a cut-off for the input force that the SB impactor could transfer to the adjacent curtain wall, given that the bag is filled with movable glass spheres.

However, the most influencing parameter for the SB description was represented by the elastic stiffness *KC*. Accordingly, the iterative calculations were carried out by setting, in the order, *KC* (with *CC* = 0), thus *FC*,*limit* (with *CC* = 0, and *KC* fix) and then *CC* (with fix *KC* and *FC*,*limit* values). The latter, being susceptible to various parameters on the side of the impacting bodies but also on the impact energy, was calculated in a variable magnitude, as a function of *Eimp* and *KF*. In any case, its final value for the SB impactor was usually optimized in the range of ≈0.1–1 Ns/mm, with limited variations for the overall experimental program.

The overall calibration process was based on the minimization of the percentage scatter of the deflection peaks from the FE model and the corresponding experimental records (LS1), under the 15 available testing configurations. The same iterative approach was thus separately repeated for the specimens with TT or SB impactor, so as to obtain a univocal simplified characterization of the two types of impactors. In Table 5, the so-calibrated values are proposed.

**Table 5.** 2-DOF input parameters for the curtain wall under SBI (with *MP* = 50 kg and *ξtest* = 12% \*).

\* = average logarithmic decrement of experimental displacements (Equation (16)).

In Figure 20, comparative examples are proposed to show the effects of iterative calibrations on the obtained 2-DOF scaled time histories, to further quantify the effects of the various input properties for the SB impactor. As in Figure 20a, the stiffness *KC* was found responsible of major variations for the corresponding impact forces, while the cut-off force *FC*,*limit* generally resulted in less pronounced variations, compared to a perfectly elastic contact (Figure 20b). In terms of damping, the effects of *CC* are proposed in Figure 20c, while Figure 20d confirms the negligible effect of the curtain wall damping C*F*. The sensitivity of *CC* to the impactor velocity or the facade stiffness is finally shown in Figure 20e, where a regular trend can be noticed with *vP*. On the other side, no clear correlation was found for the stiffness *KF* (Figure 20f), thus suggesting further explorations.

**Figure 20.** Sensitivity analysis of the 2-DOF impact time histories to selected input parameters: (**a**) contact stiffness, (**b**) contact limit force, (**c**) contact damping and (**d**) facade damping, with (**e**,**f**) variation of contact damping with impactor velocity, facade stiffness and impact energy.

#### **7. Design Outcomes**

*7.1. Reduction Coefficient RR*

The reduction coefficient in Equations (6) and (7), as earlier discussed, is implicitly accounted for in the scaled time history of impact forces, but depends on various parameters (impactor and impacted body, thus impact rigidity, impact energy, etc.) and it is a key information for design. Given that a combination of the above parameters can separately affect the corresponding reduction coefficient, some parametric calculations are discussed in this paper, in order to emphasize its sensitivity. In Figure 21, the calculated reduction coefficients are compared for the examined curtain wall system under TT (MEPLA and proposed 2-DOF+Code Aster method) or SB (proposed method). As a reference, the impact conditions of *Eimp* = 100 J, 200 J, 300 J, 450 J or 900 J (P#1) were taken into account. In the case of the SB, values for *Eimp* = 450 J and 900 J only are reported in Figure 20.

**Figure 21.** Calculated reduction coefficient *R*, as a function of impactor type and impact energy *E*imp (P#1).

As shown, a first important outcome is that the MEPLA implicit coefficient *RMepla* is mostly constant under a variable imposed impact energy, and it disregards the impactor features (TT option).

The calculated reduction coefficients *RR* from the proposed approach, otherwise, clearly modify with the assigned impact conditions, and also diverge from MEPLA estimates. From the 2-DOF+Code Aster calculations, in particular, it is possible to notice that *R*R tends to decrease as far as *Eimp* increases for both the TT and SB impactors. This behavior is in line with expectations, given that dissipative phenomena in the impacted curtain wall qualitatively increase under higher input energies. Moreover, the current 2-DOF+Code Aster data in Figure 21 for the TT are close to the unit, compared to the SB. This means that the TT approaches the "ideal" rigid elastic condition of impact. Moreover, this outcome again confirms Newman's hypothesis about the greater severity of the impact tests carried out with the TT setup (Section 2 and [34]).