Essential Stressing State Features of Laterally Loaded Masonry Wall Panels Revealed from Experimental Displacements

: This study reveals the essential and general working features of laterally loaded masonry (LLM) wall panels from their experimental displacements by applying structural stressing state theory. Firstly, the generalized work of force is proposed to express the stressing state mode and its characteristic parameter. Then, the Mann-Kendall criterion is applied to detect the mutation point in the curve of the characteristic parameter with the load increase. Correspondingly, it is verified that the evolution of the stressing state mode also embodies the mutation feature. The stressing state mutation feature is inherent and common as the embodiment of the natural law from quantitative change to qualitative change of a system. The stressing state mutation feature reveals the starting point of structural failure process, which could update the existing definition of structural failure load. Further, the elastoplastic branch (EPB) point is revealed referring to the updated failure load, which might be directly taken as the design load with the rational margin of safety. In a sense, this paper presents a new way to address the classic issue of structural load-bearing capacity uncertainty and to update the existing design codes of LLM wall panels. Highlights


Introduction
Scholars in many countries, such as Britain, Italy, America and Australia, contributed the theoretical and experimental achievements of laterally loaded masonry walls. From 1950 to 1989, Davey and Thomas, Losberg and Johnsson, Baker and Sinha proposed the methods for predicting the lateral load-bearing capacity of masonry walls [1][2][3][4]. Cajdert introduced the elastic plate theory and method [5]. Baker proposed the empirical strip analysis method and the principle stress method [6]. And Candy developed the energy method [7]. Among these theories and methods, the yield line theory could estimate the load-bearing capacity of laterally loaded masonry wall panels closer to the tested results, so that it was introduced into the British design code [8]. But the yield line theory usually underestimated and sometimes overestimated the lateral load-bearing capacity of wall panels, since the theoretical basis of the yield line method was not suitable for the brittle property of masonry [7,[9][10][11]. In 1989, Fried summarized most research results of laterally loaded masonry walls around the world and evidenced that their load-bearing capacity could not be predicted accurately due to the great variation in masonry materials and structural configuration [12].
Then, from 1990s on, the finite element method became a powerful tool to analyze the structural working behavior. The various constitutive models for the finite element simulations of masonry structures were proposed by Baker [13], Drysdale [14], Lee [15], Lourenco [16] et al. These finite element methods greatly promoted the analysis of structural working mechanism and the rational structural design. Meanwhile, artificial intelligence techniques were introduced into the working behavior prediction of laterally loaded masonry walls by Mathew [17], Zhou [18], Zhang [19] and Gopinath [20]. Besides, new materials such as fiber-reinforced polymer (FRP) material were applied to improve the mechanical properties of masonry and the working performance of masonry column, beams and walls/slabs [21][22][23][24][25][26][27].
However, both finite element methods and artificial intelligence techniques together with the applications of new materials still cannot achieve a consistent and definite prediction of structural load-bearing capacity due to the variation and uncertainty of structural ultimate working states. In fact, this was the unavoidable result based on structural ultimate working state. Thus, structural load-bearing uncertainty was considered as a basic attribute in structural analysis and the empirical/statistical judgement was as a principle in structural design. This situation naturally derived the following problems: (a) Structural failure was a process which certainly had its starting point. But the researches in structural analysis focused on the ending point of structural failure (structural ultimate state) rather than the starting point of structural failure. As well-known, the structural normal working state required by the design codes could not stay at any point in the structural failure process. In terms of this requirement, it meant that the starting point of structural failure could be more important reference to structural design than the ending point of structural failure.
(b) A new theory was expected to tell us what working law existed in structural working process instead of methods. In other words, structural analysis anticipated a scientific discovery to know the starting point of structural failure process.
(c) The past structural analysis recorded a great quantity of displacement and strain data in the tests of various structures. The experimental data were mainly applied to verify the theoretical, simulative and empirical results. But, the experimental data surely contained some unknown knowledge of structural working behavior, even the new physical law or the new physical principle.
The problems above indicated that structural analysis basically stood on structural ultimate/peak working state to pursue the accurate prediction of structural load-bearing capacity. Since the foothold had the essence of variation and uncertainty, structural analysis and design fell in the bottleneck impossible to reach a determinate and accurate prediction of structural load-bearing capacity. To a great extent, the past researches were to minimize or to avoid the negative effect of the bottleneck issue. Although the research results promoted the great engineering achievements, the cost was the overuse of materials because the design had to take the conservative measures. However, structural analysis has been anticipating to break this bottleneck in fact.
Zhou developed his structural stressing state theory to try to break the bottleneck mentioned above [28][29][30][31]. The theory thought that the working process of a structure under a load case surely presented a stressing state mutation according to the natural law from quantitative change to qualitative change of a system. The qualitative mutation point in the structural stressing state evolution defined the starting point of structural failure process. Importantly, this starting point/load of structural failure was definite, unlike the ending point of structural failure (structural ultimate point) with variation and uncertainty. Hereto, as the specific embodiment of the natural law, the stressing state mutation feature could be called as structural failure law. Accordingly, the structural failure law would derive structural elastoplastic branch (EPB) point which might be directly taken as structural design point. Indeed, the EPB feature would be a (design) principle derived from the structural failure law. The EPB design principle might lead to an update of the existing designs from the empirical and statistical basis to the physical law basis. Meanwhile, structural stressing state theory believed that the structural failure law was inevitably reflected in the experimental data (strains, displacements) of various structures or members even specimens subjected to the fully load processes. But, the proper methods and criteria were needed to model the structural stressing state by the experimental/simulative data and to detect the stressing state mutation complying with the structural failure law. This study adopted structural stressing state theory and methods to find out the stressing state features of laterally loaded masonry (LLM) wall panels. The basic intention was to investigate whether the structural failure law was embodied in the experimental displacement data or not. The investigation proposed the concept of generalized work of force (GWF) to model the experimental displacement data. The GWF values were used to build the stressing state mode and the parameter characterizing the mode. The Mann-Kendal criterion was utilized to detect the mutation features in the evolution curve of the characteristic parameter. The corresponding mutation features were verified in the evolution of the stressing state mode. Also, the other characteristic point, called as the elastoplastic branch (EPB) point, was distinguished from the evolution curve of the characteristic parameter. And a comparison was made to verify the rationality that the EPB load was taken as the design load. In a sum, this study developed the application of structural stressing state theory and provided the natural law-based reference to improve or update the existing design code of LLM wall panels. The revealed structural working features are generally reflected in the working processes of various structures.

The Experiment of LLM wall panels
This study cited the experimental displacement data of 17 LLM wall panels tested by Chong [32]. There were several reasons to select the experimental data: (a) It wanted further to reveal that the general working law or the essential working behavior features also existed in the past experimental displacement data, like those revealed by the strain data [29][30]; (b) It wanted to enrich and develop the structural stressing state theory and the relative methodologies based on the experimental displacement data; (c) It wanted to indicate that the general working features of structures should become the basic reference to structural designs or the basic principle of structural designs. Fig.1 shows the configurations of the panels which were divided into three series: Series 1, SB01-SB08 and CB01-CB02: Eight single leaf panels and two cavity panels with two panel sizes and three types of restraint conditions. The panels were constructed using perforated class B facing bricks (the mean of strength is 38.2 N/mm 2 ) set in 1:1:6 designation (ill) mortar. Four types of openings were chosen to represent window and door. The opening sizes and dimensions were typical in engineering: the opening rate of 10% for Panels SB03 and SB07; 16.5% for SB02; 12.0% for SB08; 6% for SB09. SB05 had a bituminous d.p.c in the first bed joint. CB0l and CB02 were the cavity panels with a cavity width of 50mm.
Series 2, DC01-DC02: Two single leaf panels constructed using dense concrete blocks (the mean of strength is 12N/mm 2 ) set in 1: 1: 16 designation (iii) mortar. The configurations of two panels were similar to SB01 and SB02. The bituminous d.p.c set in panel DC02 was laid in the first bed joint and no mortar was used to bind the d.p.c to the masonry. Series 3, HW01-HW04 and W0l: Five panels constructed using perforated class A engineering bricks (the mean of strength is 139.9N/mm 2 ) set in 1: 1/2:4 1/2 designation (ii) mortar. This series was designed to study the effect of support stiffness on the lateral working behavior of the panels.

Structural stressing state theory and methods.
Concept and analytical process.
Structural stressing state was defined as the response representation of a structure at a load level and could be described through modeling the experimental or simulative data such as displacements, stresses, strains and strain energy values [28][29]. Structural stressing state theory conducts the research into the general laws of material strength and structural load-bearing behavior as well as their applications. And structural stressing state analysis is to reveal the essential mutation feature in the stressing state evolution of a structural undergoing a full loading process, according to the natural law from quantitative change to qualitative change of a system [33]. Structural stressing state is expressed by the characteristic pair, the stressing state mode and the parameter characterizing the mode. The stressing state mode (matrix or vector) can be composed of structural response data. The different parameters characterizing the stressing state mode can be proposed depending on the analytical intentions. The natural law from quantitative change to qualitative change indicates that the stressing state of a structure subjected to a loading process surely presents the mutation feature around a certain load. The structural stressing state mode will change from the previous form to the other form. Meanwhile, the mutation feature is reflected in the evolution of the stressing state characteristic parameter, which can be detected by the proposed criteria such as the Mann-Kendall criterion mentioned below. The stressing state mutation point is the starting point of structural failure process with the attribute of certainty and the corresponding load is defined as the failure load. According to the starting point, the Mann-Kendall criterion can detect the other characteristic point, called as the elastoplastic branch (EPB) point, and the corresponding load is called as the EPB load. The EPB load could be taken as the structural design load.
For the stressing state analysis of a structure, the procedure can be shown as follows: The 1 st step: Transform experimental data into state variables using the concept of generalized work of force; The 2 nd step: Build the stressing state mode and its characteristic parameter using the state variable; The 3 rd step: Plot the curves of characteristic parameter and stressing state mode vs. the load increase; The 4 th step: Detect the mutation features of characteristic pair applying the Mann-Kendall criterion or the other criteria; The 5 th step: Define the failure load, i.e., the starting point of the wall panel's failure process; The 6 th step: Define the EPB load of the wall panel and verify its rationality as the design load.

State variables.
In this study, the lateral displacements at the individual measured points on the panel are modeled as state variables to express structural stressing state mode and characteristic parameter. The state variable is proposed as: in which wij is the state variable of the ith zone at the jth load; dij is the displacement at the ith point; ij F is the jth load which is the resultant force of the distributed load on the ith zone and assumed to act at the ith point. Fig. 2 below shows the zones divided around individual measured points.

Stressing state mode and characteristic parameter
Taking Panels SB01 and HW03 as examples, as shown in Fig. 2, their experimental displacements at the load-acting points are transformed into the state variables using Eq. (1). Then, the stressing state modes can be built as where the subscript of w indicates the measured point; j is the serial number of load levels. Also, the stressing state modes/submodes can be built according to the investigated zones. Panels SB01 and HW03 can be divided 9 zones and 5 zones, as shown in Fig. 3 (4) where N is the total number of the measured points; M w is the maximum state variable value among all the loading levels. Also, the parameter characterizing the stressing state submode can be formed by summing the state variables in the stressing state submode. In addition, the relevance parameter , +1 j j  can be proposed to characterize the stressing state mode, that is,

Results
The mutation feature of characteristic parameter.
The normalized state variable sum (Wj,norm) at the jth load step ( ) is proposed to characterize the stressing state mode. Fig. 4 shows the Wj,norm-curves of Panels SB01 and HW03. The Mann-Kendall criterion detects two stressing state mutation points P and F in the ,norm − curves. According to structural stressing state theory, Point P is the stressing state change from the elastic one to the elastoplastic one and Point F from the elastoplastic one to the failure one. From Point F on, the panel starts the failure process until the collapse or the ultimate state. Similarly, the other 15 panels also present the similar stressing state mutation features. This implies that the stressing state mutation could be the general or essential working behavior feature in structural working process. a) b)  of Panel HW03 also presents the mutation features, as shown in Fig. 5b. In a sum, characteristic points F and P display the following significances: (a) Until now, it can be stated that the stressing state characteristic pair evidently embodies structural failure law，i.e., the natural law from quantitative change to qualitative change of a system. From the experimental observation, characteristic point F defines the branch point where the panel's stressing state will mutate qualitatively. This suggests that Point F could be the failure starting point of the working process of the panel, which has not been revealed in the conventional structural analysis. Importantly, the structural failure starting point exhibits the significant certainty unlike the structural failure ending point (structural ultimate state) with the attribute of uncertainty. (b) Characteristic point P reflects that the working behavior of the panel changes from the elastic state to the elastoplastic state. In other words, the panel starts its plastic deformation accumulation from Point P on. When the plastic deformation develops to a certain extent, the panel starts to enter its failure state from Point F on. (c) It should be emphasized again that the experimental data of structural models certainly include the structural stressing state mutation features, which can be revealed by the proper modeling methods and the judging criteria. Even the experimental data includes the great random variation of material property and structural configuration, but the stressing state characteristic pair derived by the experimental data certainly embodies the definite mutation features at the characteristic loads.
(2) The stressing state mode expressed by the state variable sum increments. The stressing state modes of Panels SB01 and HW03 can be presented by the increment forms of state variables, i.e.,

Discussion
The stressing state analysis of the LLM wall panels reveals their starting failure points whose corresponding loads are defined as the failure loads. Around the failure load, the stressing state characteristic pair is essentially different in the developing trends. Before the failure load, the stressing state evolution basically keeps an unchangeable tendency. Then, from the failure load on, the stressing state evolution changes its tendency different from the previous one. At present, structural analysis and design basically take the ultimate or peak loads of structures as the reference. But the inherent uncertainty of ultimate or peak loads makes it the impossible to achieve the accurate formulas, so that the statistical analysis and empirical judgements are commonly adopted to determine structural design loads. This situation surely results in the irrational use of materials. Both failure load and EPB load could bring a considerable improvement on this situation, as the embodiment of structural failure law.
As well known, the structural failure load needs a reduction to be the design load based on empirical and statistical judgment. For the LLM wall panels, the yield line theory (YLT)was used in the existing design code, but it sometimes underestimated or overestimated the structural load-bearing capacity. Fig. 9a indicates that the YLT method underestimates the bearing loads of Panels S1~S7 ， C1, C2 and D1, and overestimates those of Panels H1~H4. In total, the existing structural analysis does not derive a method which could predict the load-bearing capacity more accurately than the YLT method. Here, the revealed EPB loads not only could verify the YLT inaccuracy, but also would be directly applied as the design loads, as illustrated by Fig. 9a. The EPB design loads have two margins of safety, one from the EPB load to the failure load and the other from the failure load to the ultimate load. In addition, a simply estimation of cost performance (CP) can be made by the ratio between the YLT design load or the EBP design load and the volume of the panel, as shown in Fig. 9b. The average cost performance ratios of the YLT and EPB design loads are 22.18% and 36.54% for masonry materials, respectively. The YLT design loads waste 13.36% of the load bearing capacity of the panels when compared with the EPB design loads. a) b) Figure 9. The characteristic loads and the cost performance ratios of the LLM wall panels: (a) the characteristic loads and (b) the bar graph of cost performance.
This study explores the further application of experimental displacements. The experiments of various structures recorded a great quantity of displacement data, but the data were just limitedly applied to verify the analytical and simulative results. Obviously, this needs the new analytical theory and methods to mine out the values of experimental data. Structural stressing state theory and methods lay the theoretical foundation and provide the tools to find out the working behavior features of structures from their experimental data.
In total, the achieved results in this study implies that the foothold to analyze the responses of LLM wall panels should be updated to structural failure starting point and structural EPB point. In other words, the existing design code based on structural ultimate/peak state should be updated to that based on structural failure law.

Conclusion
This study originally models and analyzes the experimental displacement data of LLM wall panels applying structural stressing state theory. The achieved results can draw the following conclusions: The proposed concept of generalized work of force can be applied to transform experimental displacements into state variables to express the stressing state mode and the corresponding characteristic parameter of the LLM wall panel. The Mann-Kendall criterion can distinguish the general and essential mutation feature in the evolution of the characteristic parameter-load curve. Accordingly, the stressing state mode also embodies the mutation feature around the characteristic load detected by the Mann-Kendall criterion. The stressing state mutation complies with the natural law from quantitative change to qualitative change of a system, which defines the starting point of the LLM wall panel's failure process. Thus, the failure load of the LLM wall panel should be defined at the starting point of its failure process. Also, the Mann-Kendall criterion can detect the EPB point of the LLM panel, which could be directly taken as the design point. Both the starting point and the EPB point could update the foothold of the existing design codes and promote the improvement of structural design.
Besides, this study further opens up the value of experimental displacement data. It shows that the experimental data still include the structural working behavior features which have not been revealed by the conventional theories and methods.