**Architectural Structure**

Luís Filipe Almeida Bernardo Edited by

Printed Edition of the Special Issue Published in *Applied Sciences*

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**Architectural Structure**

## **Architectural Structure**

Editor

**Lu´ıs Filipe Almeida Bernardo**

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*Editor* Lu´ıs Filipe Almeida Bernardo University of Beira Interior Portugal

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## **Contents**

#### **Vanessa Costalonga Martins, Sacha Cutajar, Christo van der Hoven, Piotr Baszy ´nski and Hanaa Dahy**

FlexFlax Stool: Validation of Moldless Fabrication of Complex Spatial Forms of Natural Fiber-Reinforced Polymer (NFRP) Structures through an Integrative Approach of Tailored Fiber Placement and Coreless Filament Winding Techniques Reprinted from: *Appl. Sci.* **2020**, *10*, 3278, doi:10.3390/app10093278 ................. **181**

#### **Romana Kiuntsli, Andriy Stepanyuk, Iryna Besaha and Justyna Sobczak-Piastka**

## **About the Editor**

**Luís Filipe Almeida Bernardo** (Prof.) received his Ph.D. in civil engineering (with expertise in mechanics of structures and materials) in 2004 at the University of Coimbra, Portugal. Currently, he is an Assistant Professor with Aggregation and a research member at C-MADE (Centre of Materials and Building Technologies) in the Department of Civil Engineering and Architecture at the University of Beira Interior in Covilh˜a, Portugal. His research interests include: structural analysis and design, numerical modelling and optimization of concrete structures, new structural materials, and building systems. He has been a researcher in several national research projects, and he collaborates with researchers from Brazil. He has authored/coauthored more than fifty articles in international peer-reviewed journals. He has been working as a reviewer for several international scientific journals. He has supervised several MSc and Ph.D. theses in his research field. He has also developed activities as consultant and civil engineer for the structural design of buildings and special structures in Portugal.

### *Editorial* **Special Issue on "Architectural Structure"**

#### **Luís Filipe Almeida Bernardo**

Department of Civil Engineering and Architecture, Centre of Materials and Building Technologies (C-MADE), University of Beira Interior, 6201-001 Covilhã, Portugal<EMAIL_ADDRESS>
Received: 23 July 2020; Accepted: 29 July 2020; Published: 31 July 2020

This Special Issue on "Architectural Structure" aims to gather general advances in human-made constructions which simultaneously are driven by aesthetic and structural engineering considerations. Such advances include the analysis of architectural typologies, the study of the mechanical performance of structural materials, structural systems and components, the proposal of techniques to evaluate the mechanical performance in existing structures and also new construction techniques. The aim of this Special Issue is also to inspire researchers and practitioners to share their knowledge and findings in these fields, and also to help explore new trends for the future. This Special Issue brings together twelve contributions covering the previously referred topics.

The global performance of an architectural structure strongly depends on the mechanical performance of its components and also on the used structural materials. Accordingly, the majority of the published articles in this Special Issue focus on the experimental and/or numerical behavior of structures, structural components and structural materials, including innovative ones. In [1], an innovative composite shear wall built with double steel-concrete, able to substitute classic reinforced concrete walls, was studied and a design method was proposed. A refined model, aiming to contribute to the optimum and economical design of thin-walled steel beams, nowadays widely used in architectural steel structures, was proposed in [2]. The incorporation of fly ash in both cementitious and alkali-activated concretes presents environmental advantages and allows to obtain concrete members with glossy and black surfaces, which might be aesthetically appealing for architectural structures with exposed concrete. In [3], the results of a study on the mechanical performance of reinforced beams built with mortar incorporating fly ash was presented, pointing out some important aspects to be further investigated in order to allow for the structural application of such a material. A contribution to the better knowledge of the mechanical performance of a geopolymer obtained by alkali-activation of a new binder was presented in [4], in order to, in the near future, enable the use of this environmental material in innovative architectural structures with finishes of different colors and textures. Although high strength concrete is nowadays used in practice, some particular aspects of the structural behavior of members built with this material still need to be checked for optimum design. This is the case of structural concrete members under primary torsion. In [5] a study on the mechanical performance of prestressed high strength concrete hollow beams under torsion was presented, the results of which can help in the design of box bridges. Half precast solutions have been widely used for structural applications. In [6] the mechanical performance of a recent structural system of half precast concrete slabs with inverted multi-ribs was investigated and guides for a design method were proposed. Block masonry has been used since ancient times as the main component in constructions and is still used nowadays throughout the world. Recently, the manufacture of such components has evolved based on environmental requirements. The mechanical performance and environmental benefits of recycled aggregate concrete hollow blocks were studied and guides for design were also proposed in [7]. Tall buildings are some of the most emblematic architectural structures. One of the main challenges for the designer is to control the lateral displacements. For this, in [8] the efficiency of an innovative outrigger system made of reinforced concrete wall with multiple openings was modeled and studied, and some guide rules for design were proposed.

Many existing structures must be evaluated for maintenance and rehabilitation concerns. In some projects, the structural performance of their structural materials and members must be checked to ensure the structural safety. In the past few years, self-compacting concrete has been widely used due to, for instance, its ease of placement in geometrically complicated formworks and also due to the obtained smooth and well-finished surfaces after concreting. These aspects are important to fulfill many architectural requirements. In [9], the applicability of non-destructive tests to estimate the compressive strength of self-compacting concrete was studied and useful correlations were presented for practice. Additionally, the performance of structural members in existing structures may have to be evaluated in light of current codes of practice. In this sense, in [10] a study was presented to evaluate the real cyclic load bearing of a traditional ceramic-reinforced slab incorporated in an existing building. The testing methodology and the results of the analysis were presented, which could be useful for practitioners.

Construction systems have evolved in the past few years, namely for geometrically complex structures. Two emergent moldless fabrication techniques for complex spatial forms of natural fiber-reinforced polymer structures were presented and validated in [11]. Such techniques could be, in the near future, applied to build larger building elements for more sustainable building systems.

Finally, the analysis of existing architectural typologies may help a future generation of designers to think about new typologies for architectural structures. In [12], the content of the spatial Rudolf Steiner's architecture, using reinforced concrete in architectural structures with complex geometries, and which is considered a unique case in the history of architectural heritage, is determined and discussed.

To end this editorial note, I would like to express my sincere gratitude to all the contributors of the articles submitted to this Special Issue, as well as to the editor-in-chief of *Applied Sciences*, Prof. Dr. Takayoshi Kobayashi, Mr. Melon Zhang as Managing Editor, and the editorial staff for their efforts and support.

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

**Conflicts of Interest:** The author declares no conflicts of interest.

#### **References**

© 2020 by the author. 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 (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Axial and Bending Bearing Capacity of Double-Steel-Concrete Composite Shear Walls**

#### **Peiyao Zhang 1, Quanquan Guo 1,\*, Fei Ke 2, Weiyi Zhao 3 and Yinghua Ye 1**

Received: 31 May 2020; Accepted: 15 July 2020; Published: 17 July 2020

**Abstract:** Double steel-concrete composite shear wall is a novel composite structure. Due to its good mechanical properties, it has been considered as a substitute for reinforced concrete walls in nuclear facilities, marine environmental structures, and high-rise buildings. However, the design method of the double-steel concrete composite shear wall is lacking. The purpose of this paper is to propose the bending capacity formula under large and small eccentric loads. By summarizing the test results of 49 steel-concrete composite double shear walls under cyclic loading from different studies, it was found that the bending failure of double-steel-concrete composite shear walls was featured by the concrete crushing at the bottom. A finite element model was established and it could simulate the axial and bending performance of double steel-concrete composite shear walls reasonably well. According to the experimental results and FE analysis, the primary assumptions for calculating the axial and bending bearing capacity of the double steel-concrete composite shear walls were proposed. Based on these assumptions, the bearing capacity formulas were derived according to the equilibrium theory of the cross section. The calculation results obtained by the bearing capacity formulas were in good agreement with the test results.

**Keywords:** double-steel-concrete composite shear walls; axial and bending capacity; failure characteristic

#### **1. Introduction**

Double-steel-concrete composite shear walls (SC walls) mainly consist of two surface steel plates connected by tie bars and filled with concrete. To prevent the buckling of the surface plates and ensure collaborative work between the concrete and steel plates; studs, stiffeners, and other connections are used, except for tie bars. SC walls are developed to reduce wall thickness, to enhance constructability, and to make rapid construction possible by eliminating the use of formwork and reinforcing bars. The filled concrete prevents the early buckling of steel plates, while the steel plates provide the confinement on the concrete, so the SC walls are characterized to have high strength and sufficient ductility for compressive and shear loading. Therefore, the composite effect of steel plates and concrete gives the SC walls significant advantages over the traditional shear walls, such as reinforced concrete shear walls, single steel plate, concrete composite shear walls, profile steel-reinforced concrete composite shear walls, etc.

Previously, the structural characteristics of SC walls have already been studied experimentally and numerically. Depending on the structure, SC walls can be divided into two types. The first type is the SC wall with bending stiffener attached at the edge region, which was subjected to shear failure. The experiments of the first type of SC walls were mainly finished in Japan (e.g., Usami et al. [1], Takeuchi et al. [2], Niwa et al. [3], Ozaki et al. [4], Kitano et al. [5], Funakoshi et al. [6]). The other type is the SC wall without bending stiffener was subjected to bending failure, and the experiments were mainly finished in China and Korea (e.g., Tae-Sung Eom et al. [7], Wu et al. [8], Nie et al. [9,10], Ji et al. [11], Tian et al. [12]). All the studies and experiments indicated that the SC walls had high strength, good ductility, and high energy-dissipation. In recent years, the construction of CAP1400 nuclear power plant in China has promoted the research progress of SC walls in nuclear engineering, such as Guo et al. [13], Yang et al. [14], Liu et al. [15], Li et al. [16], and Li et al. [17]. Lin et al. [18] tested 12 buckling-restrained shear panel dampers which were equipped with demountable steel-concrete composite restrainers. The influence of key design parameters on seismic behavior was studied and design equations for calculating elastic stiffness and ultimate strength were proposed. Zhao et al. [19] experimentally studied the cyclic behavior of two half-scale concrete stiffened steel plate shear wall specimens including the traditional and the innovation. Both specimens showed highly ductile behavior and stable cyclic yielding performance, and some suggestions for the design were also given based on the experiment results. Behnoosh Rassouli et al. [20] investigated experimentally and numerically the behavior of concrete stiffened steel plate shear wall using precast light weight concrete panels, and three specimens were tested under quasi-static cyclic load. The test results show that the CSPSW can reduce the seismic mass and improve the behavior of steel structures.

To date, some work has been conducted on the calculation method for the axial and bending bearing capacity of SC walls. For example, Varma et al. [21] developed a mechanics model and a detailed nonlinear finite element model, and the two modeling methods were applied to develop a conservative interaction surface in principal forces space that can be used to design or evaluate the adequacy of SC walls subjected to any combinations of in-plane forces. Tae-Sung Eom et al. [7] tested slender isolated walls and coupled walls subjected to cyclic loading. Based on their tested results, they developed the calculation method for the load-carrying capacity. Bo Wu et al. [8] calculated the lateral loading capacity of the SC walls based on the concept of the combined strength of new and old concrete. Ji et al. [11] proposed simplified formulas used to evaluate the flexure strength of the SC walls. However, all the calculation methods were derived using plastic stress distributions at the cross sections, which was the same as [22], and only suitable for calculating the compression members with large eccentricity. When applying to a small eccentric load, the methods would cause larger calculation errors, because the steel plates in tension did not reach the yield strength. Xiaowei Ma et al. [23] developed a model for the elastic and plastic analysis of the axial force-moment capacity of SC walls and analyzed the M-ϕ curve and the axial force and moment curve. They derived the formula for axial force-moment based on the key factors gained by numerical calculation and parameter analysis. However, the factors were complex and did not take into account the effects of structural measures, so the formula cannot be used widely in practical engineering. Jianguo Nie et al. [10] used the strip method to deduce the calculation formula of normal section bearing capacity, which applied to both small and large eccentric load. However, the strip method was complex and the ultimate compression strain of the concrete was taken as 0.0033, which was inconsistent with the confinement effect of steel plates on the concrete. Papanikolaou et al [24] proposed a confinement-sensitive plasticity constitutive model for concrete in triaxial compression. It incorporates a three-parameter loading surface, uncoupled hardening and softening functions, following the accumulation of plastic volumetric strain and a nonlinear Lode-angle-dependent plastic potential function. Skalomenos et al. [25] created an accurate nonlinear finite element model with the ATENA software, and the influences of different parameters were studied.

Due to the excellent strength and ductility, SC Walls have already been applied to a lot of structures requiring high resistance against severe loads since the 1980s, such as nuclear facilities, or marine environment structures. In China, SC walls are mainly used in high-rise buildings, including Yancheng TV Tower [26] and Guangzhou TV Tower. Due to the lack of detailed design codes, the above structures were mainly carried out based on the overall concept of the structure regarding other composite structures, so a reasonable and correct design method for SC Walls is urgently needed. In this paper, based on the experimental results of 49 SC walls and the results of numerical analysis, the bending failure characteristic of SC walls was studied and summarized, and a primary assumption of SC walls was put forward. Then according to the ultimate equilibrium theory of the cross section, the axial and bending capacity formula was deduced and compared with the existing experimental results.

#### **2. Experimental Results Analysis**

SC walls were conceived initially in Japan in the 1980s. Extensive research has been done in many countries to study the behavior and failure modes of SC walls. For example, Eom et al. [7] tested three isolated walls and two coupled walls subjected to cyclic lateral loading. The specimen named DSCW1C in the test failed due to outward buckling of the steel plates and subsequent fracture of the vertical weld joint, followed by the filled concrete crushing and tie bars fracture. Nie et al. [10] tested nine SC wall specimens with different shear span ratios. All the specimens failed due to the buckling of the steel plates and the filled concrete crushing. Ji et al. [11] tested five slender rectangular wall specimens subjected to axial forces and lateral cyclic loading. The specimens failed in a flexural mode, characterized by local buckling of the steel tubes and plates, fracture of the steel tube, and concrete crushing at the wall base. Besides, the tested indicated that the average strain distribution of the surface steel plates agreed with the plane assumption within the range of 300mm above the wall base. Tian [12] tested nine SC wall specimens with the rectangular cross section. The experiment showed all the specimens failed due to the steel plate buckling and concrete crushing.

The following conclusions are obtained by analyzing and summarizing the experimental results of the existing researches.

The ratio of the stud spacing (B) to the steel plate thickness (t) has a significant effect on the steel plate buckling, as shown in Table 1. When the B/t ratio is within a certain range, local buckling occurs when the specimen near reaches the ultimate bearing capacity, and the buckling is accompanied by the pull-out of the concrete, indicating that the slippage between the surface steel plate and the filled concrete is small. When calculating the axial and bending bearing capacity of the SC wall, it is assumed that the steel plate and the concrete can be well unified without relative sliding, and when the SC wall reaches the ultimate bearing capacity, the steel plate does not buckle.

**Table 1.** The influence of *B*/*t* on local buckling of steel plates.

1 *Pcr* is the local buckling load of the column steel plate; 2 *Pu* is the ultimate load; 3 buckling1 means the concrete at the embedded column is not crushed when the steel plate buckling occurs; 4 buckling2 means the steel plate buckling occurs accompanied with the concrete crushed and pushed out; 5 *B*/*t* is calculated by 600 *fy* referring to [22].

Because the concrete in the SC shear wall is wrapped inside the steel plate, variables such as the ultimate compressive strain of concrete and the average strain of concrete in a certain height range are difficult to measure. To verify whether the strain conforms to the plane assumption in more cases and obtain the strain of the concrete inside the wall under ultimate load, a finite element simulation of the specimen in [12] was conducted. Calculations under various design axial compression ratios were carried out to supplement the test results.

#### **3. Numerical Analysis**

#### *3.1. FE Model*

In the structure of SC walls, the concrete is covered by the surface steel plates, so it is difficult to measure the ultimate compressive strain of the concrete and the average strain. In this paper, the finite element analysis was done using ABAQUS (Dassault Systèmes, Providence, Rhode Island, USA, 2010) to obtain the strain distribution of the concrete, to verify the plane assumption, and to carry out the parametric analysis. There are few experiments on small eccentric compression failure, so the SC walls having small eccentric compression failure were simulated by ABAQUS.

Nonlinear finite element analysis was performed on all samples in [12] using ABAQUS. The reinforcement diagram of a specimen is shown as an example in Figure 1. The upper and lower parts of the SC wall were embedded in the loading beam and the base beam, respectively, for applying the lateral load and anchoring with the foundation. The parameters of specimens are shown in Table 2. Studs were set on the inner surface of the steel plate and tie bars were arranged between the steel plates. In the test, the specimen was fixed on the ground by two long bolts. The concrete grade was C35. The average cube compressive strength was 42.9 MPa, and the Young's modulus was 33,000 MPa. The average yield strength of the steel plate was 330 MPa, and the Young's modulus was 206 GPa. The quasi-static method was applied in the loading test. First, the vertical load was applied to the specimen until the design axial compression ratio was achieved. Then a lateral load was applied on the top of the wall. Before yielding, the lateral load increased step by step according to 1/10 of the ultimate load, which was estimated, and one cyclic loading was performed at each step.

The design axial compression ratio (μ) refers to the ratio of the representative value of the load to the design value of the material strength, which can be expressed as follow:

$$
\mu = \frac{N}{f\_{\varepsilon}A\_{\varepsilon} + f\_{\sharp}A\_{s}} \tag{1}
$$

where *N* is the design value of the axial pressure of the specimen under the action of the representative value of the gravity load. *fc* is the design value of concrete compressive strength. *fy* is the design value of the yield strength of the steel plate. *Ac* and *As* are the concrete cross section area and the steel plate cross section area, respectively.

The ABAQUS model of specimens taken from [12] is shown in Figure 2. The concrete was modeled with the C3D8R solid element; the steel plate was modeled with the S4R shell element; the tie bar was modeled with the T3D2 truss element, which is similar to [27]. The stud was modeled with the SPRING2 element.

In this paper, a three-way zero-length spring was added between the concrete slab node and the steel plate node at the actual position of the stud. The spring stiffness includes shear stiffness and axial stiffness. Aiming at the shear-slip curve, Ollgaard et al. [28] proposed a calculation model, which has been widely recognized. The shear-slip curve is expressed as:

$$V = V\_{\mathcal{U}} (1 - e^{-ns})^m \tag{2}$$

where *s* is the slippage. *m* and *n* are parameters. Different researchers used Equation (2) to fit the *m* and *n* according to their test results. The value of *m* is generally between 0.4 and 1.5, and the value of *n* is generally between 0.5 and 2.0. Gattesco and Giuriani [29] conducted two horizontal push-out shear tests of 19 mm diameter stud. Using the Equation (2) proposed by Ollgaard et al. [28] to fit the experimental value of the shear-slip curve in [29], when *m* = 0.425, *n* = 0.5, the two coincide. Another parameter that needs to be determined is the ultimate shear strength *Vu*. The test results of [29] are in good agreement with the calculation results of Eurocode-4 [30]. The equation proposed by Eurocode-4 was applied in this paper to calculate *Vu*, which is expressed as:

$$V\_u = \min\{0.8f\_uA\_\prime \: 0.29\alpha d^2 \sqrt{f\_{ck}E\_{cm}}\}\tag{3}$$

where *fu* is the ultimate tensile strength of the stud, *A* is the section area of the stud, *d* is the diameter of the stud, *fck* is the compression strength of concrete cylinder, *Ecm* is the young's modulus of concrete, α = 0.2(*h*/*d* + 1), and *h* is the height of the stud. The relationship between shear force and displacement can be obtained by Equations (1) and (2) and can be used as the nonlinear spring stiffness. According to Luis Pallarés et al. 's review [31] of the tensile failure of stud, there are generally three types of failure, namely, stud fracture, concrete cone pull-out failure, and concrete column pull-out failure. According to American Concrete Code 318-08 (ACI 318-08), when the diameter of the stud cap is greater than 1.71 times the diameter of the stud rod, there is a 95% probability that the concrete column will not be pulled out. When the length of the stud is greater than 7.5 times the diameter of the stud, the concrete cone will not be pulled out [31]. Therefore, the brittle failure of the concrete can be completely avoided by construction measures. The damage caused by the stud itself breaking is ductile. The bolt is equivalent to a tension member with one end fixed and one end freely stretched, so the stiffness of the stud can be used to calculate the stiffness of the axial spring, which is expressed as follows:

$$K = \frac{EA}{h} \tag{4}$$

where *E* is the young's modulus of the stud, *A* is the section area of the stud and *h* is the height of the stud.

The steel plates were embedded in the loading beam and the base beam. The studs and tie bars were embedded in the concrete. The contact feature was used between the elements of the concrete wall and the steel plate that was not inserted in the loading beam and the base beam. The contact properties were frictional contact in the tangential direction and hard contact in the normal direction, respectively. The boundary conditions are shown in Figure 2b. The FE model was fixed on the ground by two springs with the spring stiffness of EA/L, where A and L were the bolt section area and length, respectively. The contact between the base beam and the foundation was simulated by an elastic foundation model. The reaction force coefficient of the concrete foundation is generally 7484.6–14,715 kN/mm3 [32]. In the simulation, the reaction force coefficient was 10,000 kN/mm3. At both ends of the model foundation beam, the horizontal displacement of the wall in and out of the wall was constrained, and the vertical displacement was not constrained. The concrete damaged plasticity model was applied to simulate

the behavior of concrete. The energy equivalence principle of Sidiroff was applied to calculate the damage factor. The ideal elastoplastic stress-strain curve was applied to model the behavior of steel plates and tie bars.

**Figure 1.** Diagrams of SC walls (SCW1-2) in [33].

**Figure 2.** FE model in ABAQUS. (**a**) FE models; (**b**) Boundary conditions of FE models.

#### *3.2. FE Calculation Results*

The calculated load-displacement skeleton curves were compared with the experimental results. Comparisons between the experimental result and the calculated result are shown in Figure 3, which presents a good agreement. Besides, the average ratio of the calculated value of ultimate bearing capacity to the experimental value is 1.09; the average ratio of the peak displacement between the calculated value and experimental value is 0.9; the average ratio of the steel yield load and displacement are 0.93 and 1.04, respectively. Therefore, it could be concluded that this modeling method could effectively simulate the mechanical property of SC walls under cyclic loads.

**Figure 3.** Comparisons between the experimental result and the calculated result.

The specimens in [12] were under the same axial compression ratio. To verify whether the strain meets the plane assumption under different axial compression ratios, three representative specimens with different steel contents and shear span ratios are selected, which are specimen SCW1-1 (steel plate thickness is 3 mm, shear span ratio is 1.0), SCW1-2 (steel plate thickness is 3 mm, shear span ratio is 1.5) and specimen SCW1-4 (steel plate thickness is 4 mm, shear span ratio is 1.0). Nine FE models under different axial compression ratios were simulated using ABAQUS. The results are as follows:

#### 1. Average strain distribution of the cross section

As shown in Figure 4, when the load reaches the ultimate value, under different axial compression ratios, steel ratios and shear span ratios, the average strain at the bottom cross section of the wall is consistent with the plane assumption.

**Figure 4.** Vertical strain distribution at the wall base for specimens at ultimate load under different axial load ratios (μ is the axial compression ratio).

#### 2. Ultimate compression strain of the concrete

There are few studies on the ultimate compression strain of the confined concrete. In this paper, the ultimate compression strain was obtained by ABAQUS analysis. Figure 5 shows the concrete ultimate compression strain of the specimens with different axial compression ratios. In Figure 5, the theoretical value was calculated referring to [34], in which the ultimate compression strain of concrete was defined as the post-peak strain corresponding to the stress valued 0.5*fc* in the declining part of the stress-strain curve. For the constitutive relationship of concrete under unidirectional load, refer to the constitutive relationship of rectangular steel tube concrete proposed by Han [35]. As shown in Figure 5, the axial compression ratio, steel ratio, and shear-span ratio influence the concrete ultimate compressive strain; the theoretical value approximated to the lower limit of the results calculated by ABAQUS. Therefore, when calculating the axial and bending bearing capacity for SC walls, the confined concrete compression strain could be calculated conservatively according to the above method.

**Figure 5.** Maximum concrete compression strain versus axial compression ratio of specimens at ultimate load.

#### **4. Derivation of the Axial and Bending Bearing Capacity**

#### *4.1. Basic Assumption*

Based on the experimental results and finite element analysis, the basic assumptions for calculating the axial and bending bearing capacity of SC walls are proposed.

$$f\_{\ell\mathcal{E}} = \left[1 + \left(-0.0135\xi\_1^2 + 0.1\xi\right) \left(\frac{24}{f\_c}\right)^{0.45}\right] \tag{5}$$

where *fcc* is the compressive strength of confined concrete; *fc* is the compressive strength of the concrete; and ξ1 is the confinement effect coefficient.

The confinement effects of the column plate and the steel plate on concrete are different in SC walls. When the connectors between the surface steel plates are enough, ξ1 can be calculated conservatively using Equation (6).

$$
\xi\_1 = \frac{A\_{c0} f\_y}{A\_c f\_c} \tag{6}
$$

where *As*0 is the total cross-section area of the steel plate in the SC wall (including the vertical partition plates); *fy* is the yield strength of the steel plate; and *Ac* is the total cross-section area of the concrete in the SC wall.

#### *4.2. Section Simplification*

Figure 6a shows the configuration of the SC wall. The components used to resist the bending moment include the web plate, column plate, partition plate, and stiffening steel tube (or profile steel). To simplify the calculation, the cross section is simplified, as shown in Figure 6b. The column plate and stiffening steel are simplified to the outermost side of the wall according to the principle of providing the equivalent moment of the cross section centroid axis when the steel yielded. The yield strength of the fringe plate equals the yield strength of the web plate. The web steel plates and partition plates are simplified to web plate according to the principle of equal area.

**Figure 6.** Simplification of the cross section for SC walls. (**a**) The general section structure; (**b**) Cross section simplification.

#### *4.3. Failure Modes*

Based on the above assumptions and simplification, the ultimate equilibrium method is applied to derive the axial and bending bearing capacity formula of the SC wall. The balanced failure of the SC wall is defined so that the steel plates in tension reach the yield strength simultaneously when the concrete in the compressive zone reaches the ultimate compressive strain. The relative height of the concrete compressive zone is ξb, which can be calculated by Equation (7) referring to [36]. When ξ ≤ ξ*b*, the steel plates in tension could reach the yield strength when the wall breaks, and the failure mode of the SC wall is called large eccentric compression failure. When ξ > ξ*b*, the steel plates in tension cannot reach the yield strength when the wall breaks; the failure mode of the SC wall is called small eccentric compression failure (as shown in Figure 7).

$$
\xi\_b = \frac{\beta\_1}{1 + \frac{f\_y}{\ell\_{\rm tr}\ell\_s}} \tag{7}
$$

**Figure 7.** The strain distribution of the SC walls.

#### *4.4. Large Eccentric Compression Failure (*ξ ≤ ξ*b)*

Figure 8 shows the distribution of the cross-section stress and strain for the large eccentric SC wall. The equilibrium equation of the cross section is described as Equations (8) and (9).

$$\sum X = 0,\ N = A\_{\rm s}^{\prime} f\_{\mathcal{Y}} + \mathcal{C}\_{\rm c} + N\_{\rm sw} - A\_{\rm s} f\_{\mathcal{Y}} \tag{8}$$

$$
\sum M = 0,\\
\text{Ne} = A\_s' f\_y h\_w + M\_c + M\_{sw} \tag{9}
$$

where, *N* is the design value of the axial force; *e* is the distance from the axial force to the center of the steel plate in tension; *As* is the area of the steel plate in compression; *Nsw* is the resultant force of the web plate (stress in compressive direction is defined positive and stress in tensile direction is negative); *Msw* is the moment of the web plate normal stress about the center of the steel plate in tension (the clockwise direction is defined as positive, the counterclockwise direction is defined as negative); *Cc* is the resultant of the concrete compressive stress (stress in compressive direction is defined positive and stress in tensile direction is negative); and *Mc* is the moment of the concrete normal stress about the center of the steel plate in tension (the clockwise direction is defined as positive, the counterclockwise direction is defined as negative).

$$\text{If } \beta\_1 \mathbf{x}\_n > h\_{f'}' $$

$$\mathbf{C}\_{\mathbf{c}} = \alpha\_1 f\_{\mathbf{c}t} \beta\_1 \mathbf{x}\_{\mathbf{u}} b\_{\mathbf{w}} + \alpha\_1 f\_{\mathbf{c}t} \left( b\_f' - b\_{\mathbf{w}} \right) b\_f' \tag{10}$$

$$M\_{\rm c} = \alpha\_1 f\_{\rm c\bar{t}} \beta\_1 \mathbf{x}\_n b\_{\rm w} \Big( h\_w - \frac{\beta\_1 \mathbf{x}\_n}{2} \Big) + \alpha\_1 f\_{\rm c\bar{t}} \Big( b\_f' - b\_w \Big) h\_f' \Big( h\_w - \frac{h\_f'}{2} \Big) \tag{11}$$

If β1*xn* ≤ *h f* ,

$$\mathbf{C}\_{\varepsilon} = a\_1 f\_{cc} \beta\_1 \mathbf{x}\_n b\_f' \tag{12}$$

$$\mathcal{M}\_c = \alpha\_1 f\_{cc} \beta\_1 \mathbf{x}\_n b\_f' \Big| h\_w - \frac{\beta\_1 \mathbf{x}\_n}{2} \Big) \tag{13}$$

where, *xn* is the height of the concrete compressive zone; *bw* is the thickness of the wall web; *hf* is the height of the compression flange; and *bf '* is the width of the compression flange.

**Figure 8.** Stress and strain distribution of cross section for the large eccentric SC wall. (**a**) Strain distribution of the cross section; (**b**) Stress distribution of the cross section.

The stress distribution of the web plate is equivalent to the superimposition of one full plastic stress distribution and a triangle stress distribution, as shown in Figure 9. The triangle stress distribution can be seen as a couple. The couple has a negligible effect on the bearing capacity. For example, the horizontal bearing capacity of SCW1-5 in [12] was 1653.57 kN if the couple was considered. The result was 1653.59 kN if the couple was ignored and the difference was only 0.0012%. Therefore, the couple in Figure 9b can be ignored in the process of calculating the bearing capacity. The formula is simplified as Equations (14) and (15).

$$N\_{sw} = f\_y t\_{sw} (2x\_n - h\_w) \tag{14}$$

$$M\_{sw} = f\_y t\_{sw} \left[ \frac{1}{2} h\_w^2 - (h\_w - x\_n)^2 \right] \tag{15}$$

**Figure 9.** Equivalence of web steel plate for the large eccentric SC wall. (**a**) Full plastic stress distribution; (**b**)Triangle stress distribution.

#### *4.5. Small Eccentric Compression Failure (*ξ > ξ*b)*

Figure 10 shows the distribution of the cross-section stress and strain for the small eccentric SC wall. The equilibrium equations of the section are described as Equations (16) and (17).

$$\sum X = 0,\ N = A\_s' f\_y h\_0 + \mathcal{C}\_c + \mathcal{N}\_{sw} - T\_s \tag{16}$$

$$
\sum M = 0,\ \text{N}c = A\_s' f\_y h\_0 + M\_c + M\_{\text{sur}} \tag{17}
$$

where, *Ts* is the resultant force of the steel plate in tension. *Ts* is calculated by Equation (15), referring to [36].

$$-f\_y A\_s < T\_s = \frac{\chi\_v - h\_w}{\chi\_{nb} - h\_w} f\_y A\_s < f\_y A\_s \tag{18}$$

where, *xnb* is the height of the concrete compression zone in the balanced failure mode.

**Figure 10.** Stress and strain distribution of cross section for the small eccentric SC wall. (**a**) Strain distribution of cross section; (**b**) Stress distribution of cross section.

To simplify the calculation, the web plate stress distribution is simplified, as shown in Figure 11. η is used to revise the compression field height, to reduce the simplification error. In this paper, η was calculated using Equation (19). Therefore, *Nsw* and *Msw* can be calculated using Equations (20) and (21).

$$\eta = 1 - \frac{2\varepsilon\_y}{3\varepsilon\_{cu}}\tag{19}$$

$$N\_{sw} = f\_y t\_{sw} \eta x\_n \tag{20}$$

$$M\_{sw} = f\_y t\_{sw} \eta \mathbf{x}\_n (h\_w - 0.5 \eta \mathbf{x}\_n) \tag{21}$$

**Figure 11.** Equivalence of web plate for the small eccentric SC wall.

#### *4.6. Verification*

The ultimate bearing capacities of 31 specimens in different researches were calculated by Equations (9)–(21). The calculated results are compared with the experimental results, as shown in Figure 12. Since large eccentric compression failure occurred in all the specimens in the references, therefore, the finite element analysis for three small eccentric specimens was carried out using ABAQUS to verify the effectiveness of the calculation formula of small eccentricity, and the results were compared with the theoretical results from the calculation formula of small eccentricity. As shown in Figure 12, the calculated results agree well with the experimental and finite element analysis results. The average ratio of the calculation results to the experimental results of 31 specimens and the results of 3 FE models is 1.054, and the standard deviation is 0.086. Consequently, the above calculation method can predict the axial and bending bearing capacity of SC walls effectively.

**Figure 12.** Comparison between test results and calculated results of specimens in the existing literature.

Also, for some specimens in [12], it is clear that the calculation results are much lower than the experimental results. This is mainly because the hardening of the steel plate in tension is not considered in the formulas, but the steel plates of some specimens in [12] had reached the strain-hardening range, and even tension fracture appeared.

#### **5. Conclusions**

This paper summarized the experimental results of 49 SC walls subjected to cyclic loading. The parameter analysis using ABAQUS was carried out to supplement the experiments. Based on the analysis of the experimental results and finite element analysis results, the calculation method for the axial and bending bearing capacity of SC walls are proposed. The conclusions of this paper are summarized as follows.

**Author Contributions:** Conceptualization, Q.G.; Data curation, P.Z.; Formal analysis, P.Z. and F.K.; Funding acquisition, Q.G.; Investigation, W.Z. and F.K.; Methodology, W.Z. and Y.Y.; Project administration, Q.G. and Y.Y.; Writing—original draft, F.K.; Writing—review & editing, P.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded by National Natural Science Foundation of China (Grant No. 51778032).

**Acknowledgments:** This work was supported National Natural Science Foundation of China (Grant No. 51778032). The authors greatly appreciate the financial support.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Local Buckling and Resistance of Continuous Steel Beams with Thin-Walled I-Shaped Cross-Sections**

#### **Andrzej Szychowski \* and Karolina Brzezi ´nska**

Faculty of Civil Engineering and Architecture, Kielce University of Technology, al. Tysi ˛aclecia Pa ´nstwa Polskiego 7, 25-314 Kielce, Poland<EMAIL_ADDRESS>
**\*** Correspondence<EMAIL_ADDRESS>
Received: 8 April 2020; Accepted: 23 June 2020; Published: 28 June 2020

**Featured Application: The research results included in the paper can be widely used in designing thin-walled steel elements for use in construction. The use of the presented calculation model for the calculation of continuous beams allows for more optimal and economical design of sections of this element class under resistance conditions. In this model, the real behavior of thin-walled beams under load was taken into account much more precisely in relation to the Eurocode. This made it possible to use the resistance reserves resulting from the e**ff**ect of longitudinal stress variation and the elastic restraint of the thin-walled cross-section walls.**

**Abstract:** In modern steel construction, thin-walled elements with Class 4 cross-sections are commonly used. For the sake of the computation of such elements according to European Eurocode 3 (EC3), simplified computational models are applied. These models do not account for important parameters that affect the behavior of a structure susceptible to local stability loss. This study discussed the effect of local buckling on the design ultimate resistance of a continuous beam with a thin-walled Class 4 I-shaped cross-section. In the investigations, a more accurate computational model was employed. A new calculation model was proposed, based on the analysis of local buckling separately for the span segment and the support segment of the first span, which are characterized by different distributions of bending moments. Critical stress was determined using the critical plate method (CPM), taking into account the effect of the mutual elastic restraint of the cross-section walls. The stability analysis also accounted for the effect of longitudinal stress variation resulting from the varied distribution of bending moments along the continuous beam length. The results of the calculations were compared with the numerical simulations using the finite element method. The obtained results showed very good congruence. The phenomena mentioned above are not taken into consideration in the computational model provided in EC3. Based on the critical stress calculated as above, "local" critical moments were determined. These constitute a limit on the validity of the Vlasov theory of thin-walled bars. Design ultimate resistance of the I-shaped cross-section was determined from the plastic yield condition of the most compressed edge under the assumptions specified in the study. Detailed calculations were performed for I-sections welded from thin metal sheets, and for sections made from two cold-formed channels (2C). The impact of the following factors on the critical resistance and design ultimate resistance of the midspan and support cross-sections was analyzed: (1) longitudinal stress variation, (2) relative plate slenderness of the flange, and (3) span length of the continuous beam. The results were compared with the outcomes obtained for box sections with the same contour dimensions, and also with those produced acc. EC3. It was shown that compared with calculations acc. EC3, those performed in accordance with the CPM described much more accurately the behavior of the uniformly loaded continuous beam with a thin-walled section. This could lead to a more effective design of structures of this class.

**Keywords:** thin-walled I-section; continuous beam; local buckling; longitudinal stress variation; design ultimate resistance of the cross-section

#### **1. Introduction**

In modern metal construction, thin-walled lightweight components that are sensitive to local stability loss are increasingly being used. Local buckling of a thin-walled member causes lateral (in relation to the direction of compressive stress) displacement of the component walls of the cross-section. This reduces the load capacity of the critical cross-section, which reduces the resistance of the entire structural member. In the European Eurocode 3 (EC3) [1–3] standards, cross-sections that are subjected to local buckling (in the elastic range) are included in Class 4 cross-sections. The classification of a steel cross-section into Class 4 is based on a comparison of slenderness (b/t) of separated component walls of the section with the limit values for Class 3. In the event the limits of at least one of the component walls being exceeded, the cross-section is qualified as Class 4, which implies the need to include the effects of local buckling in the calculation model. However, an interest in thin-walled members is currently growing due to a significant reduction in the weight of structural steel cross-sections. For example, hot-formed bisymmetric sections (e.g., IPE, HEB), much lighter cold-formed open sections (monosymmetric or compound cross-sections, e.g., two-branch cross-sections), or bisymmetrical box sections can be used for supporting elements of light halls, floor beams, trusses, columns, or purlins. This approach reduces the weight of the structure. The use of steels with ever higher strength as well as modern automatic welding technologies also affects the development of thin-walled welded I-sections. Such Class 4 bisymmetrical and monosymmetrical welded cross-sections are characterized by low weight, with an analogous degree of reliability of the structural member in relation to hot-formed members.

From the point of view of the occurring instability phenomena, I-section beams may be subject to local buckling and lateral torsional buckling, including flexural–torsional loss of stability with the so-called forced rotation axis (e.g., wind suction purlin or from gravity loads in the supporting segment of continuous beams). This is due to the fact that thin-walled open cross-sections are characterized by low wall stiffness to local buckling and low rigidity of the whole cross-section to torsion [4,5]. However, a forced rotation axis at lateral–torsional buckling, e.g., with a suitably rigid cover, results in a significant increase in critical moment and, in many cases, the adverse effects of lateral–torsional buckling can be neglected.

Therefore, in the Class 4 thin-walled I-section beams, protected against lateral buckling, the influence of local stability loss remains, especially because thin-walled I-cross-sections are much less resistant to local buckling than, for example, box cross-sections [6]. This results from the fact that in such a cross-section there are cantilever walls, which are much less resistant to buckling in relation to internal walls [7,8] or cantilever walls with a stiffening bend [9]. However, the use of I-sections (in relation to box sections) allows for simpler shaping of connections and nodes due to free and double-sided access, e.g., to bolt joints. Moreover, continuous beam systems, due to the favourable longitudinal distribution of bending moments, enable the effect of longitudinal stress variation to be used in local stability analysis [10].

The resistance of a thin-walled cross-section is determined using the effective width method [3]. This method consists of determining the critical stresses of local buckling (σ*cr* = *k*σ*E*, where *k* is the plate buckling coefficient) for individual cross-section walls, assuming their hinged support and constant distribution of stresses along the member length. On this basis, the relative plate slenderness (λ*p* = *fy*/σ*cr*, where *fy* is the yield strength of steel) and effective widths of individual walls (plates) are determined *beff* = ρ λ*p b*, where ρ is the reduction factor.

Reference [6] presents an analysis of the so-called "local" critical resistance (i.e., determined from the condition of the local buckling) and the design ultimate resistance of a continuous beam with a Class 4 thin-walled box section. It was shown that the resistance of a five-span beam can be determined from the supporting segment of the first span. Calculations were made using the critical plate method (CPM) [10]. The following were analyzed with respect to their effect on the "local" critical and design ultimate resistance of box cross-sections: (1) the relative plate slenderness of the flange, and (2) the span length of the continuous beam.

In the case of transversely bent beams made of thin-walled I-sections, which are protected against lateral–torsional buckling, the possibility of local buckling of the compression flange must be taken into account along with, for adequately high cross-sections, the possibility of local buckling of the bending and shear web [11].

This study dealt with the determination of the resistance of a continuous beam (e.g., purlins) with a thin-walled I-section (sheet-welded or composed of two cold-formed channel sections). The calculations accounted for (1) the effect of the elastic restraint of the weakest cross-section wall in the stiffening wall, and (2) the effect of longitudinal stress variation caused by the variability of bending moments.

The analysis was based on the fixed I-section along the whole length of the beam (Figure 1). In this case, the resistance of the continuous beam is determined by the end span. The case where the first (extreme) span is additionally reinforced, and the resistance is determined by the intermediate spans, will be the subject of a separate work.

**Figure 1.** Static model of a continuous beam end span with thin-walled I-section.

#### **2. Computational Model for Local Cross-Section Buckling**

The Class 4 cross-section calculation model according to European standards EC3 [1–3] assumes that all component plates of athe cross-section are hinge-supported at their bonding edges. This means that the plates do not interact with each other, and the local buckling of the whole cross-section is determined by the weakest separated plate (cross-section wall).

However, in many technically important cases this model is too conservative, which has been confirmed by numerous experimental studies, e.g., References [5,12–14]; theoretical studies, e.g., References [4,7,15,16]; and numerical simulations, e.g., References [17–23]. This is due to the fact that in real thin-wall cross-sections, there is an effect of mutual, elastic interaction of component plates (walls).

Further, the standard computational model assumes that after determining the relative slenderness of the individual simply supported plates, their respective effective widths are determined, which determine the effective cross-section of the thin-walled member. In the calculation according to EC3, the effect of longitudinal stress variation, which often occurs in practice (e.g., in continuous beams), is also ignored.

In References [19,21], the effect of the mutual elastic restraint of thin-walled section walls was considered. Various load cases were taken into account, ranging from axial compression to pure bending relative to both axes of the section gravity, including interactive loads. Very large sets of simulation results were analyzed via the finite strip method using CUFSM software [18]. Calculations were made for many load cases and approximate analytical formulas were proposed for determining the plate buckling coefficient *k* [19] and the local buckling half-wavelengths [21] for various types of hot-rolled and welded sections. In Reference [22], in turn, approximation formulas of the *k* coefficient for hot-rolled sections under simple loads (axial compression, bending relative to the stronger and weaker axis of the cross-section) were given. However, the solutions obtained in References [19,21,22] relate only to constant stress distributions over the length of the structural member.

In References [7,24,25], the impact of longitudinal stress variation in the compressed cantilever plate (which is, for example, a component of an I-cross-section) was taken into account, but only for extreme support conditions (hinge or full restraint).

Meanwhile, in Reference [26], the approximation formulas of the plate buckling coefficient k for an axially compressed cantilever plate with any degree of elastic restraint of the supported edge and longitudinal stress distribution according to the linear function and 2◦ parabola were derived. At the same time, graphs of *k* coefficients were determined in References [27,28] for elastically restrained and eccentrically compressed cantilever plates with longitudinal stress variation. Many technically important load cases have been considered.

Reference [10] presents a more accurate computational model of the effect of local buckling on the resistance of thin-walled cross-sections. The critical plate method (CPM) takes into account the effect of both the elastic restraint of the cross-section plates and that of longitudinal stress variation.

#### **3. "Local" Critical Resistance and Design Ultimate Resistance of the Cross-Section**

In Reference [10], the bending moment inducing local buckling of a thin-walled cross-section was called the "local" critical moment (*ML cr*), which can be determined from the formula

$$M\_{cr}^{L} = \sigma\_{cr}^{L} \mathcal{W}\_{cl,y} / \mathcal{\gamma}\_{M0\prime} \tag{1}$$

where σ*L cr*—critical buckling stress determined according to a more accurate computational model (CPM) and related to the extreme edge of the cross-section,

*Wel*,*y* = *Iy*/*zc* – elastic section modulus,

γ*M*0—partial factor for cross-section resistance.

This moment is a limitation of the pre-buckling range of cross-section behavior and determines the limit of validity of Vlasov's thin-walled bar theory. On its basis, the critical resistance (critical load) of the structure can be determined from the condition of local stability loss of the weakest cross-section.

As regards the reliability of metal building structures, the designer's interest lies in the so-called design resistance [1], which can be determined from the formula

$$M\_{eff} = W\_{eff} f\_y / \gamma\_{M0\star} \tag{2}$$

where *Weff* = *Ieff* /*zeff*—effective section modulus.

The "local" critical moment according to (1) can in practice be treated as the resistance of a thin-walled section in the pre-buckling range, assuming unlimited material elasticity. Therefore, where *Wel*σ*L cr* > *Weff fy*, the design resistance of a Class 4 cross-section must necessarily be determined using Equation (2).

The so-called design ultimate cross-section resistance defined in References [6,10] (taking into account a more accurate model of local buckling) is a lower (conservative) estimate of the failure load determined for the mechanism of plastic hinge [29]. The resistance of a thin-walled cross-section, which is obtained during the failure phase, cannot be applied in the design of building structures. However, it can be applied, for example, in designing so-called mechanical energy absorbers [30].

#### **4. The Idea of the Critical Plate Method (CPM)**

The idea of the critical plate method [10] consists of determining the buckling stress of the weakest component plate of the cross-section, taking into account its elastic restraint in adjacent plates (walls), for which the buckling stresses are higher. In addition, the method allows the effect of longitudinal stress variation to be taken into account. In this computational model, the so-called critical plate (CP) must be identified, which determines the local buckling of the section in a given loading condition.

The elastic restraint against longitudinal edge rotation of the CP results from the action of the adjacent restraining plate (RP). For example, for certain proportions of the bending I-section [10], the critical plate may be a compressed cantilever wall of the width *bs* as a outstanding part of the flange width *bf* = 2*bs*, and the restraining plate will be a web of height *hw*. The buckling stress for the CP so determined is higher than that determined under the assumption of its simple support. This is especially true for cantilever plates, for which the elastic restraint against rotation of the longitudinal supported edge significantly affects the buckling stress values. The difference in plate buckling coefficients for an uniform compressed cantilever plate between a hinged support (*k*σ = 0.43) [3] and a fully restrained support (*k*σ = 1.25) [7,19] is almost threefold.

In References [19,21], attention was paid to the additional phenomenon of the so-called mutual constraint of the component walls of the cross-section (affecting the increase of buckling stress), despite the fact that the theoretical critical stresses of individual walls (determined under the assumption of their hinged support) are equal. This was found to be due to the different buckling lengths of the internal and cantilever plates. Therefore, in Reference [21], approximation formulas for the local buckling half-wavelengths of the full cross-section were derived. This length is between the buckling length for a separate, simply supported, and fully restrained plate, on which the critical resistance of the full cross-section depends. Knowledge of local buckling half-wavelengths is useful for the direct definition of geometric imperfections in analytical and numerical models.

On the other hand, in Reference [10], it was assumed that in a thin-walled section where thought-separated hinge-supported walls theoretically have the same buckling stresses, the effect of mutual constraint is insignificant (i.e., increases the buckling stress from 5 to 20%) and can be conservatively ignored. On this basis, so-called "zero" cross-sections were defined, in which the thought-separated hinge-supported plates have the same buckling stresses, and the resistance of such cross-sections can be calculated according to the standard procedure [1,3].

The distinction between "zero" and "non-zero" cross-sections significantly simplifies the identification of the critical plate in the latter and allows for a slightly conservative (safe) estimation of the buckling stress. Therefore, the elastic restraint coefficient of the CP in RP can be estimated using Equation (3) [10]

$$\kappa = 1/(1 + 2D\_s/b\_s\mathbb{C}\_0),\tag{3}$$

where *C*θrotational spring stiffness equal to the bending moment created during rotation by unit angle (*C*θ = *M*/θ), *bs*—CP width, *Ds*—plate flexural rigidity according to the formula

$$D\_s = \frac{Et\_s^3}{12(1 - \nu^2)}\tag{4}$$

where for *E* = 210 GPa and ν = 0.3, it can be approximately assumed that *Ds* = 19, 200*t* 3 *s*, *ts*—CP thickness.

The restraint coefficient according to Equation (3) varies from κ = 0 for simple support to κ = 1 for full restraint.

When the above-mentioned effects are taken into consideration, it is possible to determine the critical stress causing the cross-section local buckling in a more accurate way. Based on the critical stress, the following can be determined: (1) the "local" critical resistance of the cross-section *ML cr*, (2) the effective width of the critical plate, and (3) the design ultimate resistance of the cross-section *MCP eff* according to Reference [10].

#### **5. Algorithm of the CPM Method for Thin-Walled I-Cross-Sections under Bending**

For the transverse bending of I-beams of a Class 4 cross-section (Figure 2), their resistance is usually determined by local buckling of the compression flange (of width *bf* -2*bs* and thickness *tf*) or local buckling of the web (of height *hw* and thickness *tw*) under bending and shear interaction. Because of their geometric topology, those cross-sections were categorised in Reference [10] as part of the group of so-called "simple" cross-sections.

The ultimate height *h*0 of the cross-sections at which the compression flange buckling and web in-plane bending with a slight impact of shear stresses (τ ≤ 0.25τ*cr*) occur almost simultaneously can be determined using Equation (5) [10]

$$
\lambda \mu\_0 = \sqrt{\frac{k\_w^0}{k\_f^0}} \cdot \left(\frac{t\_w \mu\_s}{t\_f}\right) \tag{5}
$$

where *k*0 *i* —basic buckling coefficient for the hinged supported i-th plate at given load distribution (*k*0 *f* = 0.43 for axial cantilever plate compression, *k*0 *w* = 23.9 for bending in the plane of the web plate).

For *h* < *h*0, it is the compression flange that determines local buckling of the cross-section, whereas for *h* > *h*0, the weakest wall is the web under in-plane bending.

For the mean proportion of shear stresses (0.25τ*cr* ≤ τ ≤ 0.5τ*cr*) Equaiton (5) can be modified to the form:

$$h\_0 = \sqrt{\frac{k\_w^0}{k\_f^0}} \cdot \left(\frac{t\_w b\_s}{t\_f}\right) \left[1 - \left(\frac{\tau}{\tau\_{cr}}\right)^2\right]^{0.25} \tag{6}$$

where: τ*cr*—critical stress for the web plate in shear.

However, if there is a large proportion of transverse forces in the slender webs (plates supported at four edges) that are part of beam cross-sections, shear stresses τ may need to be considered more closely, as their contribution to local loss of web stability may be significant. Appropriate formulas for elastic strain energy and work done by external forces allowing critical stresses to be determined (using the energy method) for internal plates loaded (within their plane) with bending with high shear were derived in Reference [11].

For thin-walled I-sections, for which *h* < *h*0 according to Equation (6), the CPM assumptions are as follows: (1) the compression flange of the cross-section consists of two critical plates (CPs) which width of *bs* ≤ *bf* /2 each, which are supported on a web plate of the height *hw*, (note: dimensions *bs* and *hw* can be determined based on the rules given in Reference [10]); (2) a single CP acts as a cantilever plate, with one side elastically restrained against rotation; (3) the CP to RP connection (i.e., the web) is rigid (i.e., on the longitudinal edge of their connection, the conditions of continuity of displacements (rotation angles) and forces (bending moments), are met); (4) the transverse edges of the plates (CP and RP) are simply supported on the segment ends; and (5) the thin-walled bar segment (with the length *ls*), as in Reference [6], is defined as follows: (a) for constant longitudinal stress distribution, as the distance between the so-called buckling nodal lines, (b) for longitudinal stress variation, as the distance between transverse stiffeners (diaphragms, ribs, or supports) that maintain a rigid cross-section contour, but not longer than the range of the compression zone in the critical plate [26]. The conditions under which Assumption 3 can be adopted were discussed in Reference [10].

The case of I-sections with high and slender webs (at *h* > *h*0 acc. to Equation (6)), where the "local" critical resistance is determined by the buckling of the bent and shear web, will be discussed in a separate study.

**Figure 2.** Division of I-cross-section into critical plate (CP) and restraining plate (RP) for *h* < *h*0.

The calculation algorithm is as follows:

$$
\sigma\_{cr}^{0} = \min \{ \sigma\_{cr,i}^{0} \} \tag{7}
$$

$$
\sigma\_{cr,i}^0 = k\_i^0 \sigma\_{E,i} \tag{8}
$$

where σ*E*,*i*—Euler stress for the i-th plate according to the formula: σ*E*,*i* = 190, 000(*ti*/*bi*) 2 . Note: for *h* < *h*0 acc. to Equation (6), the critical plate (CP) is the cantilever wall of the compression flange and the rigid plate (RP) is the web;

$$l\_{\mathcal{U}} = b\_s \left(\frac{2.02 - 0.37\kappa}{\kappa^{0.25}}\right) \tag{9}$$

(5) Determination of the coefficient η [31], depending on the static scheme and the way of forcing (loading) RP by buckling CP (Figure 2) for the critical length according to Equation (9) [10]

$$
\eta = \sqrt{33.4 + 50.7(b\_r/l\_{cr})^2} - 2.78 \tag{10}
$$

where *br*—RP width;

(6) Determination of σ*L cr* according to the formula

$$
\sigma\_{cr}^L = k^\* \sigma\_{E,s} \tag{11}
$$

where *k\**—buckling coefficient according to a more accurate calculation model. The *k\** coefficient can be determined on the basis of Reference [26] from the following formulas:

• for linear stress distribution (Figure 3a),

$$\begin{aligned} k^\*(\kappa, m, \gamma) &= k\_{\infty}(\kappa) + \left[ 0.765m - 0.31m^2 + 0.227m^3 + \cdots \right] \\ \begin{pmatrix} 3.201m - 0.307m^2 - 3.724m^3 + 2.842m^4 \end{pmatrix} \kappa^2 + \\ \begin{pmatrix} -3.887m - 9.205m^2 + 21.528m^3 - 12.306m^4 \end{pmatrix} \kappa^3 + \\ \begin{pmatrix} -1.132m + 22.933m^2 - 37.767m^3 + 19.091m^4 \end{pmatrix} \kappa^4 + \\ \begin{pmatrix} 2.559m - 14.341m^2 + 20.975m^3 - 10.023m^4 \end{pmatrix} \kappa^5 \end{aligned} \tag{12}$$

• for non-linear stress distribution (according to 2◦ parabola, Figure 3b),

$$\begin{aligned} k'(\kappa, m\_{\gamma}\gamma) &= k\_{\alpha}(\kappa) + \left[1.096m - 0.808m^2 + 0.363m^3 + \right. \\ \left(20.064m - 63.78m^2 + 74.842m^3 - 30.565m^4\right)\kappa^2 + \\ \left(-58.826m + 195.88m^2 - 232.857m^3 + 95.684m^4\right)\kappa^3 + \\ \left(61.802m - 211.851m^2 + 254.031m^3 - 104.82m^4\right)\kappa^4 + \\ \left(-22.086m + 77.761m^2 - 93.977m^3 + 38.925m^4\right)\kappa^5 \left[/\gamma\right]^{(1.06+0.04m)} \end{aligned} \tag{13}$$

where *m* = 1 − σ1/σ0—longitudinal stress distribution coefficient (Figure 3), γ*s* = *ls*/*bs*. The plate buckling coefficient *k*∞(κ) of an elastically restrained and infinitely long cantilever plate can be determined according to Reference [26], using the formula

$$k\_{\rm os}(\kappa) = 0.425 + 2.893\kappa - 19.433\kappa^2 + 83.849\kappa^3 - 195.943\kappa^4 + 250.971\kappa^5 \tag{14}$$

$$-165.321\kappa^6 + 43.833\kappa^7$$

On the other hand, Reference [28] presents a simplified formula for *k*∞(κ) within the range 0.05 ≤ κ ≤ 1 in the form of

$$k\_{\rm os}(\kappa) = 0.49 + 0.974\kappa - 0.822\kappa^2 + 0.632\kappa^3 \tag{15}$$

(7) Estimation of the critical stress σ*cr*,*r* for a RP bent in its plane (width *br*) [32] for one half-wave of CP buckling length *lcr* according to Point 4.

$$
\sigma\_{cr,r} = \frac{Et\_r^2}{l\_{cr}^2 b\_r^4} \left( 11.32 l\_{cr}^4 + 1.97 b\_r^4 + 12.06 l\_{cr}^2 b\_r^2 \right) \tag{16}
$$

(8) Determination of rotational spring stiffness *C*θ according to Equation (17) and the restraint coefficient κ*i*+1 according to Equation (3) for the first (*i* = 1) and subsequent iteration steps.

$$\mathbf{C}\_{\theta} = \frac{c\_{\circ}\eta\_{\circ}D\_{r}}{b\_{r}} \left(1 - \frac{\sigma\_{cr}^{L}}{\sigma\_{crJ}}\right) \tag{17}$$

where *cj*—parameter of geometrical configuration of plates in contact with the j-th edge (for a welded I-section *cj* = 1/2 [10]; note: for an I-section made up of two channel sections *cj* = 1, i.e., one web with thickness *tw* stabilizes one CP of the flange on one edge), *Dr* = 19, 200*t* 3 *r*RP flexural rigidity, *tr*—RP thickness;

	- (a) the slenderness of the critical plate (CP) shall be determined from the buckling stress determined by Step 10 (i.e., taking into account the elastic restraint effect of the cantilever plate and the effect of longitudinal stress variation);
	- (b) for the web (rigid plate RP), a simple support shall be provided at the same edge;
	- (c) the boundary conditions at the other edge of the RP have a slight impact on the result of the calculations (conservatively, simple support can also be assumed here),
	- (d) the effect of possible longitudinal stress variation in RP is insignificant and can be ignored,

(e) the effective widths so determined "shall be folded" into an effective cross-section, for which *Weff* is determined.

**Figure 3.** Longitudinal stress distribution in a cantilever plate: (**a**) linear, (**b**) non-linear according to 2◦ parabola.

The algorithm proposed above, compared to the classical version of the effective width method, allows for a more accurate consideration of the behavior of a thin-walled section, the resistance of which is determined by the local buckling phenomenon. A more detailed comparative analysis on this issue is presented in Reference [10].

#### **6. Calculation Method for a Continuous I-Beam**

In the case of a multi-span continuous uniformly loaded beam (Figure 4), a non-linear bending moment distribution *My* occurs in the first (end) span with maximum values (*Mp* and *Ms*). For the range (*lp* + *c*), the graph *My* is convex, while for the range *ls* the graph is concave. Such longitudinal distributions of *My* cause non-linear (along the length of the beam) normal stress distributions σ*x*, which may cause local loss of stability [6].

**Figure 4.** Static scheme of a beam: (**a**) load distribution, (**b**) moment diagram for the first (extreme) span, (**c**) welded I-section, (**d**) 2C compound I-section.

Reference [33] showed that in the case of non-linear stress distribution for the full span range *lk* = *lp* + *c* (Figure 5), the *k* coefficient takes basically the same values as those determined for the reduced range *lp* (e.g., for *lk*/*bs* ≥ 4 differences not exceeding 2%). However, in Reference [26], it was shown that if the stress mark changes, the design segment length can be limited to the range of the compression zone. This is applicable to the support zone *ls* of a continuous beam. Therefore, similarly to Reference [6], in order to take into account the effect of longitudinal stress variation when

determining the span cross-section resistance, it is sufficient to assume the segment length to be *lp*, and for the support cross-section to be a segment with the length of *ls*.

**Figure 5.** Calculation model of the first (end) span of a uniformly loaded continuous beam.

For the span segment, there is a non-linear (according to 2◦ parabola) stress distribution. For the support segment, however, the concave stress distribution can be conservatively approximated by the linear distribution.

#### **7. Example of Using CPM to Determine the Resistance of a Continuous I-Beam**

#### *7.1. Statics and Cross-Sectional Forces*

The calculations were carried out, among others, for five-span continuous beams (e.g., purlins) (cf. Figure 4) with a constant span length *L* and thin-walled I-sections either (a) welded from thin sheets (I; Figure 4b,c) or (b) composed of two cold-formed channel sections (type 2C; Figure 4d). It was assumed that the above-mentioned sections after welding (I) or cold-forming (2C) were subjected to appropriate heat treatment to reduce the impact of residual stresses. In References [23,34], the influence of remaining welding stress on the behavior and resistance of welded I-sections was investigated. It was shown that this effect is greatest in the case of axial compression, but in some cases, bending elements should also be considered. Therefore, the effect of residual stresses on the local buckling of continuous beams with welded I-sections will be described in a separate study.

The beams were under continuous, uniformly distributed load. It was assumed that the elements were structurally protected against lateral–torsional buckling. In this case, the resistance of the beam depends on the resistance of the most stressed cross-section.

The relationship between *Ms* and *Mp* (Figure 4) in the first span was *u* = *Ms*/*Mp* = 1.351, irrespective of the type of cross-section and the span length. In the support cross-sections of the beam, two-sided stiffening ribs were used to transfer the support reactions without causing distortion of the thin-walled cross-section.

#### *7.2. Welded Section*

#### 7.2.1. Calculation of Critical Resistance According to CPM

Detailed calculations were carried out for a five-span continuous beam with a span length *L* = 4 *m* and a thin-walled I-section welded from sheets of dimensions I-300×tw×250×tf (I-h×tw×b×tf) for t = tf = tw = 5, 6, 7, and 8 mm made of S355-grade steel. It should be noted that in this case, adopting a smaller web thickness, i.e., tw < tf, led to a significant reduction in the elastic restraint of the compressed flange in the plate of the web. This resulted in a much greater reduction in the cross-sectional resistance than that resulting from a reduction in the elastic section modulus caused solely by a reduction in the thickness of tw.

Table 1 shows the results of the calculation of "local" critical moments for the ruling cross-sections: span *ML cr*,*p*, support *ML cr*,*s*, and the critical loads *qcr* determined from them. The symbol *qCP cr*,*s* indicates

the critical load value determined from the condition of reaching *ML cr*,*s* in the support cross-section, and the symbol *qCP cr*,*p* indicates that determined from the condition of reaching *ML cr*,*p* in the span cross-section. (Note: in Table 1, the values of the moments *ML cr*,*s* and *ML cr*,*p* are given as absolute values). The critical resistance of the beam is determined by the minimum load *qCP cr*,*min* = *min qCP cr*,*p*, *qCP cr*,*s* .

**Table 1.** Summary of critical results according to critical plate method (CPM) and European Eurocode 3 (EC3) for the I-300×t×250×t section and four wall thickness variants.

Furthermore, in Table 1, the symbol *MEC*3 *cr* indicates the critical cross-sectional moment determined according to σ*cr* calculated according to EC3 [3], and the symbol *qEC*3 *cr*,*min* stands for the minimum value of critical load determined on the basis of *MEC*3 *cr* . Due to the simplified calculation model used in the standard, *MEC*3 *cr* does not depend on the longitudinal stress distribution or on the degree of mutual elastic restraint of the cross-section component walls. It therefore determines the critical resistance of both the span cross-section (*p*) and the support cross-section (*s*).

Note: in the example under consideration, determined on the basis of the *qCP cr*,*min* parameter *h*0 according to Equation (6), *h*0 = 880 mm and was almost three times greater than the cross-section height *h* = 300 mm. Of course, the determination of the parameter *h*0 according to Equation (6) was iterative because the shear stresses τ and τ*cr* depend on the cross-section height, among other things. This check calculation allows the correct identification of the critical plate in a given cross-section loading condition to be confirmed.

A comparison of the results in Table 1 shows that *ML cr*,*s* was approx. 18.7% higher than *ML cr*,*p*. This was due to different stress distributions and different reliable lengths of the span (*lp*) and support (*ls*) segments. Despite the fact that *ML cr*,*p* < *ML cr*,*s*, the critical resistance of the beam was determined by the support zone, because: *u* = 1.351 > *ML cr*,*s*/*ML cr*,*p* = 1.187. The percentage increase of *ML cr*,*s* in relation to *MEC*3 *cr* was about 121%. The same relationship occurred for the corresponding critical loads *qmin cr* .

#### 7.2.2. Finite Element and Finite Strip Analysis

The critical stresses listed in Table 1 (σ*cr*,*s*, σ*cr*,*p*) were verified via the finite element method (FEM) using ABAQUS software [35], and the finite strip method (FSM) using CUFSM software [18].

The FEM enables the analysis of local buckling of the extreme span of a continuous beam by taking into account both the effect of mutual elastic restraint of cross-section's components plates, as well as the effect of longitudinal stress variation. However, the commonly used finite strip method (FSM) allows the effect of mutual elastic restraint of cross-section's component plates to be considered, but with the assumption of a constant stress distribution over the length of the member.

For the numerical simulations in ABAQUS v. 6.12, the first (extreme) span of a continuous beam was modeled. Due to the small wall thickness (compared to other dimensions), the cross-section's geometry was simplified to so-called centerline. S4R shell elements (four nodes with six degrees of freedom in the node) were used, and the dimensions of the finite element were assumed to be 12.5 × 12.5 mm. Such division made it possible to obtain technically sufficient accuracy while reducing calculation time. The transverse load was applied in the form of uniform pressure (p = 0.1 N/mm2) on the upper flange with a width of *b* = 250 mm, which corresponded to a uniform load of q = 25 kN/m. Conversely, the loading with the "corresponding" support moment of the value of Ms = 0.105 × qL2 = 42 kNm (where 0.105 is the coefficient from Winkler's tables) was performed by means of loading the support cross-section with a stress block. The boundary conditions in the support cross-sections were assumed to be continuous constraints perpendicular to each of the component plate. An additional tie securing the beam against displacement along the longitudinal axis was applied in the gravity axis of one of the support sections. In Figure 6b, a general view of FEM model of the I-300×5×250×5 cross-section in ABAQUS software is presented. Figure 6 shows the support ties used, and Figure 6c shows the method of loading with the support moment Ms. The critical load calculations were performed using the "buckling" procedure.

**Figure 6.** Finite element method (FEM) model of the I-300×5×250×5 beam: (**a**) boundary conditions, (**b**) division into finite elements, (**c**) method of loading with the support moment.

Figure 7 shows the local buckling mode (axonometric view and side view) corresponding to the first (smallest) eigenvalue of the critical load multiplier (qcr,s = 3.121 × q = 78.02 kN/m). Maximum local deflections of the cross-section occurred in the support zone.

**Figure 7.** Local buckling of I-300×5×250×5 beams in the support zone.

Figure 8 shows the local buckling mode (axonometric view and side view) corresponding to the third (second positive) eigenvalue of the critical load multiplier (qcr,p = 3.568 × q = 89.21 kN/m). In this case, the maximum local deflections of the cross-section occurred in the span zone.

**Figure 8.** Local buckling of I-300×5×250×5 beams in the span zone.

For numerical simulations in CUFSM v.5.04, the I-section was also simplified to centerline geometry, and then divided into 48 finite strips (16 per flange and 16 in the web). The load was applied in the form of a stress block, causing bending about the stronger axis of the cross-section.

#### 7.2.3. Comparison of Critical Stress Results

Table 2 compares the values of critical stresses for supporting σ*cr*,*s* and span σ*cr*,*p* sections, determined according to: (1) ABAQUS, (2) CPM, (3) CUFSM, (4) Reference [19], (5) Reference [22], and (6) EC3. It should be emphasised that comparison of stresses determined according to Reference [22] was for reference only, because the formulas derived in Reference [22] apply only to hot-rolled sections, and not to the welded ones.

**Table 2.** Comparison of critical stresses determined by several methods for the I-300×t×250×t section and four wall thickness variants.

From the comparison of the values given in Table 2, it follows that the new calculation model adopted in this work using the critical plate method (CPM) gave very good congruence with the results obtained from the ABAQUS software, which were adopted as a reference. This applied to both the support section (first eigenvalue, compliance approx. 0.99) and the span section (second eigenvalue, congruence approx. 0.985). The calculations obtained from CUFSM also gave very good results in the span section (congruence approx. 0.96), while in the support section, where a large stress gradient occurred, and on which the critical resistance of the beam depends, congruence was around 0.81. A similar situation occurred when comparing the results obtained from the formulas according to Reference [19] (span section about 0.91, support section about 0.77). However, a worse congruence of the results obtained according to Reference [22] (span section about 0.78, support section about 0.66) resulted from the different, in relation to hot-rolled sections, geometrical proportions of the sections considered in this study. The largest differences (span section about 0.53, support section about 0.45) were obtained from calculations made according to EC3. The local buckling half-wavelengths determined in this case were: (1) according to CPM, Equation (9) *lcr* = 307 mm; (2) according to CUFSM, *lcr*= 340 mm; and (3) according to Reference [21] *lcr* = 353 mm. It should be noted that in this case, the following rule applies: for higher critical stresses, there is a shorter local buckling half-wavelength.

The ompared computational methods showed that very good congruence of results in relation to finite element analysis (ABAQUS) was obtained by the critical plate method (CPM), which takes into account both effects. Methods that consider the effect of mutual elastic restraint of cross-section component plates give a good approximation of critical stresses in those beam zones where there is no significant variation of stresses in the longitudinal direction (e.g., span cross-section). In contrast, calculations according to EC3 are the most conservative, because they omit both effects.

#### 7.2.4. Design Ultimate Resistance

Table 3 shows the design values of the ultimate resistance for the reliable cross-sections: span *MCP eff*,*p* , support *MCP eff*,*s* , and the design ultimate loads (*qeff*) determined from them according to CPM. The symbol *qCP eff*,*s* indicates the ultimate load value determined from the condition of reaching *MCP eff*,*s* in the support cross-section, and the symbol *qCP eff*,*p* from the conditions needed to achieve *MCP eff*,*p* in the span cross-section. (Note: in Table 3, the values of the moments *MCP eff*,*s* and *MCP eff*,*p* are given

as absolute values). The ultimate resistance of the beam was determined by the minimum load *qCP eff*,*min* = *min*! *qCP eff*,*p* , *qCP eff*,*s* " .

**Table 3.** Summary of ultimate resistance results according to CPM and EC3 for the I-300×t×250×t section and four wall thickness variants.

Furthermore, in Table 3, the symbol *MEC*3 *c*,*Rd* indicates the design cross-section resistance determined according to EC3 [1,3], and the symbol *qEC*3 *eff*,*min* indicates the minimum value of ultimate load determined on the basis of *MEC*3 *c*,*Rd*.

Comparison of the results displayed in Table 3 showed that for a wall thickness of t = 5, 6, and 7 mm, the resistance *MCP eff*,*s* was approx. 4.5% higher than *MCP eff*,*p* . This was due to different stress distributions and different lengths of the span (*lp*) and support (*ls*) segment. Despite the fact that *MCP eff*,*p* < *MCP eff*,*s* , the design ultimate resistance of the beam is determined by the support zone, because *u* = 1.351 > *MCP eff*,*s* /*MCP eff*,*p* = 1.045. However, for wall thickness t = 8 mm, the difference between *MCP eff*,*s* and *MCP eff*,*p* was only 0.4%. This resulted from the fact that the cross-section loaded in this way, calculated according to CPM, can be classified as a Class 3 cross-section, for which the value of the reduction factor ρ is close to one. In contrast, the same cross-section calculated according to EC3 (due to low σcr,0) is still treated as a Class 4cross-section (in this case, ρ = 0.79). The percentage increase of *MCP eff*,*s* in relation to *MEC*3 *c*,*Rd* was about 23% for t = 5, 6, and 7 mm, and about 18% for t = 8 mm. The same relationship occurred for the corresponding ultimate loads *qmin eff* .

Figure 9 shows the effective cross-sections of the I-300×5×250×5 section determined in the span and support cross-sections according to CPM (Figure 9a,b) and EC3 (Figure 9c,d). For example, the sum of the effective widths 2*beff* of the compressed bottom flange of the support cross-section (which determines the resistance of the beam) determined according to CPM was about 39% larger than that determined according to EC3 (see Figure 9b,d). The shift of the neutral axis of the effective cross-section according to CPM in relation to the gross cross-section was e = 12.2 mm and was 51% smaller than e = 24.9 mm calculated according to EC3. This effect additionally affected the differentiation of the effective modulus *Weff* of the reliable cross-sections and, consequently, the differences in the design ultimate resistance of the beam determined according to CPM and EC3.

**Figure 9.** Effective cross-sections of the I-300×5×250×5 section according to CPM and EC3 methods. (**a**) span cross-section acc. CPM, (**b**) support cross-section acc. CPM, (**c**) span cross-section acc. EC3, (**d**) support cross-section acc. EC3.

#### *7.3. I-Section Consisting of Two Cold-Formed Channel Sections (2C)*

For compound sections (type 2C), detailed calculations were made for five-span beams with a constant span length for variants of *L* = 4, 5, 6, and 7 m made of S355-grade steel. For comparison purposes, the analyzed dimensions of 2C sections (Figure 10) with the contour of Sk250×250×t box sections (for t = 2, 3, 4, and 5 mm) presented in Reference [6] were used. In this case, the elastic gross section bending moduli *Wel* for Sk and 2C sections were equal.

**Figure 10.** Compared cross-sections of continuous beams: (**a**) box cross-section according to Reference [6], (**b**) compound cross-section type 2C.

Table 4 presents the results of the analysis of the effect of the span length on the "local" critical moment value *ML cr* and the resistance *Meff* calculated according to CPM and EC3 [1,3] for the 2C-250×250×4 (2C-h×b×t) section. For the support segment decisive of the beam resistance, the difference between the critical load value *qcr* obtained according to CPM method in relation to EC3 ranged from + 158% for *L* = 4 *m* up to + 141% for *L* = 7 *m*. The difference between the load ultimate value *qeff* ranged from + 22% for *L* = 4 *m* up to +20% for *L* = 7 *m*.

**Table 4.** Summary of results according to CPM and EC3 for the 2C-250×250×4 section and four span-length range variants.

Note: for comparison, Table 4 also includes the values of *qcr* and *qeff* determined according to CPM for the case where the resistance of the beam is decided by the span section. This would be the case, for example, in protection against local buckling of the support segment.

Table 5 compares the "local" critical resistances (*ML cr*, *qcr*) of beams with box Sk250×250×4 sections according to Reference [6] and 2C-250×250×4 compound I-sections depending on the span length of the continuous beam (*L* = 4, 5, 6, and 7 m).

Comparison of the values presented in Table 5 shows that for 2C sections, where critical resistance is determined by cantilever walls, "local" critical moments *ML cr* determined according to CPM were lower on average by about 25–26% in relation to corresponding box sections [6], and the reduction *MEC*3 *cr* was as much as 54%. The same relationship occurred when comparing the respective critical loads *qcr*. Meanwhile, the relation *qCP cr* /*qEC*3 *cr* for 2C sections in relation to Sk sections increased by as much as 62%. This effect resulted from a much larger reserve of the critical resistance of the elastically restrained cantilever plate (which determines the resistance of the 2C section) in relation to the internal plate of the compression flange, which in turn determines the resistance of the box section. In fact, this is the difference between the critical stresses for the plates (cantilever and internal) that are simply supported in relation to the same plates that are elastically restrained at one or two edges, respectively. For comparison, the relation between the buckling coefficient for the compressed cantilever plate when fully restrained (*ku* = 1.25 [7,19]) and simply supported (*kp* = 0.43 [3]) was *ku*/*kp* = 1.25/0.43 = 2.91. For an internal plate, on the other hand, the same relation was much smaller and was *ku*/*kp*= 6.97/4 =

1.74. Of course, this does not change the fact that the critical and ultimate resistance of an internal plate is much higher than that of a cantilever plate with the same geometrical and material parameters.

**Table 5.** Comparison of "local" critical resistances of box sections (Sk) and compound I-sections (2C) depending on the span length.

In turn, Table 6 compares the design ultimate resistances (*MCP eff* , *qCP eff*) of the same box section beams [6] and 2C compound I-sections, also depending on the span length. For the 2C sections, the design ultimate resistances *MCP eff* determined according to CPM were on average 4–5% lower than the corresponding box sections. The reduction, however, of *MEC*3 *eff* was nearly 14%. The same relationship occurred when comparing the respective ultimate loads *qeff* . In this case, the relation *qCP eff* /*qEC*3 *eff* for the 2C sections in relation to Sk sections increased by 11%.

**Table 6.** Comparison of design ultimate resistances of box sections (Sk) and I-sections (2C) depending on the span length.

Table 7 presents the results of the analysis of the slenderness effect (*bs*/*t*) of the compressed cantilever wall of 2C section on the value of "local" critical moment *ML cr* and the design ultimate resistance *Meff* determined according to CPM and EC3. The calculations were made for a continuous beam with a span length of *L* = 4 m. With the reduction of section wall thickness, the critical load *qcr* decreased, whereby the difference between the results obtained according to CPM and EC3 was about +158% regardless of the slenderness of the critical plate. The reduction in thickness also reduced the design ultimate resistance *Meff* and the ultimate load *qeff* , whereby the difference between the results according to CPM and EC3 was about +23% for t = 5 mm and about +16% for t = 2 mm.

**Table 7.** Summary of results according to CPM and EC3 for the 2C-250×250×t section and four wall thickness variants.

Note: for comparison, Table 7 also includes the values of *qcr* and *qeff* determined according to CPM for the case where the resistance of the beam is decided by the span section. This would be the case, for example, in protection against local buckling of the support segment.

Table 8 compares the "local" critical resistances (*ML cr*, *qcr*) of beams with box sections [6] and compound I-sections (2C) depending on the wall thickness (t = 5, 4, 3, and 2 mm).

**Table 8.** Comparison of "local" critical resistances of box sections (Sk) and compound I-sections (2C) depending on the wall thickness.

Comparison of the values presented in Table 8 shows that for 2C sections, where critical resistance is determined by cantilever walls, "local" critical moments *ML cr*,*s* determined according to CPM were lower in relation to corresponding box sections by approx. 24.3% for t = 5 mm, up to 27.3% for t = 2 mm. For *ML cr*,*p*, the differences ranged from 25.4% for t = 5 mm to 28.3% for t = 2 mm. For *MEC*3 *cr* , the reduction was 53.5% for t = 5 mm to 55.3% for t = 2 mm. The same relationship occurred when comparing the respective critical loads *qcr*. In this case, the relation *qCP cr* /*qEC*3 *cr* for 2C sections in relation to Sk box sections increased by 62.7%.

In turn, Table 9 compares the design ultimate resistances (*MCP eff* , *qCP eff*) of the same beams with box sections Sk according to Reference [6] and compound I-sections (2C), also depending on the wall thickness. For 2C sections, the design ultimate resistances *MCP eff*,*s* determined according to CPM, in relation to the corresponding box sections Sk, were reduced by 3.1% for t = 5 mm to 5% for t = 2 mm. For *MCP eff*,*p* the differences ranged from 3.8% for t = 5 mm to 5.1% for t = 2 mm. In contrast, the reduction in *MEC*3 *eff* ranged from 13.6% for t = 5 mm to 11% for t = 2 mm. The same relationship occurred when comparing the respective ultimate loads *qeff* . In this case, the relation *qCP cr* /*qEC*3 *cr* for 2C sections in relation to Sk box sections increased from 6.9% for t = 2 mm to 12.2% for t = 5 mm

**Table 9.** Comparison of design ultimate resistances of box sections (Sk) and I-sections (2C) depending on wall thickness.

Figure 11 shows the relation between the "local" critical moment *Mcr*,*s* (Figure 11a) and the design ultimate resistance *Meff* (Figure 11b) as a function of slenderness λ*CP* = *bs*/*t* of the critical cantilever plate of a compound I-section (2C).

Comparison of the graphs shown in Figure 11a showed that the "local" critical resistance and the design ultimate resistance decreased non-linearly with increasing slenderness of the cross-section wall. The difference between the value *Mcr*,*s* determined by the CPM method and the standard approach according to EC3 was constant in each case of wall thickness and was +158.1%. In contrast, the decrease of *Meff* (Figure 11b) was mildly non-linear. The percentage increase of the ultimate resistance according to CPM in relation to EC3 was basically directly proportional to the increase of the cross-section wall thickness from about 16% for t = 2 mm (λ*CP* = 61.5) to about 23% for t = 5 mm (λ*CP* = 24).

The relation between the critical load *qcr* (Figure 12a) and the ultimate load *qeff* (Figure 12b) as a function of the span length is shown in Figure 12.

**Figure 11.** Graphs of (**a**) *Mcr*,*s* and (**b**) *Meff* as a function of slenderness λ*CP*.

Comparison of the graphs shown in Figure 12 showed that an increase in the span length caused a non-linear decrease both in *qcr*,*s* and *qeff* . Moreover, it can be stated that the shorter the span, the greater the percentage increase of the critical resistance determined according to CPM in relation to EC3 (from 158% for short beams *L* = 4 m, to 141% for long beams *L* = 7 m). For ultimate resistance (Figure 12b), the differences in results ranged from 22% for *L* = 4 m to about 20% for *L* = 7 m, respectively). λ

**Figure 12.** *Cont.*

**Figure 12.** Graphs of (**a**) *qcr*,*s* and (**b**) *qeff* as a function of length *L*.

#### **8. Summary and Conclusions**

The new calculation model presented in this paper for the local stability loss of the first continuous beam span, consisting of separate local buckling analysis for the span segment (loaded with a non-linear "convex" bending moment distribution) and the support segment (in which the "concave" moment distribution was approximated by a linear distribution) was successfully verified by numerical simulations using the finite element method.

To improve the accuracy of the calculation of the critical resistance of a continuous beam with a Class 4 cross-section (in which large stress gradients in the longitudinal direction are found), it was necessary to take into account both the effect of the mutual elastic restraint of the cross-section component plates, and the longitudinal stress variation.

The use of the critical plate method (CPM) according to Reference [10] to determine the "local" critical resistance and the design ultimate resistance of a thin-walled I-section led to a more optimal design of such elements. In this method, a more accurate calculation model is used, which takes into account both the effect of mutual elastic restraint of the cross-section component plates and the effect of longitudinal stress variation. The increase in the resistance that occurs in this case, compared to the EC3 calculation, results from a more faithful representation of the behavior of the thin-walled structural member in the engineering calculation model.

At the same time, it should be noted that taking into account the above effects has only a minor impact on the complexity of calculations, which can be easily algorithmized and presented as relatively simple spreadsheets. This is due, among other things, to the simplified identification of the so-called critical plate and the definition of the so-called "zero" cross-sections [10]. Such assumptions allow for a slightly conservative assessment of the cross-section resistance. In this sense, the CPM can be used both for preliminary design or verification of FEM calculations and, in many technically important cases, for basic design.

The "local" critical resistance defines the range of pre-critical behavior of a section and constitutes the limit of validity of the thin-walled bar theory with a rigid cross-section contour (the Vlasov theory), assuming unlimited elasticity of the material.

To determine the design ultimate resistance of a thin-walled cross-section, the effective width method applied to the individual plates can be used along with the additional assumptions formulated in Section 5, Step 12. The relative slenderness of the component plates is determined from the respective critical stresses. For CP, these are stresses determined by taking into account the restraint coefficient and longitudinal stress variation. For the RP, simple support and a constant stress distribution along the length can be assumed on the same edge. Such assumptions allow for a technically precise and sufficient calculation of the resistance of a thin-walled cross-section.

In the case of a continuous beam under uniform load, the segment that determines the resistance of the structure is the support segment (*Ms*) of the first span, where the "local" critical resistance is first achieved, followed by the design ultimate resistance.

For the analyzed continuous beam made of a welded section I-300×5×250×5 and calculated according to CPM (Section 7.2), a 121% increase in the critical resistance *qcr* was determined, which translated into a 23% increase in the design ultimate resistance *qeff* in relation to the calculations done according to EC3.

In the case of the same continuous beam but made of a compound section 2C-250×250×4 and calculated according to CPM, there was a 158% increase in the critical resistance *qcr*, which translated into a 22% increase in the design ultimate resistance *qeff* in relation to the calculations done according to EC3.

The comparison of the resistance parameters of continuous beams with box sections Sk according to Reference [6] and the compound sections 2C compatible with them showed that in the calculation according to CPM, the percentage differences (to the disadvantage of open sections 2C) were as follows: (1) for critical resistance *qcr* from about 24% to 27% (more than 53% according to EC3), (2) for design ultimate resistance *qeff* only from 3 to 5%, but according to EC3, more than 13%. In this case, the use of CPM gave greater possibility to use the resistance reserve of the so-called "non-zero" cross-sections, in which the effect of mutual elastic restraint of cross-section component plates can be applied.

The advantage of 2C sections over box sections (assuming that the beam is protected against lateral buckling) is the greater ease of constructing joints and assembly nodes due to free and two-sided access to mechanical connectors (e.g., bolts). In the case of beams sensitive to lateral buckling, on the other hand, the use of closed cross-sections is more optimal due to their much higher torsional stiffness compared to open cross-sections.

If there are significant residual stresses (compressive) in a thin-walled cross-section, the critical stress of local buckling is reduced. This applies especially to axially compressed welded sections [34,35]. Therefore, further testing of welded components with a high proportion of residual stresses must take into account their adverse effects on local buckling of the cross-sections.

In open cold-formed sections (produced without welding), however, the adverse effect of residual stresses caused by bending is usually insignificant [36] and can be omitted in most cases (this effect is further reduced by the beneficial effect of increasing the strength of steel in cold-formed corners).

Beams with thin-walled I-sections, for which the ultimate resistance is determined by local buckling of the bent and shear web, and continuous beams with reinforced end spans will be subject to further study by the authors.

**Author Contributions:** Introduction was prepared by A.S. Mathematical description were written by A.S., K.B. Results were obtained by K.B., A.S. The analysis of the results and conclusions was written by A.S. All authors have read and agreed to the published version of the manuscript.

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

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

#### **References**

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Experimental Study on the Flexural Behavior of Alkali Activated Fly Ash Mortar Beams**

#### **Adelino V. Lopes 1, Sergio M. R. Lopes 2,\* and Isabel Pinto 2**

Received: 1 June 2020; Accepted: 23 June 2020; Published: 25 June 2020

**Featured Application: Fly ash is a byproduct from burning pulverized coal in the electric power industry and can be incorporated in cementitious concretes or in alkali activated concretes (AAC) with obvious environmental advantages. When part of AAC, it will be able to lead to products with glossy, black surfaces, which might be aesthetically appealing for some modern architectural solutions. A reinforced concrete structure is one type of such products. This document presents an experimental study on reinforced mortar beams, half of them made with fly ash and the other half made with Portland cement. Di**ff**erent aspects of the structural behavior are compared for possible applications of fly ash geopolymers in civil engineering structures.**

**Abstract:** This work aims to study the possibility of using alkaline activated fly ash in structural members. The work, of an experimental nature, focuses on the evaluation of the behavior of simply supported beams under two symmetrical loads (four-point tests). For such study, 10 beams were built, of which, five using fly ash and the remaining five using traditional Portland cement. The test results are compared. Conclusions on the practical application of fly ash in structures were explained and, as mention later in this document, there is room for improvement. This is one of very few works on fly ash alkali activated structures and further studies are necessary in the future. Some aspects, such as shrinkage and deformability are presented as some of the negative points concerning the potential use of fly ash. These are two aspects that need more attention in future investigations.

**Keywords:** experimental study; analytical model; reinforced concrete; beams; fly ash alkali activated; bending

#### **1. Introduction**

The cement industry is under increasing pressure from public opinion due to its contribution to CO2 emissions. To bring the cement sector in line with the 2015 Paris Agreement on climate change, its annual emissions will need to fall by at least 16 per cent by 2030 [1]. The awareness of the high level of CO2 emissions is not new, but only recent international agreements and measures really forced the cement industry to intensify the search for new alternative production technologies and materials. A vision of the producers on this issue can be found in a report by Cembureau [2].

Obviously, non-cementitious materials must be considered as an alternative route of the entire road map to low emission construction production. Various research works on alternative materials to Ordinary Portland Concrete (OPC) have been published in the last three decades, but OPC continues to dominate the market at present. However, the target limits for CO2 emissions might alter the competitive advantage of OPC and surely increases the demand for research on alternative materials. This is the case of alkali activated concretes (AAC), which show some positive points concerning the strength values, durability, or environmental impacts [3].

In fact, many works have been published on AAC in the last 2–3 decades. Almost all of them have focused on the development of the material itself, including the mechanical properties. Very few studies were concerned with the structures so far. Since the size of specimens for structural studies is bigger when compared to that of specimens for mechanical properties of the material, some difficulties arise when passing from the material to structural research. Even a recent study [4] has pointed out that there are some studies in mortars, but the need for studies in concretes still persists. This means that the step from mortar to concrete is still not entirely covered and the next step from material specimens to concrete structures is also of great necessity.

From different types of AAC, alkali activated fly ash concrete has its own particularities that need to be known. One of the most important is the workability, which is directly linked with the practical application of the material in structures. Not going deeply into the topic, it should be known that the rheological behavior observed in fly ash is different from that of OPC. The commercial admixtures for improving the workability of OPC do not have similar effects in fly ash. Further explanations on this topic are available in bibliography [5].

Shrinkage is also a characteristic of fly ash concretes, which has deserved some attention. It is known that curing conditions and the liquid-to-ash ratio are very important for shrinkage, with some researchers recommending curing temperatures of around 60–70 ◦C (Degrees Celsius) for minimizing shrinkage [6–8]. Indeed, the few experimental studies on reinforced beams made with fly ash activated concrete found in literature followed this recommendation [9–12]. Curing conditions at this temperature level are not practical for on-site applications. Some recent studies have concentrated on curing in ambient temperatures [13–15].

Curing at specific temperatures has other effects beyond shrinkage, for instance, on strength. Therefore, it should be considered as a strong option when possible, for instance, in precast plants. However, in the opinion of the authors, the on-site practical construction at ambient temperature must not be ruled out since it is a potential application of this material. The studied reported here was trying to evaluate the possibility of application on reinforced concrete members cast on-site at ambient temperature.

As explained before, the general possibility of replacing the Portland cement with alkaline activated binders, known as geopolymers, should be considered as a strong option given the increased concerns on environmental issues. Fly ash is one of the materials that can be used in alkaline activated products. The authors have tested other materials [16], but the current document reports a study on the behavior of fly ash in structures, particularly in beams under bending. It is interesting, in particular, to evaluate the behavior of these beams under bending action, for different levels of the loading, and to compare it with that of beams built with traditional Portland cement. It should be kept in mind that the behavior in service conditions is also gaining great importance in current regulations, when compared to typical ultimate strength.

To achieve this goal, an experimental procedure was carried out, in which two groups of five beams each were built. From group to group, the beams vary in material, with five built with Portland cement binder and five with fly ash based binder. Within each group of beams, only the ratio of the reinforcement was varied in order to cover an intentional range to cover specific situations explained later. The dimensions are the same for all beams, which are about half the dimensions usually used in the laboratory and in construction.

In addition to this objective, it would be also important to theoretically estimate the behavior of beams built with new materials. A good fitting of the numerical methodologies to the behavior of such new materials needs to be verified [17].

In this work, beams with reduced dimensions were used compared to those usually used in construction. Nevertheless, they are significantly bigger than the geopolymeric specimens generally presented in the bibliography for the study of the material. It is, therefore, important not to ignore the

possible scale effects. Some studies indicate that up to a scale of 1 to 3, the differences are negligible [18]. However other studies indicated that some differences in behavior could be important, especially for large size structures [19]. Kim et al. [20] presented a study with reinforced concrete specimens, using a scale of 1:5, where it was concluded that the models present equivalent results when compared to the normal size members. However, other studies indicate some differences. Since both types of beams of this investigation, fly ash and cement, were built with equal dimensions, it would be expected that the size effects, if any, would have similar consequences. Therefore, the comparison of the results can be accepted for beams of these dimensions.

#### **2. Experimental Procedure**

#### *2.1. Design*

In the construction of the beams, the following dimensions were chosen: length L = 1500 mm, width b = 100 mm, and height h = 150 mm. For the longitudinal reinforcing bars, φ6 (6 mm) and φ8 sizes were assumed with a 10 mm cover. Transversal φ4 stirrups were used (a reinforcement factory accepted to supply this special size). Taking into account the reduced covering, it was decided to use sand (maximum particle size of Dmax ≤ 2 mm) without coarse aggregates. It was intended that the reinforcement to be used, as well as the covering, would also be reduced in the same proportionality as the external dimensions of the beams.

Taking into account the objectives of the test to be carried out, it was intended that the combination of the materials allowed to obtain approximately half of the beams reaching failures by the reinforcement, and the other half by crushing of the mortar. Keeping this in mind, for the five beams to be constructed with each material, the only parameter to be varied, as uniformly as possible, would be the percentage of the reinforcement, which should cover a range of usual values in construction. That is, in addition to the dimensions, the compressed longitudinal reinforcement, the transversal reinforcement and the material in each group of beams remained constant. Obviously, the failure of the middle beam of each group could occur either by the reinforced bars or by crushing of mortar. Table 1 shows the designations adopted for the beams. In this table, φ indicates the number and diameter of the tensioned reinforcement of the beam, As the corresponding area, and ρ the percentage of tensioned reinforcement.

**Table 1.** Designation of the beams.

One of the initial concerns was to obtain failure by bending and not by shear. Since the critical beam would be the one under the greatest loads, it was decided to fix an amount of stirrups, which would prevent shear failure for such beam, and to use the same amount of transverse reinforcement in all the beams. Following this option, identical conditions of material confinement in all beams would also be guaranteed. This parameter is very important in bending beams. This led to 4 mm diameter transverse reinforcement bars (φ4) made by two branches stirrups spaced 70 mm. In the top of the beam, 2 φ 6 longitudinal bars were used. Figure 1 shows the reinforcement before casting.

**Figure 1.** Beams before casting.

#### *2.2. Materials*

Apart from the especial size 4 mm bars, which were courtesy of a factory nearby [21], the rest of the reinforcing bars were bought in the market. All of the bars were tested according to standard NP EN 10002-1 [22]. In these tests, four samples of each type of steel, about 40 cm long, were used. Table 2 shows the averages values of yield strength of reinforcement (fsy), ultimate strength of reinforcement (fsu), and strain of reinforcement steel at maximum load (εsu).

**Table 2.** Mechanical characteristics of steel bars.

The granulometry of the coarse aggregates to be used in the mixtures was conditioned by the size of the specimens to be cast, especially the cover. It was found that the size of the aggregates would not be appropriate and they were not used. A sand was used.

For the mortar production, Normal Portland Type I CEM I 42.5 R cement (Compressive Strength = 42.5 MPa) was used, from CIMPOR. According to the producing company, this material consists essentially of CaO (61.5%) and SiO2 (21%).

The fly ash came from the Sines thermal power plant and resulted from the burning of coal. The ashes are certified by the National Laboratory of Civil Engineering (LNEC); thus, fulfilling the compliance and performance requirements defined in Annex ZA of standard NP EN450-1 [23]. Its composition essentially contains calcium oxide, CaO (71%) and potassium oxide, K2O (16%), according to tests carried out at the Department of Earth Sciences of the University of Coimbra. Moreover, for these ashes, the high specific surface leads to great reactivity. An activator needs to be used; it must be dosed and concentrated by taking into account the binder, namely its chemical composition, and the degree of fineness that influence the activation reaction. In this work, it was decided to use an activator composed of sodium hydroxide (NaOH) and sodium silicate (NaSiO3), in the proportion of 1:2, respectively. The NaOH was obtained by mixing caustic soda with water in the appropriate proportions so that the 12.5 M molal concentration of NaOH could be achieved following the indications given by Pinto [24]. This decision also took into account the experience of other works carried out at the host laboratory regarding workability, and also the limits of the molar

concentration of NaOH so that the alkaline reaction occurs in full: 12 M according to Granizo [25] and 20 M according to Davidovits [26]. The second component of the activator, sodium silicate ("D40") was purchased from the company "Quimitecnica".

#### *2.3. Production of Beams*

The cement mortar was produced at Department of Civil Engineering of the University of Coimbra, as well as the tests carried out on small specimens (cubes and prisms) and on the beams. To follow the initial objectives, an average strength of 35 MPa was targeted for the beams. For this purpose, a mixture was carried out. The mix proportions were 270 kg of sand, 60 kg of cement, 30 kg of water, and 0.54 kg of superplasticizer Sika ViscoCrete 20HE. This mixture was sufficient for casting five beams, twelve 150 mm cubes, and six 40 × 40 × 160 mm prisms. After three days of curing, the beams, cubes and prisms were demolded.

The fly ash based mixture was done with: 200 kg of sand, 80 kg of fly ash, and 40 kg of compound activator. These proportions were those adopted from a base recipe suggested by Pinto [27]. With this mixture, five beams, twelve 40 × 40 × 160 mm prisms, and five 150 mm cubes were cast and demolded five days later.

Table 3 shows the weight and real dimensions, measured after the construction of the beams. The height (h) and the width (b) result from an average of three measurements made in the central area of the beam (the most important area of the tests). L represents the length of the beam. It appears that the greatest deviations occurred for the height (h).

**Table 3.** Effective dimensions of the beams.

Control tests were carried out on cubes and prisms, on different days, in order to evaluate the evolution of the strength.

Regarding the cement mortar tests, the results on cubes gave very close results to the expected average compressive strength of the concrete, an exponential curve proposed by Neville [28] (see Figure 2). In this figure, fcm indicates the average value of the compressive strength of concrete for the time T in days.

**Figure 2.** Average strength of the cement specimens.

Regarding the fly ash specimens, compression and/or tension strength tests were performed on twelve 40 × 40 × 160 mm test specimens, following NP EN 196-1 and NP EN12390-5 [29,30], to determine the mechanical characteristics of this material. The value for the modulus of elasticity of this material was about 20 GPa in non-destructive cyclic tests and about 18 GPa in failure tests. The average compressive and tensile strengths were 15.1 and 3.2 MPa at 23 days old and 23.5 and 4.0 MPa at 32 days old, respectively. The corresponding maximum stresses were respectively, 17.8 and 3.3 MPa at 23 days and 30.7 and 4.6 MPa at 32 days. From the compression tests of 150 mm cubes, an average compressive strength of 23.8 MPa, was obtained at 32 days, with a maximum value of 27.2 MPa. The standard deviation of the results was 2.6. It should be noted that all ash beams were tested between days 31 and 33 after pouring.

#### *2.4. Test Procedure*

Figure 3 shows the symmetric loading diagram adopted for the beam tests. The beams were simply supported, and the supports were placed at 50 mm from each end. Two vertical symmetrical loads P/2 were placed 450 mm apart, leading to pure flexion. A central zone of the beam was then under a constant bending moment. The self-weight was not considered in the diagrams shown in this figure.

**Figure 3.** Load set-up and shear and bending diagrams.

A beam under test is shown in Figure 4. The evaluation of the deformation of the beams (the deflection) was carried out using nine Linear Variable Differential Transformers (LVDTs). Three of them were placed on each support (to measure vertical and horizontal displacements) and three others were placed in the central zone (to measure the mid span deflection), as shown in Figure 4.

**Figure 4.** Beam under test.

The readings of the load cells and LVDTs were recorded by a Data Logger TML TDS-602, with a frequency of 2 Hz.

#### **3. Test Results**

The comparison of the behavior of the beams is based on key aspects of the evolution of the beam with load, such as the cracking point, Pcr, the yielding of steel, Py, the maximum load, Pmax, the ultimate strength, Pult, among others. The stiffness in each zone of the P-d graph is also an important aspect for comparison. The P-d diagram will be the basis for this analysis (P corresponds to the total load applied at each instant, and d to the displacement in the mid-span of the beam). The load P is obtained from the sum of the loads registered in the four load cells responsible for measuring the value of the support reactions. The value of the displacement in the mid-span, d, is obtained by subtracting the value of the LVDTs at the supports from the value measured of the LVDT placed in the mid-span of the beam.

The test results are presented in Figures 5–9. This type of beams generally presents three distinct phases in behavior, namely: State I, State II, and State III. State I, which corresponds to the Stiffness KI, extends from the origin to the point Pcr, corresponding to the beginning of cracking. The second state, Stiffness KII, goes from Pcr to the point where the reinforcement yields Py. Finally, the third and last State is the "plastic phase" and starts at Py and extends past the maximum load Pmax, and ending at the last load point Pult (Pult is assumed to be the load 15% lower than the maximum load, according to NP EN1998-1 [31]). In beams where the failure occurs due to reinforcement failure, this point generally coincides with the last point of the graph.

**Figure 5.** P-d experimental curves for beams Cem-2F6 and FA-2F6.

**Figure 8.** P-d experimental curves for beams Cem-3F8 and FA-3F8.

**Figure 9.** P-d experimental curves for beams Cem-4F8 and FA-4F8.

Figure 5 shows the P-d curves (load-deflection curves) for the insufficiently reinforced beams (ρ = 0.38%). The absence of State I in the fly ash beam curve is justified by the fact that these beams were cracked before the start of the test. This was provoked by the shrinkage due to the temperature of the curing time. This outcome was an assumed risk and could not be ruled out before casting. Since it did occur, this is an issue that needs to be further investigated. To cast these fly ash structures only under an optimized curing temperature of 60 to 70 degrees centigrade seems to be too restrictive to practical construction on site. Apart from this, it appears that there are no significant differences between both types of beams in terms of the stiffness of State II, nor in the values of Py and Pmax loads. In this case, the fly ash beam proved to be much more flexible and more ductile when compared to Cement (Cem) beam.

Figure 6 shows the P-d curves of the lightly reinforced beams (ρ = 0.57%). In State II, both types of beams have similar behavior. In terms of ductility, the beams are equivalent, showing both a long level of ductility, despite the type of steel used for reinforcement (cold hardened steel).

Figure 7 shows the P-d curve for beam CEM-4F6 (bellow normally reinforced: ρ = 0.75). Unfortunately, a rare malfunction of the data logger caused the loss of the recorded values of FA-4F6 beam. The relatively low value of the ductility of the beam stands out in this experimental curve.

Figure 8 shows the P-d diagrams of the normally reinforced beams (ρ = 1.01%). Looking at the Py and Pmax points, it is clear that the ductility of the cement beams is much higher than that of the fly ash beam.

Figure 9 shows the P-d curve of above normally reinforced beams (ρ = 1.34%). In this case, the ash beam revealed an absence of ductility. The cement beams also show a low level of ductility, which could be problematic for hyperstatic structures under seismic actions, for instance. The values of Py and Pmax are not very different when both beams are compared. However, the modulus of elasticity in State II is noticeably smaller for the fly ash beam when compared to cement beam. This means that the first beam is more flexible than the second one.

Table 4 shows some further information related to the tests. The "curing" refers to the age of the beam in the day of testing, ΔT is the duration of the test, and "failure by" means the type of failure shown in the tests, whether by the tensioned reinforcement, "As", by crushing the compressed material, "fc", or by failure of the stirrups, "V". Thus, all beams were tested in the time ranging from 31 to 37 days. In the case of insufficiently reinforced or lightly reinforced beams, the failure occurred due to the failure of the tensioned reinforcement, as expected. In the other cases, the failures occurred due to crushing of the compressed material, except for the FA-3F8 beam, where the failure occurred in a shear crack, in this case due to insufficient connection of the stirrup to the surrounding material (this was observed during the visual inspection of the zone after the test).

**Table 4.** Further information related with tests.

ΔT = duration of test; As = failure by longitudinal steel bars; fc = failure by crushing of concrete; V—failure by shear (stirrups).