Rational Outline of Timber Beams

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1. Introduction Load-bearing timber structures (TS) have found widespread application in the construction of buildings and structures due to their physical and mechanical properties, as well as their renewability, carbon neutrality, and economic efficiency [1-5]. Study [6] is devoted to comparing TS with other types of building structures. One way to improve their performance is to refine structural solutions by seeking improved cross-sectional shapes, rational axis outlines of elements, and through the use of variable rigidity, as discussed in [7]. Adopting a structurally justified solution at the design phase allows to reduce the cost of the structure and cut construction and operational expenses, as demonstrated by the design of Olympic facilities made of timber in the Russian Federation. Therefore, improving structural solutions for these load-bearing elements remains a relevant task for the construction industry. Adopting a technically justified solution at the design phase helps reduce the overall cost of the structure and lower the construction and operating costs of the entire building. This process involves modeling and complex mathematical calculations. Existing approaches include numerous variational methods, as well as the well-developed theory of uniform strength [8], and the numerical finite element methods described in [9; 10]; however, they require significant computational resources and specialized software. Variability of the properties of wood, as a natural material, poses challenges for accurately evaluating the mechanical characteristics of glued laminated beams. Therefore, it is crucial to develop and validate models for predicting the mechanical characteristics in order to fully exploit the potential of this material [11]. A stochastic model, which enables the establishment of relationships between safety factors and individual mechanical characteristics by combining structural analysis tools with probabilistic descriptions of timber planks, has become an important step in the design of timber structures, including for the purpose of enhancing the competitiveness of timber compared to other building materials [12]. The transversely isotropic model proved to be the most appropriate for describing glued structures, due to the averaging of anisotropic properties in a multilayer solid [13]. Structural optimization is closely linked to the grain distribution along the cross-section. The grain angle should depend on the magnitude of the applied load, which led to development of a nomogram for the rational lamination, in which the proportion of low-quality timber should not exceed 50%. Experimental and numerical analyses [14] indicate the advantages of using hardwood for manufacturing glued laminated beams, particularly in a combined configuration where the layers in the middle of the cross-section are made of low-grade timber, which opens up broad opportunities for resource optimization. Study [15] also demonstrates the effective use of resources in glued laminated beams by inserting lower-quality lamellas into less-loaded zones, typically in the center of the beam. The influence of geometric parameters on the stress state was numerically determined in [16]. The relationship between the section height and stress is parabolic, while the relationship between the wall thickness and modulus of elasticity is linear. The authors confirm the theory that, in the beam outline, the height axis is the most critical parameter for the structural shape. Timber lamellas in glued laminated structures contain knots formed as a result of the natural growth process of trees, which significantly affect the mechanical properties [17]. Accounting for the location of knots and finger joints in glued laminated beams, that can cause significant stress and strain concentrations, is important, as demonstrated in [18]. Special stresses are concentrated in the support zones of deep beams, during the study of which an engineering calculation of maximum stress was proposed, taking into account the rheological properties of wood, and a method of reinforcement with polymers based on glass fabric with carbon nanotubes was developed, increasing the horizontal shear strength by 10-20% [19]. The developed reinforcement method demonstrated the possibility of conserving wood resources in glued laminated beams while maintaining the design strength [20]. Study [21] demonstrates that replacing up to 34% of the lamellas with lower-quality timber (after fire exposure resulting from a forest fire) reduces the load-bearing capacity by only 2.5%. Approaches to improving beam geometry have evolved from individual formulas to comprehensive analysis and design methods [22]. In [23], a solution for the skew bending of rectangular beams using the Lagrange method was proposed, which minimizes the cross-sectional area subject to strength constraints. In [24], a variational analysis method was developed and a system of integral equations for calculating forces in all components of the structure of a multi-layer beam was obtained. A simple optimization method for “scissor” trusses, developed in [25], revealed that the slope of the upper chord minimizes forces due to scaling of the load diagrams. Finite element methods (FEM) have become the standard for analyzing complex structures. Numerical modeling of glued laminated timber beams is a complex problem; for example, the quasi-brittle nature of wood requires modeling of progressive failure mechanisms and nonlinear behavior under load [26]. Simulation in MATLAB can predict flexural rigidity with high accuracy and provide sufficiently accurate predictions of the flexural strength of glued laminated beams [27]. Developing a model in the ANSYS software allows to simulate stress states of any complexity. In the practice of large-scale design, modelling of a lattice timber dome with a diameter of 100 m allowed to reduce wood consumption by 10-15% [28]. The method for buckling analysis of glued laminated beams with variable cross-sections, proposed by the authors [29], revealed an underestimation of the design coefficients when using the Bubnov - Galerkin method. Compound timber elements connected by dowels were studied in [30], showing compliance coefficients in the range of 0.85-0.93. There is a trend toward increased use of glued laminated timber in increasingly slender structures, which requires special attention to stability issues, such as lateral buckling [31]. Lateral buckling of narrow beams with bracing, investigated in [32], was refined in an orthotropic formulation with an equation that revealed overestimation of standard values for a single bracing. Study [33] is devoted to the investigation of out-of-plane bending of glued laminated structures, presenting a comparison of the bending and shear characteristics of CLT and GLT beams. The development of the strength criteria proceeded in parallel with the refinement of standards, as shown in [34]. The authors of the study demonstrated the inadequacy of the Navier - Zhuravsky calculations for glued laminated structures and proposed Ashkenazi tensor criterion with checks for radial stresses and splitting across the grain in the near-support zones. In [35], a critical reassessment of building codes for timber structures was performed, and errors in the coefficients for lateral buckling were identified. The authors prepared tables for all types of restraints, refining the formulas and force diagram shapes. Field test results confirmed the theoretical hypotheses with a discrepancy of no more than 12%. The results demonstrate the need to refine the regulatory framework for structurally justified timber structure outlines and the validity of applying the principle of uniform strength in practical calculations. Practical implementation confirms the justification of the improvements to the TS shapes. Technically justified structures are being actively incorporated into the designs of real projects. Compound composite timber beams with mechanical connections, developed in [36], have increased the strength by a factor of 1.5-1.8 and stiffness by a factor of 2.5-3. Software [37] allows to automate the calculation of prestressed compound beams with uniform stress distribution. Having conducted the analysis of the use of timber structures in construction, including ways to improve the shape and configuration of cross-sections, it can be concluded that the study of load-bearing TS remains relevant. Contemporary researchers are exploring structural solutions aimed at conserving timber resources without compromising the overall strength. Most of the research is related to the use of composite materials, compound structures, or refinement of design codes. However, the use of technically justified structural shapes or cross-sectional configurations is not widely covered in the literature. These factors allow the authors to formulate the objective of this study, which is to determine the cross-sectional shape of a timber beam that will make the most complete use of the strength characteristics of structural timber, and to evaluate the effectiveness of the resulting designs. The object of this study is a simply supported glued laminated timber floor or roof beam with a rectangular cross-section and subjected to a uniformly distributed load. The ratio of the wood consumption of beams with a rational outline to that of beams with constant rigidity along their entire length is used as a criterion for evaluating the effectiveness of this design. The outline of a rational timber beam (RTB) is determined by constructing tangents to the function of the height of the cross-section of a beam of uniform-strength with a constant cross-sectional width. This method ensures a linear variation in rigidity in the near-support zones of the structure, while the cross-sectional height remains constant in the middle of the span. 2. Methods In the first stage of the study, the contour of a beam of uniform strength was constructed using diagrams of the relationship between the section height and the distribution of bending moments and shear forces along the axis of the member (Figure 1). The height of the cross-section of the beam of uniform strength was determined based on the strength conditions for timber beams with respect to normal and shear stresses: , (1) , (2) where M(x) = qx(L - x)/2 is the function of the bending moment distribution along the beam axis; Q(x) = q(L/2 - x) is the function of the shear force distribution along the beam axis; W(x) = bh(x)2/6 is the section modulus of the beam of variable rigidity; S(x) = bh(x)2/8 is the statical moment of the beam of variable rigidity; J(x) = bh(x)3/12 is the moment of inertia of the beam of variable rigidity; b is the width of the beam section; , are the design strengths of timber in bending and horizontal shear, respectively. The expressions for M(x) and W(x) were substituting into formula (1), the expressions for Q(x), S(x) and J(x) were substituted into formula (2), and the variable section height h(x) was then expressed: (3) (4) Next, three tangents to the curve of function were constructed. One of them ran horizontally through its extremum (point D), the vertical coordinate of which corresponds to the maximum height of the cross-section at the midpoint of the span . The other two tangents were drawn from points A and G, the vertical coordinates of which correspond to the heights of the support cross-sections . These tangents were, in turn, the extrema of function and touched the graph of function at points B and E. Solving the problem of determining the rational outline of the timber beam reduced to determining the coordinates of the nodal points of the contour A, B, C, D, E, F, G (see Figure 1). Figure 1. Diagram for determining the rational outline of a timber beam S o u r c e: made by A.V. Repin. In the second stage, half of the beam span from the origin was considered and the coordinates of points A, D, and G were determined. A coordinate system with its origin at point O was defined, which coincided with the starting point of diagram and the y-axis was oriented vertically downward. The horizontal coordinates of points A, D, and G corresponded to the beginning, middle, and end of the beam span: = 0; = L/2; = L. The vertical coordinates of points A, D, and G corresponded to the heights of the support sections and the middle of the span (maximum height): ; . The final coordinates of the points were as follows: A(0;), D(L/2;), G(L;). Thus, based on the above, the following equations were obtained: , (5) (6) In the next stage, the equations for the tangent line to curve were composed: (7) where Next, the position of point B was determined using the equation of the tangent line (7). For this, it was specified in the equation that and the coordinates of point A were substituted for x and y: . Then, using the equation given above, was expressed: . (8) The horizontal coordinate of point B is numerically equal to the distance from the support axis to the critical section in the near-support zones of variable rigidity. The vertical coordinate of point B was determined by substituting . (9) The vertical coordinates of points B and F are numerically equal to the height of the critical section in the near-support zones of variable rigidity: To determine the coordinates of the points where tangents C and E intersect, y in the equation of the tangent (7) was substituted with expression (5) for : Next, the value of from the equation given above was derived: . (10) Substituting in equation (10) with the expression for , the following is obtained: . (11) The horizontal coordinate of point C is numerically equal to the length of the undercut in the support zones of the beam: In the next stage of the study, based on the expressions given above, calculations were performed to determine the rational outline of the timber beam, and the results of the study were analyzed. During the process of constructing the trajectory of the beam of uniform strength, cross-section heights were obtained for which the ratio of internal forces to the ultimate strength was equal to one; therefore, the structural capacity was virtually exhausted. It was decided that when determining the outline of a rational beam, a safety factor of, for example, 8% is applied. The values of the design strengths will be multiplied by safety factor h = 1.08. Design parameters: ¡ of 1st-grade pine timber: - bending and tensile strength = 21.0 MPa; - shear strength along the grain of glued members in bending = 2.4 MPa. ¡ for 2nd-grade pine timber: - bending and tensile strength = 19.5 MPa; - shear strength along the grain of glued members in bending = 2.25 MPa. Values of service condition factors: - long-term strength factor = 0.66 is adopted for service conditions involving the combined action of dead, long-term temporary and occupancy loads on the floors of residential and public buildings, or the combined action of dead and short-term snow loads; - coefficient depends on the height of the rectangular cross-section of glued laminated elements in bending. In this study, the calculation was performed for beams with different cross-sectional heights; therefore, in each case, the values of , and the design strength values of timber in bending and horizontal shear will differ (see Table 1); - the values of the remaining service condition factors such as are taken as equal to one, and are not used. Rational timber beams (RTB) with spans of 6.0, 9.0, 12.0, and 15.0 m were examined, with cross-sectional dimensions relative to the span as follows: height h = (L/10; L/12; L/15); width b = (h/4; h/5; h/6). The analysis was performed in two directions: - at each step, relative cross-sectional dimensions h/L and b/h were assigned, based on which the absolute values of h and b, the geometric characteristics of the cross-section, etc., were calculated; - by varying the value of the design load q, a perfect match was achieved between the values of the extrema of the trajectory of a beam of uniform strength and the absolute value of h. Table 1. Values of design strength of timber taking into account service condition factors and the specified safety factor Section height, cm Design characteristics 50 and lower 60 70 80 100 120 and higher Coefficient 1 0.96 0.93 0.90 0.85 0.8 MPa 13.86 13.31 12.89 12.47 11.78 11.09 MPa 1.584 1.521 1.473 1.426 1.346 1.267 S o u r c e: made by M.S. Lisyatnikov. Based on the data provided above, the coordinates of the key nodal points governing the rational outline of the timber beam were determined. Next, the geometric parameters of the RTB were calculated: the height of the support section , the height of the critical section in the zones of variable beam rigidity , the length of the undercut in the near-support zones , and the undercut angle . The economic efficiency of the new structural design was evaluated by comparing the difference in wood consumption between the RTB and beams of constant rigidity. For the same cross-sectional width, the wood consumption is numerically equal to the ratio of the lateral surface areas of the compared structures: (12) where ,,, are the volumes and surface areas of the rational beam and the constant (rigidity) beam; h, are the cross-sectional heights at the midspan and at the supports, respectively; L is the span of the beam; is the length of the undercut in the near-support zones of the beam. The calculation process and verification of the results were performed using the MS Excel spreadsheet program. 3. Results and Discussion Figure 2 shows the diagrams of RTB with a span of 9.0 m for L/h = 10, 12, and 15, generated using the Chart Wizard in MS Excel. The results of the analysis are presented in Table 2. They show that the relative parameters of the rational beam outline depend on the value of the relative section height h at the midspan. Other characteristics - span L, cross-sectional width b, etc. - have virtually no effect. It has been established that as the section height h increases, the values of such relative values as the length of the undercut the height of the support section , and the coordinate of the critical section in zones of variable rigidity also increase. By performing calculations for intermediate values of the relative section height, taking into account the strength characteristics of structural timber of grades 1 and 2, the relative geometric characteristics of the RTB as a function of various L/h ratios are obtained. The calculation results are presented in Tables 3 and 4 and in Figures 3 and 4. а b с Figure 2. General view of rational beam outlines using the example of structures with a span of L = 9.0 m for: a - L/h = 10; b - L/h = 12; c - L/h = 15 S o u r c e: made by A.V. Repin. Table 2. Main results of the calculation for determining the parameters of the rational outline of timber beams made of grade 1 timber L, m Parameters and indices h = L/10 h = L/12 h = L/15 b = h/4 b = h/5 b = h/6 b = h/4 b = h/5 b = h/6 b = h/4 b = h/5 b = h/6 6.0 q, kN/m 26.61 21.29 17.74 16.04 12.83 10.69 8.21 6.57 5.48 h, m 0.600 0.500 0.400 , m 0.525 0.365 0.233 b, m 0.150 0.120 0.100 0.120 0.096 0.080 0.100 0.080 0.067 , m 2.602 2.083 1.523 , m 2.800 2.530 2.211 , о 1.534 3.064 4.312 0.875 0.729 0.583 0.434 0.347 0.254 0.467 0.422 0.368 w, % 6.2 12.9 18.1 9.0 q, kN/m 35.34 28.28 23.56 21.66 17.33 14.44 11.83 9.46 7.88 h, m 0.90 0.75 0.60 , m 0.79 0.55 0.35 b, m 0.23 0.18 0.15 0.19 0.15 0.13 0.15 0.12 0.10 , m 3.903 3.124 2.285 , m 4.200 3.795 3.316 , о 1.535 3.064 4.312 0.875 0.729 0.583 0.434 0.347 0.254 0.467 0.422 0.368 w, % 6.2 12.9 18.1 Ending of the Table 2 L, m Parameters and indices h = L/10 h = L/12 h = L/15 b = h/4 b = h/5 b = h/6 b = h/4 b = h/5 b = h/6 b = h/4 b = h/5 b = h/6 12.0 q, kN/m 44.35 35.48 29.57 27.27 21.82 18.18 14.78 11.83 9.86 h, m 1.20 1.00 0.80 , m 1.05 0.73 0.47 b, m 0.30 0.24 0.20 0.25 0.20 0.17 0.20 0.16 0.13 , m 5.203 4.165 3.047 , m 5.60 5.06 4.42 , о 1.535 3.064 4.312 0.875 0.729 0.583 0.434 0.347 0.254 0.467 0.422 0.368 w, % 6.2 12.9 18.1 15.0 q, kN/m 55.43 44.35 36.96 32.08 25.67 21.39 17.45 13.96 11.64 h, m 1.50 1.25 1.00 , m 1.31 0.91 0.58 b, m 0.38 0.30 0.25 0.31 0.25 0.21 0.25 0.20 0.17 , m 6.50 5.21 3.81 , m 7.00 6.33 5.53 , о 1.536 3.065 4.312 0.875 0.729 0.583 0.434 0.347 0.254 0.467 0.422 0.368 w, % 6.2 12.9 18.1 S o u r c e: made by M.S. Lisyatnikov. Table 3. Summary table of calculation results for beams made of grade 1 timber Parameters and indices Relative height of the section h L/10 L/10.5 L/11 L/11.5 L/12 L/12.5 L/13 L/13.5 L/14 L/14.5 L/15 , о 1.535 2.000 2.404 2.755 3.064 3.337 3.577 3.792 3.984 4.156 4.312 0.875 0.833 0.795 0.761 0.729 0.700 0.673 0.648 0.625 0.603 0.583 0.434 0.410 0.388 0.367 0.347 0.329 0.312 0.296 0.281 0.267 0.254 0.467 0.455 0.443 0.432 0.422 0.412 0.402 0.393 0.385 0.376 0.368 w, % 6.2 8.2 10.0 11.5 12.9 14.1 15.1 16.1 16.9 17.5 18.1 S o u r c e: made by M.S. Lisyatnikov. Table 4. Summary table of calculation results for beams made of 2nd grade timber Parameters and indices Relative height of the section h L/10 L/10.5 L/11 L/11.5 L/12 L/12.5 L/13 L/13.5 L/14 L/14.5 L/15 , о 1.645 2.107 2.505 2.855 3.159 3.429 3.669 3.881 4.071 4.242 4.395 0.867 0.825 0.788 0.754 0.722 0.693 0.667 0.642 0.619 0.598 0.578 0.429 0.405 0.383 0.362 0.343 0.325 0.308 0.292 0.277 0.263 0.250 0.464 0.452 0.441 0.430 0.419 0.409 0.400 0.391 0.382 0.374 0.366 w, % 6.6 8.6 10.3 11.8 13.2 14.4 15.4 16.3 17.0 17.7 18.3 S o u r c e: made by M.S. Lisyatnikov. L/h Figure 3. Diagram of the relationship between the relative parameters of the rational outline of timber beams and the relative cross-section height L/h. 1st grade timber S o u r c e: made by A.V. Repin. L/h Figure 4. Diagram of the relationship between the relative parameters of the rational outline of timber beams and the relative cross-section height L/h. 2nd grade timber S o u r c e: made by A.V. Repin. Calculations show that including a safety factor and/or changing the service condition factors has no influence whatsoever on the relative values of the rational beam parameters. The absence of influence indicates that the dominant factor is the ratio between the values of the design strengths to bending and horizontal shear, while the service condition factors and safety factors, judging from negligible influence, are similarity factors. A techno-economic evaluation of the RTB structural designs shows that their economic efficiency increases as the section height decreases from L/h = 10 to L/h = 15: from 6.2 to 18.1% for grade 1 timber and from 6.6 to 18.3% for grade 2 timber (Figure 5). At the same time, the difference between the corresponding economic efficiency values for grade 1 and grade 2 timber is within 0.4%. Thus, the influence of timber grade on the economic efficiency of RTB structural designs is insignificant. Given the presence of a large number of factors with insignificant influence on the parameters of the rational beam, it follows that the RTB analysis method will reduce to determining the relative section height L/h and subsequently selecting the RTB parameters from the corresponding table. Intermediate values are determined by interpolation. Future research will focus on developing the methodology for RTB analysis and conducting a comparative analysis with the results of numerical studies using FEM. L/h Надпись: w, %Название: Экономичность, % Figure 5. Diagram of the relationship between the economic efficiency (ω) of the RTB and the relative height of the section L/h S o u r c e: made by A.V. Repin. 4. Conclusion The following conclusions were drawn as a result of the study: 1. Theoretical studies of single-span timber beams were conducted to determine their rational outline based on the uniform strength trajectories of beams with constant cross-sectional widths in bending and horizontal shear. Expressions were derived to determine the coordinates of the nodal points of the rational beam contour, and calculations were performed to determine the absolute and relative values of the geometric parameters of rational timber beams for various spans and cross-sectional dimensions; 2. Analysis of the calculation results showed that the relative values of the shape parameters of a rational beam depend on the relative section height h at midspan. Other characteristics - length L, cross-sectional width b, etc. - have virtually no effect. It has been established that as the section height h increases, the values of the following relative values also increase: - the length of the undercut - the height of the support section - the coordinate of the critical section in zones of variable rigidity. 3. A techno-economic evaluation of structural designs for rational timber beams shows that the economic efficiency (wood consumption relative to a beam of constant rigidity) increases as the cross-sectional height decreases: from 6.2 to 18.1% for grade 1 timber and from 6.6 to 18.3% for grade 2 timber; the difference between the corresponding values is within 0.4%. The insignificant difference in wood consumption indicates that the grade of timber has a negligible effect on the economic efficiency of the studied structures.

About the authors

Mikhail S. Lisyatnikov

Vladimir State University named after Alexander and Nikolay Stoletovs

Email: mlisyatnikov@mail.ru
ORCID iD: 0000-0002-5262-6609
SPIN-code: 4089-7216

Candidate of Technical Sciences, Associate Professor of the Department of Building Structures

87 Gorky St., Vladimir, 600000, Russian Federation

Alexander V. Repin

Vladimir State University named after Alexander and Nikolay Stoletovs

Email: avlr@bk.ru
ORCID iD: 0009-0003-3011-7883
SPIN-code: 8623-6207

Student, Department of Building Structures

87 Gorky St., Vladimir, 600000, Russian Federation

Kirill M. Terentyev

Vladimir State University named after Alexander and Nikolay Stoletovs

Email: terenzeret@gmail.com
ORCID iD: 0009-0001-8678-1121
SPIN-code: 5134-7370

Postgraduate student, Assistant of the Department of Building Structures

87 Gorky St., Vladimir, 600000, Russian Federation

Svetlana I. Roshchina

Vladimir State University named after Alexander and Nikolay Stoletovs

Author for correspondence.
Email: rsi3@mail.ru
ORCID iD: 0000-0003-0356-1383
SPIN-code: 4159-8636

Doctor of Technical Sciences, Professor, Head of the Department of Building Structures

87 Gorky St., Vladimir, 600000, Russian Federation

References

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