Designing light-weight structures consisting of multiple elements loaded in bending

Abstract

The aggregation method for designing light-weight structures consists of consolidating multiple loaded elements into fewer elements with larger cross-sections under the same total load. This approach is very effective for beams with arbitrary cross-sectional shapes and leads to a significant reduction in the material required to support a given total load. For instance, in a five-to-one aggregation, the material usage is reduced by a factor of 1.71. The theoretical foundations of the method of aggregation have been rigorously developed and proven, specifically for structures composed of beams with arbitrary cross sections. Additionally, finite element simulations and experimental testing on statically indeterminate structures validated the method and confirmed its effectiveness. These investigations demonstrated that aggregating loaded elements substantially increases load-bearing capacity while reducing material consumption. It has also been established that segmenting multiple loaded beams into a larger number of beams, under the same total load, increases the total elastic strain energy of the structure. This result holds true for arbitrary cross-sectional shapes. Finally, an interpolation method has been proposed for reducing the total volume of material at a specified maximum von-Mises stress and maximum deflection.

Keywords

Light-weight design load capacity aggregation segmentation structural optimisation cantilever beams simply loaded beams frames bending

Introduction

Design optimisation is widely applied in mechanical and structural engineering to identify optimal design solutions by simultaneously considering multiple objectives.^1–3 This approach evaluates various criteria, such as strength, durability, weight, cost and performance, under specific physical, geometric, material, and economic constraints.

Multi-objective optimisation methods, for example, can generally be classified into two categories: classical and evolutionary. Classical methods typically aim to identify a single optimal solution, often through techniques such as weighted sums or constraint handling. In contrast, evolutionary methods, such as genetic algorithms and particle swarm optimisation, employ population-based search strategies to generate a diverse set of Pareto-optimal solutions, offering trade-offs among competing objectives. Ultimately, this optimisation framework supports the development of more balanced, efficient, and robust mechanical designs that better address the diverse requirements of customers and environmental considerations.

Design optimisation oriented towards the optimisation of the shape of components is the topology optimisation, originally developed as a mainstream design technique for achieving lightweight structures. In recent years, it has been widely applied to create efficient, lightweight designs.^4–7 Various topology optimisation approaches have emerged, including the evolutionary structural optimisation method, the density method and the level set method.⁴

Recent advancements in manufacturing technologies, particularly additive manufacturing (3D printing)^7–9 and advanced metal forming techniques,¹⁰ have significantly expanded the possibilities for implementing topology optimisation. Additive manufacturing enables the fabrication of complex, optimised geometries that are difficult or impossible to achieve with traditional manufacturing methods. It also facilitates the integration of multiple functions into a single component, improving load-bearing capacity while maintaining or reducing overall weight.

The combination of materials in structural components has also been leveraged to create composite beams with enhanced flexural strength, often surpassing that of the individual constituents.¹¹ Notable examples include beams made from glass fibre- and carbon fibre-reinforced composites, steel-reinforced concrete and sandwich structures. In sandwich beams, for instance, the principle of functional separation is employed: strong materials are placed away from the neutral axis to resist most of the load, while lightweight, low-strength materials near the neutral axis provide structural stability.^11–13

Aggregation techniques have already been applied in certain structural designs, where steel plates are welded together to form reinforced beams, thereby increasing the overall load-carrying capacity by enabling the beam to resist higher bending moments and shear forces.¹⁴ The structural behaviour of aggregated steel box beams filled with concrete has also been investigated.¹⁵

In this paper, we introduce a different concept of aggregation: the consolidation of multiple structural components, made of the same material, into fewer elements with larger, geometrically similar cross-sections.

For loaded components with arbitrarily shaped cross-sections, the potential of aggregation and segmentation for optimising load capacity remains largely unexplored. Although multiple loaded elements are commonly used in engineering and construction, the stress analysis literature lacks focused studies on the implications of aggregation and segmentation.^13,16–20 This is a surprising omission, given the conceptual simplicity and practical relevance of aggregation.

Moreover, structural engineering textbooks^21–23 and studies on the optimisation of loaded beams^24,25 do not address this concept. Similarly, the structural reliability literature offers no discussion on the topic.^26,27

This absence of discourse on the aggregation and segmentation of components with arbitrary cross-sections reveals a significant knowledge gap and underscores the need for further research in this area.

We emphasise that the proposed aggregation method represents a fundamentally different approach from standard topology optimisation. It has been developed independently, without relying on any topology optimisation algorithms.

Topology optimisation typically involves significant computational overhead due to the large number of iterative simulations required. In contrast, the aggregation method introduced here avoids iterative procedures entirely and involves minimal computational cost. Moreover, unlike topology optimisation, the performance of the proposed method is not influenced by the size of the individual components comprising the structure.

Traditional manufacturing techniques such as casting, forging and machining, often struggle to accommodate the intricate geometries produced by topology optimisation. These complex shapes may also introduce stress concentration zones, which can lead to premature fatigue failure. The proposed aggregation method does not suffer from these drawbacks. In fact, it offers substantial weight savings, making it a highly efficient and practical alternative for structural design optimisation.

Bending, or flexural loading, is among the most common loading conditions encountered in engineering structures. Elements with enhanced bending capacity are less prone to premature failure or fatigue due to overloading or dynamic loading, resulting in extended service life, lower maintenance requirements and improved safety.

Accordingly, this study aims to explore the effectiveness of the aggregation method through theoretical analysis, computational simulations, and experimental testing with physical prototypes. The specific objectives are as follows:

(i) To provide a theoretical foundation for the aggregation method for load-carrying elements with arbitrarily shaped cross-sections;

(ii) To compare the material needed for supporting a given total load between aggregated and non-aggregated structures with arbitrarily shaped cross-sections;

(iii) To compare the load-carrying capacities of aggregated and non-aggregated structures constructed with the same total volume of material;

(iv) To evaluate the impact of segmentation on the accumulated elastic energy.

The findings from these investigations constitute the primary contributions of this paper and offer direct implications for structural design, particularly in enhancing load capacity and reducing the risk of failure in structures subjected to bending.

Impact of aggregation and segmentation on critical properties of structures

Consider a non-aggregated and aggregated structure. The non-aggregated structure is composed of on n load-carrying elements while the aggregated structure is composed of a smaller number m of load-carrying elements (m < n). The cross sections of the aggregated and non-aggregated structure are geometrically similar, with a coefficient of proportionality p > 1 relating the geometrical dimensions of the non-aggregated and aggregated elements. Let both the non-aggregated and aggregated structure be subjected to the same total load P. Then, the load per loaded element in the non-aggregated structure is P/n while the load per element in the aggregated structure is P/m.

Segmentation is the opposite process to aggregation and is defined in a similar fashion: a number m of loaded elements are segmented into a larger number n of loaded elements with geometrically similar cross sections (m < n).

Suppose that $s$ is a particular critical property (e.g. maximum deflection, maximum tensile stress, elastic strain energy, etc.). Let $s_{1}$ be the critical property s characterising an element from the non-aggregated structure while $s_{2}$ be the critical property characterising an element from the aggregated structure.

As it will be demonstrated later, most of these critical quantities can be summaries by the equations:

s_{1} = k \frac{P^{t} / n}{G_{1}}

(1)

s_{2} = k \frac{P^{t} / m}{G_{2}}

(2)

where $k$ and t are constants, $G_{1}$ and $G_{2}$ are geometric characteristics (e.g. length, cross-sectional area, volume, second moment of area, polar moment of area, etc.) of a loaded element from the non-aggregated and a loaded element from the aggregated structure. Considering the coefficient of proportionality p between the dimensions of the non-aggregated and the aggregated elements, the relationship between $G_{1}$ and $G_{2}$ is given by

G_{2} = p^{r} G_{1}

(3)

where $r$ is a positive integer number greater than zero. Taking the ratio of expressions (1) and (2) and considering (3) gives

\frac{s_{1}}{s_{2}} = \frac{m}{n} \times \frac{G_{2}}{G_{1}} = \frac{m}{n} \times p^{r}

(4)

Let us denote $q = \frac{m}{n} \times p^{r}$ . If $q > 1$ then $s_{1} > s_{2}$ or the aggregation decreases the magnitude of the critical property in the aggregated structure while a segmentation of the structure increases the critical property. If $q < 1$ , the aggregation increases the magnitude of the property in the aggregated structure while a segmentation decreases it. If $q = 1$ , the aggregation or segmentation has no effect on the critical property. Equation (4) provides a summary formulation of the effects of aggregation and segmentation of load-carrying elements. The application of equation (4) for improving the design of structures will be demonstrated on non-aggregated and aggregated structures loaded in bending.

Theoretical justification of the aggregation method as a design optimisation method reducing the weight of structures

Let the stress permitted by the material be $σ_{cr}$ . The non-aggregated structure in Figure 1(a) includes n identical beams while the aggregated structure in Figure 1(b) includes a fewer number m of identical beams (m < n) with larger cross-sectional area.

Figure 1.

(a) Non-aggregated structure built on n cantilever beams, (b) aggregated structure built on m cantilever beams (m < n), (c) a beam cross section from the non-aggregated structure and (d) a geometrically similar beam cross-section from the aggregated structure.

The beams cross-sections from the non-aggregated and aggregated structure are with arbitrary shape and geometrically similar, with a coefficient of proportionality p > 1 (scaling factor) between the corresponding dimensions (see the example in Figure 1(c) and (d)). For the sake of simplicity in presenting the relevant theory, a uniform cross section is assumed along the length of the beam.

Let $σ_{cr}$ be the permissible tensile stress for the material. To support the same total load P without exceeding the critical stress $σ_{cr}$ , the corresponding total volumes of material for the non-aggregated and aggregated structure will be denoted by $V_{1}$ and $V_{2}$ , correspondingly. The maximum deflections of in the non-aggregated and aggregated structures at the same specified total load P, will be denoted by $δ_{1}$ and $δ_{2}$ , correspondingly. The length of the elements from the non-aggregated and aggregated structure is equal to L. An arbitrarily shaped cross-section of the non-aggregated beams has been assumed.

Theorem 1: For a non-aggregated and aggregated structure carrying the same total load, built on n and m identical cantilever or simply supported beams correspondingly (m < n), with geometrically similar cross sections, the following relationship exists:

V_{1} / V_{2} = (n / m)^{1 / 3}

(5)

δ_{1} / δ_{2} = (n / m)^{1 / 3}

(6)

where $V_{1}, V_{2}$ are the volumes of material needed to support the specified total load and $δ_{1}, δ_{2}$ are the ratios of the maximum deflections at the specified total load.

Proof: Because of the geometric similarity, an elementary surface area $ds = Δ x Δ y$ in Figure 1(c) will be transformed into the elementary area $ds' = p^{2} Δ x Δ y$ in Figure 1(d), where p is the scaling factor relating the dimensions of the geometrically similar cross sections of the aggregated and non-aggregated structure. As a result, between the cross-sectional area $S_{1} = \int \int dxdy$ of the non-aggregated structure and the cross-sectional area $S_{2} = \int \int dx' dy'$ of the aggregated structure, the following relationship exists:

S_{2} = \int \int dx' dy' = p^{2} \int \int dxdy = p^{2} S_{1}

(7)

For a bending moment M acting on a cantilever beam, the maximum tensile stress in the beam is determined from the classical formula:

σ = \frac{M}{I} \hat{y}

(8)

In equation (8), I denotes the second moment of area of the cross-section and $\hat{y}$ denotes the largest distance of a point from the cross section from the neutral axis. Because of the geometric similarity between the largest distance ${\hat{y}}_{1}$ characterising the non-aggregated cross-section and the largest distance ${\hat{y}}_{2}$ characterising the aggregated cross-section, the link:

{\hat{y}}_{2} = p {\hat{y}}_{1}

(9)

exists. The loading moment $M_{1}$ on a single element of the non-aggregated structure is $M_{1} = (P / n) L$ while the loading moment $M_{2}$ on a single element of the aggregated structure is $M_{2} = (P / m) L$ .

Now, following the definition of a second moment of area, let

I_{1} = \int \int y^{2} dxdy

(10)

be the second moment of area for the non-aggregated sections (Figure 1(c)), where y is the distance of the infinitesimally small area $dxdy$ from the neutral axis of the cross section. The second moment of area of the cross-section of an aggregated beam (see Figure 1(d)) is

I_{2} = \int \int y'^{2} dx' dy'

(11)

where $y'$ is the distance of the infinitesimally small area $dx' dy'$ from the neutral axis of the cross-section of the aggregated beam. Because of the geometric similarity between the non-aggregated and aggregated cross-sections, for the infinitesimally small area $dx' dy'$ we have: $dx' = pdx$ and $dy' = pdy$ . In addition, because of the geometric similarity between the two sections we have: $y' = py$ . The substitution in (11) results in:

I_{2} = p^{4} \int \int y^{2} dxdy = p^{4} I_{1}

(12)

As a result,

I_{2} / I_{1} = p^{4}

(13)

of the aggregated and non-aggregated cross-sections.

For a non-aggregated beam, equation (8) gives:

σ_{cr} = \frac{(P / n) L {\hat{y}}_{1}}{I_{1}}

(14)

while for an aggregated beam, equation (8) gives

σ_{cr} = \frac{(P / m) L {\hat{y}}_{2}}{I_{2}}

(15)

Taking the ratio of (14) and (15) results in

(m / n) \times \frac{I_{2} {\hat{y}}_{1}}{I_{1} {\hat{y}}_{2}} = 1

(16)

Substituting (9) and (13) in (16) results in

(m / n) \times p^{3} = 1

(17)

from which:

p = (n / m)^{1 / 3}

(18)

The total volume of the non-aggregated structure is $V_{1} = n S_{1} L$ while the total volume of the aggregated structure is $V_{2} = m S_{2} L$ , where $S_{1}$ and $S_{2}$ are the cross sections of the non-aggregated and aggregated beams, correspondingly. Considering (7), we have

\frac{V_{1}}{V_{2}} = \frac{n S_{1} L}{m S_{2} L} = \frac{n}{m} \times \frac{1}{p^{2}}

(19)

Substituting (18) in (19) finally gives:

\frac{V_{1}}{V_{2}} = {(\frac{n}{m})}^{1 / 3}

(20)

This completes the proof of assertion (5) of Theorem 1.

Note that considering (18), we also have:

\frac{V_{1}}{V_{2}} = p

In words, the ratio of the volumes necessary to support the total load, in the non-aggregated and aggregated structure, is equal to the scaling factor relating the dimensions of the geometrically similar cross sections.

For n-to-one aggregation (m = 1), from equation (20), it follows that

\frac{V_{1}}{V_{2}} = n^{1 / 3}

(21)

Figure 2 presents the volume of material reduction factor for n = 2,3,4 and 5. As can be seen from Figure 2, the reduction of material needed to support the same specified total load is impressive. For five-to-one aggregation, for example, the reduction in material used is 1.71 times!

Figure 2.

Material reduction factor for n-to-1 aggregation (n = 2,3,4,5). The aggregated and non-aggregated structures carry the same total load P.

From the classical stress analysis theory,^13,16 for the deflection at the free end of the non-aggregated cantilever beams, the value

δ_{1} = \frac{(P / n) L^{3}}{3 E I_{1}}

(22)

is obtained and for the deflection at the free end of the aggregated cantilever beam, the value

δ_{2} = \frac{(P / m) L^{3}}{3 E I_{2}}

(23)

is obtained, where E is the Young’s modulus of the material of the beams.

Now, from expression (13), for the ratio of the second moments of area of the aggregated and non-aggregated cross-sections, the following expression is obtained:

I_{2} / I_{1} = p^{4} = [(n / m)^{1 / 3}]^{4} = (n / m)^{4 / 3}

(24)

The ratio of the maximum deflections in the non-aggregated and aggregated structure is obtained by dividing equations (22) and (23):

\frac{δ_{1}}{δ_{2}} = \frac{\frac{(1 / n)}{I_{1}}}{\frac{(1 / m) p}{I_{1} {(n / m)}^{4 / 3}}} = \frac{(1 / n) {(n / m)}^{4 / 3}}{1 / m} = (n / m)^{1 / 3}

(25)

This completes the proof of assertion (6) in Theorem 1.

Note that considering (18), we also have:

\frac{δ_{1}}{δ_{2}} = p

In words, the ratio of the maximum deflections in the non-aggregated and aggregated structure is equal to the scaling factor linking the dimensions of the geometrically similar cross sections.

The same analysis and the same results (20) and (25) are also obtained for simply supported beams and to conserve space details related to the proof have been omitted.

Improving the load capacity at a constant amount of material

Suppose that the non-aggregated structure in Figure 1(a) includes n identical beams with arbitrary shape of the cross section while the aggregated structure in Figure 1(b) includes fewer number m of identical beams (m < n).

The beams from the non-aggregated and aggregated structure have geometrically similar cross sections, with a coefficient of proportionality p > 1 linking the corresponding dimensions (see the example in Figure 1(c) and (d)). The coefficient of proportionality p has been selected such that the total volume of the non-aggregated structure is equal to the total volume of the aggregated structure.

Let $σ_{t, 1}, σ_{t, 2}$ denote the maximum tensile stress in the non-aggregated and aggregated structure correspondingly, while $δ_{1}, δ_{2}$ denote the maximum deflections of the non-aggregated and aggregated structure. The length of the elements from the non-aggregated and aggregated structure is equal to L.

Theorem 2: The following relationships exist between the maximum tensile stresses and deflections of a non-aggregated structure built on n identical beams and an aggregated structure built on m identical beams (m < n), (the cross sections are geometrically similar, and the total volume of material is the same):

σ_{t, 1} / σ_{t, 2} = \sqrt{n / m}

(26)

δ_{1} / δ_{2} = n / m

(27)

Proof: Because of the geometric similarity, an elementary surface area $ds = Δ x Δ y$ in Figure 1(c) will be transformed into the area $ds' = p^{2} Δ x Δ y$ in Figure 1(d). The elementary volume $L Δ x Δ y$ from the non-aggregated structure will be transformed into the elementary volume $p^{2} L Δ x Δ y$ from the aggregated structure. The total volume of the non-aggregated and aggregated structure must be the same, which gives the condition:

n \times \int \int Ldxdy = m \times p^{2} \int \int Ldxdy

(28)

From the condition of equivalence of volumes (28), after cancelling $\int \int Ldxdy$ , a condition is obtained for the proportionality (scaling) factor p which guarantees the equivalence of volumes of the non-aggregated and aggregated structure:

p = \sqrt{n / m}

(29)

From equation (13), $I_{2} / I_{1} = p^{4}$ and considering (29) finally gives the ratio of the second moments of area of the aggregated and non-aggregated cross-sections:

I_{2} / I_{1} = (n / m)^{2}

(30)

From the classical bending theory, for the tensile stress at the fixed support of the non-aggregated cantilever beam, the value:

σ_{t, 1} = \frac{(P / n) L}{I_{1}} {\hat{y}}_{1}

(31)

is obtained while for the aggregated beam, the value

σ_{t, 2} = \frac{(P / m) L}{I_{2}} {\hat{y}}_{2}

(32)

is obtained, where ${\hat{y}}_{1}$ and ${\hat{y}}_{2}$ are the largest distances from the neutral axis for the non-aggregated and aggregated cross-section, correspondingly.

Equations (31) and (32) are special cases of equations (1) and (2), where $s_{1} = σ_{t, 1}$ , $s_{2} = σ_{t, 2}$ , $k = L$ , $t = 1$ , $G_{1} = I_{1} / {\hat{y}}_{1}$ , $G_{2} = I_{2} / {\hat{y}}_{2}$ .

Because of the geometric similarity, for the largest distances from the neutral axis ${\hat{y}}_{1}$ and ${\hat{y}}_{2}$ , the relationship

{\hat{y}}_{2} = p {\hat{y}}_{1}

(33)

holds. The ratio of the maximum tensile stresses in the non-aggregated and aggregated structure is obtained by dividing equations (31) and (32):

\frac{σ_{t, 1}}{σ_{t, 2}} = \frac{\frac{(1 / n) {\hat{y}}_{1}}{I_{1}}}{\frac{(1 / m) p {\hat{y}}_{1}}{I_{1} {(n / m)}^{2}}} = \frac{{(n / m)}^{2} (n / m)}{\sqrt{n / m}} = \sqrt{n / m}

(34)

Note that for the maximum tensile stresses, considering (29), we also have:

\frac{σ_{t, 1}}{σ_{t, 2}} = p

In words, the ratio of the maximum tensile stresses in the non-aggregated and aggregated structure is equal to the scaling factor linking the dimensions of the geometrically similar cross sections.

For the maximum deflection of the non-aggregated cantilever beams, the value

δ_{1} = \frac{(P / n) L^{3}}{3 E I_{1}}

(35)

is obtained from the standard bending theory and for the maximum deflection of the aggregated cantilever beam, the value

δ_{2} = \frac{(P / m) L^{3}}{3 E I_{2}}

(36)

is obtained, where E is the Young’s modulus of the material of the beams.

Equations (35) and (36) are also special cases of the general equations (1) and (2), where $s_{1} = δ_{1}$ , $s_{2} = δ_{2}$ , $k = L^{3} / (3 E)$ , $t = 1$ , $G_{1} = I_{1}$ , $G_{2} = I_{2}$ .

The ratio of the maximum deflections in the non-aggregated and aggregated structure is obtained by dividing the equations (35) and (36):

\frac{δ_{1}}{δ_{2}} = \frac{\frac{(1 / n)}{I_{1}}}{\frac{(1 / m) p}{I_{1} {(n / m)}^{2}}} = \frac{(1 / n) {(n / m)}^{2}}{1 / m} = n / m

(37)

This completes the proof of assertions (26) and (27) of Theorem 2.

Note that for the maximum tensile stresses, considering (29), we also have:

\frac{δ_{1}}{δ_{2}} = p^{2}

In words, the ratio of the maximum deflections in the non-aggregated and aggregated structure is equal to the square of the scaling factor linking the dimensions of the geometrically similar cross sections.

In addition, the aggregated structure has a significantly smaller total surface area along the length L compared to that of the non-aggregated structure. Because the coefficient of proportionality of the linear parameters is $p$ , between the surface area $S_{non - aggr}$ of a single element of the non-aggregated structure and the surface area $S_{aggr}$ of a single element of the aggregated structure, the geometrical relationship

S_{aggr} = p S_{non - aggr}

(38)

exists.

The ratio $\frac{n S_{non - aggr}}{m S_{aggr}}$ of the total surface area along the length L of the non-aggregated structure and the total surface area along the length L of the aggregated structure, can be determined by considering (38):

\frac{n S_{non - aggr}}{m S_{aggr}} = \frac{n}{mp} = \frac{n}{m \times \sqrt{n / m}} = \sqrt{n / m}

(39)

Since $n > m$ , in addition to the decreased stress and deflection, the aggregated structure has $\sqrt{n / m}$ times smaller surface area along the length L and proportionally smaller penetration heat loss, compared to the non-aggregated structure.

Finally, another advantage of the aggregated structure over the non-aggregated one is the smaller number of connection points (anchors) during assembly.

The same analysis and the same results (34), (37) and (39) are also obtained for simply supported beams and to conserve space, details related to the proof have been omitted.

We need to point out that a further increase in load-carrying capacity could have been achieved by tapering the cantilevered beam, which is a textbook approach to enhancing load capacity by increasing thickness at the fixed end and gradually decreasing it towards the free end. For a given total material volume, this tapered shape improves load-bearing efficiency. However, for the sake of simplicity in presenting the underlying theory, a uniform cross-section along the beam’s length was intentionally chosen to simplify the theoretical exposition of the aggregation method.

This choice in no way diminishes the advantages of the proposed method. As can be verified from Figure 2, even with a uniform cross-section, the material savings required to support the same total load are significant. For example, with a five-to-one aggregation, material usage is reduced by a factor of 1.71.

Aggregation must be applied judiciously, considering structural redundancy and operational conditions. For example, if a structure with four supporting elements must remain functional after the failure of one, a 4-to-2 aggregation may be unsuitable. However, by no means does aggregation promote structural collapse. In fact, an aggregated structure exhibits significantly increased load-bearing capacity and a much larger margin of safety than its non-aggregated counterpart. Additionally, aggregation does not necessarily imply leaving larger gaps between structural elements.

Segmentation as a design technique that increases the elastic strain energy stored in the structure

Let the aggregated structure, composed of m uniformly loaded beams, be segmented into a structure composed of n uniformly loaded beams (n > m). It is assumed that the same total volume of material has been used for the non-aggregated and the aggregated structure. The elastic strain energy U stored in a cantilever beam, loaded by a concentrated force with magnitude $P$ in the elastic region, can be obtained from the equation^13,16:

U = \frac{P^{2} L^{3}}{6 EI}

(40)

where I is the second moment of area of the beam and E is the Young’s modulus. The total load P is assumed to be the same for both the aggregated and the segmented structure. The stored elastic strain energy $u_{1}$ in each of the beams composing the segmented structure is given by $u_{1} = \frac{{(P / n)}^{2} L^{3}}{6 E I_{1}}$ . Similarly, the stored elastic strain energy $u_{2}$ in each of the beams composing the original structure is given by $u_{2} = \frac{{(P / m)}^{2} L^{3}}{6 E I_{2}}$ . In these expressions, $I_{1}$ and $I_{2}$ are the second moments of area for a beam from the non-aggregated and aggregated structure, correspondingly.

The total elastic strain energy, stored in the segmented structure, is therefore given by

U_{1} = n \times \frac{{(P / n)}^{2} L^{3}}{6 E I_{1}}

(41)

while the total elastic strain energy stored in the aggregated structure is given by

U_{2} = m \times \frac{{(P / m)}^{2} L^{3}}{6 E I_{2}}

(42)

Again, equations (41) and (42) are special cases of equations (1) and (2), where $s_{1} = U_{1}$ , $s_{2} = U_{2}$ , $k = L^{3} / (6 E)$ , $t = 2$ , $G_{1} = I_{1}$ , $G_{2} = I_{2}$ .

It will be shown that the inequality $U_{1} > U_{2}$ always holds if $n > m$ .

From equations (41) and (42), taking the ratio $U_{1} / U_{2}$ results in

\frac{U_{1}}{U_{2}} = \frac{m}{n} \frac{I_{2}}{I_{1}}

(43)

From equations (29) and (30), for equal volumes of material of the non-aggregated and aggregated structure, equation (43) becomes

\frac{U_{1}}{U_{2}} = \frac{m}{n} {(\frac{n}{m})}^{2} = \frac{n}{m}

(44)

Since n > m, the ratio $U_{1} / U_{2}$ is greater than unity:

\frac{U_{1}}{U_{2}} = \frac{n}{m} > 1

(45)

Inequality (45) shows that segmenting a beam structure increases the elastic strain energy stored in the structure.

Consider segmenting a structure containing $m = 2$ loaded elements into a structure containing $n = 8$ , loaded elements. Then, for the same magnitude of the total loading force P and for the same total volume, the segmented structure stores 4 times more elastic strain energy than the aggregated structure.

The same conclusion can also be reached for simply supported beams. To avoid repetition, the argument related to simply supported beams has been omitted.

Testing the aggregation method using finite element analysis

The verification of the aggregation method was conducted using steel beams having a Young’s modulus of 200 GPa and a Poisson’s ratio of 0.3, subjected to 8-to-2 aggregation. The structures were composed of cantilevered beams and statically indetermined $Π$ -frames. The maximum von Mises stress and the maximum deflections have been analysed using the finite element analysis software Abaqus/CAE 2021.

Testing the aggregation method on cantilevered beams

One of the ends of the cantilevered beam is fixed, while the other end is free (Figure 3). The distance between the fixed support and the end of the beam is L = 100 mm for both the non-aggregated and aggregated structures.

Figure 3.

Testing the aggregation method on cantilevered I-beams: (a) non-aggregated beams and (b) aggregated beams.

The cross sections of the non-aggregated and aggregated beams are according to Figure 4. The dimensions of the cross-sections have been selected so that the total volume of material in eight non-aggregated beams (Figure 3a) is exactly equal to the total volume of material in two aggregated beams (Figure 3b).

Figure 4.

Cross sections of the (a) non-aggregated and (b) aggregated I-beams.

The same total load P = 360 N has been applied to both the aggregated and non-aggregated structure. As a result, each beam from the non-aggregated structure in Figure 4(a) (8 beams) is loaded with a force of 360/8 = 45 N, distributed uniformly along the top surface of the beam. Each beam of the aggregated structure (2 beams) is loaded with a force of 360/2 = 180 N (8 × 45 N = 2 × 180 N = 360 N), distributed uniformly along the top surface of the beam.

Table 1 lists the maximum von Mises stress and the maximum deflection characterising non-aggregated and aggregated cantilever beams. From the table, it can be concluded that for the same total applied load (360 N), the aggregated structure experiences a significantly smaller maximum von Mises stress and maximum deflection compared to the non-aggregated structure.

Table 1.

Finite element analysis results for the maximum von Mises stress and deflection of non-aggregated and aggregated cantilever beams.

Analysed property	Non-aggregated cantilever beams	Aggregated cantilever beams
Maximum von Mises stress	350 MPa	179 MPa
Maximum deflection	1.92 mm	0.48 mm

Testing the aggregation method on statically indeterminate $Π$ -frames

Testing of the aggregation method was also conducted on centrally loaded and side loaded statically indeterminate $Π$ -frames. The aggregation selected for the $Π$ -frames was 8–2. (eight loaded elements (Figure 5(a)) with square cross sections with side 1 mm have been aggregated into two loaded elements (Figure 5(b)) with a square cross-section with side 2 mm). Figure 5 shows the dimensions of the $Π$ -frames in mm. Consequently, the same the total volume of material characterises both structures. Both structures are also subjected to the same total loading force of $P = 160 N$ . This means that the load applied on an individual non-aggregated $Π$ -frame is 160/8 = 20 N, uniformly distributed over the top section of the frame.

Figure 5.

Eight-to-two aggregation of centrally loaded $Π$ –frames: (a) non-aggregated centrally-loaded frames and (b) aggregated centrally-loaded frames.

The load applied on an individual aggregated $Π$ -frame is 160/2 = 80 N, uniformly distributed over the top section of the frame. The Young’s modulus of the material for all $Π$ -frames is 200 GPa and the Poisson’s ratio is 0.3.

The maximum von Mises stresses and deflections of the $Π$ -frames were determined by using finite element analysis conducted by using Abaqus/CAE 2021. Table 2 lists values related to the von Mises stresses and deflections. The analysis of the results in Table 2 for loaded $Π$ -frames shows that the aggregated frames experience significantly smaller deflections compared to non-aggregated frames.

Table 2.

Results for the maximum von Mises stress and maximum deflection for centrally loaded non-aggregated and aggregated $Π$ -frames.

"Eight-to-two aggregation of side-loaded $\Pi$ frames - Circuit (a) shows eight side-loaded frames feeding into two main ones, Circuit (b) demonstrates output of main $\Pi$-frame into two outputs (q${}_a$ and 2p/2)."

Analysed property	Non-aggregated $Π$ -frames	Aggregated $Π$ -frames
Maximum von Mises stress	570.7 MPa	302.3 MPa
Maximum deflection	3.39 mm	0.8 mm

The maximum von Mises stress for the aggregated $Π$ -frame is also significantly smaller than the maximum von Misses stress for the non-aggregated $Π$ -frames.

Testing of the aggregated method was also conducted on a side-loaded statically indeterminate $Π$ -frame. Figure 6 gives the dimensions of the $Π$ -frames (in mm). Both structures were subjected to the same total loading force of $P = 56 N$ . This means that the load applied on an individual non-aggregated $Π$ -frame is 56/8 = 7 N, uniformly distributed over the right vertical section of the frame.

Figure 6.

Eight-to-two aggregation of side-loaded $Π$ –frames: (a) non-aggregated side-loaded frames and (b) aggregated side-loaded frames.

The load applied on an individual aggregated $Π$ -frame is 56/2 = 28 N, uniformly distributed over the right vertical section of the frame. The Young’s modulus of the material for all $Π$ -frames is 200 GPa and the Poisson’s ratio is 0.3.

The maximum von Mises stresses and deflections of the side-loaded $Π$ -frames were determined by using finite element analysis. The results related to the maximum von Mises stresses and deflections are listed in Table 3. The analysis of the results in Table 3 for side-loaded $Π$ -frames shows that the aggregated frames also experience significantly smaller deflections compared to the non-aggregated frames.

Table 3.

Results for the maximum von Mises stress and maximum deflection for side-loaded non-aggregated and aggregated $Π$ -frames.

Analysed property	Non-aggregated $Π$ -frames	Aggregated $Π$ -frames
Maximum von Mises stress	634.5 MPa	315.4 MPa
Maximum deflection	3.94 mm	0.97 mm

The simulation studies involving statically indeterminate structures demonstrated that for statically indeterminate frames, aggregation also had a dramatic effect on their load capacity.

Reducing the total volume of material at a specified maximum von-Mises stress and maximum deflection

Tables 2 and 3 show that, for the same material volume, the aggregated structure exhibits significantly reduced stress and deflection. This offers the potential to decrease the cross-sectional area and substantially reduce the material volume, while maintaining the maximum von Mises stress and deflection within acceptable limits.

In design, both the maximum von Mises stress and the maximum deflection are critical parameters. When the total load is constant for both the non-aggregated and aggregated structures, the scaling factor can be adjusted to ensure that the von Mises stress and deflection remain below specified thresholds, while simultaneously achieving a notable reduction in material volume. The scaling factors ps* and pd*, which correspond to specific maximum permissiblevon Mises stress and deflection values, can be determined through interpolation. This process typically requires only a small number of finite element analysis simulations.

The interpolation method for determining the optimal scaling factor p* involves conducting a series of simulation experiments on an aggregated element at different scaling factor values (p1,p2,p3,p4,p5; see Figure 7a). These experiments provide a relationship between the scaling factor and the maximum von Mises stress. From this interpolated data, the scaling factor ps* corresponding to the maximum permissible von Mises stress is determined (Figure 7a).

Figure 7.

(a) The dependence maximum von-Mises stress – scaling factor and (b) the dependence maximum deflection – scaling factor.

Next, at the same scaling factor values (p1,p2,p3,p4,p5; see Figure 7b), the relationship between the scaling factor and the maximum deflection is obtained (Figure 7b). Using this interpolated graph, the scaling factor pd*, which corresponds to the maximum permissible deflection, is determined for the aggregated element (Figure 7b).

The larger scaling factors p* = max{ps*; pd*} delivers both: a von-Mises stress not exceeding the maximum permissible stress and a maximum deflection not exceeding the maximum permissible deflection. The material saving factor is given as a ratio of the total volume $V_{1}$ of the non-aggregated structure and the total volume $V_{2}$ of the aggregated structure. Denoting by $S_{1}$ and $S_{2}$ the cross-sectional areas of the non-aggregated and aggregated elements and considering also that the length of the frame L is the same for both structures, the ratio of the total volumes of the structures is determined from:

V_{1} / V_{2} = \frac{n \times S_{1} \times L}{m \times S_{2} \times L}

(46)

For a scaling factor $p^{*}$ , the link between the cross-sectional areas by $S_{1}$ and $S_{2}$ is given by:

S_{2} = (p^{*})^{2} S_{1}

(47)

Substituting (47) in (46) finally results in

V_{1} / V_{2} = \frac{(n / m)}{{(p^{*})}^{2}}

(48)

As a result, the proposed interpolation method leads to significant savings of material while maintaining the maximum von Mises stress and deflection below specified safe limits.

Table 4 and Figures 8 and 9 give the dependencies scaling factor versus maximum von Mises stress and maximum deflection determined from simulations done on Abaqus/CAE 2021 with the side-loaded aggregated frame.

Table 4.

Scaling factor versus von Mises stress for side loaded aggregated $Π$ -frame.

Scaling factor	von Mises stress, MPa	Deflection, mm
1.4	854.3	4.11
1.6	582.5	2.39
1.8	474.9	1.67
2.0	315.4	0.97

Figure 8.

Scaling factor versus von Mises stress [MPa] for side loaded aggregated $Π$ -frame.

Figure 9.

Scaling factor versus von deflection [mm], for side-loaded aggregated $Π$ -frame.

Suppose that the maximum permissible von Mises stress is 605 MPa and the maximum permissible deflection is 3 mm. From the graphs, the scaling factors that deliver maximum permissible stress not exceeding 605 MPa are bigger than 1.58 while the scaling factors that deliver maximum permissible deflections not exceeding 3 mm are bigger than 1.52. The larger scaling factor is p* = 1.58, therefore is the scaling factor that delivers both: a von Mises stress below 605 MPa and deflection below 3 mm. At a scaling factor of p* = 1.58, the calculated by Abaqus maximum von Mises stress is 604 MPa and the maximum deflection is 2.52 mm.

The material savings are calculated from Equation (48):

V_{1} / V_{2} = \frac{(8 / 2)}{{(1.58)}^{2}} = 1.6

As a result, the total volume of material of the aggregated $Π$ -frame structure can be safely reduced by a factor of 1.6.

Testing the aggregation method on physical prototypes including statically indeterminate $Π$ -frames

Testing on centrally loaded $Π$ -frames

The aggregation method was tested on physical prototypes including statically indeterminate centrally loaded $Π$ -frames (Figure 10). The type of aggregation selected for the $Π$ -frames was 8–2. This means that a non-aggregated structure containing 8 load-carrying frames (Figure 10(a)) with circular cross sections of 1 mm diameter is aggregated into a structure containing two load-carrying frames (Figure 10(b)) with a circular cross-section of 2 mm diameter. The cross-sectional area of the frames from the aggregated structure was appropriately selected so that the volume of material used for both structures is the same. The dimensions of the $Π$ -frames (in mm) are according to Figure 10. The total loading force for both structures was the same: $P = 10 N$ . As a result, the load applied on an individual non-aggregated $Π$ -frame was 10/8 = 1.25 N while the load applied on an individual aggregated $Π$ -frame was 10/2 = 5 N. The Young’s modulus of the material for all $Π$ -frames is 179 GPa.

Figure 10.

Eight-to-two aggregation of centrally loaded $Π$ –frames: (a) non-aggregated frames and (b) aggregated frames.

The physical prototype for testing the aggregation method on centrally loaded $Π$ -frames is given in Figure 11.

Figure 11.

A physical prototype for testing the aggregation method on centrally loaded statically indeterminate $Π$ -frames.

A $Π$ -frame labelled as ‘1’ was securely anchored in the mini vices ‘2’ and ‘3’. The frame’s deflection was controlled by the lever-type dial indicator ‘4’ with measurement precision of 0.01 mm. The displacement indicator was secured in the mini vice ‘5’. The lever-type indicator was chosen because of the very small measurement force, compared to plunger-type displacement indicators. The load applied was measured using the digital dynamometer ‘6’.

Testing the aggregation method on physical prototypes including side-loaded $Π$ -frames

The aggregation method was also tested on physical prototypes including statically indeterminate side-loaded frames (Figure 12). The type of aggregation selected for the side-loaded $Π$ -frames, as well as the material properties and dimensions, were the same as those used for the centrally loaded frames. The total loading force for both structures was also the same: $P = 10 N$ . As a result, the load applied on an individual side-loaded non-aggregated $Π$ -frame was 10/8 = 1.25 N while the load applied on an individual aggregated side loaded $Π$ -frame was 10/2 = 5 N.

Figure 12.

Eight-to-two aggregation of side-loaded $Π$ –frames: (a) non-aggregated side-loaded frames and (b) aggregated side-loaded frames.

The physical prototype for testing the aggregation method on side-loaded $Π$ -frames is given in Figure 13.

Figure 13.

A physical prototype for testing the aggregation method on side-loaded statically indeterminate $Π$ -frames.

A $Π$ -frame labelled as ‘1’ was securely anchored in the mini vices ‘2’ and ‘3’. The frame’s side deflection was controlled by the lever-type dial indicator ‘4’ with measurement precision of 0.01 mm. The displacement indicator was secured in the mini vice 5. The side load applied to the frame was measured using the digital dynamometer ‘6’.

Experimental results

The results from the experimental study related to the deflections of the $Π$ -frames have been summarised in Table 5 and confirm the significant decrease in the maximum deflection from applying the method of aggregation.

Table 5.

Maximum deflection [mm] for central and side loading of 8 non-aggregated versus 2 aggregated $Π$ -frames.

Type of loading	Non-aggregated $Π$ -frames		Aggregated $Π$ -frames
Centrally loaded		0.80 mm	0.20 mm
Side loaded		0.61 mm	0.15 mm

The analysis of the results in Table 5 shows that the aggregated $Π$ -frames experience significantly smaller deflections in comparison with the non-aggregated $Π$ -frames.

Conclusions

Aggregating beams with arbitrary cross-sectional shapes into fewer beams with geometrically similar cross sections leads to a dramatic reduction in the volume of material required to support a specified total load. For example, in five-to-one aggregation, the total material volume needed to support the same load is reduced by a factor of 1.71.

For a specified total load, the ratio of the minimum volumes of material required for the non-aggregated and aggregated structures is equal to the scaling factor that relates the dimensions of the geometrically similar cross sections. The ratio of deflections is also equal to this scaling factor. These results hold regardless of the shape of the beam cross sections.

A theorem has been proven that provides the theoretical foundation for the aggregation method in structures containing the same total volume of material. Under this condition, the ratio of the maximum tensile stresses is equal to the scaling factor that relates the dimensions of the geometrically similar cross sections.

For the same total material volume used in both non-aggregated and aggregated structures, the ratio of the maximum deflections is equal to the square of the scaling factor that relates the dimensions of the geometrically similar cross sections. These results are independent of the cross-sectional shape of the loaded beams.

Segmenting multiple loaded beams into a larger number of beams increases the total elastic strain energy of the structure. This result holds true for arbitrary cross-sections of the beams.

Simulation experiments confirmed that aggregating statically indeterminate cantilever beams and statically indeterminate frames leads to a significant increase in the load-carrying capacity of the structures, as measured by both maximum von Mises stress and maximum deflection.

Experiments done on physical prototypes confirmed that aggregating statically indeterminate frames under the same total load results in a significant reduction in maximum deflection.

Beam aggregation also reduces the total surface area along the length of the beams and decreases heat loss due to surface penetration.

An interpolation method has been proposed for reducing the total volume of material while maintain the von-Mises stress and maximum deflection below safe limits.

Footnotes

Declaration of conflicting interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Michael Todinov

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Designing light-weight structures consisting of multiple elements loaded in bending

Abstract

Keywords

Introduction

Impact of aggregation and segmentation on critical properties of structures

Theoretical justification of the aggregation method as a design optimisation method reducing the weight of structures

Improving the load capacity at a constant amount of material

Segmentation as a design technique that increases the elastic strain energy stored in the structure

Testing the aggregation method using finite element analysis

Testing the aggregation method on cantilevered beams

Testing the aggregation method on statically indeterminate Π -frames

Reducing the total volume of material at a specified maximum von-Mises stress and maximum deflection

Testing the aggregation method on physical prototypes including statically indeterminate Π -frames

Testing on centrally loaded Π -frames

Testing the aggregation method on physical prototypes including side-loaded Π -frames

Experimental results

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Testing the aggregation method on statically indeterminate $Π$ -frames

Testing the aggregation method on physical prototypes including statically indeterminate $Π$ -frames

Testing on centrally loaded $Π$ -frames

Testing the aggregation method on physical prototypes including side-loaded $Π$ -frames