Just de Meijer [7] introduced and examined the characteristics and features of the hexagrid pattern and compared the structural behavior of this pattern with the diagrid pattern in terms of stiffness and efficiency in his master's thesis from the University of Eindhoven. This structure is similar to a diagrid, except that its bracing grid is not triangular but hexagonal. In his research, he used a parametric method that could be applied to assess other types of innovative patterns as well. Mashhadi Ali and Kheyroddin [5] introduced the hexagrid pattern then compared the hexagrid and diagrid patterns in two cases with and without secondary bracing. The researchers concluded that secondary bracing increases the stiffness sensitivity of hexagrids three times more than diagrids. Taranath et al. [8] introduced the pentagrid pattern (made of irregular pentagons) as a new pattern, examined its structural properties in high-rise buildings, and compared it with the hexagrid pattern. In their study, pentagrid performed better than hexagrid against lateral deformation, and the optimal brace angle for it was 35 degrees. There is a lot written about the angle variation in diagrid structures, but very little on the variable density patterns along with building height except Montuori et al. [9] who assessed the effect of changing the angle and number of diagonals along with the building height for the diagrid pattern. Based on the results, the patterns with the tallest modules had a high requirement of a secondary bracing system for reducing the inter-story drifts and stabilizing the core gravity columns at intermediate floor levels. In addition, their design procedures can be helpful in exploring pattern solutions to develop diagrid potential. Zhao and Zhang [10] evaluated diagrid patterns with curved and varying-angle straight diagonals from bottom to top. They proposed empirical formulas for optimal values of bottom and top angles for each configuration. Montuori et al. [4] examined and compared the hexagrid pattern and its modeling and design with the diagrid pattern in a simulated high-rise building similar to the building in actual use, i.e. the 90 floor, 351 meter Sino Steel International Plaza. Researchers have proposed that the optimal angle in horizontal hexagrid and diagrid patterns is 60 degrees, while 30-50 degrees was proposed as an optimal angle in vertical hexagrid patterns. Ebin and Prakash [11] investigated the structural behavior of the hexagrid pattern in multi-story buildings with 15 stories and the aspect ratio H/B less than 7. For this application, a 65-degree angle was optimal. Saeedi Nejad et al. [12] introduced and compared the seismic performance of tubular structural systems of diagrid and hexagrid in high-rise buildings. According to the results, hexagrid had less weight than diagrid and also more resistance against progressive collapse. Lee and Kim [13] investigated a 60-story building with a hexagrid pattern and showed that the vertical hexagrid pattern is more stable than the horizontal hexagrid. Angelucci and Mollaioli [14] examined the stiffness-based methodology effectiveness proposed by literature by applying it to 90-story diagrids and showed that The stiffness-based design method could be used to determine optimality conditions only for diagrids with an inclination range of 60-70. The iterative optimization technique is suitable for steeper diagonal angles. Results show that local density is not the best approach to take. A variable pattern density is a more efficient solution by rarefying the diagonal elements from the base to the top of the building. Danish et al. [1] compared diagrid and hexagrid structures with the same loading and properties by the ETABS software, and finally, the hexagrid structure was more economical and stable than diagrid. Tomei et al. [15] investigated different geometric patterns in regular diagrid structures with new or complex shapes for high-rise buildings, in this study, possible combinations of rhombic and triangular diagonal components (cross-sections or with curved components) and their optimization in high-rise buildings are examined with a genetic algorithm to reduce the final weight of the structure when stiffness is constant. Angelucci and Mollaioli [3] examined the possibility of using non-regular grids for the structural systems of tall buildings. They focused particularly on periodic and non-periodic Voronoi tessellations. Using parametric modeling, they generated Voronoi-like patterns by gradually perturbing a regular honeycomb configuration. This paper focused on the impact of the grid arrangement on the lateral stiffness of the perimeter tube. As a result, at the same density rate, cell irregularity has no significant impact on lateral stiffness, and gradual reduction in density is a suitable strategy for tall buildings with Voronoi-like grids. Mele et al. [16] evaluated nonregular patterns based on the Voronoi diagram. This research was focused on assessing the mechanical properties of this grid based on the concept of the representative volume element (RVE) and the development of this concept on a statistical basis to account for the inherent irregularities, non-periodicity and randomness of the grid. In their work, they defined a framework that encompassed an almost endless variety of structural configurations, including diagrid, hexagrid, voronoi, foam trusses, and more. Mashhadi Ali and Kheyroddin [17] conducted studies on the seismic performance of hexagrid structures to develop their previous work. In this system, an attempt is made to reduce the concentration of damage on the converging braces to achieve a uniform drift distribution in the story and prevent forming a soft story mechanism. The results estimated the value of the proper coefficient to be 4. The optimal digonal angle was also obtained from 30 to 40 degrees. A recent review by Scaramozzino et al. [18] gives a comprehensive overview of the diagrid structural system, with a particular focus on the optimization of these systems based on geometrical patterns and recent research on diagrid nodes. The authors noted that diagrids could be further optimized based on geometrical features, making them suitable for sustainable design. Moreover, they declared grid-based tubes, including diagrids, were the top candidates for achieving efficient, attractive and sustainable tall buildings in the future.

The pattern of a modular structure is created by parametric duplication of an initial motif. With duplication of geometrical shapes, a kind of tessellation emerges. Tessellation is an unlimited set of polygon duplications. These polygons fit together so that they completely cover the surface. Here, each side of a polygon is the side of the surrounding polygons. Each pattern can be created horizontally or vertically. A vertical pattern is a structure that has only diagonal or vertical elements; meanwhile, the horizontal pattern is made of only diagonal or horizontal elements. The present study focused only on horizontal patterns.

In this research, a new pattern was investigated for the diagonal grid system which was called as Isometric-Cube pattern. (IC-grid) (Fig. 2).

Comparisons among patterns are conducted based on efficiency. In a grid-based structure, efficiency is obtained by dividing the grid stiffness (stiffness against lateral or vertical loading) by its structural weight (here, the grid density is representative of its weight). The stiffness of a structure is a major influence on its lateral deformation. Stiffness coefficients are classified into two categories: axial and shear stiffness.

Fig. (2). Isometric-Cube pattern.

In order to analyze the grid structure, its mechanical properties are determined by comparing it to an equivalent solid structure (with similar dimensions: same width, height, and thickness). In order to determine the stiffness of the grid structure, we apply a modifying factor to the structure stiffness. A high-rise building is assumed to be a deterministic cantilever beam [4]. The beam is simultaneously subjected to bending and shear forces. Therefore, both axial and shear stiffness must be adjusted (Fig. 3).

Fig. (3). Orthotropic membrane tube analogy: (a) structural grid; (b) equivalent solid [4].

The axial moduli (E*) is defined as the ratio of the uniaxial normal stress, σ, divided by the uniaxial strain, ε, in the loaded direction in the elastic range (Equation 1). Similarly, the shear moduli (G*) is related to the shear force (Equation 2) [4].

(1)

(2)

Where:

F₁= average normal force, which is a substitution of normal stress (σ),

F₂= average shear force, which is a substitution of shear stress (τ),

B= the width of cross-section,

b = the thickness of cross-section,

H= the height of the structure (initial length),

∆x= transverse displacement resulting from shear stress when the displacements along the y-axis of nodes on the bottom and on the top edges are constrained.

∆y= shortening or lengthening resulting from normal stress,

γ = shear strain,

ε = normal strain.

The horizontal grid (consisting of diagonal and horizontal elements) for patterns is considered here. The modulus in the hexagrid, the diagrid, and the new one are hexagonal, triangular, and isometric cube. Each of the three horizontal patterns is shown with its modulus and unit cell (Fig. 4).

Fig. (4). Modules and unit cells for three pattern: Diagrid, Hexagrid and Isometric-Cube (IC-grid).

The relative density of grids (ρ) is obtained by dividing the volume occupied by solid material (ρ*) by the total volume of cell [19]. Previous research has shown that diagrid structures' efficiency is significantly related to the angle of their diagonal elements [20]. The effect of changing the angle of diagonal elements is fully considered in the concept of grid density.

The relative grid density is calculated using Equation (3):

(3)

Where the n is the total number of elements; l_i and A_i are the length and the section area of elements, respectively; L₁ and L₂ are the dimensions of unit cell along x and y direction, respectively, and b is the thickness of unit cell.

In the case of the isometric cube pattern and according to the definition of relative density, a different unit cell type is, predictably shown in Fig. (5). As we will see later, this form of the unit cell is efficient in describing the structural performance of the grid (which is a combination of two types of nodes with different numbers of connections).

Fig. (5). Unit cell for Isometric cube grid.

According to Equation (4), the new pattern's relative density (ρ_IC) is twice the relative density of the hexagrid pattern.

(4)

Where h and d are the length of horizontal and diagonal elements, respectively, and θ is the angle between diagonal elements and horizontal axis. A_h and A_d are the area section of horizontal and diagonal elements of the grid, respectively. A comparison of the relative density of the different grids is given in Table 1.

Table 1. Comparison of the relative density of the different grids

Pattern	Grid Type
	Horizontal Diagrid [4]
	Horizontal Hexagrid [4]
	Horizontal Isometric Cube

The concept of relative density can be used to evaluate the efficiency of different grids. Dividing the grid stiffness by its relative density gives a value on which the grid's efficiency depends. This value for each grid depends entirely on the choice of the optimal angle of the diagonal elements. The optimal angle for the elements is the angle that provides the best balance between stiffness and density (in other words, it means economic expenditure) for the grid.

(Fig. 13) shows graphs that are obtained for the diagrid and hexagrid patterns based on the equations presented for axial and shear stiffness in Tables 2 and 3, and also the relative densities in Table 1. These graphs are calculated once for axial stiffness and once for shear stiffness. The obtained graphs are quite similar to those of Montouri et al. [4], which were calculated here again. By comparing the two obtained line graphs in each pattern, the range of diagonal angles where both graphs have shown the maximum efficiency is almost 50 to 70 degrees. Therefore the range of optimal angles can be considered from 50 to 70 degrees for both cases of diagrid and hexagrid patterns.

Fig. (14). The RVE of Isometric-Cube pattern (IC-grid) as a combination of hexagrid's RVE (with type-2 node) and diagrid's RVE (with type-1 node).

In the case of an Isometric-cube pattern with rigid connections, the stiffness of RVE must be obtained by combining the stiffness of two types of nodes (Type 1, 2). Under load, the formation of median hinges allows us to distinguish three separate structural parts which have one of two types of nodes. Each part has its RVE and is connected to the other at the median hinge of the horizontal elements (Fig. 14).

According to Fig. (14), this RVE can be regarded as a combination of two hexagrid’s RVE (with ‘Type 2’ nodes) located on either side of a diagrid’s RVE (with ‘Type 1’ node). As a solution, the stiffness of this RVE is calculated with the same formulas used to calculate spring set stiffness. In a set of parallel springs, each spring's force is different from and parallel to the other, so the total force (total) is obtained from the sum of each force. On the other hand, the change in the length of all springs (∆) under load is equal. Consequently, the spring constant of the set is equal to the sum of all the spring constants. Similarly, in axial loading, the RVE of an isometric cube is under such conditions. Therefore, similar to the sum of stiffness for parallel springs, the following Equation is used to calculate the total axial stiffness of the isometric cube’s RVE: (Equation 6)

(5)

To calculate the shear stiffness of the RVE shown in (Fig. 14), there are also some resemblances between RVE under shear load and spring set with series springs. In series springs, the force remains constant throughout the set of springs. Simultaneously, the amount of deformation of the set is equal to the sum of each spring's deformations. According to the lateral loading (shear force) applied to the entire isometric-cube grid, every RVE is under the same shear stress (similar to the springs in a series set), whereas each type of node has its share of the total force. It depends on the limited area of each node (as shown in Fig. 7). Like the series springs, the total deformation of this RVE is equal to the sum of deformation of nodes. Therefore, the equation of the total stiffness for calculating the shear stiffness of the isometric-cube’s RVE is as follows: (Equation 7)

(6)

As shown in Fig. (15), the isometric cube pattern's optimal angles are in the range of 50 - 70 degrees, similar to the two other patterns.

Comparison of the results from dividing the stiffness ratio to relative density for different patterns in a chart simultaneously shows that in the case of axial stiffness, the values of the graph for the isometric cube pattern with rigid connections from angles 40 to 80 degrees are significantly larger than the two previous patterns (Fig. 16).

From Fig. (17), it is obvious that the efficiency of diagrid based on shear stiffness is much higher than both the hexagrid and the cubic-grid. Consequently, the diagrid pattern is the most efficient against lateral loads. According to the significant shear stiffness of the diagrid pattern, to compare between hexagrid and new patterns, the graph related to the diagrid pattern is removed. After deleting the corresponding graph, the efficiency based on shear stiffness for the new pattern compared to the hexagrid pattern is shown in Fig. (18).

Fig. (18) shows that the behavior of the isometric cube pattern is different from the hexagrid pattern in terms of efficiency based on the shear stiffness. In the case of hexagrid pattern, this efficiency declines rapidly from 30° to 88°, whereas for the new pattern, the efficiency increases substantially in the angle range 30 °- 65 °, and decreases gradually in the range 70° -80 °. Therefore, it is predicted the shear behavior of the new pattern would be better than the hexagrid pattern in the range of the most practical angles (50 - 65 degrees).

It should be noted that if the angle of the diagonal elements increases over 70 degrees in the new pattern, the diagonal elements of the type 1 node tend towards the vertical elements. Due to the proximity of four vertical elements around the type 1 nodes, it may result in implementation problems.

Fig. (15). Axial and shear stiffness divided by relative density versus diagonal angle for Isometric-Cube pattern (IC-grid).

Fig. (16). Comparison among patterns for axial stiffness ratio to relative density.

Fig. (17). Comparison among patterns for shear stiffness ratio to relative density.

Fig. (18). Comparison between Hexagrid and Isometric-Cube patterns for axial stiffness ratio to relative density.