Numerical Study of the of Proposed Hexaloc Concrete Armor Units under Static Loading

Concrete armor units are commonly used to protect shorelines and rubble mound coastal structures. This paper presents the numerical modeling test results for the proposed Hexaloc armor unit under static load. The model shows some value of tensile stress exceeds the critical stress. It was found that the cracking occurred at the top surface of the leg. In the design, the moment due to hydrodynamic and impact loads is not permitted to exceed the cracking moment.


Introduction 1.1 Concrete Armor Units
As an archipelagic country, Indonesia requires a lot of concrete armor units for coastal structures, especially for breakwaters (Figure 1) 2 on ships at the port, is to build a coastal protection structure.This structure is a breakwater consisting of stones (natural or artificial) as armor units arranged to reach a certain height.
In designing a concrete armor unit, the weight of an individual concrete stone (W) is estimated using Hudson's formula [1].The variable of this formula is the wave height at the structure (H), the weight density of the armor unit (γ r ), the weight density of the fluid (γ w ), the slope of the structure face (1V: mH), and an empirical stability coefficient (K D ).

𝑚
Eq. 1 According to Eq. 1, the increment of K D causes a decrease in the weight of the armor unit.It means that interlocking capacity among units is better with higher K D .Various concrete armor unit that has similar shape is presented in Figure 2. The stability coefficient (K D ) of the A-Jacks armor unit is 50 and 20 for uniform and random patterns [2], whereas Hexapod has K D =8 [3], significantly lower than A-Jacks.This research investigates the geometric and structural capacity of the proposed armor unit named Hexaloc, with K D estimated as 12 for the structure head and 15 for the trunk.
Compared to the existing armor unit Hexapod and A-Jacks, the proposed Hexaloc is shown in Figure 2. The shape of the hexapod armor unit is the same as A-jack, classified as 3D-combined legs [4].Hexapod was developed in the USA in 1959 [5].The leg shape is circularly tapered.The A-Jacks consist of two identical components that fit into a single unit.The A-Jacks has six legs extending from a filleted central hub, each with a square cross-section.The Hexaloc has six legs with a hexagonal cross-section connected in one joint without a filleted central hub.a) Hexapod [5] b) A-Jacks [2] c) Proposed Hexaloc

Figure 2. Comparison Hexaloc with Hexapod and A-Jacks
The estimated weight of the Hexaloc armor unit using Hudson's formula is shown in Table 1.Recommended KD design value for the hexapod armor unit is 8.0 [3].In this study, the Hexaloc unit is assumed to have KD same as the hexapod KD 8.0.Two slope levels are 1:1.5 [6] and 1:2 [2].The waves' height varies from a minimum of 1 m to a maximum of 10 m following Indonesia's 100-yearly significant wave height [7].According to this estimation, the wave with a height of more than 4 meters, the weight on individual units is more than 5 tons.This weight is quite heavy, and the unit's dimension is difficult to produce and transport.Therefore, the Hexaloc is designed as a 5 tons unit in this study.The hydrodynamic load is assigned for seawater level (SWL) application with a wave height from 1 m to 10 m.
Table 1.The weight of Hexaloc armor unit using Hudson's formula.

Height of wave (m)
Density of armor unit, gr (ton/m3) The types of loadings subjected to the concrete armor units are static, hydrodynamic, and impact loads [8]- [11].The static loads are due to the self-weight of the unit and the superimposed dead load.The hydrodynamic loads are associated with the direct wave attack upon the unit.The impact loads are imposed from the collision of units due to either wave-induced motions or unit placement during construction.Previous research has investigated the failure pattern of armor units under impact using drop tests [12], [13], and hydrodynamic load [2].The unit's failure is initiated by cracking concrete in a critical region due to the exceeded tensile stress capacity.The critical leg area is before the connection or hub [2].Finite element (FEM) modeling of Hexaloc using Abaqus is developed to investigate the unit's capacity under static loading.

Mode of failure of armor unit
Previous research has observed the probability of the armor unit's failure mode as shown in Figure 3.During their useful life as coastal structures, concrete armor units should be designed in the ultimate limit states and serviceability limit states.In the ultimate limit state, significant damage to the armor unit may occur due to the displacement of structure toe berm units.The breaking of armor units is due to structural stresses.In a serviceability state, the armor unit should have properties of surfaces that pedestrians or fishermen safely use.The rock surfaces are not changing their ability to sustain attached life [15].
The mode of failure of the Dolosse unit has been investigated.The result is shown in Error!Reference source not found.. Micro-cracking causes damages to the concrete locally near the collision or small area contact regions or under intense shearing of contacts.[16], [17] The hydrodynamic loading subjected to the A-Jacks unit was investigated by Tedesco et al. [2].The failure occurred in the interface of the leg and filleted central hub.Due to the unreinforced A-Jacks plain concrete armor units, their strength capacity is controlled by the tensile capacity of the concrete; therefore, the maximum principal stress becomes the parameter in designing the armor unit.The higher stress occurred at the SWL location of the unit [2].

Numerical Modelling 2.1 Geometry of model
This paper presents the numerical modeling test results for the proposed Hexaloc armor unit under static load.The numerical modeling is conducted using Abaqus.The weight design of the Hexaloc unit is 5 tons.With a specific gravity of 2.4 tons/m 3 , the volume of the Hexapod is 2.1 m 3 .With this volume, the dimensions of the test object are obtained, as shown in Figure 6.The unit model has a uniform hexagonal cross-section size of the legs from the base to the tip.The length is 2,497 mm with an edge width of 416 mm and height of 721 mm.The waist ratio is 721/2,497 = 1/3.5.The element is modelled as a solid element, as shown in Figure 7a.The Abaqus program is unitless.Therefore, when creating a model, the length and force measurements do not appear in the program display, so it is essential to keep in mind the consistency of the units used with the guidelines, as shown in Table 2.

Material properties
The material properties of concrete are defined in compression and tension conditions.The compressive strength of the concrete is f'c 30 MPa.The tensile stress of concrete of 0.62√  ′ is 3.4 MPa.The coarse aggregate size is 20 mm.The constitutive compression concrete model refers to the compressive stress-strain curve proposed by Attard and Setunge [18].While the concrete model due to tension refers to the tensile stress-strain curve proposed by Hoover and Bazant [19].The compressive and tensile stress-strain curves of concrete are shown in Figure 8 and According to the American Concrete Institute Building Code Requirements for Structural Concrete [20], the modulus of elasticity, Ec, is   = 4700√  ′ Eq. 2 where   ′ is the compressive strength of concrete at 28 days.The critical stress level for plain structural concrete is the rupture modulus of concrete.

Stress analysis
The cracking occurred at the top surface of the leg with a tensile stress of 4.3 MPa.It exceeds the critical stress.
Figure 12 The stress at the top fiber cracking area of the leg under static load

Cracking Moment legs of Hexaloc
The length of the leg is 0.888 m.The cracking load is 37.6 kN.Therefore, the cracking moment capacity is 37.6 x 0.888 = 33.4kNm.In the design, the moment due to hydrodynamic and impact loads is not permitted to exceed the cracking moment.

Conclusion
This paper presents the results of modeling the proposed Hexaloc armor unit under static load.The cracking occurred at the top surface of the leg.The value of tensile stress is 4.3 MPa.It exceeds the critical stress.The length of the leg is 0.888 m.The cracking load is 37.6 kNm Therefore, the cracking moment capacity is 37.6 x 0.888 = 33.4kNm.In the design, the moment due to hydrodynamic and impact loads is not permitted to exceed the cracking moment.

Figure 1 .
Figure 1.Conventional cross-section of a rubble mound breakwater One way to reduce wave energy approaching the coast, which has the potential to erode the coast and simultaneously create a quiet area for activities on the beach and loading and unloading of goods

Figure 5 .
Figure 5.The stress of A-Jacks at failure numeric modeling results [2].

Figure 6 .
The geometry of Hexaloc

Figure 13 .
Figure 13.The tensile stress propagation until cracking occurred.

Figure 14 .Figure 15 .
Figure 14.The displacement of leg under static load

Table 2 .
The unit in Abaqus