Proposed stress block for no coarse-aggregate concrete

. Concrete is one of the materials in construction that continues to develop in strength and efficiency. Since most areas in Indonesia are vulnerable to earthquakes, the building structures must be more resistant to seismic forces. Increasing the concrete's strength might reduce the building structure's dimensions and weight, which will also reduced the earthquake force. One example of high-strength concrete is reactive powder concrete (RPC), an innovative concrete with small particle materials to fill in the space within the concrete so it can strengthen the concrete. Despite the advantage, there is still a lack of design provisions for this type of concrete. This research was conducted to analyze the stress-strain diagram curve and the compressive stress block for no-coarse aggregtae concrete. In this research, compression tests have been carried out on 100 mm × 200 mm cylinder samples with concrete compressive strength ranges from 28 to 76 N/mm 2 . Test results indicate that the actual curve of the stress-strain relationship for concrete without coarse aggregate is almost linear and shaped like a triangle with the maximum strain ranges from 0.006 to 0.008. The energy per unit volume ranges from 0.1175 to 0.3658 N/mm 2 and the average force capture point is 0.62626 units. Based on the test results, compression stress block for no-coarse aggregtae concrete are proposed.


Introduction
Concrete is one of the most used materials in construction, with its usage dating back to roughly 2000 BC.In recent years, various researches have been carried out to improve conventional concrete's mechanical properties, economic value, or other desirable properties to achieve certain conditions.One of them is high-strength concrete [1].By increasing the concrete strength, the dimensions of the building structure can be reduced.Smaller building structure dimensions, will make the building lighter which will also reduce the earthquake force.This is very important especially for earthquake-prone areas such as Indonesia.
One example of the high-strength concrete is reactive powder concrete (RPC).RPC is a type of concrete that does not use coarse aggregate but consists of cement and other powder-forming materials, such as silica fume.Silica fume is a material and technological innovation in manufacturing concrete that can help achieve high quality concrete.Silica fume, which generally has a small particle size, can fill the voids in the concrete and reduce segregation in the concrete mix to increase the compressive strength of the concrete.
Concrete has the advantage of being a solid structure.It can withstand higher pressure than other construction materials.However, because concrete is brittle, it has several weaknesses, such as low tensile and flexural strength.Testing the flexural strength of concrete is needed to determine the ability of concrete when bent.In the analysis of reinforced concrete's flexural capacity, simplified compressive stress block *Corresponding author: danielc@ft.untar.ac.id are often used to represent concrete behavior.This stress block are first introduced by Whitney [2].The coefficient values used in this simplified concrete stress block are very important [3].
A typical normal concrete stress-strain curve shows that indicates that the higher the concrete strength, the curve will form a triangular line.On the other hand, the lower concrete strength will show a parabolic shape.However, the stress-strain curve was obtained according to normal concrete experiments [4].
In this study, an analysis of the total compressive force or maximum absorbed energy for concrete without coarse aggregate was carried out.This research was conducted because the provisions for designing concrete without coarse aggregate are still being developed and have not yet to be stipulated in regulations, such as the ACI (American Concrete Institute) code.Therefore, this research objective is to obtain the shape of the stressstrain curve diagram for concrete without coarse aggregate, such as RPC.This research contributes a reference for possible improvement on the existing structural codes, such as the ACI code, in designing structural concrete without coarse aggregate.

RPC
The development of concrete technology has encouraged researchers to enhance the quality of concrete, particularly its strength, workability, durability, and cost-efficiency in making concrete, which leads to the ultra-high performance of concrete [5].The concrete strength is determined mainly by the water-cement ratio (WCR).A lower WCR reduces the porosity of the hardened concrete and thereby increases the number of interlocking solids [4].Therefore, coarse aggregate should not be used in creating a mixed design for this type of concrete.The coarse aggregate will generate a weak point in the concrete, thus limiting the concrete strength.There are various researches on how to increase the strength of concrete, with or without coarse aggregate, even stronger [6-8].

Stress-strain curve in normal concrete
The stress-strain relationship in concrete is not linear, but it is assumed that the stress-strain relationship is linear up to around 1/3 of its maximum stress value.
Fig. 1 shows a stress-strain curve for normal concrete under compression.The curve is obtained from a test that lasted about 15 minutes on a sample resembling the beam's compression zone [4].The stress-strain curve in Fig. 2 is obtained from normal concrete samples with different strengths in psi units.The shape of the curve is generated from microcracks that form gradually in the concrete structure [4].The figure shows that all curves rise to the maximum stress until it reaches a strain between 0.0015 and 0.003, followed by a decreasing branch.
The length of the descending branch can be affected by the test conditions.Often a sample concrete test cylinder subjected to an axial load will fail and explode at the point of maximum stress.The explosion occurs when the test machine's strain energy release exceeds the specimen's energy capacity [4].
The compressive strength of concrete is usually obtained from cylindrical samples with a height-todiameter ratio of 2:1.The specimen will be loaded in the longitudinal direction slowly until the maximum stress is reached in 2 or 3 minutes.In general, the compressive strength obtained by concrete at 28 days is about 80% of the ultimate compressive strength of the concrete, which is between 2000 and 8000 psi (13.3 and 55.2 N/mm2) [9].Fig. 2. Stress-strain curves for concrete with different compressive strengths [4].
In Fig. 3, various stress-strain curves are generated from a uniaxial-loaded cylinder sample test results.The curves are nearly linear, up to about half of the compressive strength.The curve's apex for highstrength concrete is relatively sharp, while for the lowstrength concrete it forms a slightly obtuse parabola.The strain at the maximum stress is about 0.002 [9].

Equivalent stress block
Several fundamental assumptions are made when obtaining a general theory for the flexural strength of reinforced concrete sections.One of these fundamental assumptions is that the stress-strain curve for concrete determines the magnitude and distribution of compressive stress, which is known.Such assumptions are necessary to assess the actual behavior of the crosssection.Since the strains in compressed concrete are proportional to the distance from the neutral axis, the shape of the stress-strain curves in Fig. 3 show the shape of the compressive stress block at various loading stages [9].
The section reaches its flexural strength (maximum moment of resistance) when the total compressive force in the concrete multiplied by the moment arm jd is maximum.The properties of the compressive stress block at the extreme moment section can be determined by the parameters k1, k2, and k3, as shown in Fig. 4. For a rectangular cross-section with width b and effective depth d, the total compressive force in the concrete becomes k1k2k3f'cbc.The arm length is equal to d -k2c, where c is the depth of the neutral axis.Many studies have been carried out to determine this parameter for unconfined concrete.
Some researchers (e.g., Whitney [2]) have suggested replacing the actual shape of concrete compressive stress block with a rectangular equivalent stress block for simplification [9].Only the magnitude coefficient (k1k3) and position coefficient (k2) of the concrete compressive force need to be known to define the stress block.ACI [10] replaces the actual stress block with an equivalent stress block, as shown in Fig. 4b.The rectangle has an average stress of 0.85f'c and a depth of a, where a/c = β1 = 0.85 for f'c ≤ 4000 psi (28 N/mm 2 ).As the concrete strength increases, the β1 coefficient decreases continuously.The reduction in β1 for highstrength concrete is mainly due to the unfavorable shape of the concrete stress-strain curve, as shown in Fig. 3.
For the resultant compressive force of the actual and equivalent stress blocks in Fig. 4 to have the same magnitude and line of action, the following Equations 1-3 must be satisfied.

Research methods
This research was compiled using literature studies and also based on tests in the laboratory.A literature study has been carried out to understand the theory related to this research.Testing in the laboratory have been carried out using a compression test with a displacement control type pressure tester.The concrete samples used consist of 15 cylinders with a diameter of 100 mm and a height of 200 mm.Tests were carried out to obtain the stressstrain diagram curve of the specimens, which can be analyzed to obtain the total compressive force of the specimens.The concrete's material composition and mix design are calculated based on the required amounts per 1 m 3 .Table 1 displays the mix design used in this study.The concrete specimens underwent a curing process by immersing them in water.The purpose of this curing is so that the hydration process does not encounter problems, e.g.cracks occur due to draining the water too quickly.The curing time is six months to ensure the concrete have developed its full capacity.

Stress-strain relationship diagram
Several stress-strain diagram obtained by analyzing the compression test results for concrete cylinder specimens without coarse aggregate are shown in Fig. 5 to Fig. 8.All specimens's stress-strain diagram shows almost linear relationship.The shape of stress-strain diagram resembles a triangular shape.
Based on the test results, the strain at peak stress for concrete without coarse aggregate ranges from 0.006 to 0.008.This values are higher than the 0.002 strain usually obtained for normal concrete.
The comparison of stress-strain diagrams are also made based on the peak stress of the concrete.Fig. 9 shows stress-strain diagrams with peak stress ranges from 20 to 40 N/mm 2 and Fig. 10 shows a stress-strain diagrams with peak stress ranges from 50 to 70 N/mm 2 .It can be seen that, the stress-strain diagram becomes more linear with the increase in concrete strength.

Peak stress
The peak stress obtained from each concrete cylinder samples can be seen in Table 2.

Energy
After obtaining the stress-strain relationship diagram, the strain energy per unit volume for each sample can be obtained.The amount of energy is obtained with the help of the AutoCAD program to calculate the area of each diagram.The area of the stress-strain diagram shows the amount of energy per unit volume that can be absorbed by the concrete during the loading process.The amount of energy per unit volume for each sample can be seen in Case study The following is a case study in calculating the nominal moment strength of the reinforced concrete crosssection made from conrete without coarse aggregate.The calculation of the nominal moment strength is carried out in 2 (two) ways, using the equivalent stress block proposed by Whitney [2] and the proposed stress block based on the results of this research.The calculation example will be carried out for sample S10.1.Table 3.

Center of gravity of energy
The center of gravity of energy (i.e. the area under stress-strain diagram) for each concrete samples without coarse aggregate were obtained using the AutoCAD program.The center of gravity of energy for each samples in strain axis (X) and stress axis (Y) can be seen

Force capture point
To determine the nominal moment strength of concrete section, the location of concrete compressive force resultant and the force capture point are needed.The force capture point (t) can be defined as the ratio of compressive force resultant to the distance of neutral axis from extreme compressive fiber.The force capture point can be calculated by dividing the center of gravity in strain axis with the maximum strain.The force capture point for each samples can be seen in Nominal moment (Mn) using Whitney's [2] equivalent stress block (Fig. 11).
To enforce section equilibrium, the sum of the compression forces must be equal to the sum of the tension forces.By using the equation ΣH = 0 with the concrete compressive force, Cc, and the tension force, T, the value of c is 62.8174 mm and a is 40.8313mm.Thus, the value of the nominal moment strength, Mn, is 164.9103kN-m.a. Nominal moment (Mn) using the proposed stress block (Fig. 12).
To obtain the depth of the neutral axis, f, the area of the stress block must be calculated.
By taking f = 1 mm, the area of the stress block is 41.3489 N/mm 2 and the volume is 8269.7853N/mm (Fig. 13).The multiplication between the stress block volume and f will result in the compressive force.By using the force balance concept, the value of the neutral axis depth, f, is 63.8972 mm.(7) where t : force capture point εmax : maximum strain X : the coordinates of the center of gravity in the strain axis The standard deviation of the force capture point from sample's test results can be seen in Table 5.The standard deviation is given to see how close the data distribution is to the average value.The standard deviation calculation can be seen in Table 6.
Based on the calculation, the deviation, s, is 0.03846.Meanwhile, the average value of the force capture points of all samples is 0.62626.The standard deviation that is smaller than the mean indicates that the data have relatively small variation.This can be said to be good because all the samples in this study were made with the same mix design and the results of the mix design do not differ too much.

Case study
The following is a case study in calculating the nominal moment strength of the reinforced concrete crosssection made from conrete without coarse aggregate.The calculation of the nominal moment strength is carried out in 2 (two) ways, using the equivalent stress block proposed by Whitney [2] and the proposed stress block based on the results of this research.The calculation example will be carried out for sample S10.1.equivalent stress block (Fig. 11).
To enforce section equilibrium, the sum of the compression forces must be equal to the sum of the tension forces.By using the equation ΣH = 0 with the concrete compressive force, Cc, and the tension force, T, the value of c is 62.8174 mm and a is 40.8313mm.Thus, the value of the nominal moment strength, Mn, is 164.9103kN-m.c.Nominal moment (Mn) using the proposed stress block (Fig. 12).
To obtain the depth of the neutral axis, f, the area of the stress block must be calculated.
By taking f = 1 mm, the area of the stress block is 41.3489 N/mm 2 and the volume is 8269.7853N/mm (Fig. 13).The multiplication between the stress block volume and f will result in the compressive force.By using the force balance concept, the value of the neutral axis depth, f, is 63.8972 mm.The nominal moment can be calculated with either the tension or the compression force multiplied by the moment arm (Fig. 14).The moment arm is obtained using the value of the force capture point.Therefore, by multiplying the force with the moment arm, the nominal moment strength is 163.0727kN-m.The nominal moment strength, Mn, obtained based on the Whitney equivalent stress block and the proposed stress block are similar, as shown in Table 7.
Table 7.The comparison of the nominal moment strength.

Conclusions
From the analysis of compression test results of 100 mm × 200 mm concrete cylinder samples without coarse aggregate, it can be concluded that: 1.The stress-strain curve of no-coarse aggregate concrete are nearly linear.The shape of stress-strain curve resembles a triangular shape.2. Based on the stress-strain curve, the maximum strain that can be reached by concrete without coarse aggregate ranges from 0.006 to 0.008, higher than the strain normally obtained for normal concrete.

The laboratory compression test results show that
the peak stress range for sample S8 is between 45 and 60 MPa, sample S9 is between 42 and 57 MPa, sample S10 is between 46 and 76 MPa, and sample S11 is between 28 and 58 MPa. 4. The highest energy per unit volume value (0.3658 N/mm 2 ) was found in sample S10.3, while the lowest was found in sample S11.1.5.The average value from the force capture point is 0.62626 units, and the standard deviation value of the force capture point in the sample is 0.03846.The smaller standard deviation compared to the mean average indicates that the data have relatively small variation, which is good because all the samples in this study were made with the same mix design.Therefore, the results of the mix design do not differ too much.6.The nominal moment strength of reinforced nocoarse aggregate concrete section computed using Whitney's [2] stress block and proposed stress block is similar.

Fig. 4 .
Fig. 4. Distribution of compressive stress in the compression zone of a rectangular concrete section.(a) The actual distribution, (b) Equivalent block distribution [9].

Table 5 ,
which is determined using the following equation.

Table 2 .
Peak stress of concrete cylinder samples without coarse aggregate.

Table 3 .
The energy on concrete samples without coarse aggregate.

Table 4 .
The energy center of the concrete sample without coarse aggregate.

Table 5 .
Force capture point on concrete samples without coarse aggregate.