Behavior of Modified Strut-and-Tie Model (STM) on Concrete Beams

Reinforced concrete components are generally designed to withhold shear and bending based on the assumption at strain varies linearly in a section where the applied force is a combination of shear with bending, torsion, or normal forces. The behavior of shear failure in reinforced concrete beams is very different from that in flexural failure. Shear failure is brittle without warning in the form of significant deflection, resulting in diagonal cracks in the beam then the shear force mechanism will be contributed by an arching action where this action can provide a reserve capacity large enough for the beam to carry the load. Nonlinear analysis using the strut-and-tie method is particularly useful for shear-critical structures where classical beam theory is not valid due to significant shear deformations. Strut-and-Tie Model research which is applied to concrete (25 MPa), also uses the optimal configuration FEM software tool from Strut-and-Tie which leads to the efficiency of Strut-and-Tie. The results of the optimization and modification of the Strut-and-Tie Model on concrete will also be applied to the experimental models tested until they fail so that the optimal conditions for numerical models will be obtained. Based on the analysis result of element model using ANSYS computational assistance program, the ultimate flexural capacity of the beam model will increase depending on the used STM model and inclination angle (Φ), at the STM type 2 high < 1000 mm, inclination angle (Φ) 45° having a decrease in ultimate shear capacity (Vu) of 49,31% against type 1, at the STM model 3 high < 1000 mm, with inclination angle (Φ) 45° having an increase ultimate shear (Vu) of 4,14% against type 1. Stress pattern performed bottle shape in line with diagonal strut. Ductility capacity will be decreased at inclination angle <45° of 27,11%, at inclination angle >45° will be decreased of 55,67%,


Introduction
The shear mechanism that acts on structural elements is a very important thing to attention to do, especially for structural components that are susceptible to shear forces, such as high reinforced concrete beams.Shear forces generally do not work alone, but occur in combination with bending, torsion, or normal forces.The behavior of shear failure in reinforced concrete beams is very different from failure caused by bending.Shear failure is brittle without warning or signs of significant deflection.Collapse that occurs in high beams is domination caused by shear forces and will result in diagonal cracks in the beam.After the crack occurs, a shear force mechanism will occur which will be contributed by arching action (SNI 2847(SNI -2019)).This action can provide a large capacity reserve for the beam to carry the load.This hypothesis applies to parts of the frame that do not experience interference from rigid areas, such as near columns, cavities or other areas of structural elements where the influence of strain due to shear cannot be ignored, thus there are elements or structural areas where the assumption of Bernoulli's hypothesis is not sufficiently representative.flexural structural behavior and stress distribution (Tuchscherer,et.al.2011) (2009).Experimental studies of high beams and comparisons with strutand-tie shear strength predictions can be found in the literature.The AASHTO LRFD3 Bridge Design Specifications were then included in Appendix A of ACI 318-021 for building design.ACI 318-144 separated the strut-and-tie model into major parts, and ACI 318-195 updated several aspects of the strut-and-tie design.This shows that the strut-and-tie modeling approach has gained attention and importance in practice.The main update in the provisions of ACI 318-195 STM is the inclusion of restraining factors that increase the effective compressive strength of struts and nodal zones when subjected to triaxial compression, updated strut coefficients, size effect factors, consistent with those determined for the new one-way shear model based on fracture mechanics, added to the maximum ultimate shear force limitation; and minimum distribution reinforcement modifications are included.In Figure 1, the crack pattern that occurs in the support area of the deep beam (deep beam), a crack pattern occurs and forms varying angles.For the largest crack pattern, a resistance model is created from the variations in angles formed and depicts a bottle-shape strut at the support of the high beam.In Figure 2, Crack patterns that occur in the deep beam area, crack patterns occur with varying angles.The large number of cases of shear cracks that occur in high beams is the background for research on shear forces, their effects and handling.
Over the past two decades, significant improvements have been made in the physical and mechanical properties of reinforced concrete.Many studies have shown the savings of using in high-rise buildings as well as low and medium-rise buildings.It is shown in a number of studies that STM failure is usually brittle and characterized by early peeling of the concrete cover.This research studies the Modified Strut-and-Tie Model (MSTM) method to determine the flexural and shear capacity of beams in reinforced concrete which is supported by efficiency factors by considering the effect of web reinforcement.It is assumed that, the total shear force carried by the beam in the reinforcement is provided by two independent resistances namely the diagonal strut-tie due to the strut-and-tie mechanism and the equivalent resisting force produced by the web reinforcement, the web reinforcement reduces the compressive softening effect of the concrete by preventing it from opening of diagonal cracks or splitting of concrete which is not found in ordinary strut-and-Tie.The unknown functions and parameters were determined from experimental results obtained by other researchers and experiments on high strength reinforced concrete carried out by the Author himself.To validate the proposed method, the results obtained were compared with several previous methods and research on existing codes such as ACI 318-05 and CSA.

Research purposes
The aims and objectives of the research are as follows: a. Analyzing the ultimate bending capacity, load, deformation and ductility which are influenced by the Strut-and-Tie Model with symmetric and a-symmetric concentrated loads on concrete beams.b.Analyzing the ultimate cross-sectional capacity using the Strut-and-Tie Model with symmetric and a-symmetric concentrated loads on concrete beams.c.Analyzing load-deformation and moment-curvature relationship curves as well as crack patterns in the Strut-and-Tie model with symmetrical and a-symmetrical concentrated loads on concrete beams.d.Evaluate the most effective STM beam configuration with concrete beam model.

Literature Review
MacGregor (2012) defines a high beam as: "a beam in which a large amount of load is carried to the support by compression" the thrust force connecting load and reaction.This occurs if the concentrated load acts closer than about 2D to the support, or for uniformly loaded beams with a span-to-height ratio (ln/d) of less than 4 to 5." (Panjehpour, et al. 2015).Since 2002, high beam design according to the requirements of ACI 318 Building Code is based on nonlinear analysis or using the strut-and-tie method (STM).The strut-and-tie approach has proven very useful for shear-critical structures where beam theory does not apply e.g., pile caps, corbels, hollow beams, anchor zones, and high beams which are the focus of this study.Classical beam theory does not apply to deep beams because the assumption that the plane cross-section remains flat after loading is invalid due to significant shear deformation.Geometry of the high beam with strut-and-tie model.
In Figure 1.3 the complex stress field of a concrete structure is represented as an equivalent frame, with elements in compression (struts), elements in tension (ties), and truss connections called nodal zones (or nodes).The strength of the struts and nodal zones controls the dimensions of the concrete structural components, while the tensile force determines the amount of reinforcement required to support the load (Panjehpour, et al. 2015).The main assumptions in the strut-and-tie model are: balance must be met, the center of gravity of each component and the line of action of the applied load must coincide at the joint, and failure occurs when the strut or node is destroyed or when the tie reaches its yield capacity, thus forming a mechanism.style (for static determinate truss models).The strut-and-tie model requires the designer to identify one possible load path for each load combination and ensure that no load path component is overloaded (Panjehpour, et al. 2015)

Research Methodology
The research was carried out by modeling reinforced concrete high beams using finite element analysis with the help of finite element computing.In FEM software, modeling will be done in 3D. by validating previous experimental results as well as secondary data from previous manual calculations.
The analysis results that will be obtained are in the form of nodal displacement, elements forces and moments, deflection, and stress contour diagrams.Apart from that, the pattern of cracks that occur will also be obtained.
Modeling was carried out to determine and predict the ability of high beams when receiving ultimate concentrated loading and the behavior of using a supporting and tie structure model (strut-and-tie model) against cracking of high beams with variations in beam height and high quality concrete.The results of this modeling will then be used to create alternative STM models (modified) to be more optimal.In type 1, the supports and ties are simple with two support rods and ties which function to transfer the load from compression reinforcement to tension reinforcement.In type 2, the supports and ties resemble a truss system with several support rods and ties which function to transfer loads from compression reinforcement to tension reinforcement.

Concrete Models
The concrete model uses SOLID65 element types which are defined in eight nodes and is an isotropic material that is able to depict deflection, cracking and destruction of concrete.This SOLID65 element can work together with other materials, for example reinforcing steel.Input SOLID65 element type data as follows: -Concrete compressive strength is obtained from previous test results.
-Poisson's ratio for concrete is used 0.20.
Tensile strength of concrete.
Stress-strain values in multilinear kinematic hardening plasticity.Isotropic elastic behavior in concrete occurs before the concrete experiences initial cracking or is in a position to experience destruction.Concrete crushing is defined as the release of an element from a material unit (ANSYS in Zhenhua, 2006).The destruction parameters on the concrete surface in a computational program on nonlinear nonmetal plasticity concrete modeling materials, namely: -Uniaxial compressive strength (fc') -Uniaxial tensile strength (ft) -Ultimate biaxial compressive strength -Ultimate compressive strength for biaxial hydrostatic pressure -Ultimate compressive strength for uniaxial hydrostatic pressure -Stress-strain curve for normal concrete using procedures for calculating reinforced concrete structures for buildings (SNI 03-2847-2002).
Reinforcing Steel Model -The reinforcing steel model uses LINK8 element types.This element is a three-dimensional element defined by 2 nodes and is an isotropic material.This material is capable of representing plastic stress and strain, creep, swelling, tensile stiffness and large deformations.-Data for the reinforcing steel model material using non-linear rate independent multilinear isotropic hardening elements and von-Mises yield criteria with Young's modulus, Poisson ratio and steel stress-strain curve values based on the procedures for calculating reinforced concrete structures for buildings (SNI 03-2847-2002).
Support of model -For the beam support model, SOLID45 is used.This element is a three-dimensional element defined by eight points with orthotropic material properties.Elements have the ability to plasticity, creep, tension stiffness, deflection and strain.-Data for the support model material uses linear isotropic by including elastic modulus and Poisson ratio data.

Model Analysis Using a Mathematical Approach
The beam model that will be analyzed manually uses an approach model with mathematical calculations.This step is carried out to obtain validation values that are close to previous experimental results.
From the results of model analysis using an approach model with mathematical calculations, data were obtained in the form of moment-curvature-ductility, load-deformation values for the beam model being analyzed.

Research Flow Chart
Flow Chart Research on strut-and-tie modifications in high strength concrete can be explained in the flow chart in Figure 5

Moment of Curvature
The calculation of moments and curvatures that occur in the beam model uses a modified stress-strain block for concrete in a confined condition with crack stages, namely initial crack condition, first yield condition and ultimate condition.Based on the calculation of the curvature ductility analysis using the Kent and Park method, a curve can be created into the relationship between the moment values and the curvature that occurs from the beam model.

Load-Deformation Model of Beam.
From the moment values obtained at the initial crack, yield and ultimate conditions, the maximum load acting and the deformation that occurs in the beam model can be determined.The maximum load applied and the deformation that occurs in the beam model in the initial crack, yield and ultimate conditions are listed in Table 2.For model BT-800.1.01;BT-800.2.02; BT-800.3.03;BT-1000.1.04;BT-1000.2.05; BT-1000.3.06; the value of the reinforcement ratio ( ) to beam height, the value decreases successively at a ratio of 1,000; 0.952; 0.904 for a beam with a height of 800 mm is 0.0063, for a beam with a height of 1000 mm it is 0.0060.Curvature ductility increases with increasing ultimate load and increasing beam height.For model BT-800.1.01;BT-800.2.02; BT-800.3.03;BT-1000.1.04;BT-1000.2.05; BT-1000.3.06,curvature ductility value to beam height, the value increases successively at a ratio of 1,000; 1,506; 1.619, for a beam with a height of 800 mm it is 6.2375, for a beam with a height of 1000 mm it is 9.3989.Based on the explanation above, it can be concluded that increasing the height of the beam increases the value of the load it can withstand and its ductility increases even though there is a decrease in the reinforcement ratio ( ).

Model Analysis Using FEM
The beam model that will be analyzed with a finite element model using the computing program, is conditioned based on variations in the support and tie (strut-and-tie) model as below; In Figure 8.a, a longitudinal section of the beam and support with Meshing Volume is visible, Figure 8.b shows a longitudinal section of the beam which illustrates the location of the reinforcement, Figure 8.c shows a perspective cross-section of the beam which more clearly depicts the location of the reinforcement.
The model that has been formed and is in accordance with the material properties conditions, is conditioned on the placement of the support for placing the beam model, and the loading.Then the program is run iteratively until it reaches the ultimate condition, to obtain optimal finite element analysis results from the model being analyzed.

Moment-Curvature Model of Beam
The moments and curvatures that occur in the beam model based on the analysis results using finite element analysis results are listed in Table 4.  4. above it can be seen that moment and curvature values at crack curvature at initial crack, yield and ultimate condition for beam result.6. above it can be seen that Curvature ductility values for beam models.Based on the curvature ductility requirements according to Park and Paulay (1974) which states that: Gravity load is μf ≥ 4 The earthquake load is μf ≥ (13 to 16).So for beam model BT-800.2.02 meet the requirements for gravity loads, for other models, BT-800.1.01,BT-800.3.03,BT-1000.1.04,BT-1000.2.05, BT-1000..3.06,not eligible.For earthquake loads, all models do not meet the requirements, so all existing models are not recommended for earthquake loads.

Load-Deformation Model of Beam.
The load and deformation values that occur in the beam model are obtained from the conversion of concrete stress-strain values in the beam model as a result of analysis as shown in Figure 9 and Figure 10.In Figure 17, there is a picture of the stress (von Misses stress) in the BT-800.1.01beam model, which is a variation of the beam model, showing that the stress that occurs depicts a Bottle-Shape-struts, the stress concentration occurs at the support and the load sharing block from the pressure load. of 10 kN.In the model, the dark blue beam shows the stress that occurs at 0.001763 kN, the closer to red the stress increases with a maximum stress of 15.096 kN.
In the BT-800.2.02 beam model, which is a variation of the beam model, shows that the stress that occurs depicts Bottle-Shape struts which are becoming increasingly vague due to the influence of the strut and tie model installed on the beam., stress concentration occurs at the supports and load sharing blocks from a pressure load of 10 kN.In the model, the dark blue beam shows the stress that occurs at 0.001474 kN, the closer to red the stress increases with a maximum stress of 18.486 kN.
In the BT-800.2.02 beam model, which is a variation of the beam model, shows that the stress that occurs depicts Bottle-Shape struts which are becoming increasingly vague due to the influence of the installed strut-and-tie model. in the beam, stress concentration occurs at the support and load sharing block from a pressure load of 10 kN.In the model, the dark blue beam shows the stress that occurs at 0.001474 kN, the closer to red the stress increases with a maximum stress of 18.486 kN.In Figure 18, a picture of the stresses (stress-von-Misses) in the BT-1000.1.04beam model, which is a variation of the beam model, shows that the stresses that occur describe a Bottle-Shape-struts which is increasingly vague due to the influence of the strut and tie model.mounted on the beam.Stress concentration occurs at the supports and load sharing blocks from a pressure load of 10 kN.
In the model, the dark blue beam shows the stress that occurs at a stress of 0.001485 kN, the closer to the red color the stress increases with a maximum stress of 21.781 kN.
In the BT-1000.2.05 beam model, which is a variation of the beam model, shows that the stress that occurs depicts a Bottle-Shape-struts which is increasingly vague due to the influence of the installed strut and tie model.on the beam.Stress concentration occurs at the supports and load sharing blocks from a pressure load of 10 kN.
In the model, the dark blue beam shows the stress that occurs at a stress of 0.001485 kN, the closer it is to the red color, the stress increases by 21.781 kN.
In the BT-1000.3.06 beam model, which is a variation of the beam model, shows that the stress that occurs depicts a Bottle-Shape-struts which is increasingly vague due to the influence of the installed strut and tie model.on the beam.Stress concentration occurs at the supports and load sharing blocks from a pressure load of 10 kN.In the model, the dark blue beam shows the stress that occurs at a stress of 0.001242 kN, the closer it is to the red color, the stress increases by 21.32 kN.

Beam Model Crack Pattern
The crack patterns in the beam models resulting from analysis based on the loads that occur are listed in Figure below,  In Figure 19, there is a picture of the overall crack pattern (all crack) in the BT-800.1.01beam model, which is a variation of the beam model showing the crack pattern that occurs, concentration stress occurs at the support and load sharing block due to a pressure load of 10 kN.In the figure, you can see a crack pattern that occurs in the middle of the beam span at the bottom.In Figure 20, a picture of the overall crack pattern (all crack) in the BT-800.2.02 beam model, which is a variation of the beam model, shows the crack pattern that occurs, the stress concentration that occurs at the support and load sharing blocks due to a pressure load of 10 kN.In the picture you can see the crack pattern that occurs in the middle of the beam span at the bottom (field).In Figure 21, a picture of the overall crack pattern (all crack) in the BT-800.3.03 beam model, which is a variation of the beam model, shows the crack pattern that occurs, the stress concentration that occurs at the support and load sharing blocks due to a pressure load of 10 kN.In the picture there is no visible crack pattern that occurs in the middle of the beam span at the bottom (field).In Figure 22, a picture of the overall crack pattern (all crack) in the BT-1000.1.04beam model, which is a variation of the beam model, shows the crack pattern that occurs, the stress concentration that occurs at the support and load sharing blocks due to a pressure load of 20 kN.In the picture you can see the crack pattern that occurs in the middle of the beam span at the bottom (field).Figure 24 Crack pattern of beam model BT-1000.3.06.Source: crack analysis results Source: crack analysis results In Figure 23, a picture of the overall crack pattern (all crack) in the BT-1000.2.05 beam model, which is a variation of the beam model, shows the crack pattern that occurs, the stress concentration that occurs at the support and load sharing blocks due to a pressure load of 20 kN.In the picture, you can see a crack pattern that occurs in the middle of the beam span at the bottom.In Figure 24, there is a picture of the overall crack pattern (all crack) in the BT-1000.3.06 beam model, which is a variation of the beam model, showing the crack pattern that occurs, the stress concentration that occurs at the support and load sharing blocks due to a pressure load of 20 kN.In the picture there is no visible crack pattern that occurs in the middle of the beam span at the bottom (field).Source: Analysis results

Comparative Analysis of Analytical Results
Table 7 shows that: A comparison of the ultimate moment, concentrated load and ultimate shear can be seen from manual calculations and FEM computations.From points 9 to 13 above, it can be concluded that the curvature ductility value at an angle <45º will decrease by 27.11%.the curvature ductility value at one angle > 45º will decrease by 55.67%, but for the two angle model (68º/71º) it will decrease by 58.62% and for three angles (68º/71º/71º) and (68º/78º/ 78º) will decrease by 55.67%.

Conclusion
Based on analysis using the finite element method using a computing program, it is concluded as follows: -Beam with height (h)= 800 mm, diagonal strut angle 45º, type 1 does not increase the ultimate moment (Mu), ultimate concentrated load (Pu) increases in type 2 respectively by 29.59% and 80.49% compared to type 1.In ultimate shear (Vu) there was an increase in type 2 of 23.09% compared to type 1. -Beam with height (h)= 1000 mm, diagonal strut angle 68º and 71º, in type 2 there is an increase of 2.90% compared to type 1.The ultimate concentrated load (Pu) in type 2 increases by 83.00% compared to type 1.In ultimate shear (Vu) there was a decrease in type 2 of 29.28% compared to type 1. -At a single angle (45º), for the beam model with a height (h) < 1000 mm there was no increase in the ultimate moment (Mu), in type 2 with three angles (68º,71º,71º) there was a decrease of 33.33% compared to type 1, the ultimate concentrated load (Pu) for type 2 increased by 80.49% compared to type 1.The ultimate shear (Vu) increased in type 2 by 23.09% compared to type 1. -At a single angle (45º), for a beam model with a height (h) = 1000 mm.there was a decrease in the ultimate moment (Mu), but in type 2 computing with three angles (68º,71º,71º) there was an increase of 2.90% compared to type 1, in type 2 there was an increase of 83.00% in type 2 computing.towards type 1.At ultimate shear (Vu) there is a decrease in type 2 relative to type 1. -At ultimate concentrated load (Pu) and ultimate shear (Vu), increasing the angle from 68º/71º/71º to 68º/78º/78º or increasing the slope by 7º indicates that the strut-and tie behavior for the 3 decreasing angles will be better than the increasing angle.larger, in the model it can be seen that a 7o angle reduction provides a minimum increase of 1.40%.-In the beam (h) = 800 mm, the curvature ductility in model 2 will decrease by 26.58% compared to model 1.In the beam (h) = 1000 mm, the curvature ductility in model 3 will decrease by 55.31% compared to model 1. -The amount of deformation that occurs in beams with a height of 800 mm and 1000 mm is respectively in the range of 0.928 mm-3.203mm; and 6.4064-10.9808mm.-The stress contours in all models show behavior parallel to the diagonal strut and take the form of a bottle shape.-The strain behavior illustrates that there is a significant increase in strain in the support area.
-The crack pattern shows a concentration of cracks in the support area and field, especially in the beam with a height of 800 mm.medium on a high beam of 1000 mm.only occurs at supports with a shear crack pattern.
From the conclusion above, for a diagonal strut angle < 68º Mu, the best is type 1 strut-and-tie beam (symmetrical diagonal reinforcement), for ultimate concentrated load (Pu) the optimal is type 2 strutand-tie beam, for ultimate shear ( The optimal Vu) is a type 2 strut-and-tie beam (diagonal truss truss).
For diagonal strut angles > 68º Mu the best is strut-and-tie beam type 1 (symmetrical diagonal reinforcement), for ultimate concentrated load (Pu) the optimal is strut-and-tie beam type 1, for ultimate shear (Vu) which is optimal is a type 2 strut-and-tie beam (diagonal truss truss).
Comparison of the curvature ductility value for a beam with strut-and-tie at an angle < 45º will decrease by 27.11%, the curvature ductility value at an angle > 45º will decrease by 55.67%, but for the two-angle model (68º/71º) will decrease by 58.62% and for three corners (68º/71º/71º) and (68º/78º/78º) it will decrease by 55.67%.

Figure 2
Figure 2 Crack and failure patterns in High Beam (Shaaban, et.al, 2020)

Figure 3
Figure 3 High Beam and strut-and-tie models (Aguilar, et.al.2022) Beam dimensions, b= 300 mm, L= 2450 mm.Variations in beam height, h1 = 800 mm, h2 = 1000 mm.Concrete force fc 25 Mpa.yield reinforcement steel fy 420 Mpa.Reinforcement diameter ; Upper longitudinal reinforcement 2 dia 13 mm.Lower longitudinal reinforcement 2 dia 16 mm.Side longitudinal reinforcement 2 dia 13 mm, left and right side.The stirrup reinforcement is 8-100 mm.Ties reinforcement dia 13 mm, The strut-and-tie reinforcement is 16 mm.Loading Variations: Symmetrical 2point load 3D software modeling uses half the span of the model Research Model The research model created is attached in Figure 4 below: (a) Type 1: Simple strut-and tie with two diagonal bars.(b) Type 2: Strut-and tie diagonal reinforcement Truss.

Figure 4
Figure 4 Variation of Research Model In Figure 2 above there are three types of Strut-and-Tie.In type 1, the supports and ties are simple with two support rods and ties which function to transfer the load from compression reinforcement to tension reinforcement.In type 2, the supports and ties resemble a truss system with several support rods and ties which function to transfer loads from compression reinforcement to tension reinforcement.

6 Figure 5
Figure 5 Flow chart of Research Model Figure 6 Flow chart of Programe Output

Figure 7 Figure 8
Figure 7.a,b,c.Longitudinal section and perspective section of Beam model BT-800.1.01,Validation, BT-800.1.01,BT-1000.1.04,with Meshing Volumes in the computational program In Figure 7. a, b, c, Figure 8. a, b, c, and Figure 9. a, b, c, are the strut-and-tie models of BT-800.1.01Beam, Validation, BT-800.1.01,and BT.AS-1000.1.04with different height variations, for BT-800.1.01Beam the beam height is 800 mm, BT-1000.1.04beam height is 1000 mm.In Figure 7.a, a longitudinal section of the beam and support with Meshing Volume is visible, Figure 7.b shows a longitudinal section of the beam which illustrates the location of the reinforcement, Figure 7.c shows a perspective cross-section of the beam which more clearly depicts the location of the reinforcement.

Figure 9 . 11 Figure 10 .
Figure 9. Deformation diagram of beam models BT-800.1.01,BT-800.2.02, and BT-800.3.03.Source: results of deformation analysis From Figure 9. for model BT-800.1.01;BT-800.2.02; and BT-800.3.03; with the deformation value that occurs tends to decrease and the behavior of the beam model becomes stronger.For variation model BT-800.1.01;BT-800.2.02; and BT-800.3.03 with the addition of the strut-and-tie model, the behavior of the variation beam model becomes stronger and the moment it can withstand is higher than the validation model, but this increase is fluctuating, depending on the type of model and the load being considered.

Figure 11
Figure 11 Load and deformation curves for BT-800.1.01Figure 12 Load and deformation curves for BT-800.1.02beam model analysis results beam model analysis results Figure 11, is the moment and deformation relationship curve that occurs in the BT-800.1.01beam model.From Curve Figure 11, the load and deformation values for model BT-800.1.01,Pu=231.69kN; deformation = 0.9256 mm. Figure 12, is the moment and deformation relationship curve that occurs in the BT-800.2.02 beam model.From Curve Figure 12, the load and deformation values for model BT-800.2.02, Pu=231.69kN; deformation = 0.9288 mm.

Figure 13
Figure 13 Load and deformation curves for Figure 14 Load and deformation curves for BT-800.3.03 beam model, analysis results BT-1000.1.04beam model, analysis results

Figure 15
Figure 15 Load and deformation curves for Figure 16 Load and deformation curves for BT-1000.2.05 beam model, analysis results BT-1000.3.05 beam model, analysis results Figure 15, is the moment and deformation relationship curve that occurs in the BT-1000.2.05 beam model.From Curve Figure 15, the load and deformation values for model BT-1000.2.05, Pu= 196.41 kN; deformation = 15.2499mm. Figure 16, is the moment and deformation relationship curve that occurs in the BT-1000.3.06 beam model.From Curve Figure 16, the load and deformation values for model BT-1000.3.06,Pu= 308.905 kN; deformation = 6.4064 mm.Stress Behavior and Crack Patterns of Beam Models The stress pattern in the beam model as a result of analysis based on the load that occurs is listed in Figure at below,

Figure 19
Figure 19 Crack pattern of beam model BT-800.1.01.Figure20Crack pattern of beam model BT-800.2.02 Source: crack analysis results Source: crack analysis results

Figure 20
Figure 19 Crack pattern of beam model BT-800.1.01.Figure20Crack pattern of beam model BT-800.2.02 Source: crack analysis results Source: crack analysis results

Figure 21
Figure 21 Crack pattern of reverse model BT-800.1.01Figure 22 Crack pattern of beam model BT-1000.1.04.Source: crack analysis results Source: crack analysis results

Table 1
. Table of manual calculation results of ultimate moment, ultimate concentrated load (Pu), and ultimate shear load (Vu) without strut-and-tie and with strut-and-tie for half-span beams.the ultimate moment for variations of the strut-and-tie model, the value increases successively.The curvature ductility value also increases with increasing beam height (h).

Table 2 .
Load and deformation values for the beam model at initial crack, yield and ultimate conditions at half span (1/2 L).The calculation results use the Kent and Park stress-strain curve.

Table 3
Values of ultimate load, reinforcement ratio and curvature ductility of the beam model at the ultimate condition of half-span view (1/2L).The calculation results use the Kent and Park stress-strain curve.

Table 4 .
Moment and curvature values of the beam model at the initial crack, yield and ultimate conditions resulting from analysis using a computing program

Table 5 .
Moment and curvature values for the beam model in the ultimate condition, results of analysis using a computing program

Table 5 .
above it can be seen that moment and curvature values at crack curvature at initial crack, yield and ultimate condition for beam result.

Table 7 .
Recapitulation table of calculation results of ultimate moment capacity (Mu), concentrated load capacity (Pu) and ultimate shear capacity (Vu) of ductility of beam models using analytical and analysis methods

Table 8 ,
recapitulation table of calculation results for ultimate moment capacity (Mu), concentrated load capacity (Pu) and ultimate shear capacity (Vu) Ductility of beam models using analytical methods and analysis