RD-E: 4702 Splitting Tensile Test (Brazilian Test)

Using a splitting tensile test to calculate input for material LAW24.



Figure 1.

A splitting tensile test is also known as a Brazilian test. It is a typical test used for concrete material characterization carried out using procedure adhering to ASTM D3967, “Standard Test Method for Splitting Tensile Strength of Intact Rock Core Specimens”. A load is applied to a concrete cylinder with its axis normal to the loading direction. A tensile stress develops in the center. The force is slowly increased until the specimen fails by an extension fracture along the loading plane. Then the tensile strength is computed from this force of failure.

The splitting tensile test aims to evaluate failure limits. It is difficult to apply uniaxial tension to a concrete specimen. Therefore, the tensile strength of the concrete material is determined by indirect test methods such as a Split Cylinder Test or even a Flexure Test. It should be noted that both methods give a higher value of tensile strength than the uniaxial tensile strength. This will be explained in the current example. This test will be modeled, and test results will be used for numerical calibration of the material law.

Options and Keywords Used

Input Files

Before you begin, copy the file(s) used in this example to your working directory.

Model Description

Since this test is quasi-static, a concrete cylinder is crushed with slow velocity. It is the standard test, to evaluate the tensile strength of concrete. This test could be performed following IS:5816-1970.

A cylinder of the concrete specimen is placed horizontally between the loading surfaces of a compression testing machine (Figure 2). The compression load is applied diametrically and uniformly along the length of the cylinder until the failure of the cylinder along the vertical diameter. A uniform distribution of the pressure load is created by using strips of plywood which are placed between the specimen and loading plates of the testing machine. This also reduces the magnitude of the high compressive stresses near the points of application of this load, concrete cylinders split into two halves along the vertical plane due to indirect tensile stress generated by Poisson's effect.


Figure 2. Diametrically loaded concrete cylinder
Assuming the concrete specimen behaves as an elastic body, a uniform lateral tensile stress acting along the vertical plane causes the failure of the specimen. This can be calculated from the following formula for the splitting tensile strength: (1) f st = 2 F max πLD MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaS baaSqaaiaadohacaWG0baabeaakiabg2da9maalaaabaGaaGOmaiaa dAeadaWgaaWcbaGaciyBaiaacggacaGG4baabeaaaOqaaiabec8aWj aadYeacaWGebaaaaaa@4378@
Where,
L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGmbaaaa@3839@
Cylinder length
D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGmbaaaa@3839@
Diameter
F max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbWaaS baaSqaaiGac2gacaGGHbGaaiiEaaqabaaaaa@3B33@
Ultimate force that leads to failure causing the specimen to split into two halves

The Radioss concrete material LAW24 is designed to work with only a few mandatory parameters. The other parameters are optional. If the optional parameters are not entered default values are calculated using typical properties of concrete material.

The numerical implementation is based on work by Han & Chen. 1 It defines an Ottosen failure envelope. 2 It can be fully determined by providing 4 failure points which are described with 5 parameters. The mandatory input is compression strength f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaam4yaaqabaaaaa@3B54@ . The four other ones are optional parameters. They are written as a ratio of compression strength which allows the following default values that are typical for concrete:
  • Direct Tensile Strength: f t = 0.05 f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaamiDaaqabaGccqGH9aqpcaaIWaGaaiOl aiaaicdacaaI1aGaamOzamaaBaaaleaacaWGJbaabeaaaaa@4159@
  • Biaxial Compression Strength: f b = 1.2 f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaamOyaaqabaGccqGH9aqpcaaIXaGaaiOl aiaaikdacaWGMbWaaSbaaSqaaiaadogaaeqaaaaa@408B@
  • Confined Compression Strength (tri-axial test): f 2 = 4.0 f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcaaI0aGaaiOl aiaaicdacaWGMbWaaSbaaSqaaiaadogaaeqaaaaa@4061@
  • Confined pressure: s 0 = 1.25 f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadohadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaaIXaGaaiOl aiaaikdacaaI1aGaamOzamaaBaaaleaacaWGJbaabeaaaaa@412A@

Splitting Tensile Test

If only splitting tensile test data is available, then all of the failure parameters f c ,   f t , f b , f 2 , s 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaam4yaaqabaGccaGGSaGaaeiiaiaadAga daWgaaWcbaGaamiDaaqabaGccaGGSaGaaGjbVlaadAgadaWgaaWcba GaamOyaaqabaGccaGGSaGaaGjbVlaadAgadaWgaaWcbaGaaGOmaaqa baGccaGGSaGaaGjbVlaadohadaWgaaWcbaGaaGimaaqabaaaaa@4B45@ are not available.


Figure 3. Splitting Tensile test illustration

Experience provides that f s t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaam4Caiaadshaaeqaaaaa@3C5D@ is related to loading force F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAeaaaa@3A20@ on the cylinder. This value is sometimes used as an estimation of the direct tensile strength f t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaamiDaaqabaaaaa@3B65@ which is one of the input parameters. It can be observed that f t < f s t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaamiDaaqabaGccqGH8aapcaWGMbWaaSba aSqaaiaadohacaWG0baabeaaaaa@3F7B@ .

Elastic theory enables to write: (2) { σ 1 ( d )= 2F πLD ( D 2 4 d 2 D 2 4 d 2 ) 2 σ 2 ( d )= 2F πLD ( 4 D 2 D 2 4 d 2 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaamaaceaabaqbaeqabiqaaaqaaiabeo8aZnaaBaaaleaacaaIXaaa beaakmaabmaabaGaamizaaGaayjkaiaawMcaaiabg2da9maalaaaba GaaGOmaiaadAeaaeaacqaHapaCcaWGmbGaamiraaaadaqadaqaamaa laaabaGaamiramaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaisdaca WGKbWaaWbaaSqabeaacaaIYaaaaaGcbaGaamiramaaCaaaleqabaGa aGOmaaaakiabgkHiTiaaisdacaWGKbWaaWbaaSqabeaacaaIYaaaaa aaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOqaaiabeo8a ZnaaBaaaleaacaaIYaaabeaakmaabmaabaGaamizaaGaayjkaiaawM caaiabg2da9iabgkHiTmaalaaabaGaaGOmaiaadAeaaeaacqaHapaC caWGmbGaamiraaaadaqadaqaamaalaaabaGaaGinaiaadseadaahaa WcbeqaaiaaikdaaaaakeaacaWGebWaaWbaaSqabeaacaaIYaaaaOGa eyOeI0IaaGinaiaadsgadaahaaWcbeqaaiaaikdaaaaaaOGaeyOeI0 IaaGymaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaaakiaa wUhaaaaa@6AA1@
Where,
F MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGmbaaaa@3839@
Loading force
D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGmbaaaa@3839@
Cylinder diameter
L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGmbaaaa@3839@
Length
d MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGmbaaaa@3839@
Position on the diameter
This formula is maximized at the center, where d MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGmbaaaa@3839@ =0 to obtain: (3) { σ 1 max ( t ) = 2 F ( t ) π L D σ 2 max ( t ) = 3 σ 1 max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaamaaceaabaqbaeqabiqaaaqaaiabeo8aZnaaDaaaleaacaaIXaaa baGaciyBaiaacggacaGG4baaaOWaaeWaaeaacaWG0baacaGLOaGaay zkaaGaeyypa0ZaaSaaaeaacaaIYaGaamOramaabmaabaGaamiDaaGa ayjkaiaawMcaaaqaaiabec8aWjaadYeacaWGebaaaaqaaiabeo8aZn aaDaaaleaacaaIYaaabaGaciyBaiaacggacaGG4baaaOWaaeWaaeaa caWG0baacaGLOaGaayzkaaGaeyypa0JaeyOeI0IaaG4maiabeo8aZn aaDaaaleaacaaIXaaabaGaciyBaiaacggacaGG4baaaaaaaOGaay5E aaaaaa@5B42@

The loading path direction and stress state in this test are different than a typical uniaxial tensile test. The material in this test undergoes both compression and tension. Compression is due to the loading force over the cylinder, and tension is due to the Poisson effect.

Another significant finding from this theoretical result is to observe that the direct tensile strength is lower than splitting tensile strength (Figure 4).


Figure 4. Splitting tensile strength versus direct tensile strength on the failure envelope

Fusco suggested for conventional concrete buildings the relation f t = 0.85 f s t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaS baaSqaaiaadshaaeqaaOGaeyypa0JaaGimaiaac6cacaaI4aGaaGyn aiaadAgadaWgaaWcbaGaam4Caiaadshaaeqaaaaa@407D@ but other authors found the relation f t = 0. 66 f s t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaS baaSqaaiaadshaaeqaaOGaeyypa0JaaGimaiaac6cacaqG2aGaaeOn aiaadAgadaWgaaWcbaGaam4Caiaadshaaeqaaaaa@406E@ . 4

Using the theoretical loading path from the elastic hypothesis with Han & Chen failure surface and the values for concrete:

f t f c = 1 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaai aadAgadaWgaaWcbaGaamiDaaqabaaakeaacaWGMbWaaSbaaSqaaiaa dogaaeqaaaaakiabg2da9maalaaabaGaaGymaaqaaiaaigdacaaIWa aaaaaa@3EE1@ ; f b f c = s 0 f c = 6 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaai aadAgadaWgaaWcbaGaamOyaaqabaaakeaacaWGMbWaaSbaaSqaaiaa dogaaeqaaaaakiabg2da9maalaaabaGaam4CamaaBaaaleaacaaIWa aabeaaaOqaaiaadAgadaWgaaWcbaGaam4yaaqabaaaaOGaeyypa0Za aSaaaeaacaaI2aaabaGaaGynaaaaaaa@4325@ ; f 2 f c =4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaai aadAgadaWgaaWcbaGaaGOmaaqabaaakeaacaWGMbWaaSbaaSqaaiaa dogaaeqaaaaakiabg2da9iaaisdaaaa@3D22@

leads to the following estimation: (4) f t = 1 2 f st 0.71 f st MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaS baaSqaaiaadshaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaWaaOaa aeaacaaIYaaaleqaaaaakiaadAgadaWgaaWcbaGaam4Caiaadshaae qaaOGaeyisISRaaGimaiaac6cacaaI3aGaaGymaiaadAgadaWgaaWc baGaam4Caiaadshaaeqaaaaa@46E7@

This estimation is of course dependent on the failure envelope which is shaped by all 5 parameters. Changing their values will change the f st f t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWccaqaai aadAgadaWgaaWcbaGaam4CaiaadshaaeqaaaGcbaGaamOzamaaBaaa leaacaWG0baabeaaaaaaaa@3C9C@ ratio.

Rigid Walls

Rigid walls are used to model the plywood used in the test to apply the load. Nodes on the cylinder the width of the plywood are included as secondary nodes of the rigid wall. A high friction value is set to prevent the concrete cylinder from sliding.

Loading Pressure

A compressive load is created by applying displacements in opposite directions to both rigid walls using the imposed displacement option /IMPDISP. The imposed displacement uses a linear function large enough to split the cylinders in half.

Solid Properties

  • qa =1e-20 and qb =1e-20
  • Isolid = 24
  • IHKT = 2
  • All other property values use the default options

Material Data

Units: mm, ms, g, MPa

The concrete material data is:
  • Initial density = 0.0024 [ g m m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada WcaaqaaiaadEgaaeaacaWGTbGaamyBamaaCaaaleqabaGaaG4maaaa aaaakiaawUfacaGLDbaaaaa@3BBC@
  • Concrete elasticity Young’s modulus E c =61000[MPa] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaS baaSqaaiaadogaaeqaaOGaeyypa0JaaeOnaiaabgdacaqGWaGaaGim aiaaicdacaaMi8UaaGjbVlaacUfaciGGnbGaaiiuaiaacggaciGGDb aaaa@4556@
  • Poisson’s ratio ν=0.17 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaIWaGaaiOlaiaabgdacaqG3aaaaa@3D00@
  • Concrete uniaxial compression strength from test f c =58[MPa] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaam4yaaqabaGccqGH9aqpcaaI1aGaaGio aiaaysW7caGGBbGaciytaiaaccfacaGGHbGaciyxaaaa@43C0@
  • Concrete uniaxial tension strength is 0.05 f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeuz0lXdbb a9frFj0=irFfea0lXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamOzamaaBaaaleaacaWGJbaabeaaaaa@3A81@ so the ratio is defined as f t f c =0.05 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaamaaliaabaGaamOzamaaBaaaleaacaWG0baabeaaaOqaaiaadAga daWgaaWcbaGaam4yaaqabaaaaOGaeyypa0JaaGimaiaac6cacaaIWa GaaGynaaaa@4175@
  • All other parameters can be left as default in LAW24 because the default values are representative of generic concrete materials.
Some measurement from splitting tensile test data:
  • D=150 mm L=300 mm F max =280100 N f st = 2 F max πLD =3.96MPa MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiaads eacqGH9aqpcaaIXaGaaGynaiaaicdacaqGGaGaamyBaiaad2gaaeaa caWGmbGaeyypa0JaaG4maiaaicdacaaIWaGaaeiiaiaad2gacaWGTb aabaGaamOramaaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaOGaeyyp a0JaaGOmaiaaiIdacaaIWaGaaGymaiaaicdacaaIWaGaaeiiaiaad6 eaaeaacaWGMbWaaSbaaSqaaiaadohacaWG0baabeaakiabg2da9maa laaabaGaaGOmaiaadAeadaWgaaWcbaGaciyBaiaacggacaGG4baabe aaaOqaaiabec8aWjaadYeacaWGebaaaiabg2da9iaaiodacaGGUaGa aGyoaiaaiAdacaWGnbGaamiuaiaadggaaaaa@61D7@
Starting with the nominal values:
  • Tensile strength f t = 1 2 f st = 1 2 ×3.96  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaS baaSqaaiaadshaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaWaaOaa aeaacaaIYaaaleqaaaaakiaadAgadaWgaaWcbaGaam4Caiaadshaae qaaOGaaeypamaalaaabaGaaGymaaqaamaakaaabaGaaGOmaaWcbeaa aaGccqGHxdaTcaqGZaGaaeOlaiaabMdacaqG2aGaaeiiaaaa@4748@ with f c =58  f t f c =0.05 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaS baaSqaaiaadogaaeqaaOGaeyypa0JaaGynaiaaiIdacaqGGaGaeyO0 H49aaSaaaeaacaWGMbWaaSbaaSqaaiaadshaaeqaaaGcbaGaamOzam aaBaaaleaacaWGJbaabeaaaaGccqGH9aqpcaaIWaGaaiOlaiaaicda caqG1aaaaa@470F@
  • Biaxial Compression Strength: f b =1.20 f c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaS baaSqaaiaadkgaaeqaaOGaeyypa0JaaGymaiaac6cacaaIYaGaaGim aiaadAgadaWgaaWcbaGaam4yaaqabaaaaa@3F58@ by default.
  • Confined Compression Strength (tri-axial test): f 2 =4.00 f c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaS baaSqaaiaabkdaaeqaaOGaeyypa0Jaaeinaiaab6cacaqGWaGaaeim aiaadAgadaWgaaWcbaGaam4yaaqabaaaaa@3F11@ by default
  • Confined pressure: s 0 =1.25 f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadohadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaaIXaGaaiOl aiaaikdacaaI1aGaamOzamaaBaaaleaacaWGJbaabeaaaaa@412A@ by default This leads to the following material card input file.

Results

The maximum force measured from the rigid wall is 275612 N which is close to the maximum force from the test of 280100 N. The elastic theory equations at d=0 predict that the principal stress P1 at the center of the cylinder should be 3.90 MPa. The principal stress P2 at the center of the cylinder should be -11.7 MPa.(5) { σ 1 max ( t )= 2F( t ) πLD 2*275612 3.14*300*150 3.90 σ 2 max ( t )=3 σ 1 max 3*3.9011.70 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaamaaceaabaqbaeqabiqaaaqaaiabeo8aZnaaDaaaleaacaaIXaaa baGaciyBaiaacggacaGG4baaaOWaaeWaaeaacaWG0baacaGLOaGaay zkaaGaeyypa0ZaaSaaaeaacaaIYaGaamOramaabmaabaGaamiDaaGa ayjkaiaawMcaaaqaaiabec8aWjaadYeacaWGebaaaiabgIKi7oaala aabaGaaGOmaiaacQcacaaIYaGaaG4naiaaiwdacaaI2aGaaGymaiaa ikdaaeaacaaIZaGaaiOlaiaaigdacaaI0aGaaiOkaiaaiodacaaIWa GaaGimaiaacQcacaaIXaGaaGynaiaaicdaaaGaeyisISRaaG4maiaa c6cacaaI5aGaaGimaaqaaiabeo8aZnaaDaaaleaacaaIYaaabaGaci yBaiaacggacaGG4baaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaGa eyypa0JaeyOeI0IaaG4maiabeo8aZnaaDaaaleaacaaIXaaabaGaci yBaiaacggacaGG4baaaOGaeyisISRaeyOeI0IaaG4maiaacQcacaaI ZaGaaiOlaiaaiMdacaaIWaGaeyisISRaeyOeI0IaaGymaiaaigdaca GGUaGaaG4naiaaicdaaaaacaGL7baaaaa@7D5C@
The stress of the center elements, Element ID 39000 and 39520, show that the results match the analytical results of 3.96 MPa.


Figure 5. Simulation rigid wall force and stress at some center elements
From the displacement plot, the Poisson's effect is shown at the top and bottom elements. The center of the cylinder is under tensile loading until the cylinder is cracked.


Figure 6. After the failure at T=1.90, the concrete has failed and the displacement is horizontal


Figure 7. Plot of principal values in the local cracking using /ANIM/BRICK/DAM1
The numerical result is consistent with the theoretical solution. However, there is a slight deviation when approaching the failure limit due to the crack opening and damage in adjacent finite elements (nonlinearities). The next diagram shows the principal stress of some elements from the simulation.


Figure 8. Failure limit

Conclusion

The splitting tensile strength f s t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaam4Caiaadshaaeqaaaaa@3C5D@ measured from the Brazilian test is bigger than the tensile strength f t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaamiDaaqabaaaaa@3B65@ from a direct tensile test f t < f s t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaamiDaaqabaGccqGH8aapcaWGMbWaaSba aSqaaiaadohacaWG0baabeaaaaa@3F7B@ . Using LAW24 with mostly default values, the tensile strength f t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaamiDaaqabaaaaa@3B65@ is about 0.71 times the splitting tensile strength f s t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaam4Caiaadshaaeqaaaaa@3C5D@ . Only the maximum force F max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbWaaS baaSqaaiGac2gacaGGHbGaaiiEaaqabaaaaa@3B33@ from the splitting tensile test and f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaam4yaaqabaaaaa@3B54@ from a cube compression test are needed as input to LAW24.

From a splitting tensile test, the parameter f t f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaamaaliaabaGaamOzamaaBaaaleaacaWG0baabeaaaOqaaiaadAga daWgaaWcbaGaam4yaaqabaaaaaaa@3D80@ in LAW24 could be determined. (6) F max exp e r i m e n t > f s t a n l y t i c a l > f t a n l y t i c a l > f t f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAeadaqhaaWcbaGaciyBaiaacggacaGG4baabaGaciyzaiaa cIhacaGGWbGaamyzaiaadkhacaWGPbGaamyBaiaadwgacaWGUbGaam iDaaaakiabgkHiTiabg6da+iaadAgadaqhaaWcbaGaam4Caiaadsha aeaacaWGHbGaamOBaiaadYgacaWG5bGaamiDaiaadMgacaWGJbGaam yyaiaadYgaaaGccqGHsislcqGH+aGpcaWGMbWaa0baaSqaaiaadsha aeaacaWGHbGaamOBaiaadYgacaWG5bGaamiDaiaadMgacaWGJbGaam yyaiaadYgaaaGccqGHsislcqGH+aGpdaWccaqaaiaadAgadaWgaaWc baGaamiDaaqabaaakeaacaWGMbWaaSbaaSqaaiaadogaaeqaaaaaaa a@66B1@
Next, compare the experiment and simulation force and strength.(7) F max exp e r i m e n t ~ F max s i m u l a t i o n f s t a n l y t i c a l ~ f s t s i m u l a t i o n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbvg9s8 qqaqFr0xc9ps0xbba9s8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO abaeqabaGaamOramaaDaaaleaaciGGTbGaaiyyaiaacIhaaeaaciGG LbGaaiiEaiaacchacaWGLbGaamOCaiaadMgacaWGTbGaamyzaiaad6 gacaWG0baaaOGaaiOFaiaadAeadaqhaaWcbaGaciyBaiaacggacaGG 4baabaGaam4CaiaadMgacaWGTbGaamyDaiaadYgacaWGHbGaamiDai aadMgacaWGVbGaamOBaaaaaOqaaiaadAgadaqhaaWcbaGaam4Caiaa dshaaeaacaWGHbGaamOBaiaadYgacaWG5bGaamiDaiaadMgacaWGJb GaamyyaiaadYgaaaGccaGG+bGaamOzamaaDaaaleaacaWGZbGaamiD aaqaaiaadohacaWGPbGaamyBaiaadwhacaWGSbGaamyyaiaadshaca WGPbGaam4Baiaad6gaaaaaaaa@6DF6@
1 D.J. Han, W.F. Chen “Plasticity model for concrete in Mechanics of Materials”, North Holland
2 Ottosen N.S. “Nonlinear Finite Element Analysis of Concrete Structures” Ris. National Laboratory DK 4000 Roskilde Denmark, May 1980
3 FUSCO, P.B. Concrete structures - Fundamentals of structural design. McGraw-Hill, 1976, São Paulo.
4 A. Ghaffar, M. A. Chaudhry and M. Kamran Ali - A new Approach for measurement of tensile strength of concrete - Journal of Research, Bahauddin Zakariya University, Vol.16, No.1, June 2005, pp. 01-0