Comparison of constitutive models for nonlinear response of cracked reinforced concrete

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Comparison of constitutive models for nonlinear response of cracked reinforced concrete

Kopitha Kirushnapillai

Rifnaz Rafeek

Kushan Kalmith Wijesundara

Abstract

Over the past few decades, many constitutive models have been proposed to analyze the behavior of cracked reinforced concrete members, taking into account the axial- shear-moment interaction. Basically, they can be classified into discrete crack models and smeared crack models. Out of these models, Modified Compression Field Theory, Disturbed Stress Field Model and Total Strain Smeared Crack Model are widely used as smeared crack models in commercial finite element codes. However, many researchers have highlighted the limitations of these models specially when predicting the shear-critical response of the cracked reinforced concrete elements. Therefore, the objective of this research is to review the selected three constitutive models and to compare their capabilities in predicting load-deformation response of wide range of concrete elements depending on different shear span to depth ratio, percentage of longitudinal and transverse reinforcement and concrete grade and hence identify best model based on coefficient of variation and mean of test results. A database including some reinforced concrete specimens was developed by collecting past experimental data. Numerical analyses were performed for all the specimens using commercial finite element software and their results were compared with the experimental data in order to validate the capability of the constitutive models to predict the overall response of shear-critical reinforced concrete elements. In addition, coefficient of variation of the predicted strength of the elements by the three constitutive models are presented. Finally, some guidelines are proposed on how to select the parameters in the constitutive models for better prediction of load -deformation response of reinforced concrete elements. It was concluded that by comparing experimental and numerical results, Modified Compression Field Theory and Disturbed Stress Field Model have some drawbacks in capacity prediction in specific situations while Total strain model give better prediction with the suitable selection of material models.

Keywords. constitutive models, axial-shear-moment interaction, shear-critical response, load-deformation response.

Сравнение определяющих моделей для нелинейного отклика потрескавшегося железобетона

Копита Кирушнапиллай, Рифназ Рафеек, Кушан Калмитх Виджесундара

Абстракт

structural model load-deformation response

За последние несколько десятилетий было предложено множество расчетных моделей для анализа поведения треснувших железобетонных конструкций с учетом взаимодействия осевого сдвига момента. В основном, их можно разделить на модели дискретных трещин и модели размытых трещин. Из этих моделей модифицированная теория поля сжатия, модель поля возмущенных напряжений и модель размытых трещин с полной деформацией широко используются в качестве моделей в коммерческих конечно-элементных кодах. Однако многие исследователи отмечают ограничения этих моделей, особенно при прогнозировании критического сдвига треснувших железобетонных элементов. Поэтому целью данного исследования является обзор выбранных трех конструктивных моделей и сравнение их возможностей в прогнозировании реакции на нагрузку-деформацию широкого спектра железобетонных элементов в зависимости от соотношения пролета сдвига к глубине, процентного содержания продольной и поперечной арматуры и марки бетона и, следовательно, определение лучшей модели на основе коэффициента вариации и среднего значения результатов испытаний. База данных, включающая некоторые железобетонные образцы, была создана путем сбора экспериментальных данных прошлых лет. Для всех образцов были проведены численные анализы с использованием коммерческого программного обеспечения для конечных элементов, результаты которых сравнивались с экспериментальными данными, чтобы подтвердить способность конструктивных моделей предсказывать общую реакцию железобетонных конструкций, критичных к сдвигу. Кроме того, представлены коэффициенты вариации прогнозируемой прочности элементов по трем расчетным моделям. Наконец, предложены некоторые рекомендации по выбору параметров в расчетных моделях для лучшего прогнозирования реакции железобетонных элементов на нагрузку и деформацию. В результате сравнения экспериментальных и численных результатов было сделано заключение, что модифицированная теория поля сжатия и модель поля нарушенных напряжений имеют некоторые недостатки в прогнозировании прочности в конкретных ситуациях, в то время как модель полной деформации дает лучший прогноз при соответствующем выборе моделей материалов.

Ключевые слова: определяющие модели, взаимодействие осевого и сдвигового моментов, сдвигово-критическая реакция, реакция нагрузки-деформации.

Introduction

Cracking of reinforced concrete members is a major issue in the structures for their service life. Basically, there are two types of cracks observed in reinforced concrete members. They are called flexural and shear cracks. As a structural engineer, we have to predict those types of failures. Prediction of shear failure is quite complex. Some constitutive models have developed to predict these type of failure modes in reinforced concrete elements. Cracked reinforced concrete constitutive models can be divided into discrete and smeared crack models. Smeared crack models can be further divided into fixed, rotating, and multi-direction crack models. In rotating crack models, it is assumed that the orientation of the crack direction and principal stress and principal strain directions in the concrete change with externally applied loading. To address the rotating crack concept Vecchio and collins (1982), Barzegar-Jamshidi and Schnobrich (1986), Ayoub and Filippou (1998), and Vecchio (2000) have formulated rotating crack models. Contrarily, in the fixed crack model, crack direction and principal stress and principal strain directions remain fixed in the direction of the first crack. Okamura and Maekawa (1998), and Kaufmann and Marti (1998) have proposed fixed crack models.

Literature review

Flexural failure and shear failure are the failure modes of reinforced concrete elements. To analyse shear failure there are many constitutive crack models. The concrete crack model can be divided into two categories: discrete crack model and the smeared crack model. The smeared crack model can be further divided into fixed, rotating, and multi-directional crack model. In the fixed crack model, orientation of the crack is kept constant and in the rotating crack model, orientation of the crack is varying with externally applied loading.

Modified Compression Field Theory. Modified compression field theory (MCFT) proposed by Vecchio and Collins (1986) tested 30 reinforced concrete panels under a variety of well-defined uniform biaxial stresses including pure shear. And from compatibility condition, equilibrium condition and the constitutive relationship they developed a constitutive model to predict the response of reinforced concrete elements when in-plane shear and axial stresses are active.

Disturbed Stress Field Model. Disturbed stress field model (DSFM) proposed by Vecchio (2000) to mainly readdress the two main weaknesses in MCFT: the assumption of principal average strain and principal average stress directions are same, and shear check at the crack. For calculation of DSFM total strain was considered instead of net strain. Total strain includes net strain, elastic strain, plastic strain offset, and average shear slip strain. Therefore, principal average strain and principal average stress directions are no longer aligned. Compatibility, equilibrium, and constitutive response are formulated in terms of average stresses and average strains. To calculate principal average tensile stress, in DSFM two constitutive models used one is tension stiffening and another one is tension softening.

A Constitutive Model for Analysis of Reinforced Concrete Solids. Vecchio and Selby used MCFT to develop this 3D model. Because of that in this model also they assumed the direction of principal stress and directions of principal strain were coinciding. Cracked concrete was considered as an orthotropic material. And for the calculation, they considered average stress and average strain terms. Local stress conditions at crack were also considered for calculation. This constitutive model subject to short-term monotonic loading.

Data collection

To analyse the limitation and compare above three theorems when the material properties, shear span and amount of longitudinal and transverse reinforcement vary on reinforced concrete beams, needed experimental results. Therefore, many research papers were reviewed, and database was created with different material properties, shear span to depth ratio and amount of longitudinal and transverse reinforcement.

To assemble a database, we could collect only 18 research papers because of availability. We couldn't include some of them due to unavailability of load-deformation curves and not enough data to proceed. Non-prestressed concrete members and not subjected to axial load members were selected. In this research, only point load at middle or known shear span members were selected. Only one fixed support beam and others are simply supported. Extracted data from experimental results are under conditions as shown in Table 1.

Table 1. Expected data extracted from experimental results

Perform sensitivity study

We have selected commercial software, which are based on the constitutive models. Response2000 based on the MCFT, VecTor5 based on the DSFM, and Midas based on the total strain model have been selected to perform the numerical analysis. We must play with the parameters which are available to change in the commercial software to get the accurate result. Therefore, before performing numerical analysis we went with parametric study for each commercial software.

Parametric study was carried out for each constitutive model. There is no option to play with selection of parameters in response2000. Therefor sensitive study was carried out for VecTor5(DSFM) and Midas (TSM).

Parametric analysis was carried out to determine the influence of the key parameters, which govern performance. The key parameters in TSM are element size, crack model (Rotating, Fixed), tension function (exponential, hordijk), compression function (Parabola, Thorenfeldt), shear function, displacement increment, lateral crack effect and confinement effect.

Results and discussion

Experimental results were compared with numerical results.

Specimen Without Transverse Reinforcement

Specimen with low shear span to depth ratio. Picture 1 shows the comparison of force-displacement response of different constitutive models with experimental result of specimen having shear span to depth ratio of 3.3 and crack pattern on the corresponding beam is shown in picture 2.

Specimen with high shear span to depth ratio. Picture 3 shows the comparison of force-displacement response of different constitutive models with experimental result of specimen having shear span to depth ratio of 5.8 and crack pattern on the corresponding beam is shown in picture 4.

Specimen With Transverse Reinforcement

Specimen with low shear span to depth ratio. Picture 5 shows the comparison of force-displacement response of different constitutive models with experimental result of specimen having shear span to depth ratio of 3.3 and crack pattern on the corresponding beam is shown in picture 6.

Picture 1. Comparison offorce-displacement response of different constitutive models with experimental result of specimen having shear span to depth ratio of 3.3

Picture 2. Crack pattern on beam with no shear reinforcement and Shear span to depth ratio of 3.3

Picture 3. Comparison offorce-displacement response of different constitutive models with experimental result of specimen having shear span to depth ratio of 5.8

Picture 4. Crack pattern on beam with no shear reinforcement and Shear span to depth ratio of 5.8

* Specimen with high shear span to depth ratio. Picture 7 shows the comparison of force-displacement response of different constitutive models with experimental result of specimen having shear span to depth ratio of 5.8 and crack pattern on the corresponding beam is shown in picture 8.

Comparison of Constitutive Models. Mean of MCFT is 1.01. Coefficient of variation of MCFT is 15%. Mean is nearest to one. Mean capacity prediction of MCFT is good. But it is 15% of variability.

Picture 5. Comparison offorce-displacement response of different constitutive models with experimental result of specimen having shear span to depth ratio of 3.3

Picture 6. Crack pattern on beam with shear reinforcement and Shear span to depth ratio of 3.3

Picture 7. Comparison offorce-displacement response of different constitutive models with experimental result of specimen having shear span to depth ratio of 5.8

Picture 8. Crack pattern on beam with shear reinforcement and Shear span to depth ratio of 5.8

Picture 9. Comparison of numerical result and experimental result of a)MCFT b)DSFM c) TSM

Mean of DSFM is 1.06. Coefficient of variation of MCFT is 19%. Even though improvements have been made for DSFM, coefficient of variation become much higher than MCFT. But mean capacity prediction is good as mean is nearest to one.

Mean of TSM is 0.94. Coefficient of variation of TSM is 9%. Shear capacity prediction of TSM is good compared to DSFM and MCFT. Mean is nearest to one and coefficient if variation is less than DSFM and MCFT (table 2).

Table 2. Comparison of mean and coefficient of variation of each constitutive model

Conclusions

When specimen having no and having shear reinforcement, it is proposed to use the following combination of parameters for the accurate prediction of shear capacity (table 3).

Table 3. TSMparameters used for specimens having no and having shear reinforcement

By comparing coefficient of variation and mean of numerical result, it can be concluded that TSM predict the shear capacity accurately compared to the MCFT and DSFM. However, it should be noted that accuracy of shear capacity prediction by TSM based on the selection of parameters.

References

1. A. Ayoub, and F.C. Filippou, “Nonlinear finite element analysis of reinforced concrete shear panels and walls”, Journal of Structural Engineering, American Society of Civil Engineers, 124(3), 1998, pp. 298-308.

2. Dino Angelakos, “The influence of concrete strength and longitudinal reinforcement ratio on the shear strength of large-size reinforcedcConcrete beams with and without transverse reinforcement”, M.Sc. Thesis, University of Toronto, 1999.

3. H. Okamura, and K.Maekawa, “Nonlinear analysis and constitutive methods of reinforced concrete”, International Standard Book Number 7655-1506-0, University of Tokyo, Tokyo, 1991.

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