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Case Study – Methods, Examples and Guide

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A case study is a research method that involves an in-depth examination and analysis of a particular phenomenon or case, such as an individual, organization, community, event, or situation.

It is a qualitative research approach that aims to provide a detailed and comprehensive understanding of the case being studied. Case studies typically involve multiple sources of data, including interviews, observations, documents, and artifacts, which are analyzed using various techniques, such as content analysis, thematic analysis, and grounded theory. The findings of a case study are often used to develop theories, inform policy or practice, or generate new research questions.

Types of Case Study

Types and Methods of Case Study are as follows:

Single-Case Study

A single-case study is an in-depth analysis of a single case. This type of case study is useful when the researcher wants to understand a specific phenomenon in detail.

For Example , A researcher might conduct a single-case study on a particular individual to understand their experiences with a particular health condition or a specific organization to explore their management practices. The researcher collects data from multiple sources, such as interviews, observations, and documents, and uses various techniques to analyze the data, such as content analysis or thematic analysis. The findings of a single-case study are often used to generate new research questions, develop theories, or inform policy or practice.

Multiple-Case Study

A multiple-case study involves the analysis of several cases that are similar in nature. This type of case study is useful when the researcher wants to identify similarities and differences between the cases.

For Example, a researcher might conduct a multiple-case study on several companies to explore the factors that contribute to their success or failure. The researcher collects data from each case, compares and contrasts the findings, and uses various techniques to analyze the data, such as comparative analysis or pattern-matching. The findings of a multiple-case study can be used to develop theories, inform policy or practice, or generate new research questions.

Exploratory Case Study

An exploratory case study is used to explore a new or understudied phenomenon. This type of case study is useful when the researcher wants to generate hypotheses or theories about the phenomenon.

For Example, a researcher might conduct an exploratory case study on a new technology to understand its potential impact on society. The researcher collects data from multiple sources, such as interviews, observations, and documents, and uses various techniques to analyze the data, such as grounded theory or content analysis. The findings of an exploratory case study can be used to generate new research questions, develop theories, or inform policy or practice.

Descriptive Case Study

A descriptive case study is used to describe a particular phenomenon in detail. This type of case study is useful when the researcher wants to provide a comprehensive account of the phenomenon.

For Example, a researcher might conduct a descriptive case study on a particular community to understand its social and economic characteristics. The researcher collects data from multiple sources, such as interviews, observations, and documents, and uses various techniques to analyze the data, such as content analysis or thematic analysis. The findings of a descriptive case study can be used to inform policy or practice or generate new research questions.

Instrumental Case Study

An instrumental case study is used to understand a particular phenomenon that is instrumental in achieving a particular goal. This type of case study is useful when the researcher wants to understand the role of the phenomenon in achieving the goal.

For Example, a researcher might conduct an instrumental case study on a particular policy to understand its impact on achieving a particular goal, such as reducing poverty. The researcher collects data from multiple sources, such as interviews, observations, and documents, and uses various techniques to analyze the data, such as content analysis or thematic analysis. The findings of an instrumental case study can be used to inform policy or practice or generate new research questions.

Case Study Data Collection Methods

Here are some common data collection methods for case studies:

Interviews involve asking questions to individuals who have knowledge or experience relevant to the case study. Interviews can be structured (where the same questions are asked to all participants) or unstructured (where the interviewer follows up on the responses with further questions). Interviews can be conducted in person, over the phone, or through video conferencing.

Observations

Observations involve watching and recording the behavior and activities of individuals or groups relevant to the case study. Observations can be participant (where the researcher actively participates in the activities) or non-participant (where the researcher observes from a distance). Observations can be recorded using notes, audio or video recordings, or photographs.

Documents can be used as a source of information for case studies. Documents can include reports, memos, emails, letters, and other written materials related to the case study. Documents can be collected from the case study participants or from public sources.

Surveys involve asking a set of questions to a sample of individuals relevant to the case study. Surveys can be administered in person, over the phone, through mail or email, or online. Surveys can be used to gather information on attitudes, opinions, or behaviors related to the case study.

Artifacts are physical objects relevant to the case study. Artifacts can include tools, equipment, products, or other objects that provide insights into the case study phenomenon.

How to conduct Case Study Research

Conducting a case study research involves several steps that need to be followed to ensure the quality and rigor of the study. Here are the steps to conduct case study research:

• Define the research questions: The first step in conducting a case study research is to define the research questions. The research questions should be specific, measurable, and relevant to the case study phenomenon under investigation.
• Select the case: The next step is to select the case or cases to be studied. The case should be relevant to the research questions and should provide rich and diverse data that can be used to answer the research questions.
• Collect data: Data can be collected using various methods, such as interviews, observations, documents, surveys, and artifacts. The data collection method should be selected based on the research questions and the nature of the case study phenomenon.
• Analyze the data: The data collected from the case study should be analyzed using various techniques, such as content analysis, thematic analysis, or grounded theory. The analysis should be guided by the research questions and should aim to provide insights and conclusions relevant to the research questions.
• Draw conclusions: The conclusions drawn from the case study should be based on the data analysis and should be relevant to the research questions. The conclusions should be supported by evidence and should be clearly stated.
• Validate the findings: The findings of the case study should be validated by reviewing the data and the analysis with participants or other experts in the field. This helps to ensure the validity and reliability of the findings.
• Write the report: The final step is to write the report of the case study research. The report should provide a clear description of the case study phenomenon, the research questions, the data collection methods, the data analysis, the findings, and the conclusions. The report should be written in a clear and concise manner and should follow the guidelines for academic writing.

Examples of Case Study

Here are some examples of case study research:

• The Hawthorne Studies : Conducted between 1924 and 1932, the Hawthorne Studies were a series of case studies conducted by Elton Mayo and his colleagues to examine the impact of work environment on employee productivity. The studies were conducted at the Hawthorne Works plant of the Western Electric Company in Chicago and included interviews, observations, and experiments.
• The Stanford Prison Experiment: Conducted in 1971, the Stanford Prison Experiment was a case study conducted by Philip Zimbardo to examine the psychological effects of power and authority. The study involved simulating a prison environment and assigning participants to the role of guards or prisoners. The study was controversial due to the ethical issues it raised.
• The Challenger Disaster: The Challenger Disaster was a case study conducted to examine the causes of the Space Shuttle Challenger explosion in 1986. The study included interviews, observations, and analysis of data to identify the technical, organizational, and cultural factors that contributed to the disaster.
• The Enron Scandal: The Enron Scandal was a case study conducted to examine the causes of the Enron Corporation’s bankruptcy in 2001. The study included interviews, analysis of financial data, and review of documents to identify the accounting practices, corporate culture, and ethical issues that led to the company’s downfall.
• The Fukushima Nuclear Disaster : The Fukushima Nuclear Disaster was a case study conducted to examine the causes of the nuclear accident that occurred at the Fukushima Daiichi Nuclear Power Plant in Japan in 2011. The study included interviews, analysis of data, and review of documents to identify the technical, organizational, and cultural factors that contributed to the disaster.

Application of Case Study

Case studies have a wide range of applications across various fields and industries. Here are some examples:

Business and Management

Case studies are widely used in business and management to examine real-life situations and develop problem-solving skills. Case studies can help students and professionals to develop a deep understanding of business concepts, theories, and best practices.

Case studies are used in healthcare to examine patient care, treatment options, and outcomes. Case studies can help healthcare professionals to develop critical thinking skills, diagnose complex medical conditions, and develop effective treatment plans.

Case studies are used in education to examine teaching and learning practices. Case studies can help educators to develop effective teaching strategies, evaluate student progress, and identify areas for improvement.

Social Sciences

Case studies are widely used in social sciences to examine human behavior, social phenomena, and cultural practices. Case studies can help researchers to develop theories, test hypotheses, and gain insights into complex social issues.

Law and Ethics

Case studies are used in law and ethics to examine legal and ethical dilemmas. Case studies can help lawyers, policymakers, and ethical professionals to develop critical thinking skills, analyze complex cases, and make informed decisions.

Purpose of Case Study

The purpose of a case study is to provide a detailed analysis of a specific phenomenon, issue, or problem in its real-life context. A case study is a qualitative research method that involves the in-depth exploration and analysis of a particular case, which can be an individual, group, organization, event, or community.

The primary purpose of a case study is to generate a comprehensive and nuanced understanding of the case, including its history, context, and dynamics. Case studies can help researchers to identify and examine the underlying factors, processes, and mechanisms that contribute to the case and its outcomes. This can help to develop a more accurate and detailed understanding of the case, which can inform future research, practice, or policy.

Case studies can also serve other purposes, including:

• Illustrating a theory or concept: Case studies can be used to illustrate and explain theoretical concepts and frameworks, providing concrete examples of how they can be applied in real-life situations.
• Developing hypotheses: Case studies can help to generate hypotheses about the causal relationships between different factors and outcomes, which can be tested through further research.
• Providing insight into complex issues: Case studies can provide insights into complex and multifaceted issues, which may be difficult to understand through other research methods.
• Informing practice or policy: Case studies can be used to inform practice or policy by identifying best practices, lessons learned, or areas for improvement.

Advantages of Case Study Research

There are several advantages of case study research, including:

• In-depth exploration: Case study research allows for a detailed exploration and analysis of a specific phenomenon, issue, or problem in its real-life context. This can provide a comprehensive understanding of the case and its dynamics, which may not be possible through other research methods.
• Rich data: Case study research can generate rich and detailed data, including qualitative data such as interviews, observations, and documents. This can provide a nuanced understanding of the case and its complexity.
• Holistic perspective: Case study research allows for a holistic perspective of the case, taking into account the various factors, processes, and mechanisms that contribute to the case and its outcomes. This can help to develop a more accurate and comprehensive understanding of the case.
• Theory development: Case study research can help to develop and refine theories and concepts by providing empirical evidence and concrete examples of how they can be applied in real-life situations.
• Practical application: Case study research can inform practice or policy by identifying best practices, lessons learned, or areas for improvement.
• Contextualization: Case study research takes into account the specific context in which the case is situated, which can help to understand how the case is influenced by the social, cultural, and historical factors of its environment.

Limitations of Case Study Research

There are several limitations of case study research, including:

• Limited generalizability : Case studies are typically focused on a single case or a small number of cases, which limits the generalizability of the findings. The unique characteristics of the case may not be applicable to other contexts or populations, which may limit the external validity of the research.
• Biased sampling: Case studies may rely on purposive or convenience sampling, which can introduce bias into the sample selection process. This may limit the representativeness of the sample and the generalizability of the findings.
• Subjectivity: Case studies rely on the interpretation of the researcher, which can introduce subjectivity into the analysis. The researcher’s own biases, assumptions, and perspectives may influence the findings, which may limit the objectivity of the research.
• Limited control: Case studies are typically conducted in naturalistic settings, which limits the control that the researcher has over the environment and the variables being studied. This may limit the ability to establish causal relationships between variables.
• Time-consuming: Case studies can be time-consuming to conduct, as they typically involve a detailed exploration and analysis of a specific case. This may limit the feasibility of conducting multiple case studies or conducting case studies in a timely manner.
• Resource-intensive: Case studies may require significant resources, including time, funding, and expertise. This may limit the ability of researchers to conduct case studies in resource-constrained settings.

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The Ultimate Guide to Qualitative Research - Part 1: The Basics

• Introduction and overview
• What is qualitative research?
• What is qualitative data?
• Examples of qualitative data
• Qualitative vs. quantitative research
• Mixed methods
• Qualitative research preparation
• Theoretical perspective
• Theoretical framework
• Literature reviews

Research question

• Conceptual framework
• Conceptual vs. theoretical framework

Data collection

• Qualitative research methods
• Focus groups
• Observational research

What is a case study?

Applications for case study research, what is a good case study, process of case study design, benefits and limitations of case studies.

• Ethnographical research
• Ethical considerations
• Confidentiality and privacy
• Power dynamics
• Reflexivity

Case studies

Case studies are essential to qualitative research , offering a lens through which researchers can investigate complex phenomena within their real-life contexts. This chapter explores the concept, purpose, applications, examples, and types of case studies and provides guidance on how to conduct case study research effectively.

Whereas quantitative methods look at phenomena at scale, case study research looks at a concept or phenomenon in considerable detail. While analyzing a single case can help understand one perspective regarding the object of research inquiry, analyzing multiple cases can help obtain a more holistic sense of the topic or issue. Let's provide a basic definition of a case study, then explore its characteristics and role in the qualitative research process.

Definition of a case study

A case study in qualitative research is a strategy of inquiry that involves an in-depth investigation of a phenomenon within its real-world context. It provides researchers with the opportunity to acquire an in-depth understanding of intricate details that might not be as apparent or accessible through other methods of research. The specific case or cases being studied can be a single person, group, or organization – demarcating what constitutes a relevant case worth studying depends on the researcher and their research question .

Among qualitative research methods , a case study relies on multiple sources of evidence, such as documents, artifacts, interviews , or observations , to present a complete and nuanced understanding of the phenomenon under investigation. The objective is to illuminate the readers' understanding of the phenomenon beyond its abstract statistical or theoretical explanations.

Characteristics of case studies

Case studies typically possess a number of distinct characteristics that set them apart from other research methods. These characteristics include a focus on holistic description and explanation, flexibility in the design and data collection methods, reliance on multiple sources of evidence, and emphasis on the context in which the phenomenon occurs.

Furthermore, case studies can often involve a longitudinal examination of the case, meaning they study the case over a period of time. These characteristics allow case studies to yield comprehensive, in-depth, and richly contextualized insights about the phenomenon of interest.

The role of case studies in research

Case studies hold a unique position in the broader landscape of research methods aimed at theory development. They are instrumental when the primary research interest is to gain an intensive, detailed understanding of a phenomenon in its real-life context.

In addition, case studies can serve different purposes within research - they can be used for exploratory, descriptive, or explanatory purposes, depending on the research question and objectives. This flexibility and depth make case studies a valuable tool in the toolkit of qualitative researchers.

Remember, a well-conducted case study can offer a rich, insightful contribution to both academic and practical knowledge through theory development or theory verification, thus enhancing our understanding of complex phenomena in their real-world contexts.

What is the purpose of a case study?

Case study research aims for a more comprehensive understanding of phenomena, requiring various research methods to gather information for qualitative analysis . Ultimately, a case study can allow the researcher to gain insight into a particular object of inquiry and develop a theoretical framework relevant to the research inquiry.

Why use case studies in qualitative research?

Using case studies as a research strategy depends mainly on the nature of the research question and the researcher's access to the data.

Conducting case study research provides a level of detail and contextual richness that other research methods might not offer. They are beneficial when there's a need to understand complex social phenomena within their natural contexts.

The explanatory, exploratory, and descriptive roles of case studies

Case studies can take on various roles depending on the research objectives. They can be exploratory when the research aims to discover new phenomena or define new research questions; they are descriptive when the objective is to depict a phenomenon within its context in a detailed manner; and they can be explanatory if the goal is to understand specific relationships within the studied context. Thus, the versatility of case studies allows researchers to approach their topic from different angles, offering multiple ways to uncover and interpret the data .

The impact of case studies on knowledge development

Case studies play a significant role in knowledge development across various disciplines. Analysis of cases provides an avenue for researchers to explore phenomena within their context based on the collected data.

This can result in the production of rich, practical insights that can be instrumental in both theory-building and practice. Case studies allow researchers to delve into the intricacies and complexities of real-life situations, uncovering insights that might otherwise remain hidden.

Types of case studies

In qualitative research , a case study is not a one-size-fits-all approach. Depending on the nature of the research question and the specific objectives of the study, researchers might choose to use different types of case studies. These types differ in their focus, methodology, and the level of detail they provide about the phenomenon under investigation.

Understanding these types is crucial for selecting the most appropriate approach for your research project and effectively achieving your research goals. Let's briefly look at the main types of case studies.

Exploratory case studies

Exploratory case studies are typically conducted to develop a theory or framework around an understudied phenomenon. They can also serve as a precursor to a larger-scale research project. Exploratory case studies are useful when a researcher wants to identify the key issues or questions which can spur more extensive study or be used to develop propositions for further research. These case studies are characterized by flexibility, allowing researchers to explore various aspects of a phenomenon as they emerge, which can also form the foundation for subsequent studies.

Descriptive case studies

Descriptive case studies aim to provide a complete and accurate representation of a phenomenon or event within its context. These case studies are often based on an established theoretical framework, which guides how data is collected and analyzed. The researcher is concerned with describing the phenomenon in detail, as it occurs naturally, without trying to influence or manipulate it.

Explanatory case studies

Explanatory case studies are focused on explanation - they seek to clarify how or why certain phenomena occur. Often used in complex, real-life situations, they can be particularly valuable in clarifying causal relationships among concepts and understanding the interplay between different factors within a specific context.

Intrinsic, instrumental, and collective case studies

These three categories of case studies focus on the nature and purpose of the study. An intrinsic case study is conducted when a researcher has an inherent interest in the case itself. Instrumental case studies are employed when the case is used to provide insight into a particular issue or phenomenon. A collective case study, on the other hand, involves studying multiple cases simultaneously to investigate some general phenomena.

Each type of case study serves a different purpose and has its own strengths and challenges. The selection of the type should be guided by the research question and objectives, as well as the context and constraints of the research.

The flexibility, depth, and contextual richness offered by case studies make this approach an excellent research method for various fields of study. They enable researchers to investigate real-world phenomena within their specific contexts, capturing nuances that other research methods might miss. Across numerous fields, case studies provide valuable insights into complex issues.

Critical information systems research

Case studies provide a detailed understanding of the role and impact of information systems in different contexts. They offer a platform to explore how information systems are designed, implemented, and used and how they interact with various social, economic, and political factors. Case studies in this field often focus on examining the intricate relationship between technology, organizational processes, and user behavior, helping to uncover insights that can inform better system design and implementation.

Health research

Health research is another field where case studies are highly valuable. They offer a way to explore patient experiences, healthcare delivery processes, and the impact of various interventions in a real-world context.

Case studies can provide a deep understanding of a patient's journey, giving insights into the intricacies of disease progression, treatment effects, and the psychosocial aspects of health and illness.

Asthma research studies

Specifically within medical research, studies on asthma often employ case studies to explore the individual and environmental factors that influence asthma development, management, and outcomes. A case study can provide rich, detailed data about individual patients' experiences, from the triggers and symptoms they experience to the effectiveness of various management strategies. This can be crucial for developing patient-centered asthma care approaches.

Other fields

Apart from the fields mentioned, case studies are also extensively used in business and management research, education research, and political sciences, among many others. They provide an opportunity to delve into the intricacies of real-world situations, allowing for a comprehensive understanding of various phenomena.

Case studies, with their depth and contextual focus, offer unique insights across these varied fields. They allow researchers to illuminate the complexities of real-life situations, contributing to both theory and practice.

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Understanding the key elements of case study design is crucial for conducting rigorous and impactful case study research. A well-structured design guides the researcher through the process, ensuring that the study is methodologically sound and its findings are reliable and valid. The main elements of case study design include the research question , propositions, units of analysis, and the logic linking the data to the propositions.

The research question is the foundation of any research study. A good research question guides the direction of the study and informs the selection of the case, the methods of collecting data, and the analysis techniques. A well-formulated research question in case study research is typically clear, focused, and complex enough to merit further detailed examination of the relevant case(s).

Propositions

Propositions, though not necessary in every case study, provide a direction by stating what we might expect to find in the data collected. They guide how data is collected and analyzed by helping researchers focus on specific aspects of the case. They are particularly important in explanatory case studies, which seek to understand the relationships among concepts within the studied phenomenon.

Units of analysis

The unit of analysis refers to the case, or the main entity or entities that are being analyzed in the study. In case study research, the unit of analysis can be an individual, a group, an organization, a decision, an event, or even a time period. It's crucial to clearly define the unit of analysis, as it shapes the qualitative data analysis process by allowing the researcher to analyze a particular case and synthesize analysis across multiple case studies to draw conclusions.

Argumentation

This refers to the inferential model that allows researchers to draw conclusions from the data. The researcher needs to ensure that there is a clear link between the data, the propositions (if any), and the conclusions drawn. This argumentation is what enables the researcher to make valid and credible inferences about the phenomenon under study.

Understanding and carefully considering these elements in the design phase of a case study can significantly enhance the quality of the research. It can help ensure that the study is methodologically sound and its findings contribute meaningful insights about the case.

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Conducting a case study involves several steps, from defining the research question and selecting the case to collecting and analyzing data . This section outlines these key stages, providing a practical guide on how to conduct case study research.

Defining the research question

The first step in case study research is defining a clear, focused research question. This question should guide the entire research process, from case selection to analysis. It's crucial to ensure that the research question is suitable for a case study approach. Typically, such questions are exploratory or descriptive in nature and focus on understanding a phenomenon within its real-life context.

Selecting and defining the case

The selection of the case should be based on the research question and the objectives of the study. It involves choosing a unique example or a set of examples that provide rich, in-depth data about the phenomenon under investigation. After selecting the case, it's crucial to define it clearly, setting the boundaries of the case, including the time period and the specific context.

Previous research can help guide the case study design. When considering a case study, an example of a case could be taken from previous case study research and used to define cases in a new research inquiry. Considering recently published examples can help understand how to select and define cases effectively.

Developing a detailed case study protocol

A case study protocol outlines the procedures and general rules to be followed during the case study. This includes the data collection methods to be used, the sources of data, and the procedures for analysis. Having a detailed case study protocol ensures consistency and reliability in the study.

The protocol should also consider how to work with the people involved in the research context to grant the research team access to collecting data. As mentioned in previous sections of this guide, establishing rapport is an essential component of qualitative research as it shapes the overall potential for collecting and analyzing data.

Collecting data

Gathering data in case study research often involves multiple sources of evidence, including documents, archival records, interviews, observations, and physical artifacts. This allows for a comprehensive understanding of the case. The process for gathering data should be systematic and carefully documented to ensure the reliability and validity of the study.

Analyzing and interpreting data

The next step is analyzing the data. This involves organizing the data , categorizing it into themes or patterns , and interpreting these patterns to answer the research question. The analysis might also involve comparing the findings with prior research or theoretical propositions.

Writing the case study report

The final step is writing the case study report . This should provide a detailed description of the case, the data, the analysis process, and the findings. The report should be clear, organized, and carefully written to ensure that the reader can understand the case and the conclusions drawn from it.

Each of these steps is crucial in ensuring that the case study research is rigorous, reliable, and provides valuable insights about the case.

The type, depth, and quality of data in your study can significantly influence the validity and utility of the study. In case study research, data is usually collected from multiple sources to provide a comprehensive and nuanced understanding of the case. This section will outline the various methods of collecting data used in case study research and discuss considerations for ensuring the quality of the data.

Interviews are a common method of gathering data in case study research. They can provide rich, in-depth data about the perspectives, experiences, and interpretations of the individuals involved in the case. Interviews can be structured , semi-structured , or unstructured , depending on the research question and the degree of flexibility needed.

Observations

Observations involve the researcher observing the case in its natural setting, providing first-hand information about the case and its context. Observations can provide data that might not be revealed in interviews or documents, such as non-verbal cues or contextual information.

Documents and artifacts

Documents and archival records provide a valuable source of data in case study research. They can include reports, letters, memos, meeting minutes, email correspondence, and various public and private documents related to the case.

These records can provide historical context, corroborate evidence from other sources, and offer insights into the case that might not be apparent from interviews or observations.

Physical artifacts refer to any physical evidence related to the case, such as tools, products, or physical environments. These artifacts can provide tangible insights into the case, complementing the data gathered from other sources.

Ensuring the quality of data collection

Determining the quality of data in case study research requires careful planning and execution. It's crucial to ensure that the data is reliable, accurate, and relevant to the research question. This involves selecting appropriate methods of collecting data, properly training interviewers or observers, and systematically recording and storing the data. It also includes considering ethical issues related to collecting and handling data, such as obtaining informed consent and ensuring the privacy and confidentiality of the participants.

Data analysis

Analyzing case study research involves making sense of the rich, detailed data to answer the research question. This process can be challenging due to the volume and complexity of case study data. However, a systematic and rigorous approach to analysis can ensure that the findings are credible and meaningful. This section outlines the main steps and considerations in analyzing data in case study research.

Organizing the data

The first step in the analysis is organizing the data. This involves sorting the data into manageable sections, often according to the data source or the theme. This step can also involve transcribing interviews, digitizing physical artifacts, or organizing observational data.

Categorizing and coding the data

Once the data is organized, the next step is to categorize or code the data. This involves identifying common themes, patterns, or concepts in the data and assigning codes to relevant data segments. Coding can be done manually or with the help of software tools, and in either case, qualitative analysis software can greatly facilitate the entire coding process. Coding helps to reduce the data to a set of themes or categories that can be more easily analyzed.

Identifying patterns and themes

After coding the data, the researcher looks for patterns or themes in the coded data. This involves comparing and contrasting the codes and looking for relationships or patterns among them. The identified patterns and themes should help answer the research question.

Interpreting the data

Once patterns and themes have been identified, the next step is to interpret these findings. This involves explaining what the patterns or themes mean in the context of the research question and the case. This interpretation should be grounded in the data, but it can also involve drawing on theoretical concepts or prior research.

Verification of the data

The last step in the analysis is verification. This involves checking the accuracy and consistency of the analysis process and confirming that the findings are supported by the data. This can involve re-checking the original data, checking the consistency of codes, or seeking feedback from research participants or peers.

Like any research method , case study research has its strengths and limitations. Researchers must be aware of these, as they can influence the design, conduct, and interpretation of the study.

Understanding the strengths and limitations of case study research can also guide researchers in deciding whether this approach is suitable for their research question . This section outlines some of the key strengths and limitations of case study research.

Benefits include the following:

• Rich, detailed data: One of the main strengths of case study research is that it can generate rich, detailed data about the case. This can provide a deep understanding of the case and its context, which can be valuable in exploring complex phenomena.
• Flexibility: Case study research is flexible in terms of design , data collection , and analysis . A sufficient degree of flexibility allows the researcher to adapt the study according to the case and the emerging findings.
• Real-world context: Case study research involves studying the case in its real-world context, which can provide valuable insights into the interplay between the case and its context.
• Multiple sources of evidence: Case study research often involves collecting data from multiple sources , which can enhance the robustness and validity of the findings.

On the other hand, researchers should consider the following limitations:

• Generalizability: A common criticism of case study research is that its findings might not be generalizable to other cases due to the specificity and uniqueness of each case.
• Time and resource intensive: Case study research can be time and resource intensive due to the depth of the investigation and the amount of collected data.
• Complexity of analysis: The rich, detailed data generated in case study research can make analyzing the data challenging.
• Subjectivity: Given the nature of case study research, there may be a higher degree of subjectivity in interpreting the data , so researchers need to reflect on this and transparently convey to audiences how the research was conducted.

Being aware of these strengths and limitations can help researchers design and conduct case study research effectively and interpret and report the findings appropriately.

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• Case Study | Definition, Examples & Methods

Case Study | Definition, Examples & Methods

Published on 5 May 2022 by Shona McCombes . Revised on 30 January 2023.

A case study is a detailed study of a specific subject, such as a person, group, place, event, organisation, or phenomenon. Case studies are commonly used in social, educational, clinical, and business research.

A case study research design usually involves qualitative methods , but quantitative methods are sometimes also used. Case studies are good for describing , comparing, evaluating, and understanding different aspects of a research problem .

Table of contents

When to do a case study, step 1: select a case, step 2: build a theoretical framework, step 3: collect your data, step 4: describe and analyse the case.

A case study is an appropriate research design when you want to gain concrete, contextual, in-depth knowledge about a specific real-world subject. It allows you to explore the key characteristics, meanings, and implications of the case.

Case studies are often a good choice in a thesis or dissertation . They keep your project focused and manageable when you don’t have the time or resources to do large-scale research.

You might use just one complex case study where you explore a single subject in depth, or conduct multiple case studies to compare and illuminate different aspects of your research problem.

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Once you have developed your problem statement and research questions , you should be ready to choose the specific case that you want to focus on. A good case study should have the potential to:

• Provide new or unexpected insights into the subject
• Challenge or complicate existing assumptions and theories
• Propose practical courses of action to resolve a problem
• Open up new directions for future research

Unlike quantitative or experimental research, a strong case study does not require a random or representative sample. In fact, case studies often deliberately focus on unusual, neglected, or outlying cases which may shed new light on the research problem.

If you find yourself aiming to simultaneously investigate and solve an issue, consider conducting action research . As its name suggests, action research conducts research and takes action at the same time, and is highly iterative and flexible. 

However, you can also choose a more common or representative case to exemplify a particular category, experience, or phenomenon.

While case studies focus more on concrete details than general theories, they should usually have some connection with theory in the field. This way the case study is not just an isolated description, but is integrated into existing knowledge about the topic. It might aim to:

• Exemplify a theory by showing how it explains the case under investigation
• Expand on a theory by uncovering new concepts and ideas that need to be incorporated
• Challenge a theory by exploring an outlier case that doesn’t fit with established assumptions

To ensure that your analysis of the case has a solid academic grounding, you should conduct a literature review of sources related to the topic and develop a theoretical framework . This means identifying key concepts and theories to guide your analysis and interpretation.

There are many different research methods you can use to collect data on your subject. Case studies tend to focus on qualitative data using methods such as interviews, observations, and analysis of primary and secondary sources (e.g., newspaper articles, photographs, official records). Sometimes a case study will also collect quantitative data .

The aim is to gain as thorough an understanding as possible of the case and its context.

In writing up the case study, you need to bring together all the relevant aspects to give as complete a picture as possible of the subject.

How you report your findings depends on the type of research you are doing. Some case studies are structured like a standard scientific paper or thesis, with separate sections or chapters for the methods , results , and discussion .

Others are written in a more narrative style, aiming to explore the case from various angles and analyse its meanings and implications (for example, by using textual analysis or discourse analysis ).

In all cases, though, make sure to give contextual details about the case, connect it back to the literature and theory, and discuss how it fits into wider patterns or debates.

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Case Study Research Method in Psychology

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Case studies are in-depth investigations of a person, group, event, or community. Typically, data is gathered from various sources using several methods (e.g., observations & interviews).

The case study research method originated in clinical medicine (the case history, i.e., the patient’s personal history). In psychology, case studies are often confined to the study of a particular individual.

The information is mainly biographical and relates to events in the individual’s past (i.e., retrospective), as well as to significant events that are currently occurring in his or her everyday life.

The case study is not a research method, but researchers select methods of data collection and analysis that will generate material suitable for case studies.

Freud (1909a, 1909b) conducted very detailed investigations into the private lives of his patients in an attempt to both understand and help them overcome their illnesses.

This makes it clear that the case study is a method that should only be used by a psychologist, therapist, or psychiatrist, i.e., someone with a professional qualification.

There is an ethical issue of competence. Only someone qualified to diagnose and treat a person can conduct a formal case study relating to atypical (i.e., abnormal) behavior or atypical development.

 Famous Case Studies

• Anna O – One of the most famous case studies, documenting psychoanalyst Josef Breuer’s treatment of “Anna O” (real name Bertha Pappenheim) for hysteria in the late 1800s using early psychoanalytic theory.
• Little Hans – A child psychoanalysis case study published by Sigmund Freud in 1909 analyzing his five-year-old patient Herbert Graf’s house phobia as related to the Oedipus complex.
• Bruce/Brenda – Gender identity case of the boy (Bruce) whose botched circumcision led psychologist John Money to advise gender reassignment and raise him as a girl (Brenda) in the 1960s.
• Genie Wiley – Linguistics/psychological development case of the victim of extreme isolation abuse who was studied in 1970s California for effects of early language deprivation on acquiring speech later in life.
• Phineas Gage – One of the most famous neuropsychology case studies analyzes personality changes in railroad worker Phineas Gage after an 1848 brain injury involving a tamping iron piercing his skull.

Clinical Case Studies

• Studying the effectiveness of psychotherapy approaches with an individual patient
• Assessing and treating mental illnesses like depression, anxiety disorders, PTSD
• Neuropsychological cases investigating brain injuries or disorders

Child Psychology Case Studies

• Studying psychological development from birth through adolescence
• Cases of learning disabilities, autism spectrum disorders, ADHD
• Effects of trauma, abuse, deprivation on development

Types of Case Studies

• Explanatory case studies : Used to explore causation in order to find underlying principles. Helpful for doing qualitative analysis to explain presumed causal links.
• Exploratory case studies : Used to explore situations where an intervention being evaluated has no clear set of outcomes. It helps define questions and hypotheses for future research.
• Descriptive case studies : Describe an intervention or phenomenon and the real-life context in which it occurred. It is helpful for illustrating certain topics within an evaluation.
• Multiple-case studies : Used to explore differences between cases and replicate findings across cases. Helpful for comparing and contrasting specific cases.
• Intrinsic : Used to gain a better understanding of a particular case. Helpful for capturing the complexity of a single case.
• Collective : Used to explore a general phenomenon using multiple case studies. Helpful for jointly studying a group of cases in order to inquire into the phenomenon.

Where Do You Find Data for a Case Study?

There are several places to find data for a case study. The key is to gather data from multiple sources to get a complete picture of the case and corroborate facts or findings through triangulation of evidence. Most of this information is likely qualitative (i.e., verbal description rather than measurement), but the psychologist might also collect numerical data.

1. Primary sources

• Interviews – Interviewing key people related to the case to get their perspectives and insights. The interview is an extremely effective procedure for obtaining information about an individual, and it may be used to collect comments from the person’s friends, parents, employer, workmates, and others who have a good knowledge of the person, as well as to obtain facts from the person him or herself.
• Observations – Observing behaviors, interactions, processes, etc., related to the case as they unfold in real-time.
• Documents & Records – Reviewing private documents, diaries, public records, correspondence, meeting minutes, etc., relevant to the case.

2. Secondary sources

• News/Media – News coverage of events related to the case study.
• Academic articles – Journal articles, dissertations etc. that discuss the case.
• Government reports – Official data and records related to the case context.
• Books/films – Books, documentaries or films discussing the case.

3. Archival records

Searching historical archives, museum collections and databases to find relevant documents, visual/audio records related to the case history and context.

Public archives like newspapers, organizational records, photographic collections could all include potentially relevant pieces of information to shed light on attitudes, cultural perspectives, common practices and historical contexts related to psychology.

4. Organizational records

Organizational records offer the advantage of often having large datasets collected over time that can reveal or confirm psychological insights.

Of course, privacy and ethical concerns regarding confidential data must be navigated carefully.

However, with proper protocols, organizational records can provide invaluable context and empirical depth to qualitative case studies exploring the intersection of psychology and organizations.

• Organizational/industrial psychology research : Organizational records like employee surveys, turnover/retention data, policies, incident reports etc. may provide insight into topics like job satisfaction, workplace culture and dynamics, leadership issues, employee behaviors etc.
• Clinical psychology : Therapists/hospitals may grant access to anonymized medical records to study aspects like assessments, diagnoses, treatment plans etc. This could shed light on clinical practices.
• School psychology : Studies could utilize anonymized student records like test scores, grades, disciplinary issues, and counseling referrals to study child development, learning barriers, effectiveness of support programs, and more.

How do I Write a Case Study in Psychology?

Follow specified case study guidelines provided by a journal or your psychology tutor. General components of clinical case studies include: background, symptoms, assessments, diagnosis, treatment, and outcomes. Interpreting the information means the researcher decides what to include or leave out. A good case study should always clarify which information is the factual description and which is an inference or the researcher’s opinion.

1. Introduction

• Provide background on the case context and why it is of interest, presenting background information like demographics, relevant history, and presenting problem.
• Compare briefly to similar published cases if applicable. Clearly state the focus/importance of the case.

2. Case Presentation

• Describe the presenting problem in detail, including symptoms, duration,and impact on daily life.
• Include client demographics like age and gender, information about social relationships, and mental health history.
• Describe all physical, emotional, and/or sensory symptoms reported by the client.
• Use patient quotes to describe the initial complaint verbatim. Follow with full-sentence summaries of relevant history details gathered, including key components that led to a working diagnosis.
• Summarize clinical exam results, namely orthopedic/neurological tests, imaging, lab tests, etc. Note actual results rather than subjective conclusions. Provide images if clearly reproducible/anonymized.
• Clearly state the working diagnosis or clinical impression before transitioning to management.

3. Management and Outcome

• Indicate the total duration of care and number of treatments given over what timeframe. Use specific names/descriptions for any therapies/interventions applied.
• Present the results of the intervention,including any quantitative or qualitative data collected.
• For outcomes, utilize visual analog scales for pain, medication usage logs, etc., if possible. Include patient self-reports of improvement/worsening of symptoms. Note the reason for discharge/end of care.

4. Discussion

• Analyze the case, exploring contributing factors, limitations of the study, and connections to existing research.
• Analyze the effectiveness of the intervention,considering factors like participant adherence, limitations of the study, and potential alternative explanations for the results.
• Identify any questions raised in the case analysis and relate insights to established theories and current research if applicable. Avoid definitive claims about physiological explanations.
• Offer clinical implications, and suggest future research directions.

5. Additional Items

• Thank specific assistants for writing support only. No patient acknowledgments.
• References should directly support any key claims or quotes included.
• Use tables/figures/images only if substantially informative. Include permissions and legends/explanatory notes.
• Provides detailed (rich qualitative) information.
• Provides insight for further research.
• Permitting investigation of otherwise impractical (or unethical) situations.

Case studies allow a researcher to investigate a topic in far more detail than might be possible if they were trying to deal with a large number of research participants (nomothetic approach) with the aim of ‘averaging’.

Because of their in-depth, multi-sided approach, case studies often shed light on aspects of human thinking and behavior that would be unethical or impractical to study in other ways.

Research that only looks into the measurable aspects of human behavior is not likely to give us insights into the subjective dimension of experience, which is important to psychoanalytic and humanistic psychologists.

Case studies are often used in exploratory research. They can help us generate new ideas (that might be tested by other methods). They are an important way of illustrating theories and can help show how different aspects of a person’s life are related to each other.

The method is, therefore, important for psychologists who adopt a holistic point of view (i.e., humanistic psychologists ).

Limitations

• Lacking scientific rigor and providing little basis for generalization of results to the wider population.
• Researchers’ own subjective feelings may influence the case study (researcher bias).
• Difficult to replicate.
• Time-consuming and expensive.
• The volume of data, together with the time restrictions in place, impacted the depth of analysis that was possible within the available resources.

Because a case study deals with only one person/event/group, we can never be sure if the case study investigated is representative of the wider body of “similar” instances. This means the conclusions drawn from a particular case may not be transferable to other settings.

Because case studies are based on the analysis of qualitative (i.e., descriptive) data , a lot depends on the psychologist’s interpretation of the information she has acquired.

This means that there is a lot of scope for Anna O , and it could be that the subjective opinions of the psychologist intrude in the assessment of what the data means.

For example, Freud has been criticized for producing case studies in which the information was sometimes distorted to fit particular behavioral theories (e.g., Little Hans ).

This is also true of Money’s interpretation of the Bruce/Brenda case study (Diamond, 1997) when he ignored evidence that went against his theory.

Breuer, J., & Freud, S. (1895).  Studies on hysteria . Standard Edition 2: London.

Curtiss, S. (1981). Genie: The case of a modern wild child .

Diamond, M., & Sigmundson, K. (1997). Sex Reassignment at Birth: Long-term Review and Clinical Implications. Archives of Pediatrics & Adolescent Medicine , 151(3), 298-304

Freud, S. (1909a). Analysis of a phobia of a five year old boy. In The Pelican Freud Library (1977), Vol 8, Case Histories 1, pages 169-306

Freud, S. (1909b). Bemerkungen über einen Fall von Zwangsneurose (Der “Rattenmann”). Jb. psychoanal. psychopathol. Forsch ., I, p. 357-421; GW, VII, p. 379-463; Notes upon a case of obsessional neurosis, SE , 10: 151-318.

Harlow J. M. (1848). Passage of an iron rod through the head.  Boston Medical and Surgical Journal, 39 , 389–393.

Harlow, J. M. (1868).  Recovery from the Passage of an Iron Bar through the Head .  Publications of the Massachusetts Medical Society. 2  (3), 327-347.

Money, J., & Ehrhardt, A. A. (1972).  Man & Woman, Boy & Girl : The Differentiation and Dimorphism of Gender Identity from Conception to Maturity. Baltimore, Maryland: Johns Hopkins University Press.

Money, J., & Tucker, P. (1975). Sexual signatures: On being a man or a woman.

Further Information

• Case Study Approach
• Case Study Method
• Enhancing the Quality of Case Studies in Health Services Research
• “We do things together” A case study of “couplehood” in dementia
• Using mixed methods for evaluating an integrative approach to cancer care: a case study

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Methodology

• What Is a Research Design | Types, Guide & Examples

What Is a Research Design | Types, Guide & Examples

Published on June 7, 2021 by Shona McCombes . Revised on September 5, 2024 by Pritha Bhandari.

A research design is a strategy for answering your   research question  using empirical data. Creating a research design means making decisions about:

• Your overall research objectives and approach
• Whether you’ll rely on primary research or secondary research
• Your sampling methods or criteria for selecting subjects
• Your data collection methods
• The procedures you’ll follow to collect data
• Your data analysis methods

A well-planned research design helps ensure that your methods match your research objectives and that you use the right kind of analysis for your data.

You might have to write up a research design as a standalone assignment, or it might be part of a larger   research proposal or other project. In either case, you should carefully consider which methods are most appropriate and feasible for answering your question.

Table of contents

Step 1: consider your aims and approach, step 2: choose a type of research design, step 3: identify your population and sampling method, step 4: choose your data collection methods, step 5: plan your data collection procedures, step 6: decide on your data analysis strategies, other interesting articles, frequently asked questions about research design.

• Introduction

Before you can start designing your research, you should already have a clear idea of the research question you want to investigate.

There are many different ways you could go about answering this question. Your research design choices should be driven by your aims and priorities—start by thinking carefully about what you want to achieve.

The first choice you need to make is whether you’ll take a qualitative or quantitative approach.

Qualitative research designs tend to be more flexible and inductive , allowing you to adjust your approach based on what you find throughout the research process.

Quantitative research designs tend to be more fixed and deductive , with variables and hypotheses clearly defined in advance of data collection.

It’s also possible to use a mixed-methods design that integrates aspects of both approaches. By combining qualitative and quantitative insights, you can gain a more complete picture of the problem you’re studying and strengthen the credibility of your conclusions.

Practical and ethical considerations when designing research

As well as scientific considerations, you need to think practically when designing your research. If your research involves people or animals, you also need to consider research ethics .

• How much time do you have to collect data and write up the research?
• Will you be able to gain access to the data you need (e.g., by travelling to a specific location or contacting specific people)?
• Do you have the necessary research skills (e.g., statistical analysis or interview techniques)?
• Will you need ethical approval ?

At each stage of the research design process, make sure that your choices are practically feasible.

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Within both qualitative and quantitative approaches, there are several types of research design to choose from. Each type provides a framework for the overall shape of your research.

Types of quantitative research designs

Quantitative designs can be split into four main types.

• Experimental and   quasi-experimental designs allow you to test cause-and-effect relationships
• Descriptive and correlational designs allow you to measure variables and describe relationships between them.

With descriptive and correlational designs, you can get a clear picture of characteristics, trends and relationships as they exist in the real world. However, you can’t draw conclusions about cause and effect (because correlation doesn’t imply causation ).

Experiments are the strongest way to test cause-and-effect relationships without the risk of other variables influencing the results. However, their controlled conditions may not always reflect how things work in the real world. They’re often also more difficult and expensive to implement.

Types of qualitative research designs

Qualitative designs are less strictly defined. This approach is about gaining a rich, detailed understanding of a specific context or phenomenon, and you can often be more creative and flexible in designing your research.

The table below shows some common types of qualitative design. They often have similar approaches in terms of data collection, but focus on different aspects when analyzing the data.

Your research design should clearly define who or what your research will focus on, and how you’ll go about choosing your participants or subjects.

In research, a population is the entire group that you want to draw conclusions about, while a sample is the smaller group of individuals you’ll actually collect data from.

Defining the population

A population can be made up of anything you want to study—plants, animals, organizations, texts, countries, etc. In the social sciences, it most often refers to a group of people.

For example, will you focus on people from a specific demographic, region or background? Are you interested in people with a certain job or medical condition, or users of a particular product?

The more precisely you define your population, the easier it will be to gather a representative sample.

• Sampling methods

Even with a narrowly defined population, it’s rarely possible to collect data from every individual. Instead, you’ll collect data from a sample.

To select a sample, there are two main approaches: probability sampling and non-probability sampling . The sampling method you use affects how confidently you can generalize your results to the population as a whole.

Probability sampling is the most statistically valid option, but it’s often difficult to achieve unless you’re dealing with a very small and accessible population.

For practical reasons, many studies use non-probability sampling, but it’s important to be aware of the limitations and carefully consider potential biases. You should always make an effort to gather a sample that’s as representative as possible of the population.

Case selection in qualitative research

In some types of qualitative designs, sampling may not be relevant.

For example, in an ethnography or a case study , your aim is to deeply understand a specific context, not to generalize to a population. Instead of sampling, you may simply aim to collect as much data as possible about the context you are studying.

In these types of design, you still have to carefully consider your choice of case or community. You should have a clear rationale for why this particular case is suitable for answering your research question .

For example, you might choose a case study that reveals an unusual or neglected aspect of your research problem, or you might choose several very similar or very different cases in order to compare them.

Data collection methods are ways of directly measuring variables and gathering information. They allow you to gain first-hand knowledge and original insights into your research problem.

You can choose just one data collection method, or use several methods in the same study.

Survey methods

Surveys allow you to collect data about opinions, behaviors, experiences, and characteristics by asking people directly. There are two main survey methods to choose from: questionnaires and interviews .

Observation methods

Observational studies allow you to collect data unobtrusively, observing characteristics, behaviors or social interactions without relying on self-reporting.

Observations may be conducted in real time, taking notes as you observe, or you might make audiovisual recordings for later analysis. They can be qualitative or quantitative.

Other methods of data collection

There are many other ways you might collect data depending on your field and topic.

If you’re not sure which methods will work best for your research design, try reading some papers in your field to see what kinds of data collection methods they used.

Secondary data

If you don’t have the time or resources to collect data from the population you’re interested in, you can also choose to use secondary data that other researchers already collected—for example, datasets from government surveys or previous studies on your topic.

With this raw data, you can do your own analysis to answer new research questions that weren’t addressed by the original study.

Using secondary data can expand the scope of your research, as you may be able to access much larger and more varied samples than you could collect yourself.

However, it also means you don’t have any control over which variables to measure or how to measure them, so the conclusions you can draw may be limited.

As well as deciding on your methods, you need to plan exactly how you’ll use these methods to collect data that’s consistent, accurate, and unbiased.

Planning systematic procedures is especially important in quantitative research, where you need to precisely define your variables and ensure your measurements are high in reliability and validity.

Operationalization

Some variables, like height or age, are easily measured. But often you’ll be dealing with more abstract concepts, like satisfaction, anxiety, or competence. Operationalization means turning these fuzzy ideas into measurable indicators.

If you’re using observations , which events or actions will you count?

If you’re using surveys , which questions will you ask and what range of responses will be offered?

You may also choose to use or adapt existing materials designed to measure the concept you’re interested in—for example, questionnaires or inventories whose reliability and validity has already been established.

Reliability and validity

Reliability means your results can be consistently reproduced, while validity means that you’re actually measuring the concept you’re interested in.

For valid and reliable results, your measurement materials should be thoroughly researched and carefully designed. Plan your procedures to make sure you carry out the same steps in the same way for each participant.

If you’re developing a new questionnaire or other instrument to measure a specific concept, running a pilot study allows you to check its validity and reliability in advance.

Sampling procedures

As well as choosing an appropriate sampling method , you need a concrete plan for how you’ll actually contact and recruit your selected sample.

That means making decisions about things like:

• How many participants do you need for an adequate sample size?
• What inclusion and exclusion criteria will you use to identify eligible participants?
• How will you contact your sample—by mail, online, by phone, or in person?

If you’re using a probability sampling method , it’s important that everyone who is randomly selected actually participates in the study. How will you ensure a high response rate?

If you’re using a non-probability method , how will you avoid research bias and ensure a representative sample?

Data management

It’s also important to create a data management plan for organizing and storing your data.

Will you need to transcribe interviews or perform data entry for observations? You should anonymize and safeguard any sensitive data, and make sure it’s backed up regularly.

Keeping your data well-organized will save time when it comes to analyzing it. It can also help other researchers validate and add to your findings (high replicability ).

On its own, raw data can’t answer your research question. The last step of designing your research is planning how you’ll analyze the data.

Quantitative data analysis

In quantitative research, you’ll most likely use some form of statistical analysis . With statistics, you can summarize your sample data, make estimates, and test hypotheses.

Using descriptive statistics , you can summarize your sample data in terms of:

• The distribution of the data (e.g., the frequency of each score on a test)
• The central tendency of the data (e.g., the mean to describe the average score)
• The variability of the data (e.g., the standard deviation to describe how spread out the scores are)

The specific calculations you can do depend on the level of measurement of your variables.

Using inferential statistics , you can:

• Make estimates about the population based on your sample data.
• Test hypotheses about a relationship between variables.

Regression and correlation tests look for associations between two or more variables, while comparison tests (such as t tests and ANOVAs ) look for differences in the outcomes of different groups.

Your choice of statistical test depends on various aspects of your research design, including the types of variables you’re dealing with and the distribution of your data.

Qualitative data analysis

In qualitative research, your data will usually be very dense with information and ideas. Instead of summing it up in numbers, you’ll need to comb through the data in detail, interpret its meanings, identify patterns, and extract the parts that are most relevant to your research question.

Two of the most common approaches to doing this are thematic analysis and discourse analysis .

There are many other ways of analyzing qualitative data depending on the aims of your research. To get a sense of potential approaches, try reading some qualitative research papers in your field.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

A research design is a strategy for answering your   research question . It defines your overall approach and determines how you will collect and analyze data.

A well-planned research design helps ensure that your methods match your research aims, that you collect high-quality data, and that you use the right kind of analysis to answer your questions, utilizing credible sources . This allows you to draw valid , trustworthy conclusions.

Quantitative research designs can be divided into two main categories:

• Correlational and descriptive designs are used to investigate characteristics, averages, trends, and associations between variables.
• Experimental and quasi-experimental designs are used to test causal relationships .

Qualitative research designs tend to be more flexible. Common types of qualitative design include case study , ethnography , and grounded theory designs.

The priorities of a research design can vary depending on the field, but you usually have to specify:

• Your research questions and/or hypotheses
• Your overall approach (e.g., qualitative or quantitative )
• The type of design you’re using (e.g., a survey , experiment , or case study )
• Your data collection methods (e.g., questionnaires , observations)
• Your data collection procedures (e.g., operationalization , timing and data management)
• Your data analysis methods (e.g., statistical tests  or thematic analysis )

A sample is a subset of individuals from a larger population . Sampling means selecting the group that you will actually collect data from in your research. For example, if you are researching the opinions of students in your university, you could survey a sample of 100 students.

In statistics, sampling allows you to test a hypothesis about the characteristics of a population.

Operationalization means turning abstract conceptual ideas into measurable observations.

For example, the concept of social anxiety isn’t directly observable, but it can be operationally defined in terms of self-rating scores, behavioral avoidance of crowded places, or physical anxiety symptoms in social situations.

Before collecting data , it’s important to consider how you will operationalize the variables that you want to measure.

A research project is an academic, scientific, or professional undertaking to answer a research question . Research projects can take many forms, such as qualitative or quantitative , descriptive , longitudinal , experimental , or correlational . What kind of research approach you choose will depend on your topic.

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Single Case Research Design

• First Online: 10 November 2021

Cite this chapter

• Stefan Hunziker 3 &
• Michael Blankenagel 3  

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This chapter addresses the peculiarities, characteristics, and major fallacies of single case research designs. A single case study research design is a collective term for an in-depth analysis of a small non-random sample. The focus on this design is on in-depth. This characteristic distinguishes the case study research from other research designs that understand the individual case as a rather insignificant and interchangeable aspect of a population or sample. Also, researchers find relevant information on how to write a single case research design paper and learn about typical methodologies used for this research design. The chapter closes with referring to overlapping and adjacent research designs.

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Baškarada, S. (2014). Qualitative case studies guidelines. The Qualitative Report, 19 (40), 1–25.

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• Published: 10 September 2024

Elastoplastic analysis on deformation and failure characteristics of surrounding rock of soft-coal roadway based on true triaxial loading and unloading tests

• Chongyang Jiang 1 ,
• Lianguo Wang 1 ,
• Jiaxing Guo 1 &
• Shuai Wang 1  

Scientific Reports volume  14 , Article number:  21103 ( 2024 ) Cite this article

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• Civil engineering

Since accidents such as roof caving, rock fragmentation, and severe deformation are particularly likely to occur during roadway excavation in soft and thick coal seams, grasping the range and distribution of deformation and fracturing of surrounding rock is of crucial for evaluating roadway stability and optimizing support design in such coal seams. In this study, based on the stress paths encountered during roadway excavation, true triaxial loading and unloading tests were carried out on soft coal, and the deformation and strength evolutions of soft coal under different intermediate principal stress conditions were analyzed. The test results show that the stress–strain relationship in the pre-peak plasticity-strengthening and post-peak plasticity-weakening stages follows a quadratic function, and the strengeth evolution conforms to the Mogi–Coulomb criterion. Moreover, analytical solutions for the displacement of surrounding rock, the radius of the broken zone, and the radius of the plastic zone of soft-coal roadways under excavation stress paths were derived after taking the nonlinear hardening and softening characteristics of the strain of soft coal, the Mogi–Coulomb criterion, the intermediate principal stress, and the dilatancy characteristics of surrounding rock into comprehensive consideration. Finally, in accordance with a practical engineering case, the influences of the intermediate principal stress coefficient, the lateral pressure coefficient, and the support force on the deformation and failure characteristics of the soft-coal roadway were analyzed. The analysis reveals that an increase in intermediate principal stress aggravates the deformation of surrounding rock and enlarges the plastic and broken zones; variations in the lateral pressure coefficient alter the shape of the broken zone and the distribution of surface displacement; and an increase in the support force effectively reduces the plastic zone, broken zone, and surface displacement of the roadway. The research results can provide valuable theoretical basis for the stability evaluation and support design of soft-coal roadways.

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Introduction.

With the depletion of shallow coal resources, deep mining has become the norm in the coal industry 1 , 2 . Affected by high ground stress in the deep, coal seams tend to become soft and fractured. During the excavation of a roadway in such soft and thick coal seams, large broken and plastic zones will emerge in the surrounding rock, posing safety hazards such as roof caving, rock fragmentation, and severe deformation 3 . Given the above fact, grasping the range and distribution of deformation, broken zone, and plastic zone in the surrounding rock is crucial for evaluating roadway stability and optimizing support design in such coal seams 4 . However, different constitutive models and strength criteria can lead to significant differences in the theoretical calculation results of the deformation and failure characteristics of the surrounding rock. Therefore, aming at accurately calculating the deformation and failure characteristics of the surrounding rock, it is meaningful to select a constitutive model and a strength criterion that are consistent with actual engineering.

Roadway excavation is actually a loading and unloading process for coal in a true triaxial stress state 5 , 6 . In this process, the stress of the surrounding rock of a roadway gradually transitions from being triaxial to being biaxial or unidirectional 7 , 8 , and the surrounding rock undergoes changes of vertical stress loading and horizontal stress unloading 9 . Numerous scholars have conducted plentiful loading and unloading tests on coal and rock according to the excavation and unloading process. Feng et al. 10 , 11 , 12 carried out loading and unloading tests on hard rock (e.g., marble) under different stress paths and explored the strength, deformation, and failure mechanisms of hard rock under these paths. Li et al. 13 conducted true triaxial loading and unloading tests on sandstone with different intermediate principal stresses and revealed the influences of the intermediate principal stress and the unloading action on sandstone dilatancy. Through true triaxial loading and unloading tests on columnar jointed rock specimens with different inclinations, Que et al. 14 uncovered the impact of excavation and unloading on the anisotropy strength and energy evolution of columnar jointed rock. Liu et al. 15 studied the changes in acoustic emission parameters of red sandstone during its true triaxial unloading failure and established the relationship between the intermediate principal stress and the acoustic emission information. Wang et al. 16 performed unloading tests on coal samples with the aid of a true triaxial fluid–structure interaction testing system, and obtained their deformation, damage, and failure characteristics. Liang et al. 17 conducted true triaxial loading and unloading tests on coal under different stress paths and analyzed the changes in deformation parameters, energy distribution, and fracture characteristics of coal under these paths. Currently, experimental research on true triaxial loading and unloading is mainly carried out on marble, granite, sandstone, hard coal, etc., while that on soft and fractured coal is scarcely reported.

In terms of elastoplastic analysis, relavent scholars have conducted extensive research on the deformation and failure characteristics of surrounding rock of roadways using different constitutive models and yield criteria of coal and rock. Sharan 18 , Sofianos and Nomikos 19 , Lee and Pietruszczak 20 , and Lv et al. 21 considered different constitutive models of rock based on the Hoek–Brown strength criterion and the non-correlated flow rule, and made elastoplastic analysis on the surrounding rock of roadways. Moreover, they derived analytical solutions for the deformation and failure range of surrounding rock. Zareifard and Fahimifar 22 , Park 23 , and Ranjbarnia et al. 24 derived analytical solutions for the stress and displacement of surrounding rock under hydrostatic pressure fields based on the linear strain softening model and the Mohr–Coulomb criterion. They also analyzed the influence of softening parameters on the stress and displacement of surrounding rock through numerical examples. Ghorbani and Hasanzadehshooiili 25 , Zhang et al. 26 , Yuan et al. 27 , and Wang et al. 28 integrated the intermediate principal stress, strain softening characteristics, rock shear dilation parameters, and Young’s modulus variations. Based on the Drucker–Prager criterion or the unified strength theory, they derived analytical solutions for the stress, displacement, and plastic zone of surrounding rock of circular roadways. Based on the Mohr–Coulomb and Drucker–Prager strength criteria, Jing et al. 29 derived the elastoplastic solution for the surrounding rock by combining rock rheology and long-term strength tests, and verified it with engineering examples. Overall, scholars have conducted abundant research on the analytical solutions for deformation and failure of roadways. However, most of the existing elastoplastic solutions for roadways are theoretically derived based on the constitutive relationship and strength criteria of coal and rock masses under compressive loading conditions, which differ from the stress changes encountered during roadway excavation and have certain limitations. The above research status nessesiates performing true triaxial loading and unloading tests based on the stress change characteristics of surrounding rock during the excavation of soft-coal roadways and establishing corresponding elastoplastic analysis models.

In this study, the evolutions of deformation and strength characteristics of soft coal under true triaxial loading and unloading conditions were investigated, and the constitutive relationship and strength criterion of soft coal under real stress paths during roadway excavation were determined. On this basis, the analytical solutions for the displacement of surrounding rock, the radius of the broken zone, and the radius of the plastic zone of soft-coal roadways under excavation stress paths were derived by taking the intermediate principal stress and the dilatancy characteristics of surrounding rock into account. Finally, in accordance with a practical engineering case, the effects of different influencing factors on the deformation and failure characteristics of the soft-coal roadway were analyzed. The research results can provide essential theoretical basis for the stability evaluation and support design of soft-coal roadways.

True triaxial loading and unloading tests on soft coal

Sample preparation.

The coal samples used in this experiment were collected from #8 coal seam in Huaibei Mining Area, China. This coal seam is of an extremely low strength and basically appears in a fragmented and loose state. Since such soft and fractured coal is difficult to be sampled and directly used for tests, it was processed into equivalent soft coal samples after being collected on site. The loose raw coal has a moisture content of approximately 3.25%. Prior to sample preparation, the raw coal particles were screened using sieves with pore sizes of 0.075 mm, 0.25 mm, 0.5 mm, 1 mm, 2 mm, 5 mm, 10 mm, and 15 mm, respectivly, based on which the raw coal samples were then classified. The mass proportions and particle size distribution curves for the raw coal were calculated (Fig.  1 ). As illustarted in Fig.  1 , the particle size of the raw coal predominantly ranges from 0 to 15 mm. Particles within this size range account for 97.92% of the total mass, while those larger than 15 mm, mostly consisting of small gangue, account for only 2.08% and are considered unrepresentative. Therefore, raw coal particles within the size range of 0–15 mm were selected for testing, and samples were prepared based on their proportional distribution. This approach not only replicates the grading composition of the raw coal but also avoids test errors arising from uneven packing density and particle arrangement, thereby ensuring sample uniformity.

Size distribution of raw coal particle.

Considering that the actual buried depth of the coal seam was 850 m, the forming pressure of the coal samples was set to 22 MPa. Following several preliminary tests, it was determined that 1480 g of loose raw coal was required for preparing a coal sample. The coal particles were weighed according to their respective size proportions, uniformly mixed (Fig.  2 a), and then added to the briquette pressing mold shown in Fig.  2 b, where it was subjected to slowly increasing loads through a testing machine until the forming pressure was reached. Afterwards, the pressure was maintained for 20 min before the cubic soft coal sample with a size of 100 mm × 100 mm × 100 mm was taken out (Fig.  2 c). Finally, the obtained samples were wrapped with cling films for later use. To ensure homogeneity of the samples, wave velocity tests were carried out to remove highly dispersed ones. The test results show that the wave velocities of the selected samples range from 0.279 to 0.364 km/s, with an average of 0.331 km/s.

Sample mold and soft coal samples: ( a ) raw coal: ( b ) sample mold; ( c ) part of coal samples.

Experimental equipment

The true triaxial loading and unloading tests were carried out with the aid of a true triaxial electro-hydraulic servo loading test system from China University of Mining and Technology. This system mainly consists of a three-way servo control loading system, an automatic acquisition system, a true triaxial pressure chamber, and an acoustic emission monitoring system 30 . As displayed in Fig.  3 a, the three-way servo control loading system comprises three mutually perpendicular and independent loading subsystems \(\sigma_{1}\) , \(\sigma_{2}\) , and \(\sigma_{3}\) , which can achieve independent servo loading control and simulate the real stress state of rock masses in underground engineering. The maximum servo loading pressures in the three directions are 1,600 kN, 500 kN, and 300 kN, respectively. The measurement accuracies of the test system for force and deformation are 0.01 kN and 0.002 mm, respectively. As shown in Fig.  3 b, the true triaxial pressure chamber, located at the center of the three-way servo control loading system, is composed of a pressure box, a base, and loading plates. The sample and the loading plates are interlocked, which allows for loading on the sample in three main stress directions.

True triaxial electro-hydraulic servo loading test system: ( a ) true triaxial testing machine; ( b ) true triaxial pressure chamber.

Experimental scheme

According to field measurements, the maximum principal stress ( \(\sigma_{1}\) ) is approximately 22 MPa, the intermediate principal stress ( \(\sigma_{2}\) ) ranges from 13.57 MPa to 21 MPa, and the minimum principal stress ( \(\sigma_{3}\) ) is around 10 MPa. Before roadway excavation, the surrounding rock is in a three-dimensional stress equilibrium state. During excavation, the stress state of the surrounding rock changes. To be specific, stress concentration and stress unloading occur, typically characterized by an increase in \(\sigma_{1}\) and a gradual decrease in \(\sigma_{2}\) and \(\sigma_{3}\) . Affected by stress changes, soft coal fractures and becomes unstable, which results in deformation and failure of the surrounding rock. In addition, the intermediate principal stress significantly influences the deformation and failure of the surrounding rock. Therefore, in the hope of exploring the deformation and failure characteristics of soft coal in the presence of excavation-dincued disturbance, five sets of true triaxial loading and unloading tests under different intermediate principal stress conditions were designed in this experiment. The test scheme is disclosed in Table 1 , and the specific stress path is illustrated in Fig.  4 . Considering the discreteness of the experimental results, each group of tests was repeated three times.

Schematic diagram of the stress path in the true triaxial loading and unloading tests.

As illustrated in Fig.  4 , a tested coal sample was first loaded to the preset initial true triaxial stress state. Then, \(\sigma_{1}\) , \(\sigma_{2}\) , and \(\sigma_{3}\) were synchronously applied to the sample at a rate of 0.2 MPa/s until the target hydrostatic pressure state (10 MPa) was reached. Subsequently, \(\sigma_{3}\) remained unchanged, and \(\sigma_{2}\) was loaded to the preset value at a rate of 0.2 MPa/s. After that, \(\sigma_{2}\) and \(\sigma_{3}\) remained constant, and \(\sigma_{1}\) was loaded to the preset value at a rate of 0.2 MPa/s and then remained unchanged. At this time, the initial true triaxial stress state was reached. Based on relevant studies 31 , 32 and multiple pre-experiments, it was concluded that a lower loading and unloading rate conduces to effectively preventing sudden failure or instability of the sample, allowing for a more accurate capture of the stress–strain relationship in soft coal. Acorddingly, \(\sigma_{1}\) was loaded at a rate of 0.1 MPa/s, while \(\sigma_{2}\) and \(\sigma_{3}\) were simultaneously unloaded at a rate of 0.1 MPa/s until the specimen failed.

Analysis and discussion of test results

Deformation characteristics of soft coal.

To facilitate comparison, the deformation of the samples in the initial stress state after loading was taken as the starting point, and only the data of subsequent loading and unloading processes were analyzed. Figure  5 presents the variation curves of axial stress \(\sigma_{1}\) of soft coal with axial strain \(\varepsilon_{1}\) , lateral strains \(\varepsilon_{2}\) and \(\varepsilon_{3}\) , and volumetric strain \(\varepsilon_{v}\) under true triaxial loading and unloading conditions, and Fig.  6 displays the variation curves of peak strains with the intermediate principal stress when the peak stress of soft coal is reached.

Stress–strain curves of soft coal under true triaxial loading and unloading conditions: ( a ) \(\sigma_{2}\)  = 14 MPa; ( b ) \(\sigma_{2}\)  = 16 MPa; ( c ) \(\sigma_{2}\)  = 18 MPa; ( d ) \(\sigma_{2}\)  = 20 MPa; ( e ) \(\sigma_{2}\)  = 22 MPa.

Variation curves of peak strains of soft coal under different intermediate principal stress conditions: ( a ) axial strain ε 1 ; ( b ) lateral strains ε 2 and ε 3.

It can be observed from Fig.  5 that the stress–strain curves of soft coal under these five intermediate principal stress conditions can be divided into four stages, namely the elastic stage (Stage I), the pre-peak plasticity-strengthening (hardening) stage (Stage II), the post-peak plasticity-weakening (softening) stage (Stage III), and the instability failure stage (Stage IV). In Stage I, the stress of coal changes roughly linearly with the strain. As the loading and unloading test proceeds, the coal enters Stage II where the axial stress \(\sigma_{1}\) rises nonlinearly at a gradually decelerated rate. After reaching the peak, it starts to decline progressively. In Stage III, the axial stress \(\sigma_{1}\) decreases nonlinearly, while the strains in the \(\sigma_{2}\) and \(\sigma_{3}\) unloading directions surge. The dilatancy and deformation in the \(\sigma_{3}\) direction are more notable than those in the \(\sigma_{2}\) direction. Eventually, the coal loses its bearing capacity in Stage IV. In Stages II and III, the stress–strain relationship basically follows a quadratic function, whose maximum value is exactly the peak stress of coal. Besides, the volumetric strain tends to increase first and then decrease throughout the loading and unloading process, indicating that the deformation pattern of coal transitions from axial compression in the initial stage to lateral dilatancy later stage.

It can be found from Fig.  6 that at the initial \(\sigma_{2}\) of 14 MPa, the values of \(\sigma_{2}\) and \(\sigma_{3}\) differ slightly, so do the values of \(\varepsilon_{2}\) and \(\varepsilon_{3}\) when the peak strength is reached. At this time, the lateral deformation of coal remains coordinated, and its bearing capacity is relatively strong. As the initial \(\sigma_{2}\) increases, the difference between values of \(\sigma_{2}\) and \(\sigma_{3}\) enlarges gradually. When the peak strength coal is reached, \(\varepsilon_{1}\) gradually decreases; \(\varepsilon_{2}\) gradually declines and tends to level off; and \(\varepsilon_{3}\) progressively rises. That is, the difference between \(\varepsilon_{2}\) and \(\varepsilon_{3}\) enlarges correspondingly. These results indicate that an increase in \(\sigma_{2}\) can accelerate the failure of coal, which makes the coal more prone to dilatancy and deformation in the \(\sigma_{3}\) direction.

Discussion on the applicability of strength criteria

The variation of the peak strength of coal with the intermediate principal stress under true triaxial loading and unloading conditions is exhibited in Fig.  7 . Clearly, the peak strength gradually decreases with the increase in initial \(\sigma_{2}\) . Specifically, it decreases from 35.37 MPa at the \(\sigma_{2}\) of 14 MPa to 26.3 MPa at the \(\sigma_{2}\) of 22 MPa, a decrease of 25.64%. The reason for this phenomenon is that an increase in \(\sigma_{2}\) restrains dilatancy in the \(\sigma_{2}\) direction and promotes dilatancy in the \(\sigma_{3}\) direction, thereby accelerating the failure of coal.

Variations of the peak strength of coal under true triaxial loading and unloading conditions.

The Mohr–Coulomb criterion, Drucker–Prager criterion, and Mogi–Coulomb criterion, three common criteria for coal strength, were selected for investigating the applicability of different strength criteria in true triaxial loading and unloading tests on soft coal.

According to the Mohr–Coulomb criterion, the principal stress can be written as:

where \(\phi\) is the friction angle of coal; and c is the cohesion.

Based on the Drucker–Prager criterion, the principal stress can be expressed by:

where \(I_{1}\) is the first invariant of stress; \(J_{2}\) is the second invariant of stress bias; and \(\alpha\) and \(K\) are experimental constants related to the friction angle \(\phi\) and the cohesive force c , which can be calculated as:

The Mogi–Coulomb criterion is an empirical criterion based on plentiful true triaxial test results 33 . According to this criterion, a specimen has yielded or failed if the octahedral shear stress \(\tau_{{{\text{oct}}}}\) on any side of it reaches the limit value, as expressed by:

where \(\tau_{{{\text{oct}}}}\) is the octahedral shear stress; \(\sigma_{m,2}\) is the average stress; and \(a_{1}\) and \(a_{2}\) are experimental constants related to the friction angle \(\phi\) and the cohesion c , which can be calculated as:

The strength of soft coal under the true triaxial loading and unloading path was fitted based on the above three strength criteria, and the fitting results are given in Fig.  8 .

Fitting results based on the three strength criteria: ( a ) Mohr–Coulomb criterion; ( b ) Drucker–Prager criterion; ( c ) Mogi–Coulomb criterion.

As can be seen from the fitting results (Fig.  8 ), the coefficient of determination ( R 2 ) of the Mohr–Coulomb criterion is 0.76, which represents a low fitting degree. As this criterion does not consider the influence of the intermediate principal stress, it fails to accurately reflect the strength characteristics of soft coal under the true triaxial loading and unloading path. Meanwhile, the R 2 of the Drucker–Prager criterion is only 0.73, which represents a low fitting degree, so this criterion is also not suitable for describing the strength characteristics of soft coal under loading and unloading conditions. In contrast, the R 2 of the Mogi–Coulomb criterion reaches 0.92, demonstrating its excellent fitting effect. Hence, the Mogi–Coulomb criterion is the most effective in describing the strength characteristics of soft coal under the true triaxial loading and unloading path. Finally, the fitting parameters were utilized to calculate the cohesion and internal friction angle of soft coal, which turned to be 0.858 MPa and 38.95°, respectively.

Elastoplastic analysis on surrounding rock of soft-coal roadway during its deformation and failure

Mechanical model of soft-coal roadway.

The geological conditions of roadways are complex in practical engineering. To facilitate the theoretical elastoplastic analysis on soft-coal roadways, the “equal-area method” was adopted when constructing the mechanical model, and the circular arch-shaped roadway was equivalent to a circle with a radius of \(R_{0}\) . As displayed in Fig.  9 , the surrounding rock of the roadway is divided into a broken zone, a plastic zone, and an elastic zone. In practical engineering, the roadway excavation direction is typically parallel to the horizontal principal stress direction in order to minimize roadway deformation and damage. In view of this fact, this paper assumes that the three principal stress directions are either orthogonal or parallel to the roadway axis. The vertical and horizontal stresses applied to the roadway model are p 0 and λp 0 , respectively; \(\lambda\) is the lateral pressure coefficient; and \(p_{i}\) is the support force of the roadway.

Mechanical model of the roadway.

It has been uncovered from the stress–strain curves of soft coal in the true triaxial loading and unloading tests that coal stays in an elastic state in the early stage of loading and unloading. When the limit elastic strength is reached, the coal enters the plastic stage, and its strain strengthens nonlinearly. After the peak stress is reached, the strain tends to weaken nonlinearly. Finally, the coal enters the failure stage. As depicted in Fig.  10 , the stress is linearly correlated with the strain in both the elastic stage and the failure stage, while their correlation follows a quadratic function in the plastic phase (i.e., the pre-peak hardening and post-peak softening stages). In Fig.  10 , \(\varepsilon_{E}\) is the ultimate elastic strain of coal; \(\varepsilon_{0}\) is the peak strain of coal; \(\varepsilon_{b}\) is the failure strain of coal; \(\sigma_{E}\) is the ultimate elastic stress of coal; \(\sigma_{c}\) is the peak stress of coal; and \(\sigma_{b}\) is the failure stress of coal.

Stress–strain relationships in different stages.

It is assumed that the stress \(\sigma_{p}\) and the strain \(\varepsilon_{p}\) of coal in the plastic phase (both hardening and softening stages) satisfy the quadratic function:

where D 1 , D 2 , and D 3 are constants.

At the peak strain \((\varepsilon_{0} ,\sigma_{c} )\) in Fig.  10 ,

In Fig.  10 , the elastic and plastic curves are smoothly connected and have the same slope E . At the ultimate elastic strain \((\varepsilon_{E} ,\sigma_{E} )\) , the stress and strain in the plastic stage follow the quadratic relationship:

By combining Eqs. ( 11 – 13 ), this quadratic relationship can be rewritten as:

It has been verified in the above section that the Mogi–Coulomb criterion can effectively describe the strength characteristics of soft coal under true triaxial loading and unloading conditions. Thus, this criterion is employed for identifying surrounding rock failure in theoretical calculation here. In practical engineering, the intermediate principal stress coefficient b is often introduced to denote the relationship between the three principal stresses, which is defined as:

The elastoplastic problem of the surrounding rock can be solved as a plane strain problem. Then, the stress state of the surrounding rock follows:

where \(\sigma_{\theta }\) , \(\sigma_{z}\) , and \(\sigma_{r}\) are the tangential stress, axial stress, and radial stress of the surrounding rock, respectively.

By combining Eqs. ( 6 ), ( 7 ), ( 8 ), ( 15 ), and ( 16 ), the Mogi–Coulomb criterion can be expressed as:

where \(M_{i} = \frac{{3a_{2} + 2\sqrt {2b^{2} - 2b + 2} }}{{2\sqrt {2b^{2} - 2b + 2} - 3a_{2} }}\) ; \(N_{i} = \frac{{6a_{1} }}{{2\sqrt {2b^{2} - 2b + 2} - 3a_{2} }}\) ; the subscript i can be replaced by “ p ” and “ b ”, which represent the plastic zone and the broken zone of the surrounding rock, respectively.

The surrounding rock will fracture and dilate when it fails. Based on the experimental results, the relationship between the dilatancy coefficient and the strain was plotted (Fig.  11 ).

Dilatancy model of surrounding rock.

In Fig.  11 , the tangential and radial strains in the plastic and broken zones of the surrounding rock satisfy the non-correlated flow rule:

where \(\eta_{1}\) and \(\eta_{2}\) are the dilatancy coefficients of these two zones, respectively.

Elastoplastic mechanical analysis on soft-coal roadway

Basic equations

According to elastoplastic mechanics, the equilibrium differential equation of the surrounding rock in various zones of the roadway can be described as:

where the subscript “ i ” can be replaced by “ e ”, “ p ”, and “ b ” which represent the elastic zone, the plastic zone, and the broken zone, respectively.

The geometric equations can be written as:

The physical equations are given:

The boundary conditions for various zones of the roadway are:

Analysis on the elastic zone

As depicted in Fig.  12 , the non-isobaric stress field of the circular roadway can be regarded as a superposition of two situations, namely a uniform compressive stress field and a stress field that is tensile on two sides and compressive on the other two sides 34 , 35 .

Analysis on stress fields of the surrounding rock.

In Fig.  12 ,

By using elastic mechanics to solve the stress fields in two different situations and combining their solutions, the stress of the surrounding rock in the non-isobaric stress field can be obtained:

where \(\sigma_{R}\) is the radial stress at the elastic–plastic interface. When \(r = R_{p}\) , \(\sigma_{re} = \sigma_{R}\) , and the radial and tangential stresses of the surrounding rock meet the Mogi–Coulomb criterion:

The stress at \(r = R_{p}\) is:

By substituting Eq. ( 25 ) into the physical equations, the strain in the elastic zone can be calculated:

Based on the geometric equations, the displacement of the elastic zone can be obtained:

Analysis on the plastic zone.

By combining the equilibrium equation and the Mogi–Coulomb criterion, the integral solution is calculated:

where C is an undetermined coefficient. When \(r = R_{p}\) , \(\sigma_{rp} = \sigma_{re}\) , and then the expression for C can be obtained:

Thus, the stress in the plastic zone can be expressed by:

The strain in the plastic zone consists of two parts, namely the elastic strain and the plastic strain:

By combining Eqs. ( 18 ) and ( 33 ) with the geometric equations and utilizing the continuous displacement conditions at the elastic–plastic interface, the displacement of the softening zone can be obtained:

where \(A_{1} = \varepsilon_{re} \left| {_{{r = R_{p} }} = } \right.\frac{1 + \mu }{E}[p_{1} (1 - 2\mu ) - p_{1} + \sigma_{R} + 4p_{2} \cos 2\theta (\mu - 2)]\)

According to the geometric equations, the strain in the plastic zone is:

Analysis on the broken zone.

Likewise, in the broken zone, \(\sigma_{r}\) and \(\sigma_{\theta }\) satisfy the equilibrium differential equation and the Mogi–Coulomb criterion, and when \(r = R_{0}\) , \(\sigma_{rb} = p_{i}\) . The stress in the broken zone is calculated as:

The displacement of the broken zone can be obtained in the same manner as the plastic zone:

where \(A_{2} = \varepsilon_{rp} \left| {_{{r = R_{b} }} } \right.\) , \(B_{2} = \varepsilon_{\theta p} \left| {_{{r = R_{b} }} } \right.\) , and \(C_{2} = u_{p} \left| {_{{r = R_{b} }} } \right.\) .

Determination of radii of the plastic zone and the broken zone.

In the plastic zone, Eq. ( 38 ) can be obtained on the basis of Eq. ( 35 ):

As mentioned earlier, in the plastic zone, \(\sigma_{p}\) and \(\varepsilon_{p}\) satisfy Eq. ( 14 ), then

When \(r = R_{b}\) , by substituting Eq. ( 38 ) into Eq. ( 39 ), the ratio of the radius of the broken zone to that of the plastic zone can be calculated by:

Additionally, when \(r = R_{b}\) , \(\sigma_{rp} = \sigma_{rb}\) . By combining Eqs. ( 32 ), ( 36 ), and ( 40 ), the calculation formulas for the radii of the plastic zone and the broken zone is obtained:

Analysis on influencing factors on deformation and failure of soft-coal roadway

The practical engineering of the #842 ventilating roadway in Guobei Coal Mine in Huaibei Mining Area was investigated as a case study. The surrounding rock of this roadway was identified as soft coal, and its geological conditions were obtained: the buried depth around 850 m, the stress of original rock \(p_{0}\)  = 22 MPa, the lateral pressure coefficient λ  = 1.2, the equivalent radius R 0  = 2.5 m, and the support force \(p_{i}\)  = 0.4 MPa. According to the true triaxial loading and unloading tests, the mechanical parameters of soft coal were measured: the elastic modulus E  = 270 MPa, Poisson’s ratio μ  = 0.31, the internal friction angle φ  = 38.95°, the cohesion c  = 0.858 MPa, the plastic-zone dilatancy coefficient η 1  = 1.18, and the broken-zone dilatancy coefficient η 2  = 1.32. The control variable method was adopted to successively analyze the effects of the intermediate principal stress coefficient b , the lateral pressure coefficient λ , and the support force \(p_{i}\) on the surface deformation of the surrounding rock, the radius of the plastic zone, and the radius of the broken zone.

Influence of intermediate principal stress coefficient on roadway failure

To investigate the influence of the intermediate principal stress coefficient b on the radius of the plastic zone R p , the radius of the broken zone R b , and the surface displacement of the surrounding rock \(u_{b}\) , based on the true triaxial loading and unloading test scheme, the values of b were determined to be 1/3, 1/2, 2/3, 5/6, and 1, respectively, while the other parameters remained unchanged. With these settings, the values of R p , R b , and \(u_{b}\) under different intermediate principal stress conditions were calculated and plotted (Figs. 13 , 14 and 15 ).

Radius of the plastic zone R p under different intermediate principal stresses.

Radius of the broken zone R b under different intermediate principal stresses.

Surface displacement of the surrounding rock under different intermediate principal stresses.

In Figs. 13 , 14 and 15 , under different intermediate principal stress coefficients, the plastic zones are ellipse; the broken zones are elongated ellipse; and the surface displacements are distributed in the shape of a spindle. The maximum values of R p , R b , and \(u_{b}\) all appear at the roof of the roadway ( θ  = 90°). As b increases, R p , R b , and \(u_{b}\) at the roof tend to slowly increase. Specifically, when b  = 1/3, R p , R b , and \(u_{b}\) at the roof are 5.28 m, 3.92 m, and 73.6 mm, respectively. When b increases to 2/3, their values rise to 5.69 m, 4.17 m, and 85.3 mm, a rise of 7.76%, 6.37%, and 15.90%, respectively. When b  > 2/3, their values start to surge. As b increases to 5/6, their values are 6.14 m, 4.34 m, and 95.5 mm, which are 7.90%, 4.07%, and 11.96% higher than those at b  = 2/3, respectively. When b increases to 1, their values grow to 6.68 m, 4.58 m, and 113.7 mm, 8.79%, 5.53%, and 19.06% higher than those at b  = 5/6, respectively. These results indicate that during excavation-induced unloading of surrounding rock, an increase in the intermediate principal stress will exacerbate the deformation of the surrounding rock and the dilatancy of the plastic and broken zones, exerting a noticeable impact on roadway failure.

Influence of lateral pressure coefficient on roadway failure

The distributions of R p , R b , and \(u_{b}\) under different lateral pressure coefficients are presented in Figs. 16 , 17 , and 18 , respectively. When λ  = 0.9, the plastic zone, the broken zone, and the surface displacement distribution are all ellipse. The minimum values of R p , R b , and \(u_{b}\) all appear at the roof of the roadway ( θ  = 90°), which are 4.98 m, 2.94 m, and 64.3 mm, respectively. Their maximum values all appear on the sides of the roadway ( θ  = 0°), which are 5.20 m, 3.52 m, and 70.4 mm, being 4.41%, 19.7%, and 9.5% higher than those at the roof, respectively. When λ  = 1, the plastic zone, the broken zone, and the surface displacement distribution are all circular, R p , R b , and \(u_{b}\) being 5.15 m, 3.32 m, and 65.1 mm, respectively. When λ  > 1, the shapes of the broken zone and the displacement distribution begin to change with the increase in λ . The broken zone gradually changes from an ellipse to an elongated ellipse, while the displacement distribution gradually transitions from an ellipse to a spindle. The maximum values of R p , R b , and \(u_{b}\) appear at the roof and gradually increase, while their values on the sides gradually decrease. When λ  = 1.1, their values at the roof are 5.30 m, 3.69 m, and 70.0 mm, respectively. When λ  = 1.3, these values grow to 5.60 m, 4.36 m, and 94.6 mm, a growth of 5.67%, 18.2%, and 35.1%, respectively. The radius of the broken zone and the surface displacement of the surrounding rock increase notably at the roof, suggesting that an increase in the lateral pressure coefficient will lead to pronounced deformation and failure of the roadway roof.

Radius of the plastic zone R p under different lateral pressure coefficients.

Radius of the broken zone R b under different lateral pressure coefficients.

Surface displacement of the surrounding rock under different lateral pressure coefficients.

Influence of support force on roadway failure

The distributions of R p , R b , and \(u_{b}\) under different support force conditions are displayed in Figs. 19 , 20 , and 21 , respectively. On the whole, the plastic zone, the broken zone, and the surface displacement distribution are in the shape of an ellipse, an elongated ellipse, and a spindle, respectively. The maximum values of R p , R b , and \(u_{b}\) all appear at the roof, and the minimum values all appear on the sides. As the support force p i increases, the plastic zone, the broken zone, and the surface displacement of the roadway gradually shrink. When p i  = 0.1 MPa, the R p , R b , and \(u_{b}\) at the roof are 5.91 m, 4.37 m, and 104.2 mm, respectively. As p i increases to 0.3 MPa, their values at the roof drop to 5.59 m, 4.13 m, and 84.8 mm, by 5.41%, 5.49%, and 18.6%, respectively. When p i increases to 0.5 MPa, these values decline to 5.32 m, 3.93 m, and 71.1 mm, a further decline of 4.83%, 4.85%, and 16.2%, respectively. Hence, it can be concluded that during the excavation of a soft-coal roadway, the deformation and failure of its surrounding rock can be controlled by increasing the support force.

Radius of the plastic zone R p under different support forces.

Radius of the broken zone R b under different support forces.

Surface displacement of the surrounding rock under different support forces.

Conclusions

The deformation and strength characteristics of soft coal under true triaxial loading and unloading conditions were investigated, and the constitutive relationship and strength criterion of soft coal under true stress paths in roadway excavation were determined. On this basis, analytical solutions for the displacement of the surrounding rock, the radius of the broken zone, and the radius of the plastic zone were derived. Finally, in accordance with a practical engineering case, the effects of different influencing factors on the deformation and failure characteristics of a soft-coal roadway were analyzed. The following main conclusions were drawn.

The stress–strain curves of soft coal in the true triaxial loading and unloading tests can be divided into four stages, namely the elastic stage, the pre-peak plasticity-strengthening (hardening) stage, the post-peak plasticity-weakening (softening) stage, and the instability failure stage. The stress–strain relationship in the pre-peak hardening and post-peak softening stages follows a quadratic function. As the initial \(\sigma_{2}\) increases, the difference between strains in the two unloading directions gradually enlarges, making the coal more prone to dilatancy and deformation in the \(\sigma_{3}\) direction and thereby decreasing its peak strength.

Compared to the Mohr–Coulomb criterion and the Drucker–Prager criterion, the Mogi–Coulomb criterion can more accurately describe the strength characteristics of soft coal under true triaxial loading and unloading paths.

Analytical solutions for the displacement of the surrounding rock, the radius of the broken zone, and the radius of the plastic zone in soft-coal roadways under excavation stress paths were derived after taking the nonlinear strain strengthening and softening characteristics of soft coal, the Mogi–Coulomb criterion, the intermediate principal stress, and the surrounding rock dilatancy characteristics into account.

An increase in the intermediate principal stress coefficient b will aggravate the deformation of the surrounding rock and the dilatancy of the plastic and broken zones. Meanwhile, an increase in the lateral pressure coefficient λ can bring about a gradual increase in the deformation degree of the plastic and broken zones at the roof and a decrease on the sides. The shape of the broken zone and surface displacement distribution will change correspondingly. Moreover, as the support force p i increases, the plastic zone, broken zone, and the surface displacement of the roadway all gradually shrink.

Data availability

The data used to support the findings and results of this study are available from the corresponding author upon request.

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Chongyang Jiang, Lianguo Wang, Jiaxing Guo & Shuai Wang

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C. J.: Conceptualization, Methodology, Formal analysis, Investigation, Data curation, Visualization, Writing—original draft; L. W.*: Conceptualization, Project administration, Resources, Supervision, Formal analysis; J. G.: Investigation, Validation, Visualization; S. W.: Investigation, Validation.

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Jiang, C., Wang, L., Guo, J. et al. Elastoplastic analysis on deformation and failure characteristics of surrounding rock of soft-coal roadway based on true triaxial loading and unloading tests. Sci Rep 14 , 21103 (2024). https://doi.org/10.1038/s41598-024-72052-4

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Received : 16 July 2024

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DOI : https://doi.org/10.1038/s41598-024-72052-4

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