Review of Hysteresis Curves of Steel and Concrete Members
This article explores the hysteresis curve in steel and concrete members, focusing on behaviors such as EPP, SD, and SSD, and their significance in energy dissipation analysis.
 What is a hysteresis curve?
 The relationship between stress and strain or force and displacement.
 The Concept of Energy Dissipation in the Hysteresis Curve
 Stiffness Degradation and Energy Dissipation
 Types of hysteresis curves and structural behaviors
 Description of the EPP (Elastoplastic Perfectly Plastic) behavior curve
 Review of SD (Stiffness Degrading) Behavior
 Behavior SSD (Strength and Stiffness Degrading)
In this article, we will conduct a comprehensive review of hysteresis curves in steel and concrete structures. The Hysteresis Curves represents the behavior of a structure or materials under cyclic loads, such as earthquakes, strong winds, or other alternating forces. These behaviors play a crucial role in the design and analysis of structures resistant to dynamic forces. This article will address fundamental questions about hysteresis curves and their application in civil and structural engineering. We will also examine various hysteresis behaviors, such as EPP, SD, and SSD, and explain examples of the behavior of steel and concrete members using these curves.
What is a Hysteresis Curves?
The Hysteresis Curves represents the relationship between stress and strain or force and displacement during a loading and unloading cycle. This curve is particularly important in analyzing the behavior of structures under
dynamic loads such as earthquakes. During loading, materials initially deform elastically and then enter the plastic region as the load increases. The hysteresis curve shows how the structure absorbs and dissipates energy. The shape of this curve can indicate ductility, energy absorption, and the extent of damage sustained by the structure.
Example: In a steel beam subjected to cyclic loads, the Hysteresis Curves initially shows linear elastic behavior and then plastic behavior with a larger enclosed area, indicating the high energy dissipation capacity of steel materials. source of this image.
Examination of the Hysteresis Curves of the Connection Above
To conduct a more detailed examination of the hysteresis curves in the four cases presented in the images, attention must be paid to the key parameters that represent the cyclic behavior of steel connections. These parameters include stiffness, stiffness degradation, ductility, energy dissipation, and the stability of the connection during cyclic loading.
1 Examination of hysteresis behavior in state a
Shape of the loops: The curve has relatively large and open loops. This openness indicates that the connection in this state absorbs and dissipates a significant amount of energy in each loading cycle.
Stiffness Degradation: There is a noticeable reduction in stiffness in this curve. Initially, the slope of the curve (indicating stiffness) is greater, but in subsequent cycles, this slope decreases, signifying a reduction in stiffness and a change in the behavior of the connection over time.
Energy Absorption Capacity: This connection has a high capacity for energy absorption, indicating good ductility.
Overall, it can be said that the connection in state (a) has high ductility and adequate energy absorption capacity; however, the gradual reduction in stiffness may indicate the occurrence of damage over the cycles.
2 Examination of hysteresis behavior in state b
 Shape of the loops: The hysteresis loops in this state are smaller compared to state (a). This indicates that this connection absorbs less energy.
 Stiffness degradation: Similar to state (a), a decrease in stiffness is observed here as well, but the amount is less.
 Energy absorption capacity: The connection in this state absorbs less energy compared to state (a), suggesting a decrease in ductility or a reduced ability to absorb higher energies in subsequent cycles.
 Conclusion: This state may indicate weaker performance of the connection in energy absorption compared to state (a). Stiffness decreases, but to a lesser extent than in state (a).
3 Examination of hysteretic behavior in state c
 Shape of loops: The curve in this case shows greater ductility and stability. The loops are almost symmetrical and uniform, with less opening.
 Stiffness degradation: The stiffness reduction appears to be less compared to previous states, which can indicate greater stability of the connection.
 Energy absorption capacity: Given the smaller loop openings, less energy is absorbed, but the connection has greater stability over the cycles.
Overall, the connection in state (c) shows acceptable ductility and experiences less stiffness reduction. This state may be considered as an optimal one in terms of stability.
4 Analysis of behavior in case d
The shape of the loops: The hysteresis loops in this case are larger than in cases (b) and (c), and similar to case (a). This indicates greater energy absorption in each cycle.
Stiffness degradation: The stiffness reduction in this case is similar to case (a), but more significant stiffness degradation may be observed in higher cycles.
Energy absorption capacity: The connection absorbs a high amount of energy and demonstrates high ductility.
Therefore, this state may be suitable for conditions that require greater energy absorption, but the reduction in stiffness and the potential for damage in higher cycles need to be addressed.
Overall Comparison
States (a) and (d): Both states indicate high energy absorption, but the reduction in stiffness during loading cycles is more significant. These connections are suitable for conditions where greater energy absorption is desired, but they may require repair or reinforcement after several loading cycles.
States (b) and (c): These two states have lower energy absorption, but they appear to offer greater stability in terms of reduced stiffness and lack of behavioral change after several cycles. These connections may be more suitable for conditions where moderate ductility and stability are more important. As you can see, a complete interpretation can be made from the behavior of the hysteresis curve.
The relationship between stress and strain or force and displacement.
1 Definition of Stress and Strain
Stress (Ïƒ): Stress is defined as the internal force created per unit area and is usually represented by the symbol $\sigma$. The formula for calculating stress is as follows:
$\sigma = \frac{F}{A}$
where $F$ is the applied force and $A$ is the crosssectional area.
Strain (Ïµ): Strain refers to the relative deformation of a body due to the applied force and is represented by the symbol $\epsilon$. The formula for strain is as follows:
$\epsilon = \frac{\Delta L}{L_0}$
where $\Delta L$ is the change in length and $L_0$ is the original length.
2. Relationship Between Stress and Strain
Hooke’s Law: Hooke’s Law states that, within the elastic limit, stress is proportional to strain. This relationship is represented as:
$\sigma = E \cdot \epsilon$
where $E$ is the modulus of elasticity (or Young’s modulus) that indicates the material properties. This law applies to elastic materials.
3. Force and Displacement
Force: Force is defined as an external factor that causes a change in the state of motion or rest of an object. Force can be either compressive or tensile.
Displacement: Displacement refers to the change in position of an object and is usually represented as $\Delta x$. Displacement typically refers to the amount of change in position over time.
4. Relationship Between Force and Displacement
Hooke’s Law for Force: Similar to stress and strain, Hooke’s Law also applies to force. The force applied to an object can lead to its displacement. The relationship between force and displacement is given as:
$F = k \cdot \Delta x$
where $k$ is the spring constant or stiffness of the object.
5. StressStrain and ForceDisplacement Diagrams
StressStrain Diagram: This diagram represents the relationship between stress and strain for a specific material and usually includes elastic and plastic phases. In the elastic region, there is a linear relationship between stress and strain.
ForceDisplacement Diagram: This diagram shows the relationship between the applied force and the displacement of an object. In the elastic region, this diagram is also linear.
6. Applications of Stress and Strain Curves
In civil engineering and structural analysis, understanding the relationship between stress and strain as well as force and displacement is crucial for analyzing the behavior of structures under various loads. These concepts are essential for designing safe and earthquakeresistant structures.
The Concept of Energy Dissipation in the Hysteresis Curve
One of the most important characteristics of the hysteresis curve is its ability to represent the amount of energy absorbed and dissipated during the loading and unloading cycles. In the curve, the area enclosed between the loading and unloading lines indicates the amount of energy lost. This energy is released as heat within the materials and is associated with permanent deformation.
In earthquakeresistant structures, the amount of energy absorbed and dissipated in each cycle provides vital information about the stability and ductility of the structure. The larger the enclosed area of the curve, the greater the structure’s capacity to absorb and dissipate energy.
Example: In a concrete structure, the hysteresis curve becomes smaller after each cycle, indicating a reduction in the structure’s ability to absorb energy.
Explanation of the hysteresis curve of a concrete member in the image:
The hysteresis curve presented for a concrete member is compared with the results of finite element analysis (FEA) and experimental tests. This curve illustrates the cyclic behavior of the concrete member under lateral loads and relative displacements (drift). Below, details and analysis of this curve are provided:
1 Shape of the Hysteresis Curve in the Concrete Member
The curve consists of two categories of lines: the red dashed lines represent the finite element analysis (FEA), while the blue solid lines indicate the results of experimental tests. Both curves exhibit a nearly similar cyclic behavior, illustrating energy absorption and dissipation during lateral loading cycles.
2 Correlation of Experimental and Numerical Results
According to the image above showing the hysteresis curve of a concrete member, it can be inferred that the laboratory results and FEA align closely. Although there are some discrepancies in the values of displacements and lateral forces in the higher loading ranges, a similar trend is generally observed. The laboratory results (blue line) indicate slightly more instability than the FEA analysis in the regions of higher displacement. This could be due to the nonlinear behavior of concrete materials or cracking in the experimental tests that is less accurately modeled in the numerical analysis.
Stiffness Degradation and Energy Dissipation
Stiffness degradation is observed in both curves (laboratory and FEA) as the number of cycles increases. This reduction in stiffness indicates that with an increasing number of loading cycles, the resistance of the concrete member to further displacements decreases. Additionally, the opening of the hysteresis loops reflects the energy dissipation in each cycle. The concrete member dissipates a portion of the applied energy as internal dissipation (due to cracking and material degradation) in each cycle. This energy dissipation is greater in the laboratory curve than in the FEA curve.
Load Capacity and Displacement
The curve shows that the concrete member experiences up to 5% drift under cyclic loads. The maximum lateral load in both curves is approximately 80 to 90 kN, indicating the loadbearing capacity of the member against lateral forces. In regions of larger displacements, the curves exhibit a drop in load capacity, meaning a decrease in the member’s resistance under higher loads.
Differences in High Displacement Regions
In regions with a drift greater than 3%, the FEA curve (red) begins to show a lower load reduction compared to the laboratory curve. This may be due to a simpler modeling of the concrete behavior in FEA compared to reality, where cracks and minor material failures are not accurately simulated in the numerical model. In contrast, the laboratory curve reflects a more significant load reduction in the higher displacement regions, which results from actual failures in the concrete member, such as crack propagation or internal material degradation.
The FEA curve has shown good performance in predicting the cyclic behavior of the concrete member, especially in the regions of small to medium displacements. This curve has been able to largely simulate the cyclic behavior and stiffness degradation.
The experimental curve reflects the actual behavior of the concrete member, where real failures such as cracking, stiffness degradation, and greater energy dissipation are observed. This curve shows more instability at higher displacements compared to the FEA curve.
Types of hysteresis curves and structural behaviors
EPP (Elastoplastic Perfectly Plastic) behavior curve
In this type of behavior, materials initially deform elastically and then enter the plastic region. When the structure reaches the yield point, deformation continues without any additional stress being applied to the materials. The hysteresis curve in EPP behavior appears as a horizontal straight line after the yield point, indicating that the force remains constant while the deformation increases. This behavior is more commonly observed in steel members with high ductility, demonstrating their high resistance to dynamic forces such as earthquakes.
Example: A steel column that enters the plastic region during loading cycles and continues to deform without an increase in force demonstrates a hysteresis curve with a large area, indicating high energy dissipation.
Description of the EPP (Elastoplastic Perfectly Plastic) behavior curve
The EPP diagram in the image above shows two material behavior curves with the characteristic of perfect elastoplasticity (EPP) drawn under different loading and unloading conditions. The left diagram describes the relationship between stress (Ïƒ) and strain (Îµ), while the right diagram displays the relationship between moment and curvature. Letâ€™s examine both diagrams in detail:
StressStrain Diagram (Left Side)
This diagram illustrates the elastoplastic behavior of a material during the loading and unloading process. Initially, when the material is subjected to a load (Loading), the straight initial path moves with a slope of E (elastic modulus). This linear section indicates that the material is in its elastic region, and any strain created is completely recoverable upon unloading. When the stress reaches the yield stress Ïƒy, the material enters the plastic region. From this point onward, increasing strain does not result in an increase in stress, as the material remains in the yielding state and undergoes plastic deformation.
MomentCurvature Diagram (Right Side)
This diagram similarly shows the bending behavior of a section or structure under loading and unloading. In this diagram, the bending moment is plotted on the vertical axis and curvature (Î¦) on the horizontal axis. The section from O to A in this diagram corresponds to the elastic behavior of the system, where changes in curvature linearly accompany an increase in moment. When reaching point B (plastic moment Mp), the section enters the plastic region, and similarly to the stressstrain diagram, further changes in curvature occur without a significant increase in moment.
When the load is unloaded (Unloading), the section returns elastically, but with a residual strain, meaning some curvature remains even after the moment is completely removed. If reloading occurs (Reloading), the unloading and loading paths will again be elastic until the section reaches the yield point again.
Additional Details on the EPP Curve
At the top of the momentcurvature diagram, two different models of plastic behavior are displayed:
 The perfectly elastoplastic model shows the loading path up to point B, after which curvature increases with a constant moment.
 The refined plastic model, where the loading path after yielding may exhibit more complex changes and a subtler nonlinear behavior. This behavior is illustrated in regions such as Bâ€² and Aâ€².
Ultimately, these curves indicate that after exceeding the yield limit, the material or structure undergoes plastic deformation and does not return to its original state after unloading; instead, a residual deformation (Plastic Deformation) remains.
In SD behavior or stiffness degradation, the structure loses its initial stiffness after each loading cycle, gradually having less capacity to resist forces in subsequent cycles. In the hysteresis curve representing this behavior, the slope of the curve decreases gradually, indicating a reduction in the stiffness of the structure. This behavior is more commonly observed in concrete structures or components that have experienced cracking and damage.
Example:
In a concrete shear wall that has experienced cracking, the hysteresis curve becomes smaller with each cycle, and its slope decreases, indicating that the structure gradually loses its resistance and stiffness.
Review of SD (Stiffness Degrading) Behavior
The SD (Stiffness Degrading) behavior model represents the behavior of systems that experience a reduction in stiffness under repeated loading. This model is typically used for systems like concrete or steel structures subjected to dynamic loads such as earthquakes or repeated loadings. This diagram allows for the examination of the changes in applied force (F) and displacement (d) in the system.
Analysis of Diagram (a) in the SD (Stiffness Degrading) Behavior Curve
This diagram illustrates the relationship between the applied force F and displacement d in a system subjected to cyclic loading. It shows several stages of the system’s mechanical behavior, each of which is explained below:

Beginning of Elastic (Linear) Behavior in SD (Stiffness Degrading)
In the elastic region, the system behaves linearly, and the applied force is directly proportional to the displacement. The slope of this line equals the initial stiffness $k_0$, which represents the stiffness of the system in the elastic phase.

Yield Point in SD (Stiffness Degrading)
At this point, the system reaches the yield force $f_y$ and yield displacement $d_y$. After crossing this point, the system exits the elastic state and enters the plastic region.
Maximum Point:
With further loading, the system reaches the maximum point where the maximum force $f_m$ and displacement $d_m$ occur. At this point, the system’s stiffness gradually decreases, and the diagram enters a nonlinear state.
Stiffness Degradation:
After reaching the maximum point, the system’s stiffness gradually decreases, indicated by the reduced slope in the diagram. The extent of this stiffness reduction is modeled by a parameter $r k_0$, where $r$ represents the percentage reduction in stiffness compared to the initial stiffness.
Subsequent Loading Stages (Reversed Loading):
After unloading and returning to the zero point, the system is loaded again. However, the stiffness of the system at this stage is less than its initial state, represented by a new reduced stiffness $k_u$. The difference between the displacements $d_p$ and $d_y$ indicates the plastic displacement of the system. The relationship between $k_u$ and $k_0$ is derived from the equation $k_u = k_0 \left(\frac{d_y}{d_m}\right)^a$, where $a$ is a coefficient.
Analysis of Diagram (b) in the SD (Stiffness Degrading) Behavior Curve
This diagram illustrates the system’s behavior during loading and unloading cycles in the form of a hysteresis loop. The hysteresis behavior is represented as a closed loop indicating the system’s delayed response to loading.
In this diagram:
 Area A1: Represents the energy absorbed by the system during loading and deformations.
 Area A2: Indicates the energy dissipated or hysteresis energy in the system, which occurs due to the plastic behavior and stiffness degradation of the system during loading.
These two areas represent the amount of mechanical energy in the system and its plastic deformation. Systems exhibiting hysteresis behavior typically respond to repeated loadings with reduced stiffness, which can lead to permanent deformations and ultimately failure.
SD (Stiffness Degrading) Behavior at a Glance
SD behavior (Stiffness Degrading) indicates that the system loses some of its stiffness with each cyclic loading. This phenomenon occurs due to plastic deformation or ongoing damage in structural materials, gradually reducing the system’s ability to withstand subsequent loads. This type of behavioral model is crucial for the design and analysis of structures subjected to cyclic loads, such as earthquakes, as it allows for the prediction of stiffness reduction and the remaining capacity of the structure.
Behavior SSD (Strength and Stiffness Degrading)
The SSD behavior is a combination of both strength and stiffness degradation. In this type of behavior, not only does the stiffness decrease, but the resistance of the structure also diminishes with each loading cycle. The hysteresis curve in this case becomes smaller and narrower with each cycle, indicating a reduction in the structure’s ability to withstand applied loads. This behavior is commonly observed in concrete structures that have been subjected to severe loads and have suffered significant damage.
Example:
In a concrete bridge, after several heavy loading cycles, the hysteresis curve gradually becomes smaller and narrower, indicating a decrease in loadbearing capacity and energy dissipation of the structure.
Comparison of Behavioral Models According to Material Hysteresis Behavior
The behavioral models EPP (Elastoplastic Perfectly Plastic), SD (Stiffness Degrading), and SSD (Strength and Stiffness Degrading) each illustrate how structures respond to cyclic loads (such as earthquakes) differently and have various applications. In the EPP model, the structure behaves without any loss of stiffness or strength after entering the plastic region. That is, after surpassing the yield point, there is no reduction in loadbearing capacity or stiffness. This model is mainly used for structures that require recoverability and loadbearing capacity after significant plastic deformations. Such models are typically employed in the design of earthquakeresistant structures that must retain at least some of their loadbearing capacity after severe loading.
The SD (Stiffness Degrading) and SSD (Strength and Stiffness Degrading) models gradually show a decline in stiffness and strength with each loading cycle. In the SD model, a gradual reduction in system stiffness is observed, which is more suitable for structures subjected to repeated cyclic loads, such as prolonged earthquakes or repeated loads. This model is useful for predicting the behavior of structures that gradually lose stiffness and undergo permanent plastic deformations. In the SSD model, not only does stiffness decrease, but strength is also reduced, meaning that with each loading cycle, the loadbearing capacity of the structure is compromised. This model is suitable for analyzing more vulnerable structures and assessing the potential for failure under severe loads or multiple earthquakes.
In summary, EPP is appropriate for structures requiring recoverability, SD for those experiencing gradual stiffness loss, and SSD for structures that rapidly lose both stiffness and strength, thus posing a high risk of failure. The choice of model depends on the loading conditions, type of structure, and design objectives.
Examples of Hysteresis Behavior in Steel Members
Steel members typically exhibit stable hysteresis behavior due to their ductile behavior and ability to endure large deformations. Under cyclic loading, these members initially display elastic behavior followed by plastic behavior. After yielding, steel continues to deform without breaking, leading to a bounded hysteresis area that remains constant or even increases with each cycle. This characteristic makes steel a suitable material for earthquakeresistant structures.
Example:
A steel frame shows a large hysteresis curve after several loading cycles, indicating high energy dissipation and the ability to absorb repeated loads.
Examples of Hysteresis Behavior in Concrete Members
Concrete structures typically exhibit hysteresis with reduced stiffness and strength due to the brittleness of the material. Concrete experiences cracking under repeated loading cycles and gradually loses its stiffness and loadbearing capacity. The hysteresis curve for concrete members usually becomes smaller and narrower with each cycle, indicating a reduction in the structure’s ability to dissipate energy and resist cyclic loads.
Example:
A concrete shear wall subjected to cyclic loading shows a smaller hysteresis curve with a decreasing slope, indicating reduced capacity for energy absorption.
Conclusion and Summary of Hysteresis Curves
In this article, we examined hysteresis curves and their role in analyzing the behavior of steel and concrete members. Understanding hysteresis behavior in structures, particularly in earthquakeresistant designs, is crucial for optimal and safe design. Various hysteresis behaviors such as EPP, SD, and SSD can provide accurate information about energy absorption capacity and the ability of structures to withstand cyclic loads. Considering the different behaviors of steel and concrete members, appropriate design and material selection for each type of structure will significantly impact its performance against dynamic loads.