Tension Stiffening in Reinforced Concrete Analytical Models, Experimental Insights, and Seismic Implications

Concrete tension stiffening refers to the phenomenon where, after cracking, the concrete between cracks still transfers part of the tensile force through bond action with the reinforcement. As a result, the effective stiffness of the member does not immediately drop to zero after cracking. This behavior influences service deflection, curvature distribution, and the onset of plasticity, making it important in nonlinear analyses and performance evaluations of structures.

In practice, design codes take two main approaches. Some, such as ACI, include the effect of tension stiffening in a simplified engineering way by introducing the concept of an effective moment of inertia in deflection calculations. Others adopt fracture-based or σ–w Concrete tension stiffening  models, which are more suitable for detailed numerical analysis. Therefore, the choice of modeling approach depends on the purpose of the analysis  whether it is for service-level deflection estimation or nonlinear behavior assessment.

In performance-based design frameworks such as FEMA P-58 and the ASCE 41 seismic evaluation guidelines, the material model and stiffness variations have a direct influence on performance outputs like predicted displacements and damage levels. In fact, if concrete tension stiffening affects the analytical results, it should be explicitly modeled or at least evaluated through sensitivity analysis.

The technical concept of concrete tension stiffening

As shown in the figure, when a reinforced concrete member is subjected to tension, the concrete initially carries the entire tensile force up to its own tensile strength. Once the stress reaches the cracking limit $f_{ct}$, cracks begin to form. However, the concrete between the cracks still transfers part of the load through its bond with the reinforcing bars. This is the phenomenon known as concrete tension stiffening.

As a result, the stress–strain curve of a reinforced concrete member under tension does not drop suddenly after cracking; instead, it gradually decreases with a mild slope until the concrete can no longer transfer any effective force. This descending branch represents the gradual softening of cracked concrete and provides a more realistic depiction of the interaction between steel and concrete.

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General Stress–Strain Relationships for Modeling Concrete Tension Stiffening

Several models have been proposed to describe the tensile behavior of cracked concrete. Examples include:

1. Goto Model (1971):
The average concrete stress between cracks decreases exponentially:

$$\sigma_c = f_{ct} e^{-k\left(\frac{\varepsilon – \varepsilon_{cr}}{\varepsilon_{cr}}\right)}$$where $k$ is an empirical parameter (typically between 1 and 5), and $\varepsilon_{cr}$ is the strain corresponding to $f_{ct}$.
Reference: Goto, Y. (1971). “Cracks formed in concrete around deformed tension bars.” ACI Journal.

2. Bischoff Model (2001):
Concrete stress between cracks is expressed as a function of steel strain and reinforcement ratio:

$$\frac{\sigma_c}{f_{ct}} = \frac{1}{1 + \alpha \left(\frac{\varepsilon – \varepsilon_{cr}}{\varepsilon_{cr}}\right)}$$where $\alpha$ is determined experimentally from direct tension tests.
Reference: Bischoff, P.H. (2001). “Tension stiffening and cracking of reinforced concrete.” Journal of Structural Engineering, ASCE.

As shown in the figure, when a reinforced concrete member is subjected to tension, the concrete initially carries the entire tensile force up to its own tensile strength. Once the stress reaches the cracking limit $f_{ct}$, cracks begin to form. However, the concrete between the cracks still transfers part of the load through its bond with the reinforcing bars. This is the phenomenon known as concrete tension stiffening.

As a result, the stress–strain curve of a reinforced concrete member under tension does not drop suddenly after cracking; instead, it gradually decreases with a mild slope until the concrete can no longer transfer any effective force. This descending branch represents the gradual softening of cracked concrete and provides a more realistic depiction of the interaction between steel and concrete.

General Stress–Strain Relationships for Modeling Concrete Tension Stiffening

Several models have been proposed to describe the tensile behavior of cracked concrete. Examples include:

1. Goto Model (1971):
The average concrete stress between cracks decreases exponentially:

$$\sigma_c = f_{ct} e^{-k\left(\frac{\varepsilon – \varepsilon_{cr}}{\varepsilon_{cr}}\right)}$$where $k$ is an empirical parameter (typically between 1 and 5), and $\varepsilon_{cr}$ is the strain corresponding to $f_{ct}$.
Reference: Goto, Y. (1971). “Cracks formed in concrete around deformed tension bars.” ACI Journal.

2. Bischoff Model (2001):
Concrete stress between cracks is expressed as a function of steel strain and reinforcement ratio:

$$\frac{\sigma_c}{f_{ct}} = \frac{1}{1 + \alpha \left(\frac{\varepsilon – \varepsilon_{cr}}{\varepsilon_{cr}}\right)}$$where $\alpha$ is determined experimentally from direct tension tests.
Reference: Bischoff, P.H. (2001). “Tension stiffening and cracking of reinforced concrete.” Journal of Structural Engineering, ASCE.

4. Hognestad Model (for comparison):
Although the Hognestad model was originally developed for compressive concrete, modified versions have been proposed for the tensile region. These include a linear ascending branch up to $f_{ct}$, followed by a parabolic softening curve that represents the post-cracking reduction in stiffness.

Behavioral Interpretation

In practice, the effect of tension stiffening in concrete leads to the following outcomes:

• An increase in the number of cracks and a reduction in crack spacing.

• A decrease in beam deflection and wall displacement.

• A more realistic stress distribution between concrete and reinforcing steel.

This behavior is essential for accurate modeling in nonlinear analysis of reinforced concrete structures, such as shear walls and deep beams.

Analytical and Empirical Models for Considering Concrete Tension Stiffening

In modern concrete design codes, Concrete tension stiffening is not directly incorporated into design equations, but its effects are implicitly considered in the formulations for deflection, effective stiffness, and crack distribution. The two main references in this field are ACI 318-19 and fib Model Code 2010, which address this phenomenon in different ways.

In ACI 318-19

In the ACI 318 code, Concrete tension stiffening is considered through the concepts of effective cracked stiffness and member deflection. The code assumes that even after cracking, concrete still carries a small portion of the tensile stress. This effect is included in the effective moment of inertia equation:

$$\frac{1}{I_e} = \frac{M_{cr}}{M_a} \cdot \frac{1}{I_g} + \left(1 – \frac{M_{cr}}{M_a}\right) \cdot \frac{1}{I_{cr}}$$

where:

• $I_e$: Effective moment of inertia

• $I_g$: Gross (uncracked) section moment of inertia

• $I_{cr}$: Cracked section moment of inertia

• $M_{cr}$: Cracking moment

• $M_a$: Applied moment

This relationship, developed by Branson (1968), indirectly accounts for the tension stiffening effect, leading to reduced deflection and increased effective stiffness in cracked members.
Reference: Branson, D.E. (1968). “Design procedures for computing deflections.” ACI Journal.

In numerical models based on ACI provisions (for example, in ETABS or SAP2000), the flexural stiffness reduction factor is typically assumed between 0.35 and 0.7, which inherently reflects the influence of Concrete tension stiffening.
Reference: ACI 318-19, Section 24.2.3.5 & Commentary R24.2.3.5.

In fib Model Code 2010

In my view, fib Model Code 2010 provides a more advanced and physically consistent approach by explicitly incorporating the tension stiffening behavior through Tension Chord Models.

In these models, the tensile stress of concrete is expressed as a function of crack opening $w$:

$$\sigma_t = f_{ctm} \left(1 – \frac{w}{w_1}\right) \quad \text{for } 0 < w < w_1$$$$\sigma_t = 0 \quad \text{for } w \ge w_1$$where:

• $f_{ctm}$: Mean tensile strength of concrete

• $w$: Crack opening width

• $w_1$: Critical crack opening (typically 0.3–0.5 mm)

In this formulation, tension stiffening is captured more realistically, since the stress–crack opening relationship directly influences both effective stiffness and stress transfer between concrete and reinforcement.
Reference: fib Model Code for Concrete Structures 2010, Section 7.3.2.

Complementary Empirical Models

Beyond code-based formulations, several empirical models are widely used to represent Concrete tension stiffening in concrete, including:

1. Gilbert Model (2007):

$$\frac{\sigma_c}{f_{ct}} = e^{-0.8 \frac{\varepsilon – \varepsilon_{cr}}{\varepsilon_{cr}}}$$This model provides accurate results for reinforced concrete beams with typical reinforcement ratios.
Reference: Gilbert, R.I. (2007). “Tension stiffening in reinforced concrete slabs.” Journal of Structural Engineering, ASCE.

2. CEB-FIP Model Code 1990:

$$\sigma_c = f_{ct} \left(1 – \frac{\varepsilon – \varepsilon_{cr}}{\varepsilon_{cu} – \varepsilon_{cr}}\right)$$where $\varepsilon_{cu}$ is the ultimate tensile strain of concrete. This model formed the basis for later developments in fib Model Code 2010.

In summary, ACI 318-19 considers Concrete tension stiffening implicitly through effective stiffness formulations, while fib Model Code 2010 models it explicitly via stress–crack opening relationships. Empirical models such as Gilbert (2007) and CEB-FIP 1990 allow for even more refined numerical modeling and realistic simulation of reinforced concrete tension behavior.

Implementation of Concrete Tension Stiffening in Numerical Models

In numerical modeling, tension stiffening in concrete is usually simulated either by modifying the tensile branch of the concrete stress–strain curve or by using specialized material models that can capture post-cracking tension behavior. Below, I’ll outline the common approaches used in three widely adopted software packages OpenSees, ETABS Nonlinear, and ABAQUS.

Modeling in OpenSees or OpenSeesPy

In OpenSees, several concrete material models are available. However, to account for concrete tension stiffening, the Concrete02 model is most commonly used because it allows defining softening behavior in tension.

The Concrete02 model defines the tensile behavior as follows:

$$\sigma_t = \begin{cases} E_t \, \varepsilon_t, & \varepsilon_t \le \varepsilon_{cr} \\ f_t \left(1 – \frac{\varepsilon_t – \varepsilon_{cr}}{\varepsilon_u – \varepsilon_{cr}}\right), & \varepsilon_{cr} < \varepsilon_t < \varepsilon_u \\ 0, & \varepsilon_t \ge \varepsilon_u \end{cases}$$

where:

• $f_t$: Tensile strength of concrete (≈ 0.33√f’c)

• $\varepsilon_{cr} = f_t / E_c$: Cracking strain of concrete

• $\varepsilon_u$: Ultimate tensile strain (typically 10–15 times $\varepsilon_{cr}$)

This formulation allows the tensile stress to gradually decrease after cracking, effectively reproducing the tension stiffening phenomenon observed in experiments.

References:

• Mazzoni et al. (2007), “OpenSees Command Language Manual.”

• fib Model Code for Concrete Structures 2010, Section 7.3.2.

where:

• $f_t$: Tensile strength of concrete (≈ 0.33√f’c)

• $\varepsilon_{cr} = f_t / E_c$: Cracking strain of concrete

• $\varepsilon_u$: Ultimate tensile strain (typically 10–15 times $\varepsilon_{cr}$)

This formulation allows the tensile stress to gradually decrease after cracking, effectively reproducing the Concrete tension stiffening phenomenon observed in experiments.

References:

• Mazzoni et al. (2007), “OpenSees Command Language Manual.”

• fib Model Code for Concrete Structures 2010, Section 7.3.2.

In OpenSeesPy, the same model can be implemented using the following command syntax:

uniaxialMaterial('Concrete02', matTag, fpc, epsc0, fpcu, epsU, ft, Ets)

Here, ft and Ets define the tensile strength and softening stiffness, respectively — parameters that directly control the tension stiffening response of the concrete element.

Modeling in ETABS

In ETABS and SAP2000, users don’t have direct access to define the tensile properties of concrete. Therefore, the effect of concrete tension stiffening is applied indirectly through flexural stiffness reduction factors.

• For flexural members:
  $I_{eff} = 0.35\,I_g$ to $0.7\,I_g$, depending on the level of cracking different for beams, columns, and concrete walls.

• For shear walls in nonlinear analysis:
  The effective flexural stiffness is typically taken as 0.5 of the linear stiffness to account for cracking and tension stiffening effects in concrete.

These coefficients are, in fact, an engineered equivalent of the Concrete tension stiffening phenomenon, as recommended in ACI 318-19 Commentary R24.2.3.5.

References:

• CSI Analysis Reference Manual, 2023

• ACI 318-19, Commentary R24.2.3.5

Modeling in ABAQUS

In ABAQUS, this behavior can be modeled more explicitly and accurately, since the software allows defining stress–strain or stress–crack opening relationships for concrete in tension.

There are two main approaches:

1. Concrete Damaged Plasticity (CDP)
   In this model, the tensile behavior of concrete is defined using two key parameters:

   • $f_t$: Tensile strength of concrete

   • $\varepsilon_t\!-\!\sigma_t$ or $w_t\!-\!\sigma_t$: Softening curve

   For example, using the fib Model Code relationship:

   $$\sigma_t = f_t \left( 1 – \frac{w}{w_1} \right)$$
   

   where $w_1$ is about 0.3–0.5 mm.

2. Crack Band Model (Hillerborg)
   Here, the stress is defined as a function of fracture energy (Gf):

   $$\sigma_t = f_t \left( 1 – \frac{\varepsilon_t}{\varepsilon_{cr} + \frac{G_f f_t}{l_{ch}}} \right)$$
   

   where $l_{ch}$ is the characteristic element length.

References:

• ABAQUS Documentation 2024, Section 23.5.4: Tension stiffening for concrete

• Hillerborg, A. et al. (1976). “Analysis of crack formation and growth in concrete by means of fracture mechanics.” Cement and Concrete Research.

Effect of Concrete Tension Stiffening on the Effective Flexural Capacity of Shear Walls in Seismic Design

In seismic design of reinforced concrete shear walls, it’s often assumed that concrete in the tensile zone contributes nothing after cracking. In my view, that’s a conservative but not always realistic assumption. Recent tests show that within service-level drifts and even up to near-yield conditions of reinforcement, cracked concrete still carries part of the tensile load. This is exactly the concrete tension stiffening phenomenon, which increases both the effective flexural capacity and initial stiffness of walls compared to classical predictions.

According to ASCE 41-17, the effective stiffness of shear walls is estimated as $I_{eff} = \alpha I_g$, where $\alpha$ ranges from 0.35 to 0.7 depending on the member type. In reality, this factor is a simplified way to include tension stiffening implicitly. Studies by Gilbert and Vecchio (2013) found that if tension stiffening is modeled explicitly, the actual $\alpha$ lies between 0.6 and 0.8, meaning the wall is stiffer than what the code assumes. This difference can notably affect drift and damage predictions in FEMA P-58 performance assessments.

In nonlinear FEMA P-58 analysis, the seismic response is derived from the force–displacement curve. If tension stiffening is ignored, the initial slope is lower and yielding occurs earlier—making the structure appear softer and more vulnerable than it truly is. Lim et al. (2020, Journal of Structural Engineering, ASCE) confirmed this, recommending that the initial wall stiffness in nonlinear models be adjusted to include tension stiffening for accurate hysteretic energy and response simulation.

Mechanically, tension stiffening in concrete increases the effective moment of inertia in cracked regions because tensile stresses still exist between cracks. In tall walls with low reinforcement ratios, this effect can raise the effective flexural capacity by up to 20% (Paulay & Priestley 1992; Palermo et al. 2015). In contrast, in boundary-confined or heavily reinforced walls, its influence is smaller since concrete exits tension sooner.

Though most codes don’t explicitly include this effect in design equations, newer guidelines like ASCE 41-23 allow engineers to use realistic material-based stiffness models. When using concrete material models that include a tensile softening branch (e.g., Concrete02 or ConcreteCM in OpenSees), the concrete tension stiffening behavior naturally emerges, removing the need for empirical stiffness factors.

Simply put, if we model a concrete wall assuming fully cracked, tension-free concrete, we overestimate flexibility and drift. Including tension stiffening aligns the force–displacement curve with experimental results yielding higher effective stiffness and more realistic flexural capacity.

In summary, for performance-based seismic design under FEMA P-58 and ASCE 41, concrete tension stiffening should be incorporated at the material-model level, not through constant reduction factors. Doing so leads to more accurate predictions of damage, capacity, and stiffness, especially in walls with low reinforcement ratios or high-strength concrete, where this effect is most pronounced.

If you’ve worked with concrete tension stiffening in OpenSees, ABAQUS, or ETABS, I’d love to hear about your experience in the comments real-world insights are what truly help our structural engineering community grow.