Figure 1. Numerical shear failure

The nonlinear behavior of deep beams makes them an ideal case study to illustrate the strengths and limitations of NLFEA in structural analysis.

Balancing the Pros and Cons of NLFEA

NLFEA brings significant advantages, especially for structures like masonry and reinforced concrete (R.C.), where nonlinear effects become critical after cracking and during reinforcement’s post-yield phase. Unlike linear models, which often fail to capture these complexities, NLFEA accurately simulates behaviors such as concrete crushing, nonlinear stress-strain relationships, and bond-slip effects. These factors are essential for a realistic assessment of structural performance under challenging conditions.

However, NLFEA presents challenges. It requires a strong foundation in structural analysis, careful modeling techniques, and informed assumptions. Proper implementation demands detailed sensitivity analyses to evaluate the impact of material properties, boundary conditions, and loading scenarios. Poorly executed models or incorrect assumptions can lead to significant errors, making it crucial to rely on experienced practitioners who can interpret results accurately.

Despite its benefits, the complexity of NLFEA discourages many engineers. The steep learning curve, along with the need for specialized knowledge, leads many to prefer more straightforward linear methods. As a result, NLFEA is often approached cautiously, particularly by those less familiar with its intricacies.

The Growing Accessibility of NLFEA

Fortunately, advancements in computational resources and software tools are making NLFEA more accessible. Modern computers can handle the complex calculations required for nonlinear analysis more efficiently, reducing the time and effort needed. Additionally, user-friendly interfaces and automation are streamlining the modeling process, allowing engineers to focus on interpreting results.

As more professionals gain experience with nonlinear methods, and as education places greater emphasis on these techniques, NLFEA is likely to be viewed not as an intimidating tool but as an essential part of structural analysis. Training programs and resources are helping to build the necessary expertise to implement NLFEA effectively.

Nonlinear Plasticity for R.C. and Masonry Members

R.C. and masonry members are essential for capturing material and geometric nonlinearities that linear methods cannot address. This type of analysis becomes critical after concrete cracks, as traditional linear elastic approaches fail to predict structural response accurately. Two main approaches are employed to model nonlinear behavior: lumped plasticity and distributed plasticity.

In lumped plasticity models, the nonlinear behavior is concentrated at specific points, such as the ends of beams or columns. This method is computationally efficient and relatively simple to implement, but it may lack accuracy in more complex scenarios where the material’s behavior needs detailed modeling.

In contrast, distributed plasticity models account for nonlinear behavior throughout the entire structure. This approach provides a more detailed and accurate simulation of material responses by distributing the nonlinearity across the length of the structural elements. However, this accuracy comes at the cost of increased computational effort.

Both approaches lumped and distributed plasticity, can use fiber sections to simulate material behavior. In fiber section modeling, structural elements are divided into individual fibers, each assigned specific material properties, such as concrete or steel. Stress and strain distributions across these fibers are calculated, offering a detailed understanding of the element’s response under loading conditions.

For more complex structures, like slabs and walls, layered shell elements are used to represent different layers of material properties. Solid elements are employed with distributed plasticity for three-dimensional components to capture various types of nonlinear behavior in Masonry and R.C. structures. These advanced techniques allow for a comprehensive analysis of reinforced concrete behavior under diverse loading scenarios.

This document explores, in particular, state-of-the-art methods for 2D and 3D modeling. Though not yet widely adopted in practice, these techniques are poised to become more prevalent in the coming years. As simulation technologies advance, the growing demand for high-fidelity simulations across industries is expected to drive the adoption of these sophisticated modeling approaches, making them integral to future workflows.

Application example: shear failure of an R.C. deep beam

Problem statement

D-Regions, also known as “Disturbed” or “Discontinuity Regions,” are areas in concrete structures where stress distribution becomes nonlinear due to factors like abrupt changes in loading, support reactions, or significant variations in cross-sectional dimensions. Unlike B-regions (Bernoulli Regions), where stresses follow a more uniform and linear pattern, D-regions require a more sophisticated analytical approach.

Beyond the B-region, structural elements should be treated as D-regions. The Strut-and-Tie Model (STM) and NLFEA are two common methods for analyzing these regions, each offering distinct advantages and limitations.

STM is well-suited for regions with clearly defined stress paths, such as deep beams and corbels. However, it has limitations in handling more intricate stress patterns. STM tends to simplify stress distributions, relies heavily on engineering judgment, and can lead to variable results depending on the analyst’s experience. Additionally, it may struggle to predict local stress concentrations or nonlinear behaviors accurately, and it often provides conservative capacity estimates due to its foundation in lower-bound plasticity theory.

In contrast, NLFEA offers a detailed and versatile analysis of stress distributions, providing deeper insights into structural behavior. As previously discussed, this method demands significant computational resources and a meticulous setup, including refined meshing and precise boundary conditions. NLFEA generates large volumes of data that can be difficult to interpret, and nonlinear models often face convergence issues that require time-consuming adjustments. The accuracy of NLFEA is highly dependent on the correct input of material properties and modeling assumptions, necessitating expertise in both the software and structural analysis.

The following sections will compare the results obtained using STM and NLFEA, emphasizing the strengths and weaknesses of each method in analyzing D-regions. The calculations are by Eurocode 2[1], now referred to as EC2. Unless otherwise specified, the material characteristic properties are used, and no safety factors are applied in the calculations to compare the analytical capacity with the FEM results.

Structures analyzed

Geometry and reinforcement

The reinforced concrete (RC) beam has a shear span of  l = 743 mm, with overall dimensions L = 3600 mm in length, h = 400 mm in height, and b = 200 mm in width, as shown in Figure 2. Given that the shear span-to-height ratio l/h is 1.86, the beam can be classified as a deep beam. The beam is reinforced with four 24 mm diameter rebars at the bottom and two 24 mm diameter rebars at the top. Only two bottom rebars are fully anchored, while the others are partially anchored. Additionally, the beam is reinforced in shear with 6 mm diameter stirrups placed at 320 mm intervals. The steel has a characteristic yield strength of f_yk = 500 MPa, while the concrete has a characteristic compressive strength f_ck = 20 MPa.

Figure 2 Beam geometry and reinforcement (mm)

STM Solution

The first step is to identify the D-Region in the structure where the Strut-and-Tie Model will be applied, shown in the following Figure 3.

Figure 3. D-Region and nodes

The Figure 3 illustrates the Tie, responsible for carrying the tensile forces resisted by the reinforcement, along with the nodes where struts and ties intersect: “Compression-Compression” nodes (three compressive struts, CCC) and “Compression-Tension” nodes (two compressive struts and one tie, CCT, and one compressive strut and two ties, CTT). In a CCC node, all forces converging at the node are compressive, whereas, in a CCT and CTT node, forces in both compression and tension converge.

The beam capacity can be determined by applying the material properties and equilibrium conditions, assuming it corresponds to the lowest capacity among the resisting mechanisms.

The strut capacity is calculated with EC2 (6.5.1). The design strength for a concrete strut in a region subjected to transverse compressive stress, or in the absence of transverse stress, can be calculated as σ_Rd,max = f_cd

Note. It should be noted that EC2 states that it may be appropriate to assume a higher design strength in regions where multi-axial compression exists. Thus, when multiaxial stress states are acting, the effect of lateral confinement should be considered, using more refined approaches such as FEM.

The design strength of concrete struts should be reduced in cracked compression zones and, unless a more rigorous approach is applied, can be calculated using:

σ_Rd,max = 0.6v’f_cd, v’ = 1 – 250/f_ck

Figure 4. Strut capacity

Assuming γ_c = 1, f_cd = γ_cf_ck = 20 MPa,, v’ = 0.92.

Thus, σ_Rd,max = 20 MPa with transverse or no compression and σ_Rd,max = 0.6 × 0.92f_cd = 0.55 f_cd = 11.0 MPa with transverse tension.

It is helpful to examine the multiaxial concrete yield surface to understand how the strut capacity is determined, as illustrated in Figure 5.

Figure 5. Multiaxial concrete yield

In the region subjected to transverse compressive stress (σ₂ < 0), or in the absence of transverse stress (σ₂ = 0), the minimum uniaxial concrete strength is σ₁ = -f_cd. However, the Figure 5 indicates that in this region, the uniaxial compressive strength. f_cd increases by approximately 16% under conditions of equal biaxial compression and roughly 25% for σ₁/σ₂ = 2.

Note: the capacity of the concrete strut may be higher than that determined using EC2, even in simple plane stress conditions.

Under triaxial conditions where σ₃≠0, the strength of concrete can vary significantly, remarkably increasing when σ₃>0 and decreasing when σ₃ < 0.

Assuming a reduction in the design strength of concrete struts in cracked compression zones with transverse tensile stresses (σ₂ > 0), equal to 0.55 f_cd, as in this case, implies that the strut is subjected to transverse tensile stresses approximately equal to 60% of the concrete tensile strength f_ct.

Note: EC2 significantly reduces the compressive strength of struts in the presence of tensile transverse stresses using a very simplified approach, regardless of the amount of transverse stress applied. However, to accurately estimate the compressive capacity of struts under transverse tension, a multiaxial state of stress should be considered.

In many cases, particularly for complex biaxial and triaxial states, the assumptions made by EC2 may be overly conservative. In such instances, FEM could provide a more accurate prediction of the concrete strut compressive strength.

Figure 6 shows a drawing of the D-Region highlighting the CCT and CCC nodes. The strut angle is calculated as:

θ = tan^-1(h/l) = tan^-1((400-35)/743) = 26.2°

The height of the CCT node’s back face is typically assumed to be twice the distance from the near face of the beam to the centroid of the tension reinforcement, h_s=70 mm. The strut width a₂, at the CCT node interface is calculated as a₂= hₛ cos θ + l_bp sin θ = 151 mm.

The strut capacity is calculated with EC2 by taking into account the tensile transverse stresses induced by the Tie is σ_Rd,max = 11.0 MPa, therefore, the strut capacity at the strut-to-node CCT interface is C_CCT = b × a₂ × σ_Rd,max = 315 kN.

Figure 6. CCC and CCT Nodes

The depth of the back face of a CCC node, h_cc (see Figure 6) corresponds to the depth of the equivalent compressive stress block derived from a standard flexural analysis and is given by:
h_cc = (A_sσ_s – A’_sσ’_s) / (bf_cd)

where A_s and A’_s are the areas of the steel in tension and compression, respectively, σ_s and σ’_s are the corresponding steel stresses. If the steel in compression is neglected and the steel in tension is assumed to have yielded, the depth simplifies to:

h_cc = (A_s f_yd) / (bf_cd) = 113 mm

This value is close to the exact depth obtained by applying equilibrium, h_cc=140 mm
The strut width a₃ at the CCC node interface is calculated as a₃ = hₛ cos θ + l_lp sin θ = 151 mm
The design values for compressive stresses within CCC nodes, as specified in Section 6.5.4 of EC2, can be determined in compression using the formula: σ_Rd,max = k_₁v’f_cd

where v’ = 1 – 250/f_ck, k_₁ = 1, v’ = 0.92.

Thus, σ_Rd,max = 18.4 MPa

The strut capacity at the strut-to-node CCC interface is C_ccc = b × a_₃ × σ_Rd,max= 556 kN.

The following Figure 7 illustrates the force equilibrium diagram.

Figure 7. Equilibrium at the nodes

The force in the strut is C = F / sin⁡θ.

At the CCC interface, the maximum attainable shear demand is F=C_CCC sin⁡θ=245 kN. Similarly, at the CCT interface, the maximum attainable shear demand is F = C_CCT sin⁡θ = 147 kN, and the force in the Tie is T = F tan⁡(90-θ) = 299 kN.

The Tie capacity is determined by the area of reinforcement anchored in the CCT node. Since only two of the four 24 mm rebars are anchored, the Tie capacity is T=452 kN.

Therefore, the beam shear capacity calculated with STM is governed by the failure in compression of the CCT node, Vcd=147 kN.

FEM Solution

Nonlinear model

Commercial finite element analysis (FEA) software uses various constitutive models to simulate the nonlinear behavior of R.C. and masonry under different loading conditions. Each program offers specific models designed to accurately represent materials like concrete, masonry, steel reinforcement, and bond-slip behavior.  Each model is tailored for different applications, balancing accuracy with computational efficiency.

The analyses are conducted using a detailed 3D nonlinear model, as illustrated in Figure 8. The model is developed with the combined capabilities of STKO (Scientific ToolKit for OpenSees) and OpenSees, providing a robust platform for simulating complex structural behaviors.

STKO is used to build and visualize the model, while OpenSees performs advanced nonlinear computations, enabling accurate representation of material and geometric nonlinearities. Each feature is carefully selected to enhance the accuracy and robustness of the analysis, ensuring that the model can effectively simulate the complex behaviors of the structure.  The model’s key features are outlined below:

• Concrete: ASDConcrete3D is a plastic-damage model for concrete based on continuum-damage mechanics. It calculates the stress tensor directly from the total strain tensor, which enhances both efficiency and robustness, particularly for large-scale structural models. It uses a simplified plasticity to incorporate inelastic effects without demanding excessive computational resources. ASDConcrete3D employs the IMPL-EX integration scheme to improve strain-softening simulations’ accuracy. This hybrid approach combines the stability of implicit methods with the efficiency of explicit ones, correcting material-level errors at the structural level. As a result, it provides fast and stable computations for highly nonlinear problems. For more details, refer to the ASDConcrete3D OpenSees documentation. The 3D element used is F
• Rebars: The rebars are modeled with 1D truss elements (truss) with fiber sections (Fiber.), and the elastoplastic steel02 constitutive model is used for this analysis. Steel reinforcement is “embedded” in the concrete matrix. ASDEmbedded, developed by ASDEA Software and integrated into STKO for OpenSees, is a method to model the interaction between embedded elements (such as rebars or fibers) and a host material (like concrete, masonry, or soil). This technique imposes kinematic constraints to ensure the embedded elements deform compatibly with the host material without requiring mesh alignment, making it computationally efficient. For more information, refer to the ASDEmbeddedNodeElement documentation.
• Bond-slip: ASDEmbedded assumes a perfect bond between the embedded and host materials. However, this may limit accuracy when significant slip or separation occurs at the interface. To address this, STKO provides a Beam-to-Solid Bar Slip This tool allows modeling the interaction between beam elements (such as reinforcements) and solid elements by incorporating slip behavior. It simulates the relative movement between the beam and surrounding material, capturing bond-slip effects that are critical for realistic analysis of reinforced concrete, masonry, composite, and soil structures. The nonlinear behavior is simulated with the uniaxial material Pinching4
• Conditions:
  • Fix Uy, Uz, Rx at the support
  • Fix Ux, Ry condition of symmetry
  • The bond slip is modelled with this condition
  • An edge force represents the applied load
    

Figure 8. FEM model

Results

The Figure 9 below presents the load-displacement curve, with the vertical displacement measured at midspan.

Figure 9. Load-displacement curve

The predicted beam capacity is 200 kN, 36% higher than the capacity calculated using the STM model. The main reasons for this difference are that the STM does not account for the partially anchored rebars, and the 3D stress state of the concrete in the CCT node is simplified compared to the more detailed representation in the FEM model.

Figure 10 illustrates the crack pattern. At 100 kN, the beam exhibits flexural cracking. When the load reaches 147 kN (STM failure), shear cracks develop. Finally, at the peak load (Vmax), the CCT nodes fail, leading to the collapse of the beam.

Figure 10. Crack pattern

At the peak load, the maximum crack width is around 2.4 mm. The scalar of the negative damage shows that the failure starts propagating at the CCT node.

Figure 11. Crack width and negative damage d- at Vmax

Figure 11 presents the strut width and minimum principal stress components. In the CCT node, the minimum principal stresses are mostly below -12 MPa, with some localized variations and an average stress of roughly 8 MPa. Unlike the oversimplified linear STM model, this approach accurately captures the concrete triaxial stress state.

Figure 12. Strut width and principal minimum stress components

Figure 12 illustrates the bond-slip behavior of the reinforcement bars.

Figure 13. Slip (mm)

As expected, the rebars that are not fully anchored within the node slip significantly, limiting the force they can transfer to the surrounding concrete. The lack of sufficient anchorage prevents the mechanical interlock and friction from fully developing between the rebar and the concrete, thereby reducing the bond strength and lowering the load-carrying capacity of the rebars. This limited force transfer impacts the overall structural performance, especially in areas where the reinforcement needs to resist significant tensile forces. The distribution of stresses and strains along the rebar becomes uneven, affecting the structure’s response under load. This situation may require additional reinforcement detailing or anchorage solutions to meet the intended structural capacity. Evaluating bond-slip behavior using simplified approaches like STM cannot fully capture the complex interactions and non-linearities associated with bond-slip phenomena in reinforced concrete structures.

Another critical aspect of this analysis is evaluating the beam’s capacity when confined. The confinement effect, while extensively documented in the literature, is not accounted for in standard design codes and cannot be accurately assessed using simple linear analysis methods. When a beam begins to crack, it undergoes expansion. If this expansion is restrained—such as when the beam is connected to adjacent structural elements like columns or forms part of a statically indeterminate (hyperstatic) system—an axial compressive force is generated within the beam. As the cracking progresses and the deformations increase, the restraint imposed by the surrounding structure becomes more significant, leading to a higher axial force. This built-up axial load enhances the beam’s flexural and shear capacities. The increased capacity is due to the additional compression within the concrete, which improves its resistance to further deformation and crack propagation.

However, linear elastic analysis, which assumes a direct proportionality between stress and strain, can only capture these beneficial effects after considering the complex interactions after cracking. A sophisticated nonlinear analysis is required to account for the confinement effects. Such an approach can incorporate the material nonlinearity, the crack development, and the interaction between the axial forces and bending moments, allowing for a more realistic representation of the beam’s actual behavior under load.

The dashed curve represents the capacity of the beam when lateral confinement is applied, as illustrated in the following Figure 14.

Figure 14. Confined load-midspan displacement curve

The static configuration remains unchanged; only the addition of lateral confinement modifies the beam’s conditions. The cracks are prevented from opening, which enhances the stiffness and overall capacity of the beam after cracking occurs. The beam reaches a capacity of 243 kN, which is 22% higher than in the condition without confinement. At peak load, the section near the support experiences compression. The axial load rises due to the lateral restraint, which counteracts the beam’s expansion resulting from cracking. At the CCT node, the minimum principal stresses shift with the introduction of confinement, as shown in Figure 15. This change reduces the transverse stresses within the compressive strut, enhancing the beam’s shear capacity. Consequently, the confinement effectively increases the overall load-carrying capacity of the structure.

Figure 15. Minimum principal stresses at the CCT node: left image shows results with confinement, while the right image shows results without confinement.

As extensively documented in the published literature, the confinement effect plays a crucial role in enhancing the flexural, shear, and punching-shear capacities of bridge deck slabs, beams, and slabs. Confinement increases the strength and ductility of concrete by limiting lateral expansion, particularly under compressive loads, which significantly increases overall load-bearing capacity.

When assessing the structural safety of these elements, neglecting the confinement effect could lead to conservative or inaccurate evaluations. Incorporating this effect into safety assessments ensures a more realistic representation of the structural behavior, allowing engineers to understand the load-carrying capacity and deformation limits better.

Conclusion

NLFEA offers substantial advantages over conventional linear methods, especially in analyzing structures where nonlinear behaviors, material cracking, and yielding occur. By accurately modeling complex phenomena such as bond-slip effects, large deformations, and stress redistribution, NLFEA provides a more precise and realistic assessment of structural behavior. This enhanced understanding enables engineers to optimize design and improve safety, particularly in demanding applications such as earthquake-resistant structures, deep beams, and masonry elements where traditional approaches often fall short.

One benefit of NLFEA is its ability to incorporate the effects of membrane confinement, which significantly influences the structural capacity. When a structure is laterally restrained, such as through membrane action, the confinement increases the internal axial forces, enhancing both shear and flexural capacity.

Despite the complexities associated with NLFEA, such as the need for specialized expertise, significant computational resources, and detailed modeling practices, its valuable insights make it indispensable for modern structural engineering. Accurately predicting load-deflection behavior, crack propagation, and capacity improvements due to confinement effects contribute to safer and more reliable structural designs. Embracing NLFEA is crucial for addressing the demands of contemporary structural analysis, ensuring that engineers can design and assess structures with the highest standards of accuracy and performance. As structural engineering advances, the use of NLFEA will become increasingly essential for achieving design excellence and maintaining safety standards in complex and high-performance structures.

References

[1] European Committee for Standardization (CEN). (2004). Eurocode 2: Design of Concrete Structures—Part 1-1: General Rules and Rules for Buildings (EN 1992-1-1:2004). Brussels: CEN.

Eurocode 2-1 Design of concrete structures Part 1: Buildings;

Eurocode 2 Part 2: Concrète Bridges – Design and detailing rules

Federation Internationale du Beton (International Federation for Structural Concrete – FIB) Model Code 2010 (MC2010);

Federation Internationale du Beton (International Federation for Structural Concrete – FIB) Model Code 2020 (MC2020) ;

Max A.N. Hendriks and Marco A. Roosen (editors), “Guidelines for Nonlinear Finite Element Analysis of Concrete Structures”, Rijkswaterstaat Centre for Infrastructure, Report RTD:1016- 1:2022, Dutch Ministry of Infrastructure and Water Management.

Belletti, Beatrice & Walraven, Joost & Trapani, Francesco. (2015). Evaluation of compressive membrane action effects on punching shear resistance of reinforced concrete slabs. Engineering Structures. 95. 10.1016/j.engstruct.2015.03.043.

Vidaković, Aleksandar & Majtánová, Lucia & Halvonik, Jaroslav. (2021). Effect of Membrane Forces on Punching Shear Capacity of Flat Slabs. Solid State Phenomena. 322. 136-141. 10.4028/www.scientific.net/SSP.322.136.

Pang, Bo & Wang, Feiliang & Yang, Jian & Nyunn, Sandy & Azim, Iftikhar. (2021). Performance of slabs in reinforced concrete structures to resist progressive collapse. Structures. 33. 4843-4856. 10.1016/j.istruc.2021.04.092.