Nonlinear FE Modeling & Validation
of Partial Composite Beams Under Fire

1.1 Mechanics of Composite Floor Systems

Composite steel-concrete floor beams represent one of the most structurally efficient configurations in commercial multi-storey construction. The structural synergy arises from coupling the compressive capacity of a reinforced concrete floor slab with the tensile and flexural capacity of a structural steel I-section. The force transfer between these two distinct structural media occurs across a discrete mechanical interface comprising headed shear stud connectors welded to the top flange of the steel beam and embedded within the concrete slab.

In standard design office practice, composite beams are categorized as either fully composite (where sufficient shear studs are provided to develop the full plastic flexural capacity of the steel section or concrete slab) or partially composite (where the number of studs limits the horizontal force transfer capacity). Under partial composite action, strain discontinuity occurs across the steel-concrete interface, manifesting as relative horizontal displacement termed interfacial shear slip $s(x)$.

1.2 Benchmark Problem Specification — AISC Example I-1

To establish an authoritative baseline, the physical problem investigated in this thesis is adapted from the American Institute of Steel Construction (AISC) Steel Construction Manual (15th Edition), Design Example I-1. The structural system consists of a simply supported wide-flange W21×55 beam spanning $L = 45.0\text{ ft}$ ($13.716\text{ m}$) supporting a $7.5\text{ in.}$ ($190.5\text{ mm}$) thick normal-weight concrete slab ($f'_c = 4.00\text{ ksi} = 27.58\text{ MPa}$) with an effective width $b_{\text{eff}} = 120.0\text{ in.}$ ($3.048\text{ m}$). Interface shear transfer is supplied by 34 headed shear stud connectors ($3/4\text{ in.}$ diameter) distributed uniformly along the span (17 studs per shear span), establishing a Partial Composite Connection ratio ($PCC$) of $36\%$.

1.3 Research Objectives & Scientific Methodology

The primary research objectives of this investigation are structured into four scientific domains:

1. Cross-Sectional Fiber Discretization: Develop a high-resolution Python fiber section solver to compute nonlinear moment-curvature ($M-\chi$) behavior and plastic stress block redistribution at $20^\circ\text{C}$.
2. Dual-Beam Finite Element Kinematics: Formulate an advanced 2D structural model in SAFIR utilizing offset node lines tied vertically via kinematic constraint equations, with explicit diagonal X-bracing truss elements representing stud shear slip.
3. Multi-Tier Analytical Cross-Validation: Mathematically validate simulated elastic service deflections against two classical closed-form mechanics benchmarks: Newmark's exact differential equation (Newmark et al., 1951) and Nie & Cai's ASCE equivalent flexural rigidity solution (Nie & Cai, 2003).
4. Coupled High-Temperature Fire Response: Execute 2D transient thermal analyses under ISO 834 fire exposure and couple the resulting thermal fields into nonlinear structural analyses to evaluate elevated-temperature structural endurance and plastic runaway collapse.

2.1 Structural Steel Constitutive Model (Eurocode 3)

Structural steel (Grade S355 / ASTM A992) is modeled using the Eurocode 3 (EN 1993-1-2) elevated-temperature elastoplastic uniaxial constitutive relationship. At ambient conditions ($20^\circ\text{C}$), steel exhibits an elastic modulus $E_s = 210\text{ GPa}$ ($30,458\text{ ksi}$) and yield strength $f_y = 355.0\text{ MPa}$ ($51.49\text{ ksi}$). Under thermal exposure, reduction factors $k_{y,\theta} = f_{y,\theta}/f_y$ and $k_{E,\theta} = E_{s,\theta}/E_s$ govern strength and stiffness degradation, leading to severe thermal softening beyond $400^\circ\text{C}$.

2.2 Concrete Constitutive Model (Eurocode 2)

The concrete slab is discretized using the Eurocode 2 (EN 1992-1-2) uniaxial compression-only non-linear parabolic model for siliceous aggregate concrete. Ambient compressive strength is set to $f'_c = 27.58\text{ MPa}$ ($4.00\text{ ksi}$) with an initial secant modulus $E_c = 23.5\text{ GPa}$ ($3,408\text{ ksi}$). Tensile strength is conservatively neglected ($f_{ct} = 0$). Peak compressive strain $\varepsilon_{c1,\theta}$ increases with temperature while ultimate strength $f_{c,\theta}$ degrades per Eurocode thermal curves.

2.3 Transient Thermal Heat Transfer Governing Equations

The transient thermal field within the composite cross-section is governed by 2D non-linear heat conduction:

Thermal boundary conditions follow ISO 834 standard fire exposure on three sides of the steel section (bottom flange, web, under-flange) and the bottom surface of the concrete slab ($h_c = 25\text{ W/m}^2\text{K}, \varepsilon = 0.7$). The top flange of the steel beam is in unexposed contact with the concrete slab. The top slab surface experiences ambient cooling to $20^\circ\text{C}$ ($h_c = 4\text{ W/m}^2\text{K}, \varepsilon = 0.7$).

Prior to conducting elevated-temperature simulations, the structural mechanics pipeline was verified against the published AISC Example I-1 calculations using a custom Python cross-sectional fiber solver.

3.1 Fiber Section Moment-Curvature Analysis

The W21×55 steel shape and $120\text{ in.}$ concrete slab were discretized into 1,000 horizontal fibers. Incrementing curvature $\chi$ allows tracking of neutral axis migration and stress integration across the section:

Figure 3.1: Mid-span deflection profiles for construction dead load and service live load conditions overlaid with AISC code limits.

Figure 3.2: Moment-curvature ($M-\chi$) relationship demonstrating plastic moment capacity convergence at $M_n = 836.6\text{ kip-ft}$.

3.2 SAFIR Ambient Bending Verification ($t = 20\text{ s}$)

At the conclusion of the 20-second ambient load ramp, SAFIR achieved static mid-span moment $M_z = 487.5\text{ kip-ft}$ ($661.0\text{ kN}\cdot\text{m}$), matching theoretical statics ($wL^2/8 = 488.5\text{ kip-ft}$) within 0.20% error.

The standard SAFIR structural model enforces full Euler-Bernoulli strain compatibility across the steel-concrete section (no slip). Thermal cross-sections are meshed with 1,166 nodes and 678 fibers, generating temperature histories under ISO 834 fire.

Figure 4.1: Mid-span deflection history for the standard full-composite SAFIR model under ISO 834 fire. Structural serviceability limit ($\text{Span}/20 = 685.8\text{ mm}$) is breached at $11.70\text{ min}$.

To explicitly model interfacial shear stud slip, an advanced dual-beam finite element model was established in SAFIR. Steel beam elements are positioned at $y = 0.0\text{ m}$ and concrete slab elements at $y = 0.35941\text{ m}$ (the physical centroidal distance). Vertical separation is prevented using `SAME` vertical DOF displacement constraints, while horizontal movement remains unconstrained.

Figure 5.1: Finite element topology comparison: Standard single-line beam formulation (left) vs. Advanced vertically offset dual-line beam formulation with diagonal X-bracing truss shear connectors (right).

5.1 Decoupled Two-Parameter Truss Connector Calibration

The 34 physical shear studs distributed across 20 element intervals yield an average density of $1.700\text{ studs/interval}$. To eliminate discretization errors, a decoupled two-parameter mathematical calibration was derived to achieve exact 0.00% numerical discrepancy for both elastic shear stiffness and plastic shear strength:

5.1.1 Target Mechanical Properties per Interval

• Target Shear Stiffness: $K_{h,\text{target}} = 1.700 \times 100.0\text{ kN/mm} = \mathbf{170.00\text{ kN/mm}}$
• Target Shear Plastic Strength: $V_{h,\text{target}} = 1.700 \times 76.507\text{ kN} = \mathbf{130.062\text{ kN}}$

5.1.2 Mathematical Calibration Derivation

1. Calibrated Cross-Sectional Area ($A_{\text{truss}}$): Solved from the horizontal stiffness equilibrium equation of the diagonal X-brace pair:

Substituting $A_{\text{truss}} = 3.9947\text{ cm}^2$ reproduces $K_h = 170.00\text{ kN/mm}$ with **0.00% discrepancy**.

2. Calibrated Yield Strength ($f_{y,\text{truss}}$): Solved from plastic shear equilibrium using the calibrated area $A_{\text{truss}}$:

Assigning a dedicated steel material (`S367.58`) to the truss elements yields $V_h = 130.06\text{ kN}$, achieving exact plastic capacity with **0.00% discrepancy**.

5.2 Fire Performance Under Partial Composite Action

Figure 5.2: Deflection history of the advanced dual-beam model with stud slip under ISO 834 fire. Structural limit reached at $11.17\text{ min}$.

Figure 5.3: Fire deflection curve comparison: Full composite (red) vs. Partial composite with slip (blue). Slip reduces structural limit survival time by $0.53\text{ min}$ ($4.5\%$).

To independently validate the accuracy of SAFIR's advanced non-linear slip formulation ($1.85\text{ in.}$ deflection), the simulated serviceability response was benchmarked against two classical closed-form elasticity solutions from composite beam literature: Newmark's Elastic Differential Theory (Newmark et al., 1951) and Nie & Cai's ASCE Equivalent Flexural Rigidity Closed-Form Solution (Nie & Cai, 2003).

6.1 Newmark (1951) Classical Elastic Partial-Interaction Governing Formulation

In their seminal investigation (*Tests and Analysis of Composite Beams with Incomplete Interaction*, 1951), Newmark, Siess, and Viest derived the exact linear elastic second-order differential equation governing partial interaction in composite beams under axial force $F(x)$ and shear slip $s(x)$:

For a simply-supported composite beam of span $L$ subjected to a uniform distributed load $w$, integration of the governing differential equation yields the exact closed-form mid-span elastic deflection $\Delta_{\text{Newmark}}$:

Where the key stiffness parameters are evaluated as:

• Sum of Component Inertias ($I_0$): $I_0 = I_s = 1,140\text{ in.}^4 = 4.745 \times 10^{-4}\text{ m}^4$
• Full Transformed Rigid Inertia ($I_{\text{tr}}$): $I_{\text{tr}} = I_0 + A_{\text{eq}} h^2 = 3,933.8\text{ in.}^4 = 1.6374 \times 10^{-3}\text{ m}^4$
• Continuous Connector Modulus ($k$): $k = \frac{34 \times 100\text{ kN/mm}}{13.716\text{ m}} = \mathbf{247.89\text{ MPa}} \quad (5,177.2\text{ kips/in.}^2)$
• Stiffness Parameter ($\alpha L$): $\alpha = \sqrt{\frac{k}{E_s} \left[ \frac{1}{A_{\text{eq}}} + \frac{h^2}{I_0} \right]} \implies \alpha L = \mathbf{9.2262}$
• Shear Connection Discount Factor ($\Phi(\alpha L)$): $\Phi(\alpha L) = \mathbf{0.07401}$

Evaluating Newmark's exact closed-form differential solution yields a mid-span elastic service deflection of **$\Delta_{\text{Newmark}} = 1.76\text{ in.}$** (**$44.6\text{ mm}$**).

6.2 Nie & Cai (2003) ASCE Equivalent Bending Rigidity Formulation

In their ASCE Journal of Structural Engineering publication (*Steel-Concrete Composite Beams Considering Shear Slip Effects*, 2003), Nie and Cai simplified Newmark's differential formulation into an explicit reduced flexural rigidity $(EI)_{\text{eff}}$ using a single-term Fourier sine series approximation for simply supported beams under UDL:

Substituting $\alpha_{\text{slip}} = 0.11748$ yields an effective equivalent moment of inertia $I_{\text{eff,Nie}} = \mathbf{3,054.4\text{ in.}^4}$ ($1.2713 \times 10^{-3}\text{ m}^4$). Evaluating the deflection under UDL yields **$\Delta_{\text{Nie \& Cai (2003)}} = 1.91\text{ in.}$** (**$48.6\text{ mm}$**).

6.3 Multi-Tier Quantitative Benchmark Cross-Validation

A rigorous quantitative comparison of SAFIR against classical elasticity research solutions and AISC design code baselines is presented below:

6.3 Comparative Mechanics: Newmark Elastic Theory vs. Python Fiber Solver

To establish full theoretical clarity, it is essential to highlight the fundamental formulation differences between Newmark's Analytical Approach (1951) and the Python Fiber Section Solver:

Figure 6.1: Mid-span serviceability deflection profile comparison — SAFIR advanced FEM model (blue) vs. Nie & Cai (2003) ASCE closed-form (purple) vs. Newmark (1951) classical elastic solution (green) vs. AISC code equations (amber/slate).

6.4 Mechanics Reconciliation of Code Conservatism

The gap between SAFIR / Newmark / Nie & Cai ($1.76\text{--}1.91\text{ in.}$) and the AISC code equations ($I_{\text{eff}} = 2.43\text{ in.}, I_{\text{LB}} = 2.52\text{ in.}$) is fully reconciled by five distinct mechanical factors:

1. Non-Uniform Interfacial Slip Distribution: AISC's $I_{\text{eff}}$ empirical formula assumes a uniform stiffness reduction across the entire span. In reality (and as modeled in SAFIR and Newmark), interface slip is zero at mid-span (symmetry boundary condition) and reaches maximum magnitude at the supports. Because mid-span experiences maximum bending moment where slip is near zero, the mid-span region maintains near-full composite action.
2. Independent Concrete Slab Flexural Rigidity ($E_c I_{\text{slab}}$): AISC hand calculations model the slab purely as an axial flange carrying a composite couple, neglecting its independent bending stiffness ($E_c I_{\text{slab}} = 41.3\text{ MN}\cdot\text{m}^2$). SAFIR's dual-beam model allows the slab beam elements to carry flexural load directly, reducing steel beam deflections.
3. Sensitivity Mismatch (Elastic Stiffness vs. Plastic Strength Ratio): The 36% composite ratio ($PCC = 0.36$) in AISC Example I-1 is defined by ultimate plastic shear strength ($\sum Q_n / A_s f_y = 0.36$). Under service gravity loads ($M_{\text{service}} = 488.5\text{ kip-ft} \ll M_n = 836.6\text{ kip-ft}$), bending stresses remain well within the elastic range. Because the shear studs possess high elastic shear stiffness ($K_{\text{stud}} = 100\text{ kN/mm}$), SAFIR predicts that elastic deflection softens by only +10.1% ($1.68\text{ in.} \rightarrow 1.85\text{ in.}$). In contrast, the AISC empirical equation applies $\sqrt{PCC} = \sqrt{0.36} = 0.60$ to discount elastic stiffness across the full span, predicting a +36.5% deflection increase ($1.78\text{ in.} \rightarrow 2.43\text{ in.}$).
4. Conservative Lower-Bound Screening Nature of Code Equations: The AISC $I_{\text{eff}}$ formula was historically calibrated as a simplified conservative design office screening tool to account for long-term creep, shrinkage, and worst-case stud slip under full design live loads.
5. Material Property Alignment ($f'_c = 27.58\text{ MPa}$): Compressive strength and elastic modulus strictly follow the AISC Example I-1 specification across all thermal and structural fiber elements.

7.1 Graphical Representation in Diamond 2016

In SAFIR's structural post-processor (Diamond 2016), the steel beam and concrete slab appear as 1D line segments positioned along their respective centroidal axes ($y = 0.0\text{ m}$ for steel, $y = 0.35941\text{ m}$ for slab), linked by diagonal X-bracing truss elements. The 2D fiber meshes stored in the thermal `.tem` files internally integrate axial forces, bending moments, and thermal degradation at every step.

7.2 Sequential 6-Step Simulation Pipeline

• 1
  2D Thermal Mesh Generation — Steel Profile (steel_beam.IN)

  W21×55 section discretized into 238 steel fibers, exposed to 3-sided ISO 834 fire. Generates temperature field history steel_beam.tem.

• 2
  2D Thermal Mesh Generation — Concrete Slab (concrete_slab.IN)

  Concrete slab discretized into 238 concrete fibers, exposed to fire on bottom surface and ambient air on top. Generates temperature field history concrete_slab.tem.

• 3
  Structural Mesh Generation — Offset Dual Node Lines

  Parallel node lines established at $y = 0.0\text{ m}$ and $y = 0.35941\text{ m}$ (41 main nodes + 40 mid-nodes per line, 3-node quadratic beam formulation).

• 4
  Explicit Shear Connector Mesh — Calibrated Truss Elements

  40 truss elements (20 diagonal X-braces) assigned calibrated area $A_{\text{truss}} = 3.9947\text{ cm}^2$ and steel grade `S367.58`, mapped to ambient temperature file truss_temp.tem.

• 5
  Boundary Conditions & Kinematic DOF Constraints

  Pinned-roller supports applied to steel nodes; rollers applied to slab nodes. Vertical displacements tied using `SAME` constraint equations, permitting free horizontal slip.

• 6
  Coupled Non-Linear Ramped Thermal-Structural Execution

  Service UDL ($28.16\text{ kN/m}$) ramped over 20 s via `RAMP.fct`. At $t = 20\text{ s}$, ISO 834 thermal fields activate, executing non-linear Newton-Raphson iterations until structural failure.

8.1 Master Summary of Key Results

1. American Institute of Steel Construction (AISC). (2016). Steel Construction Manual, 15th Edition, Example I-1: Composite Beam Design. Chicago, IL: AISC.
2. Newmark, N. M., Siess, C. P., & Viest, I. M. (1951). Tests and Analysis of Composite Beams with Incomplete Interaction. Proceedings of the Society for Experimental Stress Analysis, 9(1), 75–92.
3. Nie, J., & Cai, C. S. (2003). Steel-Concrete Composite Beams Considering Shear Slip Effects. ASCE Journal of Structural Engineering, 129(4), 495–506.
4. Franssen, J.-M., & Gernay, T. (2017). SAFIR 2017: A Software for Modelling Structures Under Fire. Journal of Structural Fire Engineering, 8(3), 300–323. University of Liège, Belgium.
5. European Committee for Standardization (CEN). (2004). EN 1994-1-2: Eurocode 4 — Design of Composite Steel and Concrete Structures, Part 1-2: General Rules — Structural Fire Design. Brussels: CEN.
6. European Committee for Standardization (CEN). (2004). EN 1993-1-2: Eurocode 3 — Design of Steel Structures, Part 1-2: General Rules — Structural Fire Design. Brussels: CEN.
7. European Committee for Standardization (CEN). (2004). EN 1992-1-2: Eurocode 2 — Design of Concrete Structures, Part 1-2: General Rules — Structural Fire Design. Brussels: CEN.
8. International Organization for Standardization (ISO). (1999). ISO 834-1: Fire Resistance Tests — Elements of Building Construction, Part 1: General Requirements. Geneva: ISO.
9. Ollgaard, J. G., Slutter, R. G., & Fisher, J. W. (1971). Shear Strength of Stud Connectors in Lightweight and Normal-Weight Concrete. AISC Engineering Journal, 8(2), 55–64.
10. Ranzi, G., Bradford, M. A., & Uy, B. (2003). A Direct Stiffness Analysis of a Composite Beam with Partial Shear Interaction. International Journal for Numerical Methods in Engineering, 61(5), 657–672.