LittleDef Theory Manual

Mathematical Foundation and Structural Mechanics

1. Linear Static Analysis

The core of LittleDef is the Direct Stiffness Method (DSM). The structure is represented as a global stiffness matrix K, nodal displacement vector d, and force vector F:

K · d = F

Elements are modeled as Bernoulli-Euler beams with three degrees of freedom per node (u_x, u_y, φ). The stiffness matrix for each element includes axial, shear, and flexural terms.

2. Sign Conventions

• Nodal Displacements: Positive in global X (right) and Y (up). Rotations are positive counter-clockwise.
• Internal Forces:
  • Axial Force (N): Tension is positive (+), Compression is negative (-).
  • Shear Force (V): Positive according to beam convention (upward on left face).
  • Bending Moment (M): Plotted on the tension side of the member.

3. Advanced Element Modeling

3.1 Hinges and Semi-rigid Connections

Internal hinges are implemented by modifying the element stiffness matrix. For a semi-rigid connection with rotational stiffness k, we define dimensionless coefficients:

A = 2 + 6EI / (L · k_a)

Where k_a is the stiffness at the start of the element. If k → ∞, the connection is rigid; if k → 0, it is a perfect hinge.

3.2 Second-Order Effects (P-Delta)

Geometric nonlinearity is captured by adding a geometric stiffness matrix k_g to the elastic stiffness matrix. This accounts for the additional moments generated by axial loads acting on the deformed geometry of the structure.

3.3 Advanced Plastic Analysis

An incremental event-to-event solver predicting the structural collapse load multiplier. It incorporates a quadratic N-M yield interaction surface (limiting bending resistance based on axial force levels: |M|/M_pl + (N/N_pl)² = 1.0) and geometric stiffness updates to capture instability-triggered collapse.

4. Eurocode Implementation Details

4.1 EN 1992 (Concrete)

Verification is based on the strain compatibility method. N-M interaction diagrams are generated using a parabolic-rectangular stress block for concrete and a bilinear model for reinforcement. Crack widths are calculated per EN 1992-1-1, 7.3.4, considering tension stiffening effects.

Second Generation Preview (EN 1992-1-1:2023): Implements Critical Shear Crack Theory (CSCT) shear capacity formulation for members without shear reinforcement using aggregate roughness parameters ($d_{dg}$) and minimum shear stress capacity ($\tau_{Rdc,min}$); punching shear capacity checking at $0.5d$ perimeter with the $k_{pb}$ factor; 2nd Gen SLS crack width spacing parameters ($k_b=0.8$, $k_{fl}=0.5$, $k_w=1.3$); and dynamic column slenderness limit criteria ($\lambda_{lim}$) under second-order nominal stiffness methods.

4.2 EN 1993 (Steel)

The software performs cross-section classification (Class 1-4) based on local buckling limits (c/t ratios). Stability checks for flexural buckling and lateral-torsional buckling use the χ and χ_LT reduction factors according to the European buckling curves.

Second Generation Preview (EN 1993-1-1:2022): Integrates elasto-plastic section classifications and $W_{ep}$ modulus interpolation for Class 3 webs/CHS sections, dynamic moment distribution gradients ($f_M$) for lateral-torsional buckling, and support for high-strength steel grades up to S700.

4.3 EN 1995 & 1996 (Timber & Masonry)

Timber checks include the k_mod factor to account for load duration and service class. Masonry verification uses the slenderness reduction factor Φ calculated according to Annex G of EN 1996-1-1.

4.4 EN 1994 (Composite)

Composite steel-concrete beam checking evaluates the plastic moment capacity based on plastic neutral axis depth, vertical shear connections via shear studs, and minimum longitudinal reinforcement checking.

4.5 EN 1997 (Geotechnical)

Integrated geotechnical checks include pad footing bearing capacity and sliding resistance (ULS GEO) per Design Approaches DA1, DA2, and DA3, alongside settlement estimations using Schmertmann's serviceability limit state (SLS) method.

5. Dynamic Analysis

Natural frequencies are determined by solving the eigenvalue problem:

(K - ω²M) · φ = 0

Where M is the consistent mass matrix. This ensures a more accurate distribution of inertial forces compared to simplified lumped mass models.