🏗️ Understanding Axial Force Restraint in Post-Tensioned Slabs: A Practical FEM-Based Approach for Ground Floor Analysis

Introduction

In post-tensioned (PT) concrete slab design, one subtle yet critical phenomenon is the axial restraint that develops at the ground floor level. When tendons are stressed, compressive forces are induced into the slab — but if the slab edges are restrained (by columns, walls, cores), axial forces accumulate.

If not carefully evaluated, these restraint effects can:

• Cause unexpected slab cracking,

• Generate additional moments,

• Compromise serviceability and long-term durability.

Thus, modeling axial restraint is essential for safe and efficient PT slab design — especially at the ground floor, where boundary restraint is significant.

Scope of the Study

In this study, we focused on:

• Modeling the lateral restraint effect due to ground floor columns against post-tensioned slab shortening.

• Simplifying the problem into a 2D FEM frame:

  • The slab was represented as axially loaded beam elements.

  • Columns were modeled as vertical springs offering lateral restraint through bending stiffness.

The goal:✅ To simulate how the post-tensioning axial force distributes along the slab,✅ How columns resist and redistribute the compression,✅ And to evaluate the resulting axial force and moment profiles.

Our Modeling Approach

To capture the essence of the phenomenon, we developed a custom FEM solver tailored for this problem.

Key Modeling Assumptions:

Item	Approach
Slab (Beam)	Modeled with axial and bending stiffness
Columns	Modeled as vertical cantilevers offering lateral bending stiffness (no vertical DOF needed)
Nodes	Represent column-slab intersections
Degrees of Freedom	Only 2 per node: Horizontal translation (uₓ) and Rotation (θ_z)
Loading	Pure axial compression applied at both slab ends

Why We Eliminated Vertical DOFs

• In pure axial shortening analysis, vertical displacements are irrelevant.

• Eliminating vertical DOFs makes the system simpler, faster to solve, and focused purely on lateral effects.

  
  

Finite Element Model Construction

• Beam elements connect adjacent column nodes, carrying both axial force and bending moments.

• Column elements at each node provide lateral stiffness through their flexural resistance.

• The system is restrained naturally by column stiffness — no artificial boundary conditions were applied.

  
  

We solved:

K⋅U=F

• K = Assembled global stiffness matrix (beam + column contributions)

• U = Displacement vector (horizontal translation and rotation at nodes)

• F = Global force vector (compressive forces at slab ends)

Results Achieved

✅ Symmetric Axial Force Distribution

• With symmetric span lengths, column sizes, and loading, the axial force distribution along the slab was symmetric.

• Axial force smoothly decreased from end to center due to column restraint.

  
  

✅ Column Shear Forces and Moments

• Each column resisted part of the slab compression by developing:

  • Lateral shear force (balancing the shortening force),

  • Bending moment at the column top.

The columns near the ends resisted more shear and moment, while center columns resisted less — matching structural intuition.

Force distribution along the span without end column release

Force distribution along span with end column release (Note that forces at both end columns are zero)

✅ Clear Visualization

• Axial force distribution along the slab plotted cleanly (compression positive upward).

• Column shear and moment bar charts plotted for quick understanding.

All outputs confirmed the correctness of the model and the physical realism of the setup.

Key Technical Highlights

Feature	Description
DOFs per Node	2 (uₓ, θ_z)
Vertical DOFs	Eliminated
Frame Model	Slab + Columns combined naturally
Boundary Condition	No artificial supports; columns restrain naturally
Solver	Full K×U=F direct solve
Postprocessing	Axial force, shear force, and moment extraction

Insights Gained

• Columns play a crucial role in restraining post-tensioned slab shortening at ground floor.

• Even without vertical displacement freedom, the frame can resist loads by axial shortening and bending.

• Removing unnecessary DOFs simplified the system without loss of accuracy.

• Consistent stiffness modeling (both beam and column) is essential for symmetric and realistic output.

Conclusion

Through careful FEM modeling:

• We successfully captured the restraint mechanism of ground floor columns resisting slab axial shortening.

• We developed a simple but powerful FEM solver that is fully transparent, expandable, and based on first principles.

• This work serves as a foundation for future developments in:

  • More advanced PT slab restraint analysis,

  • Nonlinear behavior,

  • Progressive tendon stressing simulation,

  • Cracking prediction under restraint.