Nonlinear Analysis in RF-/CONCRETE

Technical Article

When designing reinforced concrete components according to EN 1992‑1‑1 [1], it is possible to use nonlinear calculation methods to determine internal forces for the ultimate limit state and the serviceability limit state. In this case, the internal forces and deformations are determined with respect to their nonlinear behaviour. The analysis of stresses and strains in cracked state usually provides the deflections, which clearly exceed the linearly determined values.

One of the previous articles explains the general methods for the calculation and modelling of downstand beams, ribs and T‑beams in cracked state. This article describes the design process of a continuous beam made of reinforced concrete. The calculation can be carried out in the add‑on modules CONCRETE and RF‑CONCRETE Members in combination with the licenses for EC2 and RF‑CONCRETE NL.

System and Loading

A continuous beam consists of a rectangular cross‑section of 20/35 cm and the concrete class C30/37.

The permanent loads and traffic loads are organised in three load cases. To determine the design combinations according to EN 1990, the automatic combinatorics for the ultimate limit state and the serviceability limit state (usual design situation) of RFEM/RSTAB are used.

Figure 01 - System and Loading

Linear Calculation of Reinforcement in ULS

First, the reinforcement is determined for the ultimate limit state. The calculation is performed, taking into account the moment redistribution and reduction for the internal forces of the result combination RC1. Furthermore, the following reinforcement parameters are specified:

  • Reinforcement diameter of 16 mm
  • Curtailment of reinforcement for three areas
  • Concrete cover of 30 mm
  • Minimum reinforcement of 2 Ø 12 for the upper and bottom positions
  • Secondary reinforcement for the maximum reinforcement distance of 15 cm with Ø 12

Based on these entries, the program determines a reinforcement concept according to the linear-elastic approach. In Window 3.1, it is possible to check the reinforcement, which is the basis for the nonlinear analysis.

Figure 02 - Window ‘3.1 Provided Longitudinal Reinforcement’ in CONCRETE

Nonlinear Calculation of Crack Widths and Deformations in SLS

The nonlinear calculation of the serviceability limit state is performed for the load combinations LC6 to LC8 (result combinations do not allow for any clear stress‑strain relations). In the nonlinear analysis, the tension stiffening effects should be integrated. For this, the method with the modified characteristic curve for steel according to [2] is applied.

Figure 03 - Window ‘1.1 General Data’ for Serviceability Limit State with Settings for Nonlinear Calculation According to [2]

In addition, the creep and shrinkage effects are considered. These can be set in Window 1.3.

Figure 04 - Window ‘1.3 Cross‑Sections’ with Settings for Creep and Shrinkage

Results

A physical and geometric nonlinear calculation is performed. The iteration of the strain state is done on the cross‑section plane. Based on the distribution of internal forces within the iteration cycle, new current strain‑stress states are always calculated. The convergence is achieved when the state of equilibrium is set.

As expected, the maximum deformations occur in Field 1 for the loading of LC6 (LC1 + 0.5 ⋅ LC2). The crack widths are small.

Figure 05 - Window ‘6.2.3 Serviceability Limit State for Nonlinear Calculation by Member’

The deformation resulting from the nonlinear calculation with regard to the creep effect is significantly greater than the deformation from the pure linear elastic calculation without the creep effect. This is obvious when comparing the deformations.

Figure 06 - Comparison of Deformations

The stiffness diagram shows that a large area of Field 1 is cracked in the serviceability state.

Figure 07 - Distribution of Stiffness Iym ⋅ E

Summary

In comparison with the linear-elastic calculation of reinforced concrete components, the nonlinear stiffness and stress analysis provides deformation values that can be considerably higher when considering the crack formation. This effect can be resolved by using the nonlinear analysis methods implemented in the add‑on modules for structural analysis and design of reinforced concrete structures by Dlubal Software. It is also possible to consider the creep and shrinkage effects here.

Reference

[1]   Eurocode 2: Design of concrete structures - Part 1‑1: General rules and rules for buildings; EN 1992‑1‑1:2004 + AC:2010
[2]   DAfStb. (2003). DAfStb-Heft 525 - Erläuterungen zu DIN 1045‑1. Berlin: Beuth.
[3]   Manual CONCRETE. (2012). Tiefenbach: Dlubal Software. Download.

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RFEM Main Program
RFEM 5.xx

Main Program

Structural engineering software for finite element analysis (FEA) of planar and spatial structural systems consisting of plates, walls, shells, members (beams), solids and contact elements

RSTAB Main Program
RSTAB 8.xx

Main Program

The structural engineering software for design of frame, beam and truss structures, performing linear and nonlinear calculations of internal forces, deformations, and support reactions

RFEM Concrete Structures
RF-CONCRETE 5.xx

Add-on Module

Design of reinforced concrete members and surfaces (plates, walls, planar structures, shells)

RSTAB Concrete Structures
CONCRETE 8.xx

Add-on Module

Linear and nonlinear analysis of reinforced concrete members with reinforcement concept

RFEM Concrete Structures
RF-CONCRETE NL 5.xx

Add-on Module

Physical and geometrical nonlinear calculation of beam and plate structures consisting of reinforced concrete

RFEM Concrete Structures
EC2 for RFEM 5.xx

Module Extension for RFEM

Extension of the modules for reinforced concrete design by the Eurocode 2 design