User:Hip2008/Biaxial

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Reinforced Concrete Biaxial Bending with Axial Force

Reinforced Concrete Biaxial Bending with Axial Force is analyzed on this page, starting with a description of reinforced concrete material, biaxial bending definition and ending with a summary of tools for analysis.

Reinforced Concrete

Reinforced concrete is a structural material used worldwide. It's composed of concrete and steel bars.

Concrete has a good behavior under compression but is week under tension.
As a approximation, concrete characteristic tension resistance is about 1/15 to 1/10 of its characteristic compression resistance.
Eurocode 2 (EN1992-1-1:2004)^[1] relates this two resistances in the following manner:

Characteristic compression resistance: f_ck
Mean tension resistance: fctm=0.30 x fck ^(2/3), if f_ck < 50 MPa (C50/60)
Characteristic tension resistance: fctk(0.05%)=0.70 x fctm

As so, we can relate tension and compression characteristic resistances with the following expression:
fctk(0.05%)=0.21 x fck ^(2/3), if f_ck < 50 MPa (C50/60)

Steel bars have a good tension resistance and a good bound to concrete.

Connecting steel bars and concrete leads to structural elements that have a good behavior under compression and bending.

Steel reinforcement also gives concrete ductility and removes britle behavior if minimum reinforcement area is respected.

Reinforced concrete is usually named RC, in short.

Reinforced Concrete Behavior

RC behavior must be analyzed taking into account concrete behavior, steel behavior and cracking.

Steel has a linear behavior until yielding and a non linear behavior above that point. Steel is usually modeled with a elastic-plastic or bi-linear stress-strain relationship.

Concrete has a non-linear behavior as cracks under small tension stresses which increases his non-linearity.

Taking the above into account, reinforced concrete is a non-linear material and must be analyzed as such.

Biaxial bending

When a beam or a column is subjected to a transversal load it bends. When the load is applied in two orthogonal directions the structural elements suffers bending in two orthogonal directions. This effect is called biaxial bending.

Eurocode 2 (EN1992-1-1:2004)^[2] has the following definition for biaxial bending:

Biaxial bending: simultaneous bending about two principal axes

Biaxial bending with axial force

When a structural element is subjected to an axial force simultaneously with biaxial bending we reach a stress state that is called biaxial bending with axial force.
In short, biaxial bending with axial force is a stress state that has non null axial effort, and two orthogonal bending moments.

Tools for analyze biaxial bending with axial force

The need of tools for biaxial bending with axial force analysis started as soon as the use of reinforced concrete columns.

The non-linear behavior of RC leads this problem to not have an analytic solution.

The first methods developed where based in tables and abacus written with computer help.

Those tables and abacus where also transformed into simplified analytic expressions using numerical approaches.

Eurocode 2 (EN1992-1-1:2004) has a paragraph reserved to this simplified methods^[3].

With personal computer development, commercial software applications where constructed which incorporates solving of biaxial bending with axial effort through fast and accurate methods.

Nowadays, almost every RC commercial software as a routine dedicated to this issue.

The advent of new web page formats allowed online softwares to be developed, and online applications appeared to solve biaxial bending with axial force.

References

^ EN1992-1-1:2004: Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings (Table 3.1)
^ EN1992-1-1:2004: Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings (§5.8.1))
^ EN1992-1-1:2004: Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings (§5.8.9))

External links

Online Biaxial Bending Software Sample

[1] EN1992-1-1:2004: Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings (Table 3.1)

[2] EN1992-1-1:2004: Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings (§5.8.1))

[3] EN1992-1-1:2004: Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings (§5.8.9))

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