[★★★☆☆] Reinforced Concrete Section Analysis

In this page, the analysis of a rectangular reinforced concrete section of a 2D beam is performed to compute the full plastic moment. This is achieved by using the SingleSection2D element. There is no need to create a larger model.

The model can be downloaded. rc-section-analysis.supan

Section Definition

The section configuration is shown as follows.

Model Development

The SingleSection2D element is NOT a connector element. Only one node is required to define the element. First, we define an arbitrary node.

node 1 0 0

For material models, we use a simple concrete model that adopts Tsai's backbone ConcreteTsai and the MPF steel model.

material ConcreteTsai 1 30. 3. 2. 2. 2. 2. .2 2E-3 1E-4
material MPF 2 2E5 400 .01

With the above definition, we have \(f_c=30~\text{MPa}\), \(f_t=3~\text{MPa}\), \(\varepsilon_c=0.002\), \(\varepsilon_t=0.0001\), \(E=200~\text{GPa}\) and \(f_y=400~\text{MPa}\). For detailed material definitions, please refer to the corresponding pages.

Now we define a rectangular concrete section with the dimension of \(400~\text{mm}\times500~\text{mm}\) and nine integration points along section height. Since it is a 2D section, it is meaningless to define multiple integration points along \(z\) axis. All 2D sections only use 1D integration schemes along \(y\) axis.

section Rectangle2D 2 400. 500. 1 9

Now define some rebars. The eccentricities are \(\pm220~\text{mm}\) and \(0~\text{mm}\).

section Bar2D 3 900. 2 220.
section Bar2D 4 900. 2 -220.
section Bar2D 5 600. 2 0.

To combine those independent sections into a whole, we use the Fibre2D section. It is a wrapper that wraps all valid sections into one piece. Accordingly, a SingleSection2D element can be defined.

section Fibre2D 1 2 3 4 5
element SingleSection2D 1 1 1

Before defining steps, we first create two recorders to record nodal reactions and displacements.

hdf5recorder 1 Node RF 1
hdf5recorder 2 Node U 1

If the axial deformation shall be suppressed, the first DoF needs to be restrained. Here, instead of doing that, we apply an axial force of \(10\%\) section capacity, which is \(600~\text{kN}\).

step static 1
set ini_step_size 1E-1
set fixed_step_size 1
set symm_mat 0

cload 1 0 -6E5 1 1

converger AbsIncreDisp 1 1E-10 20 1

Now in the second step, a rotation of \(10^{-4}\) is applied on the second DoF.

step static 2
set ini_step_size 1E-2
set fixed_step_size 1
set symm_mat 0

displacement 2 0 1E-4 2 1

converger AbsIncreDisp 2 1E-10 20 1

Result

Perform the analysis, the rotation versus moment can be plotted. The maximum moment under such a loading configuration is about \(350~\text{kNm}\).

Users with a relative background may help to justify the result.

Asymmetric Layout

If the layout is asymmetric, say, for example, the rebars at \(y=-220~\text{mm}\) are removed.

section Rectangle2D 2 400. 500. 1 9
section Bar2D 3 900. 2 220.
# section Bar2D 4 900. 2 -220.
section Bar2D 5 600. 2 0.

In this case, a positive moment makes the unreinforced region in tension. This decreases the moment capacity. However, a negative moment does not change the moment capacity significantly.

Interested readers can try to apply both positive and negative rotation and verify the results.