Practically applicable models for rotation capacity

Link zur deutschen Version: Praxistaugliche Bestimmung des Rotationsvermögens

One focus of my research was the deformation capacity of structures loaded predominantly in bending, where the central question was how to assess whether a structural member possesses sufficient ductility. This is of fundamental importance for the redistribution of internal forces in statically indeterminate structures and for accommodating residual stress states. Both the potential for moment redistribution and the ability to accommodate residual stresses are restricted if the ductility of a member is limited because only small cross-sectional curvatures, and hence insufficient rotation capacity, develop in the plastic hinge prior to failure.

The verification of deformation capacity is typically carried out by comparing the rotation capacity at potential plastic hinge locations with the rotation demand. Note that residual stress states must be accounted for when determining the rotation demand, as they can increase it. In this blog post, I start by explaining some underlying principles for determining of rotation capacity:

I explain how high concrete compression zone depths and low reinforcement ductility reduce the rotation capacity and how this is covered in current codes.
Then I present two general concepts for determining rotation capacity based on curvatures.

Next, I compare practically applicable methods for determining the rotation capacity of reinforced concrete cross-sections:

I compare the method taught in our chair’s “Advanced Structural Concrete” lecture (originally by Sigrist and Marti) side-by-side with the new method in the EN 1992-1-1:2023, providing details for their practical use, and
compare the calculated rotation capacities with data from over 100 experiments.

Influence of the concrete compression zone depth and the ductility class of the reinforcement

The attainable cross-section curvatures are strongly influenced by the strains in the reinforcement: As the steel strain increases, cracks open further, allowing rotations to develop in the plastic hinge region. Figure 1 shows that for small relative compression zone depths $x/d$ , failure occurs by rupture of the reinforcement (blue). If the ultimate strain of the reinforcing steel is small (red), the achievable steel strains, and thus the cross-sectional curvatures, are restricted. This results in a low rotation capacity $\theta_{plu}.$

For large relative compression zone depths (green), the rotation capacity decreases as $x/d$ increases. As the depth of the concrete compression zone increases, the concrete failure criterion is reached at progressively smaller steel strains. Consequently, the achievable curvatures and the associated rotation capacity are reduced.

Figure 1: Rotation capacity $\theta_{plu}$ as a function of the relative compression zone depth $x/d$ ( $\varepsilon_{cu}$ = ultimate concrete strain, $\varepsilon_{ud}$ = ultimate steel strain)

These relationships are reflected in both the Swisscode and the Eurocode: SIA 262:2025 requires the verification of deformation capacity for the redistribution of internal forces in statically indeterminate systems when $x/d$ > 0.35 or when the reinforcing steel does not belong to ductility classes B500B or B500C. Sufficient deformation capacity is also required to neglect residual stress states, as is commonly done in practice. In the new Eurocode EN 1992-1-1:2023, such a verification is already required for $x/d$ > 0.25 when plastic analysis methods are applied.

General concepts for modelling rotation capacity

The rotation capacity can be calculated by integrating the mean curvature profile over the plastic length $L_y$ . The plastic length is defined as the length over which plastic deformations occur (see Figure 2).

(1) $\begin{equation*}\theta_{plu}=\int_{x_{gov}-L_y/2}^ {x_{gov}+L_y/2} \left( \chi_{m,u}(x) - \chi_{m,y}(x) \right) dx \end{equation*}$

Note: In this blog post, the symbol $x$ is used both for the depth of the compression zone and for the coordinate along the longitudinal axis of the structural element. The respective meaning is given by the figure captions or the context.

In a simplified manner, the rotation capacity can be determined by multiplying an idealised plastic hinge length $L_{pl,id}$ by the mean curvature in the governing cross-section.

(2) $\begin{equation*} \theta_{plu}=L_{pl,id} \left( \chi_{m,u}(x_{gov}) - \chi_{m,y}(x_{gov}) \right) \end{equation*}$

This has the advantage that the curvature profile does not need to be known. However, the idealised plastic hinge length must then be defined. By comparing the two approaches (Figure 2), it becomes clear that $L_{pl,id}$ can be interpreted as the product of an integration factor, which depends on the curvature distribution, and the plastic length $L_y$ . $L_{pl,id}$ is generally determined empirically by comparing calculated and experimentally measured rotation capacities.

Figure 2: (a) Zone with plastic deformations; (b) Mean curvature as a function of the coordinate $x$ ( $x$ = 0 at the intermediate support = plastic hinge location).

In both approaches, the mean curvatures at the onset of yielding (subscript $y$ ) are subtracted from those at failure (subscript $u$ ), since only the plastic contribution is considered. In both approaches, the mean curvature over a crack element must be used, as tension stiffening significantly influences rotation capacity. If tension stiffening is not taken into account, the rotation capacity may be overestimated by a factor > 2. The next section explains in more detail how tension stiffening can be incorporated in practice through application of the Tension Chord Model.

Comparison of Two Calculation Methods

This section presents two methods that follow the principle of Equation (2) and are suitable for hand calculations:

the method proposed by Sigrist and Marti (see the paper, the publicly available course Structural Behaviour of Reinforced Concrete (Chapter 9.3, German), or the lecture course Advanced Structural Concrete (Chapter 2.3)); and
the new method according to EN 1992-1-1:2023.

\theta_{plu} = d \left( \min \left( \dfrac{\varepsilon_{smu}}{d-x}, \dfrac{\varepsilon_{cmu}}{x} \right) - \dfrac{\varepsilon_{smy}}{d-x} \right)

\begin{align*}\theta_{plu} = & \dfrac{1}{\gamma_\theta} \cdot 1.3d \biggl(TS_{Mu} \min \left(\dfrac{\varepsilon_{ud}}{d-x_u}, \dfrac{\varepsilon_{cu,d,\rho_w}}{x_u} \right) \\ & - TS_{My} \dfrac{\varepsilon_{yd}}{d-x } \biggr) \quad \text{Equation (7.18)}\end{align*}

The expressions for the mean curvature at failure, $\chi_{m,u}(x_{gov})$ , distinguish between failure due to rupture of the reinforcement (the first term is smaller than the second term, blue curve in Figure 1) and failure due to crushing of the concrete compression zone (the second term is smaller than the first term, green curve in Figure 2). Failure due to concrete crushing is modelled using simplified failure criteria:

\varepsilon_{cu,d,\rho_w} = 0.002 + \dfrac{1.35}{d}+3\rho_w \leq 0.015

Tension stiffening is accounted for with the Tension Chord Model in both methods:

Sigrist und Marti

EN 1992-1-1:2023

$\varepsilon_{smu}$ and $\varepsilon_{smy}$ can be determined using formulas (101)-(103) in the dissertation of M. Alvarez (assumption: bilinear constitutive relationship for the reinforcement).
To determine $\varepsilon_{smu}$ , the steel stress can be set to $\sigma_{s,max} = f_{td}$ ; to determine $\varepsilon_{smy}$ , it can be set to $\sigma_{s,max} = f_{yd}$ .
The crack spacing is assumed to be $s_{rm} = \dfrac{Ø f_{ct} (1-\rho_{eq})}{2 \tau_{b0}\rho_{eq}}$ with nominal bond shear stresses $\tau_{b0}= 2f_{ct}$ und $\tau_{b1}= f_{ct}$ according to the Tension Chord Model.
These formulas are also implemented in our group’s teaching application “Tension Chord Model”.
The equivalent reinforcement ratio can be determined as:

$\begin{align*} \rho_{eq} & = \left( \dfrac{M_r (d-x) E_s}{f_{ct} EI^{II}} + 1 - n \right)^{-1} \\& =\left( \dfrac{M_r }{A_s (d-x/3) f_{ct}} + 1 - n \right)^{-1}\end{align*}$

( $n=E_s/E_c$ , $M_r$ = cracking moment)

The Tension Chord Model was used to define the factors $TS_{Mu}$ and $TS_{My}$ .
To determine these factors, the crack spacing according to EN 1992-1-1:2023 and the steel stress $f_{s,ef}$ and strain $\varepsilon_{ud,ef}$ at the bending resistance, accounting for strain hardening (bilinear constitutive relationship), need to be determined first. In the case of a failure by rupture of the reinforcement, the latter two can be set to $f_{s,ef} = f_{td}$ and $\varepsilon_{ud,ef} = \varepsilon_{ud}$ .
In the case of a failure by concrete crushing, the steel stress $f_{s,ef}$ depends on $\varepsilon_{cu,d,\rho_w}$ and on the depth of the concrete compression zone. The steel stress $f_{s,ef}$ and strain $\varepsilon_{ud,ef}$ can be determined iteratively, assuming a linear strain distribution in the cracked cross-section.

Comparison with experimental data

Figure 3 compares the experimentally measured rotation capacity $\theta_{plu,exp}$ (obtained from tests on continuous (a)-(c) and simply supported (d)-(f) beams and slab strips) with the calculated rotation capacity $\theta_{plu,calc}$ , according to Sigrist and Marti (a), (d), EN 1992-1-1:2023 (b), (e), and EN 1992-1-1:2023 with shear reinforcement neglected ( $\rho_w=0$ , see previous section) (c), (f). Failure due to reinforcement rupture is indicated by filled symbols, while failure due to concrete crushing is indicated by empty symbols.

Figure 3: Comparison of experimental and calculated rotation capacity: (a)-(c) Experiments on continuous beams and slab strips; (d)-(f) Experiments on simply supported beams and slab strips.

Figure 3 shows that the method proposed by Sigrist and Marti is generally conservative. The method according to EN 1992-1-1:2023 is also conservative for failure due to reinforcement rupture, but in many cases it is not conservative for failure due to concrete crushing. This can be partially remedied by neglecting the shear reinforcement when calculating the concrete compressive strain (Figure 3 (c), (f)).

TL;DR

If required, verification of deformation capacity can be carried out by comparing the rotation demand with the rotation capacity of the plastic hinge.
The rotation capacity can be determined in a simplified manner by multiplying an idealised plastic hinge length by the mean plastic curvature in the governing cross-section.
For this purpose, either the method proposed by Sigrist and Marti or the method according to EN 1992-1-1:2023 (using $\rho_w = 0$ in Equation (7.20)) may be applied.

Sigrist und Marti	EN 1992-1-1:2023
$\theta_{plu} = d \left( \min \left( \dfrac{\varepsilon_{smu}}{d-x}, \dfrac{\varepsilon_{cmu}}{x} \right) - \dfrac{\varepsilon_{smy}}{d-x} \right)$	$\begin{align}\theta_{plu} = & \dfrac{1}{\gamma_\theta} \cdot 1.3d \biggl(TS_{Mu} \min \left(\dfrac{\varepsilon_{ud}}{d-x_u}, \dfrac{\varepsilon_{cu,d,\rho_w}}{x_u} \right) \\ & - TS_{My} \dfrac{\varepsilon_{yd}}{d-x } \biggr) \quad \text{Equation (7.18)}\end{align}$
Thus: idealised plastic hinge length $L_{pl,id} = d$	$L_{pl,id} = 1.3 d$
Mean curvature at failure $\chi_{m,u}(x_{gov}) = \min \left( \dfrac{\varepsilon_{smu}}{d-x}, \dfrac{\varepsilon_{cmu}}{x} \right)$	$\chi_{m,u}(x_{gov}) = TS_{Mu} \min \left(\dfrac{\varepsilon_{ud}}{d-x_u}, \dfrac{\varepsilon_{cu,d,\rho_w}}{x_u}\right)$
Mean curvature at the onset of yielding $\chi_{m,y}(x_{gov}) = \dfrac{\varepsilon_{smy}}{d-x}$	$\chi_{m,y}(x_{gov}) = TS_{My} \dfrac{\varepsilon_{yd}}{d-x }$

Sigrist und Marti	EN 1992-1-1:2023
$\varepsilon_{cmu} = 0.0035$	$\varepsilon_{cu,d,\rho_w} = 0.002 + \dfrac{1.35}{d}+3\rho_w \leq 0.015$ In these expressions, $d$ must be entered in mm. Recommendation: Set the shear reinforcement ratio to $\rho_w = 0$ , even if shear reinforcement is present, as otherwise the rotation capacity is overestimated in many cases (see Figure 3)

Influence of the concrete compression zone depth and the ductility class of the reinforcement

General concepts for modelling rotation capacity

Comparison of Two Calculation Methods

Comparison with experimental data

TL;DR

Further reading