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Overview

1 Introduction

2 Theoretical Background

2.4.5.2 Longitudinal, Shear and Torsional Stiffness

Longitudinal, Shear and Torsional Stiffness

The determination of the bending stiffness as the input parameter for the nonlinear calculation is described in previous chapters. The remaining stiffness parameters can be determined as follows.

Longitudinal stiffness

Similar to the procedure for bending, the longitudinal stiffness E ⋅ A is determined from the ratio of the strain ε₀ to the acting axial force. When bending moment and axial force occur at the same time, it is no longer possible to apply this relation directly because this would result in negative stiffnesses in particular areas, provided that the approach is performed consistently. This results from the simplified analysis not considering the shifting of the neutral axis for strain. When performing nonlinear calculations, this axis no longer matches the centroid of the cross-section. Generally, it is possible to take this fact into account by uncoupling the stiffness matrix from the centroid. However, this will result in a direct correlation between moment and axial force in the terms of the stiffness matrix. RF-CONCRETE Members does not consider axis strain due to crack formation or physical nonlinearity.

Looking at the relation between axial force and bending moment, we can see a direct correlation between both stiffness terms. To clarify, imagine a column with a constant compression force: If an increasing moment acts in addition to the axial force, a curvature leading to a displacement of the resulting axial force from the centroid is added to the pure constant strain diagram. Seen from a plastic point of view, the effective area of the resultant force is thus reduced as well, which by necessity leads to larger strains and thereby to decreasing stiffnesses. Therefore, the approximate consideration for an affinity between bending stiffness and strain stiffness in case of bending with axial force is a practical solution.

Shear stiffness

Determining the shear stiffness in detail is very difficult for the design of reinforced concrete structures, and is an endeavor that is barely manageable in regards to various geometry and load arrangements. The beam theory quickly reaches its limits because the bearing capacity should be determined by the truss effect in order to represent the stiffness for moderate shear loading. In the past, such models were used to develop different methods which are generally not or only partially sufficient in their application.

In a simplified method, Pfeiffer [8] reduces the shear stiffness in accordance with the available bending stiffness. Even if this approach seems to be somewhat strange at first, it is the result of a basic idea that is quite simple and plausible. Imagine that bending load and shearing stress are independent values. When looking at the modified loading of moment and axial force, the bending stiffness changes according to the strain and curvature diagram. However, this does not only affect the stiffness in the beam's longitudinal direction, but also in transversal direction used to transfer shear forces.

This approach is meant as an approximation, which assumes a sufficient shear capacity, but does not (or only roughly) determine slanting cracks, increase of tension force, etc. In spite of these simplifications, the method according to Pfeiffer for moderately slender beams can be considered a sufficiently accurate approach. Alternatively, we can also take the linear elastic shear stiffness as a basis for the calculation in RF-CONCRETE Members.

Torsional stiffness

Compared to bending stiffness, torsional stiffness is reduced strongly in case of cracking. On one hand this is positive, as torsion moments from restraint which frequently occur in building construction are almost completely reduced for load increments until failure is reached. On the other hand, there is the so-called equilibrium torsion where the strong decrease of the torsional stiffness may already lead to remarkable torsions in the serviceability state and thus to a reduction of the serviceability.

There are two different approaches for considering torsional stiffness available for the calculation with RF-CONCRETE Members.

Torsional stiffness according to Leonhardt [9]

Torsional stiffness in uncracked sections (state I)

For the torsional stiffness in state I, the program takes into account that the stiffness is reduced by 30 and 35 % until the crack moment is reached. Reasons indicated by Leonhardt are the following: The concrete core escapes the loading and the stresses are displaced to the outside. A micro crack formation is also involved in the reduction to some extent.

${(G_{c} \cdot I_{T} (x))}_{I} = 0.8 \cdot G_{c} \cdot I_{T, 0} (x) as mean value$

where

I_T : torsional constant
G_c : shear modulus

Torsional stiffness in cracked sections (state II)

The torsional stiffness in state II is derived from a spatial truss model. For simplification, we can assume the inclination of the compression strut to be below 45°. According to Leonhardt, this assumption is also true when the ratios of the longitudinal and transverse reinforcement are not equal. Minor strut inclinations result from the equilibrium analysis or from the design assumption, if the reinforcement ratio of the links is less than the one of the longitudinal reinforcement. However, tests showed that the assumed planer inclination of cracks only occurs for high stress.

Tests also showed that the truss model provides a good algorithm for determining the torsional stress for the limit of failure. However, for the serviceability state we can observe that the steel stresses in the shear and longitudinal reinforcement do not reach the values according to the truss analogy even after several load repetitions.

Link inclinations of 90°:

${(G_{c} \cdot I_{T} (x))}_{I I} = \frac{4 E_{c} \cdot A_{k}^{3}}{u_{k}^{2}} \cdot \frac{1}{k_{T} (\frac{1}{μ_{L}} + \frac{1}{μ_{Li}}) + \frac{4 α \cdot A_{k}}{u_{k} \cdot t} \cdot (1 + φ)}$

Link inclinations of 45°:

${(G_{c} \cdot I_{T} (x))}_{I I} = \frac{E_{c} \cdot A_{k}^{2} \cdot t}{u_{k}^{2}} \cdot \frac{1}{\frac{k_{T}}{μ_{Li}} + \frac{α}{4} \cdot (1 + φ)}$

where

Table 2.4
$k_{T} = 1 - \frac{T_{e d} - 0.7 \cdot T_{c r}}{T_{R d, s y} - 0.7 \cdot T_{c r}}$	for compression strut inclination of 90°
$k_{T} = 1 - \frac{T_{E d} - 0.9 \cdot T_{c r}}{T_{R d, s y} - 0.9 \cdot T_{c r}}$	for compression strut inclination of 45°
$μ_{L} = \frac{A_{s l}}{A_{k}}$	longitudinal reinforcement ratio related to kern
$μ_{Li} = \frac{α_{s w} \cdot u_{k}}{A_{k}}$	transverse reinforcement ratio related to kern

$T_{R d, s y} = min \{\begin{cases} A_{s w} / s_{w} \cdot f_{y} \cdot 2 \cdot A_{k} \\ A_{s l} / u_{k} \cdot f_{y} \cdot 2 \cdot A_{k} \end{cases}$

Determination of crack moment for solid cross-section:

Table 2.4
Start:	$f_{c t r 1} = 0.55 \cdot f_{c k}^{2 / 3}$
End:	$f_{c t r 2} = 0.65 \cdot f_{c k}^{2 / 3}$

Determination of crack moment for hollow cross-section:

Table 2.4
Start:	$f_{c t r 1} = 0.45 \cdot f_{c k}^{2 / 3}$
End:	$f_{c t r 2} = 0.55 \cdot f_{c k}^{2 / 3}$

Table 2.4
T_Rd,sy	torsional moment for which steel stress in truss model reaches yield point (torsional moment that can be absorbed)
T_cr	torsional moment for transition to state II (crack moment)
	$min \{\begin{cases} W_{T} \cdot f_{c t r 1} \\ 2 \cdot A_{k} \cdot t \cdot f_{c t r 1} \end{cases}$
A_k	area enclosed by center line of walls
A_sl	cross-sectional area of longitudinal reinforcement
A_sw	cross-sectional area of shear reinforcement
α	ratios of moduli of elasticity E_s / E_c
u_k	perimeter of area A_k
s_w	spacing of links
t	effective thickness of wall
φ	creeping coefficient to consider

A mutual influence of torsional and bending stiffness is not effected.

Global reduction of torsional stiffness

As an alternative, it is possible to calculate with a linear elastic torsional stiffness that is reduced on a percentage basis in the cracked area.

Literature

[8]	Pfeiffer, Uwe. Die nichtlineare Berechnung ebener Rahmen aus Stahl- oder Spannbeton mit Berücksichtigung der durch das Aufreißen bedingten Achsendehnung. Cuviller Verlag, Göttingen, 2004.
[9]	Leonhardt, Fritz. Vorlesungen über Massivbau - Teil 1 bis 4. Springer Verlag, 3. Auflage, 1984

Parent section

2.4.5 Determination of Element Stiffnesses