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Stiffness of reinforced concrete slabs subjected to torsion

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Abstract

The study of torsion in reinforced concrete members began in the early twentieth century but it has become more widespread in the past 60 years. Even today, this topic has not been deeply researched with respect to slabs. This work reports a study on the behavior of reinforced concrete slabs subjected to torsion, with particular attention to the stiffness in state I and state II (uncracked and cracked phase) and to the relation T–θ (torsional moment-twist). It is known that torsional forces lead to a much larger loss of stiffness in state II, due to the strong cracking, than the loss of flexural stiffness. When the member is predominantly subjected to bending moments the K I/K II ≈ 3–5 is usually admitted, but when torsion has a preponderant role this relationship is not valid. This study may alert the scientific community to a seemingly unknown relation: torsional stiffness of reinforced concrete slabs, in the cracked phase, is about 1/17–1/15 of the stiffness in the elastic phase. For the analysis of slab deformation, particularly under service conditions, accurate knowledge of the slab stiffness in state II is important. The knowledge of T–θ relation is also important to perform a numerical nonlinear analysis. An experimental program was designed and carried out in which 9 slabs were tested until failure. To submit the slabs predominantly to torsional moments, vertical movement at 3 corners was restricted and the action was applied to the fourth corner. Based on the results obtained, the following relations were defined: load–displacement (P–d), torsional moment-twist (T–θ) and K T,I/K T,II.

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Abbreviations

d :

Vertical displacement

\( f_{\text{c}}^{\prime } \) :

Concrete compressive strength

f cm :

Average concrete compressive strength

h :

Wall thickness of the hollow section

m xu :

Flexural resistance in the x direction

m yu :

Flexural resistance in the y direction

m x , m y :

Flexural design moments in the reinforcement directions (x and y)

m xy :

Torsional design moment

n :

Equal to E s/E c

s :

Longitudinal distance between stirrups

t :

Average thickness of the slab

x,y :

Minor and major dimension of the plain section, respectively

x 1, y 1 :

Shorter and the longer sides of the rectangular area defined by the centre lines of the stirrups, respectively

A :

Area bounded by the centre line of the wall of the hollow section and is equal to x 1 . y 1

A c :

Equal to x . y

A l :

Steel reinforcement cross section area

A t :

Stirrup (transversal reinforcement) cross section area

C :

Stiffness factor computed from St. Venant’s theory: for rectangular plain sections C = β · x 3 y and for rectangular hollow sections C = 4 · A 2  · h/u

E c :

Young’s modulus of concrete

E s :

Young’s modulus of steel

K :

Correcting factor and is almost 0.7

K T,I :

Torsional stiffness in state I

K T,II :

Torsional stiffness in state II

L :

Length of the beam or slab span; and diagonal length of the slab

T c :

Torque level supported by concrete, \( \frac{{x^{2} \cdot y}}{3} \cdot 2,4\sqrt {f_{\text{c}}^{'} } \) (\( f_{\text{c}}^{\prime } \) in psi, x and y in in)

T cr :

Cracking torque (torsional moment applied when the first crack appears)

T y :

Yielding torque (torsional moment applied when the steel start yield)

(GC)I :

Torsional stiffness of stage I, G (shear modulus) is E c/[2 · (1 + ν)]

(GC)II :

Torsional stiffness at stage II

α :

Deformation parameter considered, which may be a strain, a curvature, or a rotation

α I, α II :

Values of the parameter calculated for the uncracked and fully cracked conditions respectively

β:

St. Venant’s coefficient

η :

Equal to 0.57 + 2.86 · h/x, for plain sections η = 2 since h/x = 0.5

ρl :

Longitudinal reinforcement ratio and is equal to A l/A c

ρt :

Transversal reinforcement ratio and is equal to A t . u/A c

θ :

Twist

θ Icr :

Twist corresponding to T cr (twist when the first crack appears)

μcr :

Torsional cracking moment

μy :

Torsional yielding moment

ν:

Poisson coefficient and is 0.2, in state I

U :

Perimeter of area defined by the centre line

ζ :

Is a distribution coefficient, assumes a value between 0 (uncracked sections) and 1 (full cracked sections), and allows to evaluate the tension stiffening effect

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Acknowledgments

The authors would like to express their gratitude to the Department of Civil Engineering of the University of Coimbra (FCTUC) for providing the conditions to carry out this study, especially Pedro Miguel Gonçalves, Rita Carina Menoita, Rui Manuel Pinto, Paulo César Rodrigues and Pedro Filipe Ferreira.

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Authors and Affiliations

  1. Department of Civil Engineering, F.C.T.U.C, Polo II, University of Coimbra, 3030-290, Coimbra, Portugal

    Adelino V. Lopes & Sérgio M. R. Lopes

  2. Department of Civil Engineering, Instituto Superior de Engenharia de Coimbra, Rua Pedro Nunes, Quinta da Nora, 3030-199, Coimbra, Portugal

    Ricardo N. F. do Carmo

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  1. Adelino V. Lopes
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  2. Sérgio M. R. Lopes
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  3. Ricardo N. F. do Carmo
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Correspondence to Ricardo N. F. do Carmo.

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Lopes, A.V., Lopes, S.M.R. & do Carmo, R.N.F. Stiffness of reinforced concrete slabs subjected to torsion. Mater Struct 47, 227–238 (2014). https://doi.org/10.1617/s11527-013-0057-x

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