Abstract
Using strain-induced orthotropic material concepts, a new numerical constitutive model is demonstrated by predicting the ultimate capacity of a unique simple test specimen and two cylindrical shell structures. Only the uniaxial compressive strength of the concrete and the yield stress of the reinforcing steel are needed as input to predict both constitutive behavior and structural strength. The approach is simple, reliable and computationally efficient.
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Abbreviations
- 1, 2:
-
local directions of principal strain in the concrete or the principal material directions of an orthotropic material
- a, b:
-
local directions of the steel reinforcing rods
- A i,j :
-
membrane (extensional) components of shell constitutive matrix
- B i,j :
-
membrane-bending coupling components of shell constitutive matrix
- [B]:
-
strain displacement matrix
- D i,j :
-
bending components of shell constitutive matrix
- {ΔDE}:
-
incremental global nodal displacements induced by {ΔRE}
- E c :
-
modulus of elasticity of the concrete
- E i :
-
modulus of elasticity of an orthotropic material relative to theith direction
- E ij :
-
reduced stiffness of shell constitutive matrix
- E a :
-
modulus of elasticity of the reinforcing steel taken as 200 GPa (2900 ksi)
- [E] ij :
-
tangential constitutive matrix relative to thei, j directions
- f ′ c :
-
uniaxial compressive strength of the concrete
- G ij :
-
shear modulus relative to thei,j coordinate system
- [k] i :
-
tangenital elemental stiffness matrix due to materiali (global coordinates)
- [k]:
-
tangential elemental stiffness matrix due to concrete and steel (global coordinates)
- [K]:
-
global tangential stiffness matrix
- N :
-
number of layers
- p :
-
external hydrostatic pressure
- Δp :
-
increment in load
- {re}:
-
elemental nodal load vector calculated from stress state in element
- {R}:
-
global nodal load vector of externally applied loads
- {RE}:
-
global nodal load vector calculated from the internal stresses in elements
- {ΔRE}:
-
equal to {R}—{RE}
- t k :
-
thickness of thekth shell layer
- u, v, w :
-
global displacements in the x, y, z directions
- V i :
-
elemental volume of materiali
- x, y, z :
-
global coordinate system
- z k :
-
distance from the shell midsurface to the midsurface of thekth shell layer
- ε i :
-
normal strain in theith coordinate direction
- ε ys :
-
uniaxial yield strain of the reinforcing steel
- ν:
-
effective Poisson's ratio for concrete (set equal to 0.2)
- σ i :
-
normal stress in theith coordinate direction
- σ ys :
-
uniaxial yield stress of the reinforcing steel
- {σ}:
-
vector of elemental stresses
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Rule, W.K., Rowlands, R.E. A simple orthotropic elasticity-based constitutive model for reinforced concrete. Experimental Mechanics 29, 448–454 (1989). https://doi.org/10.1007/BF02323866
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DOI: https://doi.org/10.1007/BF02323866