THEORY
Table of Contents
1. Types of Analyses
1.1 Eigenvalue Analysis
1.2 Linear Static Analysis
1.3 Nonlinear Static Analysis
1.4 Nonlinear Dynamic Analysis
2. Elements
2.1 Linear Elastic Frame Element
2.2 Nonlinear (Flexural) Frame Element
2.3 Nonlinear (Flexure+Shear) Frame Element
2.4 Membrane Finite Element
2.5 Nonlinear Shell Finite Element
3. Materials
3.1 Uniaxial steel and concrete
3.2 Biaxial model for reinforced concrete.
3.3 Models for joint (contact) elements.
4. References
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1. Types of Analysis
1.1 Eigenvalue Analysis.
Calculation of mode shapes and corresponding natural periods. The following eigenvalue problem is solved:
(K-ω2M)Φ=0
where K is the stiffness matrix, M the mass matrix, ω the angular frequency (rad/s) and Φ the modes shapes of the structure. The program also calculates modal participation factors and effective masses for each mode according to:
Γ=ΦTMI/(ΦTMΦ)
m*=(ΦTMI)2/(ΦTMΦ)
where I is the vector of influence coefficients.
1.2 Linear Static Analysis.
Calculation of nodal displacements and element forces from the solution of the static system of equations:
Kd=F
where d is the vector of nodal displacements and F the vector of applied forces. From displacements, element forces can be recovered using the element stiffness matrix.
1.3 Nonlinear Static Analysis (Force Control)
Solves the equilibrium equations, as in the previous case, but for a nonlinear system. The solution is obtained iteratively using the standard Newton-Raphson method (see Figure). The external forces are applied incrementally as defined by the load factor λ. The incremental system of equations is given as:
KtΔd=λF-Fint
where Kt is the tangent stiffness matrix, Δd is the displacement increment, F is now a shape vector defining the incrementally applied forces and Fint the vector of internal forces. Once Δd is obtained, the total displacements can be d updated. Because the program uses a fiber-discretization approach, material strains ε are calculated at each section fiber from d . Stresses are obtained from ε using constitutive models for each corresponding material, and then integrated to obtain section forces, element forces and finally the vector of internal forces of the structure Fint.
1.3 Nonlinear Static Analysis (Displacement Control)
For structures with softening behavior, the above method (Force Control) fails to convergence beyond the post-peak range. The displacement-control iterative method can be used to overcome this problem. In addition to the incremental equation of static equilibrium, a constrain equation is required which is imposed on a selected degree of freedom (d.o.f.):
dc=dimp
where dc is the control d.o.f. , and dimp the imposed value of displacement. The load factor λ is now unknown and has to be determined at each load step from the constrain equation.
1.4 Nonlinear Dynamic Analysis
The nonlinear dynamic system of equations is given as:
Md''+Cd'+Fint=Fext
where C is the damping matrix, d,d' and d'' are the displacement vector and its derivatives (velocity and acceleration). As in the case of linear dynamics, this system can be solved using implicit or explicit integration schemes. In the present case, the program uses the average acceleration Newmark method, whereby the velocity and displacement at time step n+1 are approximated as:
d'n+1=d'n+Δt(γd''n+1+(1-γ)d''n)
dn+1=dn+Δtd'n+1/2Δt2(2βd''n+1+(1-2β)d''n)
where Δt is the time step, and γ=0.5, β=0.25. As in the case of static equilibrium, at each load step additional Newton-Raphson iterations are performed in order to satisfy dynamic equilibrium.
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2. Elements
2.1 Linear Elastic Frame Element
This is the classical elastic frame element with 3 dof per node: two translations and one rotation. The corresponding element forces are the axial force Fx, shear Fy and bending moment Mz. The element is contained in the xy plane, z being the out-of-plane axis. The stiffness matrix is defined from the bending and axial stiffnesses EA and EIz, and the element length L. In the program, this element corresponds to type=1.
2.2 Nonlinear (Flexural) Frame Element
This is actually a displacement-based beam-column finite element, based on the Euler-Bernoulli beam assumption (i.e. plane sections remain plane and orthogonal to the beam axis):
u(x,y)=uo-θzy
vo'=θz
where u(x,y) is the axial cross-section displacement, uo is the axial displacement of the beam axis, θz the cross-section rotation, and v' the derivative of the vertical displacement. Note that the shear strain is zero in this formulation. The element stiffness matrix is found from numerical integration along the element at selected integration points (Gauss Points). Cubic shape functions are used to interpolate the transverse displacement, and linear ones for the axial displacement. The element has the same number of dof as the elastic frame element, i.e. 3 per node. The cross section is discretized into fibers (or layers in 2D). At each layer, the axial strain εx is found from section compatibility and the corresponding stress, σx, from the uniaxial constitutive model:
εx=Bd
σx=f(εx)
Hence there is no need to define sectional properties (EA or EIz) since these are obtained from numerical integration over the cross-section
In the program, this element corresponds to type=2, and it can be used to model beams, columns, slender walls, etc..., dominated by flexure.
2.3 Nonlinear (Flexure+Shear) Frame Element
As opposed to the typical flexural beam element, which is based on the Euler-Bernoulli beam assumption and uniaxial constitutive laws, an enhanced beam-column element was implemented which effectively couples axial and shear stresses. The element is based on the Timoshenko beam assumption (plane sections remain plane but not necessarily orthogonal to the beam axis) and biaxial constitutive laws, namely an orthotropic smeared-crack constitutive model for reinforced concrete.
The shear strain at the cross-section does not vanish, since the section rotation θz and the first derivative of the vertical displacement vo' are not the same:
γoy=vo'-θz
The above value is constant at a given (cross-section) integration point. However, for the purpose of cross-section state determination, a parabolic shear strain distribution is assumed as:
γxy(y)=1.5γoy(1-(2y/h)2)
This element is available in the program under element type=3
2.4 Membrane Finite Element
This is a quadrilateral (4-node) plane membrane finite element with rotational (drilling) dofs. The element has 3 dofs per node, two translations and one rotation, thus being fully compatible with beam elements. This element is originally obtained from a quadratic membrane element with intermediate nodes. Upon imposing the following kinematic link between the intermediate and end nodes, the 4-node element with a rotational dof. is obtained:
(u,v)T=(L-s)/L(ui,vj)T+s/L(ui,vj)T+(L-s)s/(2L)(ωj-ωi)(cosα sinα)T
For the case of elastic material, the response is uniquely characterized by the Young Modulus E and Poisson coefficient ν (or Shear Modulus G):
The element stiffness matrix and vector of internal forces are obtained through numerical integration using a 2x2 Gaussian quadrature rule. This element is available in the program as element type=4.
2.4 Nonlinear Membrane Finite Element
The finite element formulation of this element is analogous to the previous one. The nonlinearity stems from the material behavior. Now at each integration point the material response will change, affecting the element stiffness matrix and vector of internal forces. The linear-elastic stress-strain relationship (shown above) does not apply. Hence a different method is used, which is based on working in the axes of anisotropy (crack directions for reinforced concrete). Details are given in the next section.
2.5 Nonlinear Shell Finite Element
This element is obtained from combination of the membrane element presented before and a plate finite element based on the Kirchhoff kinematic assumption. The plate element is the so colled Discrete Kirchoff Element (DKT), which is obtained after condensation of the intermediate nodes in a six-node plate element.
The finale shell element has four nodes with six degrees of freedom at each node: three translations (ux,uy,uz) and three rotations (θx,θy,θz). The thickness is discretized into several layers, each one subjected to in-plane stress conditions, similar to the membrane element. Different reinforcement ratios can be assigned to the layers, such that top and bottom reinforcement can be defined for slab related problems for example.
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3. Materials
3.1 Uniaxial steel and concrete
For steel, four uniaxial constitutive models are available: elasto-plastic, elastic with kinematic strain hardening, elastic with isotropic strain hardening and the Menegotto-Pinto, which takes into account Baushinger effect. Typical material parameters are the Young Modulus (E), Yield Strength (σy), and strain hardening parameter (b). For the Menegotto-Pinto, additional parameters defining the Baushinger effect are required (R0,a0,a1).
The model used for concrete is the one from Mander, Park and Paulay (1988), which takes into account strength and stiffness degradation, as well as confinement effects due to transverse reinforcement. The latter is defined with a confinement factor (k). These uniaxial models are used at the fiber-level of the nonlinear flexural frame element (type=2). For the Timoshenko element (type=3) and the nonlinear membrane element (type=5) biaxial stress models are used, as explained next.
3.2 Biaxial model for reinforced concrete.
This model effectively couples axial and shear stress components, thus being useful for analyzing members (or regions) under bi-axial stress conditions such as deep-beams, beam-column joints, RC walls, etc. The main idea is to work in the crack-directions (axes of anisotropy), instead of the xy reference system (which greatly simplifies the problem of shear), and to use equivalent uniaxial constitutive laws in these directions. These laws are similar to those previously described, however they are coupled with the orthogonal direction. This approach is referred to as orthotropic, smeared-crack model, and it is analogous to the Compression Field Models, e.g. the Modified Compresion Field Theory (Vecchio and Collins 1986), Softened Truss Models (Hsu and co-workers), among others.
For instance, the response of concrete in compression takes into account softening due to the presence of orthogonal tensile strain (βd), and also effects of transverse confinement in the case of biaxial compression (βσ). The response in tension has to consider the tension-stiffening effect (contribution of concrete between cracks), as well as crack-closing/crack-opening phenomenon. Before cracking, the response in tension is linear. Thereafter, the tension-stiffening branch is given by the following equation proposed by Vecchio and Collins 1986:
σ2=fct/(1+(C1ε2)0.5)
where fct is the concrete tensile strength, and C1 a decay parameter suggested as 500 for large-scale elements and 200 for small-scale elements
An important aspect is that the crack-directions (1-2) are not coincident with the principal stress (or strain) directions, and therefore there will be shear stress/strain components acting on the crack plane. A simple relationship is used in this case, which is based on the shear retention factor βsh:
τ12=βshGγ12
where τ12, γ12 are the shear stress and shear strain acting on the crack, and G is the uncracked shear stiffness. Suggested values for βsh can be found in the literature. An upper limit is set on τ12 due to aggregate interlock failure (Walraven 1980, Vecchio and Collins 1986) :
τ12<vcimax=(fc)0.5/(0.31+24w/(agg+16))
where fc is the concrete compressive strength, w the crack width and agg the maximum aggregate size. The crack width, w, is estimated from the crack spacing, s, and the tensile strain ε2:
w=sε2
Longitudinal and transverse reinforcement is assumed to be smeared. Hence the reinforcement ratios have to be defined as:
ρx=Asx/Ac
ρy=Asy/Ac
Stresses in the reinforcement are determined from the corresponding strains, obtained after transformation of the total strains to the reinforcement directions, and the constitutive models for steel presented in the previous section.
3.3 Models for joint (contact) elements.
Several uniaxial models are available for modeling joints: linear, bilinear symmetric and asymmetric, contact with and without gap, etc..., New models can be easily implemented for specific cases.
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