Elastic Beam Column Element with Stiffness Modifiers

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This command is used to construct a ModElasticBeam2d element object. The arguments for the construction of an elastic beam-column element with stiffness modifiers is applicable for 2-D problems. This element should be used for modelling of a structural element with an equivalent combination of one elastic element with stiffness-proportional damping, and two springs at its two ends with no stiffness proportional damping to represent a prismatic section. The modelling technique is based on a number of analytical studies discussed in Zareian and Medina (2010) and Zareian and Krawinkler (2009) and is utilized in order to solve problems related to numerical damping in dynamic analysis of frame structures with concentrated plasticity springs.

element ModElasticBeam2d $eleTag $iNode $jNode $A $E $Iz $K11 $K33 $K44 $transfTag <-mass $massDens> <-cMass>


$eleTag unique element object tag
$iNode $jNode end nodes
$A cross-sectional area of element
$E Young's Modulus
$Iz second moment of area about the local z-axis
$K11 stiffness modifier for translation
$K33 stiffness modifier for translation
$K44 stiffness modifier for rotation
$transfTag identifier for previously-defined coordinate-transformation (CrdTransf) object
$massDens element mass per unit length (optional, default = 0.0)
-cMass to form consistent mass matrix (optional, default = lumped mass matrix)


Element Formation:


For structural elements with time invariant moment gradient, a two-dimensional, prismatic beam element with six degrees of freedom is to be replaced with a two-dimensional, prismatic beam element composed of semi-rigid rotational springs at the ends and an elastic beam element in the middle. The rotational stiffness at the end of the original beam element is Ke = 6EIz/L (where E is the modulus of elasticity, Iz the moment of inertia, and L the length of the beam), and the ratio of the rotational spring stiffness, Ks, to the elastic beam stiffness, Ke, of the modified beam element is defined as n = Ks/Ke.


Elastic Element with 2-end Springs:

1. The elastic element in between the two springs should have an elastic moment of inertia equal to Iz,mod = (n+1)/n * Iz
2. The "n" times stiff rotational springs should have an elastic stiffness of Ks=n*6*EIz,mod/L
3. the elastic element should have an elastic stiffness coefficient K44 = 6*(1+n)/(2+3*n)
4. The elastic element should have an elastic stiffness coefficient K11 = K33 = (1+2*n)*K44/(1+n)
5. The modified stiffness coefficient bmod for stiffness proportional damping of the elastic element is: "bmod" = 1+(1/2n)*b

Elastic Element with 1-end Spring:

1. The elastic element in between the two springs should have an elastic moment of inertia equal to Iz,mod = (n+1)/n * Iz
2. The "n" times stiff rotational springs should have an elastic stiffness of Ks=n*6*EIz,mod/L
3. The elastic element should have an elastic stiffness coefficient K44 = 6*n/( 1+3*n )
4. the elastic element should have an elastic stiffness coefficient K11 = (1+2*n)*K44/(1+n)
5. the elastic element should have an elastic stiffness coefficient K33 = 2*K44
6. The modified stiffness coefficient bmod for stiffness proportional damping of the elastic element is: "bmod" = 1+(1/2n)*b

EXAMPLE:

element ModelasticBeam2d 1 2 4 5.5 100.0 1e6 4.0 4.0 2.0 1; # elastic element tag 1 between nodes 2 and 4 with area 5.5, E 100 and IZ 1e6 with K11=K33=4.0, K44=2.0 which uses transformation 1
Note: For n = 1 rigid spring, the stiffness coefficients of the elastic springs are as follows: K11=K33 = 4.0 and K44 = 2.0

References:

[1] Zareian, F. and Medina, R. A. (2010). “A practical method for proper modeling of structural damping in inelastic plane structural systems,” Computers & Structures, Vol. 88, 1-2, pp. 45-53.
[2] Zareian, F. and Krawinkler, H. (2009). "Simplified performance-based earthquake engineering" Technical Report 169, The John A. Blume Earthquake Engineering Research Center, Department of Civil Engineering, Stanford University, Stanford, CA. [electronic version: https://blume.stanford.edu/tech_reports]

Code Developed by: by Dr. Dimitrios G. Lignos (McGill University)