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E.2 Material Nonlinear Beam Element

E.2.1 Uniaxial Concrete Material Model

To verify the uniaxial concrete model, a hysteresis behavior test was performed for a compressive strength of 27 MPa, a tensile strength of 3 MPa, and an elastic modulus of 25,000 MPa.

Figure E.2.1 Compressive Envelope (UConcreteTest1.inp)

Figure E.2.2 Loading and Reloadign of Mander model and Maekawa model (UConcreteTest1.inp)

Figure E.2.3 Loading and Reloadign of Mander model and Maekawa model (UconcreteTest2.inp)

Figure E.2.4 Loading and Reloadign of Mander model and no tensile strength model (UconcreteTest3.inp)

Figure E.2.5 Loading and Reloadign of Mander model and tensile cut-off model (UconcreteTest4.inp)

UGeneric, Usteel, vonMises 모델을 상용하여 철근용 모델을 검토하였다. Figure E.2.6에서 완전탄소성 철근은 UGeneric을 사용하여 모사할 수 있지만, hardening이 있는 경우 Ugeneric을 사용하면 비현실적이다. 이는 UGeneric이 isotropic hardening만 모사할 수 있기 때문이다.

Figure E.2.6 Perfectly-Plastic Steel and Hardening Steel using UGeneric (USteelTest.inp)

Figure E.2.7은 Menegoto-Pinto 모델을 적용한 Usteel과 kinematic hardening을 도입한 von Mises 모델의 이력곡선을 보여주고 있다.

Figure E.2.7 USteel and vonMises material

Input file

  • UConcreteTest1.inp: Compression side – Mander, Hognestad, and CEB models; Tension side – Maekawa model + plastic behavior; Loading in compression direction, including loading and reloading tests.

  • UConcreteTest2.inp: Compression side – Mander, Hognestad, and CEB models; Tension side – Maekawa model + damage elastic behavior; Loading and reloading tests.

  • UConcreteTest3.inp: Compression side – Mander, Hognestad, and CEB models; Tension side – no strength, damage elastic behavior; Loading and reloading tests.

  • UConcreteTest4.inp: Compression side – Mander, Hognestad, and CEB models; Tension side – cut-off type damage elastic behavior; Loading and reloading tests.

  • USteelTest.inp: Steel rebar test with UGeneric, Usteel, vonMises

E.2.2 Cyclic Loaded Bar with Material Softening

An analysis is performed for the case where two bars with different cross-sections and material properties are connected, as shown in the figure. The left bar uses a material model with material softening, while the right bar is modeled as elastic. This problem is intended to verify the validity of the theoretical approach for determining the internal forces of fiber elements (element state determination).

Figure E.2.6 Steel Bar with Material Softening

Figure E.2.7 Analysis Result ; (a) Loading History, (b) Response at the End Node

Input file

  • softbar.inp

E.2.3 Nonlinear Analysis of Steel Cantilever (Elasto-perfect plastic material)

An analysis was performed on a steel cantilever beam with elastic–plastic material properties. First, the case with an elasto–perfectly plastic material was analyzed and compared with the analytical solution. For a steel cantilever beam with a rectangular cross-section, an analysis was conducted under bending loads using an elastic–plastic bilinear material model (Figure E.12.1). This example illustrates the difference between the force-based formulation and the displacement-based formulation. In the case of the force-based formulation, accurate results close to the analytical solution can be obtained with just one element, whereas for the displacement-based element, approximately 20 elements are required to achieve accurate results.

Figure E.2.8 Analysis Model

The cross-section was discretized into 40 fibers.

Figure E.2.9 Analysis Results According to Element Type and Number

Input file

  • cant-beam2d-1.inp: B2D2H, 1 element

  • cant-beam2d-10.inp: B2D2H, 10 elements

  • cant-beam3d-1.inp: B3D2H, 1 element

  • cant-beam3d-10.inp: B3D2H, 10 elements

  • cant-beam2d-1-rebar.inp: B2D2H, 1 element + Tendon elements (end layers replaced with Tendon elements)

  • cant-beam2d-10-rebar.inp: B2D2H, 10 elements + Tendon elements (end layers replaced with Tendon elements)

  • cant-beam3d-1-rebar.inp: B3D2H, 1 element + Tendon elements (end layers replaced with Tendon elements)

  • cant-beam3d-10-rebar.inp: B3D2H, 10 elements + Tendon elements (end layers replaced with Tendon elements)

  • cant-beam3d-10-ec.inp: B3D2H, 10 elements + Truss elements + Embedded Constraint applied (end layers modeled with Truss elements and connected via embedded constraint)

E.2.4 Nonlinear Analysis of Steel Cantilever (Elasto-plastic material)

An analysis is performed for the model in Section E.2.3 using a material model with slight hardening.

Figure E.2.10 Analysis of a Cantilever with a Hardening Material

Input file

  • cant-beam2d-10-h.inp : B2D2H, 10 elements

  • cant-beam3d-10-h.inp : B3D2H, 10 elements

References

  1. F. Taucer, E. Spacone and F. C. Filippou, "A fiber beam-column element for seismic response analysis of reinforced concrete structures", p.86

  2. Wane-Jang Lin, "Modern computational environments for seismic analysis of highway bridge structures", p.89

  3. J. M. Gere and S. P. Timoshenko, "Mechanics of Materials", 3rd editition, p.519.