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Moment-curvature Relation for a Beam Using Shell Finite Element Model

In many practical applications it is advantageous to use a hierarchical modeling approach. In general, it involves using a higher-order mathematical model (e.g., solids, shells) to obtain an accurate response for a subsystem and then embedding this information into a simplified model (e.g., beams, springs) and using this simplified model for analysis of the entire structure.

In this News, we present an application of this approach to a civil engineering problem. The aim is to model the response of a large structure mainly made of unsymmetric I-beam steel girders with horizontal and vertical stiffeners that undergo elasto-plastic deformation and also experience local buckling of the lower flange and parts of the stiffeners. A simple beam element model can’t capture all these complex behaviors mainly due to the kinematic assumptions of the beam. Hence, we use a detailed MITC4 shell finite element model to study the behavior of a representative cell of the beam (Figure 1) and to establish its moment-curvature response. Later, we can embed this generalized response into the moment-curvature beam element in ADINA for analysis of the entire structure. A similar approach was adopted in the simulation of buckling of bridge braces where shell elements were used to construct the generalized force-displacement response of the beams. Such an approach has several advantages over using shell/solid elements for modeling the entire structure:

• The actual structure usually consists of many such beams so the computational cost of such detailed models can grow very rapidly with the number of beams involved

• The modeling effort involved in constructing such detailed shell/solid models for the entire structure is prohibitive

• The beam representation will provide analysis results that are more readily usable for the design purposes as opposed to those provided by detailed shell/solid models.

Figure 1 shows the schematic of the model. The left and right ends of the shell model are each constrained to the centroid of the corresponding cross-section using rigid links and each are subject to equal and opposite incremental rotations up to 0.02 rad. Note that due to symmetry, we could have used only one half of the model.

In both animations above, the elasto-plastic behavior of the material is taken into account. In the left animation, the displacements are assumed to be small. Hence, no local buckling is detected while in the second analysis (animation on the right) nonlinear kinematics are included and as can be seen, towards the end of loading, the lower flange and some of the stiffeners experience buckling. Note that to highlight local buckling, displacements are magnified in both cases.

The moment-rotation responses of both models are depicted in Figures 2 and 3 below. Figure 2 represents the behavior when kinematic nonlinearity is not taken into account. The response consists of 3 distinct phases — I:linear response (material is still elastic), II:partial plasticity of the section, III:section becomes fully plastic and there is a plateau in the response with small slope due to the hardening.

Figure 3 represents the behavior when kinematic nonlinearity is included. It consists of 4 distinct phases — I, II, and III are similar to the above while IV is where the local buckling happens and the moment bearing capacity of the beam drops as the end rotations increase beyond a critical value.

Figure 2 Moment-rotation relationship when kinematic nonlinearity is not included

Figure 3 Moment-rotation relationship when kinematic nonlinearity is included

Of course, there are other ways for obtaining moment-curvature response of a beam. One can use physical experiments or even use introductory strength of material-type calculations when the local buckling effect is not important and the cross-section is relatively simple.

Another example of such analyses in civil engineering applications is encountered in modeling steel beam/column connections in buildings. There, a detailed finite element model involving solids or shell elements can be used to study the elasto-plastic response of these connections. Later, the result can be used in defining non-linear rotational springs in the frame-element model of the building for further analyses of the entire structure.

The moment-curvature beam element in ADINA provides a powerful tool for modeling slender structures, especially when nonlinear elastic or elasto-plastic response is involved, local buckling is of interest or the cross-section of the beam is rather complex. ADINA Primer Problem 14 gives a tutorial on using moment-curvature beams in ADINA.

Keywords:
Moment-curvature beam, shell model, MITC4, hierarchical modeling, local buckling, elasto-plastic beam, steel structures, bridges, buildings