Planar Frame Analysis Using the Finite Element Method

Implementation of the Newton Method to Analyze a Plane Frame Structure with Nonlinear, Hyperelastic Material

Robert J Firman III1, Nathaniel H McNichols1

1 Graduate Student, School of Civil Engineering, Structural GroupPurdue University West Lafayette, IN

AbstractFinite element analysis was performed on a planar frame structure with nonlinear, hyperelastic material to determine the effect that certain parameters have on the overall load bearing capacity and deformation of the structure. To simplify the solution process, members were designed as planar beams rigidly connected at the joints and fixed at the base. Each element was subdivided into either three or four nodes and polynomial Lagrangian finite element interpolation functions were implemented. Matlab was then utilized to analyze the structure by means of Newtons Method and the material properties presented. Many different loading cases and material sets were explored but only a few are presented herein. A first-order analysis was also performed in a structural analysis program in order to confirm the results obtained by the authors implementing Newtons Method.

Based on the relationships determined between stress and strain and varying parameters ADD RESULTS

IntroductionThe analysis of complex structural systems is often done by using computer programs that are able to perform iterative calculations quickly. Before these programs can be utilized, several steps need to be taken to ensure that the output from the analysis is accurate based on the properties of the structural system. For simplicity, this article references the analysis of a simple plane frame structure shown in Figure 1 but the concepts can be applied to more complex systems. A nonlinear, hyperelastic material is also used with the constitutive law shown in Equation 1.

Figure 1: Planar frame used for analysis with possible loading scenario shown.

(1)

Planar Beam TheoryThe first step in understanding the solution process is to model each member as a planar beam element. Using the Kinematic Hypothesis, the assumption is made that plane sections remain plane even after deformation. From here, a deformation map is ascribed to the displacement and rotation of each element. Once this map is found, the strains can be computed from the strain tensor shown in Equation 2. It should be noted that the strain tensor is not linearized because the elements being modeled do not stay in the elastic region of deformation (as can be seen from the nonlinearity of the constitutive law).

(2)

The constitutive relations imply stresses through the use of strains, and using these, the resultant tractions on each face of the element are found. The relationship between stress and strain can be seen in Equation 3 as calculated from Equation 1. These constitutive relations and corresponding tractions are used in conjunction with the general equilibrium equations to find the equations for stress resultants. The stress resultants are functions of material properties (Youngs Modulus and Poissons ratio) as well as member cross-sectional dimensions and are shown in Equations 4-6. The values from these equations refer to the normal stress, shear stress, and in-plane bending moment respectively. When forces are applied to an element, the resultant stresses describe how the element will react.

(3)

where:

C = Youngs Modulus = Poissons Ratio

(4)

(5)

(6)

where:d = depth of section in out-of-plane direction (x2)h = height of section in vertical direction (x1)

Boundary conditions must also be considered. In the same way as before, tractions and equilibrium equations can be used to describe displacements. If these displacements are known, either a prescribed force or a reaction force can be used to ensure that the displacement is satisfied. In other words, if two different elements share a common node, the displacements and rotations of that node must be the same for each member.

Although they dont come up in this project, there are also some limitations to the beam theory that should be noted. The first is that there are inconsistencies relating to equilibrium. The theory considers equilibrium over the cross section only in an average sense. This means that locally, equilibrium may not be satisfied. This problem can be fixed by introducing a shear coefficient, which changes the cross sectional area so that when added, the beam theory will yield better results. This is often necessary when considering the effects of shear deformations. The plane surface assumption from the Kinematic Hypothesis can also cause issues with stiffness. For most cross sectional shapes, out-of-plane warping must accompany displacement and rotation of the cross section in order to satisfy traction conditions on the surface. This problem can be solved by multiplying the shear modulus by the torsional stiffness, as opposed to using the shear modulus and the polar moment of the cross sectional area. Since only a planar frame was used in this study, these corrections were not implemented.

Newtons MethodNonlinear problems such as the one presented here cannot be solved directly. For this reason, Newtons Method is used to iteratively solve a linearized version of a nonlinear equation. A specific virtual-work functional is considered in terms of the load parameter, displacement field, and arbitrary virtual displacement field. The principal of virtual work suggests that if this functional is equal to zero for all boundary conditions and virtual displacements, equilibrium holds at the given load level. The goal is to find parameters that allow this to happen. The virtual work equation is linear within incremental changes in displacement field and load level. Different increments of these parameters are taken until the functional is equal to zero or is close to a specified tolerance. The displacement fields introduced are discussed in greater detail in the next section.

Lagrangian Finite Element Interpolation FunctionsEach planar beam element of the frame was discretized into smaller sections in order to more accurately represent the behavior. This is done primarily to be able to enforce the principals of virtual work along the length of each element. Essentially, each element is divided into several elements with additional nodes along the length. Doing this allows for displacements, rotations, and stresses to be computed at points along the length of each element and not just at the element ends. Lagrangian polynomial functions were then used to estimate the displacements and rotations along the member length. For this study, two different discretizations were used, three nodes per element and four nodes per element.

For the case considering three nodes per element, quadratic functions were used to estimate the displacements and rotations. The equations corresponding to this case can be seen in Figure 2. Similarly, for four nodes per element, cubic functions were used to estimate the displacements and rotations and these equations can be seen in Figure 3. The equations shown in these two figures were determined based on assuming the displacement or rotation at any node is equal to one at that node and the other components are equal to zero.

Figure 2: Quadratic Lagrangian finite element interpolation functions.

Figure 3: Cubic Lagrangian finite element interpolation functions.

Matlab ProgrammingIn order to implement Newtons Method for various scenarios relating to the planar frame structure, a Matlab program was created. This program requires the user to input material properties, member dimensions, and loading. The program then processes this information based on the inputs and member orientations in order to formulate a stiffness matrix that can be solved iteratively with Newtons Method. Within the processing of the user defined information, Equations 4-6 are used to compute stresses and the equations shown in Figures 2-3 are used to approximate the displacement field. The initial program was developed by Dr. Ghadir Haikal at Purdue University and was modified by the authors to incorporate the specifics of the problem studied.

In addition to the simple implementation of the program in Matlab, a version of the program that is more user-friendly was created to allow for faster, less-cumbersome analysis and visual outputs. This was accomplished by using Mastan2, an interactive structural analysis program that operates in conjunction with Matlab. Mastan allows for user-defined analysis to be applied to structural systems that are created within its graphical user interface. The initial program was altered so that it was able to function within Mastan but only for the simplistic planar frame structure and with an understanding of the element discretization. By using Matlab to solve Newtons Method for the hyperelastic planar frame structure, the results can easily be compared to those from a first-order linear elastic analysis that is inherent in Mastan. This will prove to be important in understanding the accuracy of the results from Newtons method and will be discussed in greater detail later.

Analysis and ResultsStructural Systems StudiedThe planar frame system was varied with many different loading cases, material properties, and dimensions. The inclusion of all of these scenarios that were explored is beyond the scope of this study. However, some of the general results from these additional examples are discussed briefly in the next section.

The specific cases shown include three loading cases with two sets of material properties. Each of these initial frames was modeled with one element per member and with either three or four nodes per element, for a total of twelve different scenarios. During the analysis process, it was determined that using only one element per member did not produce reasonable results. For this reason, the third loading case was expanded to include as many as five and ten elements per member.

The first loading case included a compression point load on the top of each column. The second loading case was the same as the first but with varied dimensions of the frame. The third loading case included point loads at the midpoint of the beam and the left column. This case was designed to create a non-uniform loading scenario. The loading cases are shown in Figure 4. The material properties and other information for each scenario are shown in Tables 1-2.

Case 1Case 3

Case 2

Figure 4: Specific loading cases presented in this study.

Table 1: Loading cases used in analysis.Loading caseCase 1Case 2 Case 3

Frame

Height (in)120120120

Length (in)120480120

Table 2: Material properties used in analysis.Material setSet 1Set 2

Beam

Modulus (ksi)2900014500

Poisson's0.30.2

Depth (in)1010

Width (in)510

Column

Modulus (ksi)2900029000

Poisson's0.30.3

Depth (in)1010

Width (in)55

Summary of ResultsAs stated, there were many different loading cases and material sets that were explored during analysis. The overall results from each of these cases were as expected. For example, increasing the stiffness by means of material properties or cross-sectional dimensions decreases the displacements of the loaded structure. Similarly, by applying greater loads, displacements increased as expected. Different loading patterns also produced expected results. By loading the columns in pure compression, there was very little to no displacement in horizontal direction. Also, symmetric loading cases produced symmetric results. When loading the frame with more than one individual load, the results were similar to the combined loading had the individual loads been applied separately.

The dimensions of the frame also impacted the results. For taller, slender frames the displaced shape was dominated by bending deformations. Conversely, for shorter, deep frames the displaced shape was dominated by shear deformations. This was observed by varying material properties for each case and determining the overall displacements when considering the contributions from bending and shear separately. Problems arose when applying distributed loads to elements. With distributed loads, the solution process and a simplistic version Newtons Method was unable to converge to a solution within the maximum number of iterations specified. One way to correct this issue is to include an arc-length constraint and solve Newtons Method by incrementing the distributed load. Although this can be done and was explored, no distributed load cases are shown in detail in the following sections.

The selected results shown are only for the first material set presented previously. This is merely done for simplicity, as the results for different material properties are as expected.

Selected Results from Sample CasesThe deflected shapes from implementing the program for each loading case are shown in Figures 5-7. Figure 5 compares using three or four nodes per element with only one element per member. This was done to determine which yields a more accurate response. As can be seen in Cases 2 and 3, when using four nodes per element, the force was divided equally between the two middle nodes. For this reason, the deflected shapes using three and four nodes per element are slightly different.

It should also be noted that the use of the cubic interpolation functions leads to more flexible results. This is evident mainly in Figure 5, Case 3. Although the loading pattern is slightly different, the displacements of the structure appear to have a much different curvature. The four nodes per element example has several exaggerated curves along the length of the members. This suggests that the interpolation functions are working as intended but they may actually be too flexible for this loading case.

Expanding upon this, Figure 6 looks at using five or ten elements per member while also varying between three or four nodes per element. Shown are only the results considering Case 3 and material set 1. The reason for expanding the number of elements per member was to allow for a more accurate solution that could more closely represent the actual displacements of the structure. Although it is possible to increase the number of nodes per member to achieve this instead, there is a limit on the number of nodes per member where the resulting displacements become inaccurate. This is due largely to the increased flexibility and curvature that would exist by using higher order interpolation functions as briefly mentioned previously.

Verification of ResultsIn order to ensure that the results obtained from implementing the solution method outlined in this project were reasonable, a built-in analysis within Mastan was performed for each case for comparison. Figure 7 shows the results obtained from a first-order elastic analysis performed in Mastan. Although the material properties for this analysis were linear and not identical to the properties used previously, the resulting displacements, reactions, and member stress should be similar.

Appendix C shows a comparison of the outputs, beyond simply the deflected shape, of the two different analyses to be acceptably close to one another.

Case 1, 3 nodes per elementCase 1, 4 nodes per element

Case 2, 3 nodes per element

Case 2, 4 nodes per element

Case 3, 3 nodes per elementCase 3, 4 nodes per element

Figure 5: Deflected shape for Cases 1-3 with varied number of nodes per element.

3 nodes per element, 5 elements per member4 nodes per element, 5 elements per member

3 nodes per element, 10 elements per member4 nodes per element, 10 elements per member

Figure 6: Deflected shape for Case 3 with 5 or 10 elements per member.

Case 1, Material 1Case 1, Material 2

Case 2, Material 1

Case 2, Material 2

Case 3, Material 1Case 3, Material 2

Figure 7: Deflected shape from first-order, elastic analysis from Mastan used for comparison.

ConclusionsThe purpose of this study was to develop a computer program that accurately describes the responses of a frame with nonlinear hyperelastic material by understanding the fundamental mechanics that describe the material and relate strains, stresses, and displacement. Once the program was developed and the resulting outputs were determined satisfactory, many loading cases were explored but only a few of these cases were presented. From the cases shown previously and an overall understanding of the results from other cases, several conclusions can be made.

First, there are constraints placed on the type of loading that can be modeled based solely on the number of nodes per element and elements per member. Consider the scenario with one element per member. The three nodes per element allows for a load to be placed at the midspan, whereas the four nodes per element does not allow for this. It is easy to see how complications can arise when attempting to analyze a structure when the loading does not conform to the node locations. For this reason, it is suggested that the number of nodes per element and elements per member be designed not only to achieve the most accurate displaced shape but also to best represent the actual loading scenario.

Related to the previous idea, the solution becomes more accurate by adding more nodes per element or by adding more elements per member. This is shown by comparing Figure 6 to the results shown for Case 3 in Figure 7. The deflected shape appears to more accurately represent the one shown in Figure 7 as the number of nodes per element and elements per member increases. However, there is a limit on this level of accuracy. The ten elements per member is very similar to the five elements per member. Similarly, as previously stated, increasing the number of nodes per element may not be the best solution for all loading cases because it could potentially introduce an erroneous flexibility.

Aspect ratio also becomes important as illustrated in load case two presented previously. The aspect ratio refers to the ratio between the lengths of each element in different members. Because the beam length in case two is four times the column height, the displaced shape is much less accurate for the beam. An aspect ratio close to unity is ideal and the results become less accurate as the aspect ratio diverges in either direction.

The results and conclusions presented represent the understanding of applying the Newton Method to a plane frame structure with nonlinear, hyperelastic material. The formulation used to determine displacements, stresses, and reactions is concurrent with the planar beam theory presented and the results are similar to those obtained from performing an analysis with a readily available software program.

References:1. Hjelmstad, Keith D. Fundamentals of Structural Mechanics. Second ed. New York, NY: Springer, 2005. Print.

11Firman & McNicholsDecember 10, 2010