Verification Using Simple-Geometry Models¶

Elastic Static Analysis¶

This verification uses a cantilever beam meshed as shown in Figure 9.1.1. Seven cases, exA through exG, were analyzed using the loading conditions shown in Figure 9.1.2. Case exG uses the same loading conditions as exA, but with a direct solver.

Verification results for each loading condition are shown in Tables 9.1.1 through 9.1.7.

Figure 9.1.1 Example of a Cantilever-Beam Mesh (Hexahedral Elements)

	(a) exA and exG: Concentrated load
	(b) exD: Gravitational load
	(c) exB: Distributed surface load
	(d) exE: Centrifugal force
	(e) exC: Body load
	(f) exF: Thermal load

Figure 9.1.2 Verification Conditions for the Cantilever-Beam Model

Table 9.1.1 exA: Verification Results for the Concentrated-Load Problem

Table 9.1.2 exB: Verification Results for the Distributed-Surface-Load Problem

Table 9.1.3 exC: Verification Results for the Body-Load Problem

Table 9.1.4 exD: Verification Results for the Gravitational-Load Problem

Table 9.1.5 exE: Verification Results for the Centrifugal-Force Problem

Table 9.1.6 exF: Verification Results for the Thermal-Stress Load Problem

Table 9.1.7 exG: Direct-Solver Verification Results (Concentrated-Load Problem)

Nonlinear Static Analysis¶

(2-1) exnl1: Geometric Nonlinear Analysis¶

The verification model for case exI is identical to the models for cases exA through exG. Figure 9.1.3 shows a schematic of the verification model. Geometric nonlinear analysis is performed on this model. The verification results are shown in Table 9.1.8.

The nonlinear calculation uses a reference load of \(P = 1.0\ \mathrm{kgf}\), with an increment of \(0.1P\) over 10 load steps.

Figure 9.1.3 Verification Model

Table 9.1.8 exI: Verification Results (Maximum-Deflection History)

(2-2) exnl2: Elastoplastic Deformation Analysis¶

This verification problem is based on NAFEMS Test NL1 from the National Agency for Finite Element Methods and Standards (U.K.), and evaluates elastoplastic deformation with geometric nonlinearity and multiple hardening laws. Figure 9.1.4 shows the analysis model.

Figure 9.1.4 Elastoplastic Deformation Analysis Model

(1) Verification Conditions

(2) Boundary Conditions

All nodes not listed here are fully constrained. The theoretical solution for this problem is as follows.

The calculation results are as follows.

Contact Analysis (1)¶

This verification problem is based on contact patch test CGS-4 from the National Agency for Finite Element Methods and Standards (U.K.), and tests the finite-sliding contact capability with friction. Figure 9.1.5 shows the analysis model.

Figure 9.1.5 Contact Analysis Model

The equilibrium condition for this problem is as follows.

In the sticking-friction stage, the friction force is

whereas in the sliding-friction stage, it is

This gives the relation above.

The calculated results are compared with the analytical solution below.

Contact Analysis (2): Hertz Contact Problem¶

This verification analyzes the Hertz contact problem between an infinitely long cylinder and an infinite plane. The cylinder radius is \(R = 8\ \mathrm{mm}\), and the Young's modulus \(E\) and Poisson's ratio \(\mu\) of the deformable body are \(1100\ \mathrm{MPa}\) and \(0.0\), respectively. The contact area is assumed to be sufficiently small relative to the cylinder radius. Accounting for symmetry, the analysis uses a quarter-cylinder model.

Figure 9.1.6 Analysis Model for the Hertz Contact Problem

(1) Verification Results for the Contact Radius¶

The theoretical equation for calculating the contact radius is as follows.

where

This gives the expression above. For the present calculation, when the pressure is \(F=100\), the contact radius is \(a=1.36\).

Figure 9.1.7 shows the equivalent nodal forces at the contact points. The contact radius is obtained by extrapolating this nodal-force distribution.

Figure 9.1.7 Equivalent Nodal-Force Distribution at the Contact Points

(2) Verification Results for the Maximum Shear Stress¶

In the theoretical solution, at the contact position

the maximum shear stress is

This is the theoretical maximum shear stress. For the present calculation conditions,

This is the theoretical value under the present conditions. The calculation, however, gives

This is the calculated result.

Figure 9.1.8 Shear-Stress Distribution (Maximum Value = 15.6)

(3) Modal Analysis¶

The verification models for cases exJ and exK are identical to those for cases exA through exG. Figure 9.1.9 shows a schematic of the verification model. Modal analysis is performed on this model. The first through third natural frequencies are calculated. Case exJ uses an iterative solver, whereas case exK uses a direct solver. The verification results are shown in Tables 9.1.9 through 9.1.12.

Figure 9.1.9 Verification Model

The natural frequencies of the cantilever beam are given by the following equations.

First natural frequency

Second natural frequency

Third natural frequency

The properties of the verification model are

Thus, the first three natural frequencies are as follows.

Table 9.1.9 exJ: Iterative-Solver Verification Results (First Natural Frequency)

Table 9.1.10 exJ: Iterative-Solver Verification Results (Second Natural Frequency)

Note: In three-dimensional models, the first two modes are degenerate; therefore, the third computed natural frequency is reported as the second natural frequency in the table.

Table 9.1.11 exK: Direct-Solver Verification Results (First Natural Frequency)

Table 9.1.12 exK: Direct-Solver Verification Results (Second Natural Frequency)

Note: In three-dimensional models, the first two modes are degenerate; therefore, the third computed natural frequency is reported as the second natural frequency in the table.

(4) Heat Conduction Analysis¶

The common conditions for steady-state heat conduction analysis are shown in Figure 9.1.10. The individual conditions for verification cases exM through exT are shown in Figure 9.1.11. The same mesh as for exA is used.

Tables 9.1.13 through 9.1.20 show the temperature-distribution results for each case.

Temperature Dependence of Thermal Conductivity

Figure 9.1.10 Verification Conditions for Steady-State Heat Conduction Analysis

exM: Linear material
exN: Prescribed-temperature problem
exO: Concentrated heat-flux problem
exP: Distributed heat-flux problem
exQ: Convective heat-transfer problem
exR: Radiative heat-transfer problem
exS: Volumetric heat-generation problem
exT: Internal-gap problem

Figure 9.1.11 Analysis Conditions for Each Verification Case

Table 9.1.13 exM: Verification Results for Steady-State Calculation with a Linear Material

Table 9.1.14 exN: Verification Results for the Prescribed-Temperature Problem

Table 9.1.15 exO: Verification Results for the Concentrated-Heat-Flux Problem

Table 9.1.16 exP: Verification Results for the Distributed-Heat-Flux Problem

Table 9.1.17 exQ: Verification Results for the Convective-Heat-Transfer Problem

Table 9.1.18 exR: Verification Results for the Radiative-Heat-Transfer Problem

Table 9.1.19 exS: Verification Results for the Volumetric-Heat-Generation Problem

Table 9.1.20 exT: Verification Results for the Internal-Gap Problem

Linear Dynamic Analysis¶

In exW, linear dynamic analysis is performed on the same cantilever beam as in Section (1). Figure 9.1.12 shows the verification conditions. The effect of the time increment is verified using the same mesh. Both implicit and explicit methods are used, with element types 361 and 342. The verification results are shown in Table 9.1.22 and Figures 9.1.13 through 9.1.15.

Analysis Model

Time History of External Force F

The theoretical solution for displacement at the excitation point is as follows:

where

Figure 9.1.12 Verification Conditions for Linear Dynamic Analysis

Verification Conditions:

Table 9.1.21 Verification Conditions for Linear Dynamic Analysis (Continued)

Table 9.1.22 exW: Verification Results for Linear Dynamic Analysis of a Cantilever Beam

Figure 9.1.13 Deformed Shape and Equivalent Stress Distribution of the Cantilever Beam

(a) Element Type 361: Implicit Method

(b) Element Type 361: Explicit Method

Figure 9.1.14 Time History of Excitation-Point Displacement \(u_z\)

(a) Element Type 342: Implicit Method

(b) Element Type 342: Explicit Method

Figure 9.1.15 Time History of Excitation-Point Displacement \(u_z\)

Frequency Response Analysis¶

This verification performs frequency response analysis of a cantilever beam and compares the results with those from general-purpose commercial software. The analysis model and verification conditions are shown below.

Analysis Conditions:

Figure 9.1.16 Analysis Model (First-Order Tetrahedral Elements: 126 Elements, 55 Nodes)

The first five natural frequencies obtained by modal analysis, together with the frequency response at the excitation point, are shown below.

Figure 9.1.17 Frequency Dependence of Displacement Magnitude at the Excitation Point