E.1 Beam Element

E.1.1 Cantilever

To verify the B2D2H, B2D2MH, B3D2H, and B3D2MH elements provided in Hyfeast, analyses were conducted on a cantilever beam, as shown in the figure, subjected to an end load, uniform distributed load, triangular distributed load, and temperature load. In addition, the boundary condition was changed to a simply supported configuration to examine the natural frequencies. The material was steel with an elastic modulus of 200 GPa, Poisson’s ratio of 0.3, and density of 7800 kg/m³. The member was discretized into either one or four elements, and the results were compared.

Figure E.1.1 Analysis Model

Figure E.1.2 Analytic solution (Friedman and Kosmatka (1993))

The solutions for the end load, uniform distributed load, and triangular distributed load are presented in Figure E.1.1. For a Bernoulli beam, the coefficient \(\small \phi\) is zero, while for a Timoshenko beam, it is given as follows.

\[ \small \begin{array}{r} \phi = \frac{12EI}{kGAL^{2}}\tag{E.1.1} \end{array} \]

For rectangular sections, a shear correction factor of 1/1.2 = 0.83333 is commonly used; however, when considering the effect of Poisson’s ratio, it is given as follows (Friedman and Kosmatka, 1993).

\[ \small \begin{array}{r} k = \frac{10(1 + \nu)}{12 + 11\nu}\tag{E.1.2} \end{array} \]

Table E.1.1 Analysis Results under Different Load Conditions

Load	Analytic solution			Analysis Results in Hyfeast
	Bernoulli	Timoshenko		1 element		4 elements
		Conventional k	Exact k	Bernoulli	Timoshenko	Bernoulli	Timoshenko
Tip Load	1.25E-05	1.29E-05	1.29E-05	1.2500E-05	1.2890E-05	1.2500E-05	1.2890E-05
Uniform Load	4.69E-05	4.88E-05	4.88E-05	4.6875E-05	4.8825E-05	4.6875E-05	4.8825E-05
Triangular Load	1.25E-05	1.32E-05	1.31E-05	1.2153E-05	1.2803E-05	1.2499E-05	1.3149E-05

For the case of a temperature load, the calculated displacements are 0.001 in the axial direction and 0.05 in the vertical direction, regardless of the element formulation.

The natural frequencies of a simply supported beam were compared between the Bernoulli beam and the Timoshenko beam. For the Timoshenko beam, which accounts for shear deformation, the results are compared with the solutions presented by Friedman and Kosmatka (1993).

Table E.1.2 Analysis Results (Natural Frequencies)

Mode	Analytic solution		4 Timoshenko elements in F&K (1993). Ratio to analytic solution	4 Bernoulli elements		4 Timoshenko elements in Hyfeast
	Bernoulli (Hz)	Timoshenko (Hz)		Frequency (Hz)	Ratio to analytic solution	Frequency (Hz)	Ratio to analytic solution
	1	45.9226505	43.1856605	1.0024	45.1971	0.984201	43.2589	1.001695921
2	183.690602	150.038484	1.0281	173.358	0.94375	153.927	1.025916793
3	413.303855	287.080857	1.0952	369.538	0.894107	313.461	1.091890985
4	734.762408	436.127412	1.5101	658.597	0.89634	658.597	1.510102282
5	1148.06626	589.449365	1.4682	955.606	0.832361	863.391	1.46474159
6	1653.21542	743.859685	1.2714	1364.58	0.82541	934.81	1.256702062

Input File

• E1.inp: Input using one element each of B2D2H, B2D2MH, B3D2H, and B3D2MH
• E4.inp : Input using four elements each of B2D2H, B2D2MH, B3D2H, and B3D2MH

E.1.2 Portal Frame

Analyses were performed for the portal frame shown in the figure under three loading conditions, resulting in five analysis cases. In CASE 4 and CASE 5, the same load condition LC4 was analyzed using BeamDistributedLoad and BeamTractionLoad, respectively. CASE 6 was conducted to examine step-by-step analysis: after performing CASE 1 (self-weight analysis) and maintaining that state, load condition LC1 was applied. Since the system is linear, the solution for CASE 6 is identical to the superposition of the results from CASE 1 and CASE 2. The elements used in the analysis were B2D2H and B3D2H, with either a Rectangle or Mesh as the cross-section cell. Shear deformation was neglected in all cases.

Figure E.1.3 Analysis Model

Table E.1.4 Load Cases

Load case	Load condtions	Remark
Case 1	Self-Weight(LC1)
Case 2	Nodal Loads(LC2)
Case 3	Concentric Load within Element(LC3)
Case 4	Distributed Load within Element (LC4)	*Load, TYPE= BeamDistributed
Case 5	Distributed Load within Element (LC4)	*Load, TYPE=BeamTraction
Case 6	LC 1 → LC 2	Step-by-Step Analysis
Case 7		Frequency Analysis

Table E.1.5 Analyss Result (Case 1)

Result Type	Node	DOF			Remark
		X	Y	RZ
Displacement (m)	2	1.49815E-6	-3.597E-5	-0.000374837
	3	1.49815E-6	-3.597E-5	0.000374837
Reaction (N)	1	2516.89	45322.2	-8386.29
	4	-2516.89	45322.2	8386.29

Table E.1.6 Analyss Result (Case 2

Result Type	Node	DOF			Remark
		X	Y	RZ
Displacement (m)	2	0.0532197	5.09971E-5	-0.00320049
	3	0.0531602	-5.09971E-5	-0.00319454
Reaction (N)	1	-50020	-42837.6	285945
	4	-49980	42837.6	285679

Table E.1.7 Analyss Result (Case 3)

Result Type	Node	DOF			Remark
		X	Y	RZ
Displacement (m)	2	7.4375E-06	-5.95238E-5	-0.00186086
	3	-7.4375E-06	-5.95238E-5	0.00186086
Reaction (N)	1	12495	50000	-41633.3
	4	-12495	50000	41633.3

Table E.1.8 Analyss Result (Case 4 and Case 5)

Result Type	Node	DOF			Remark
		X	Y	RZ
Displacement(m)	2	4.95833E-6	-5.95238E-5	-0.00124058
	3	-4.95833E-6	-5.95238E-5	0.00124058
Reaction(N)	1	8330	50000	-27755.6
	4	-8330	50000	27755.6

Table E.1.9 Analyss Result (Case 6)

Result Type	Node	DOF			Remark
		X	Y	RZ
Displacement(m)	2	0.0532212	1.50271E-5	-0.00357533
	3	0.0531587	-8.69671E-5	-0.00281971
Reaction(N)	1	-47503.1	2484.64	277559
	4	-52496.9	88159.8	294065

Table E.1.10 Analyss Result (Case 7) – Natural Frequencies

Mode 1	Mode 2	Mode 3	Mode 4	Remark
2.78045Hz	83.116Hz	83.1692Hz	117.567Hz

Input File

• portalEB2.inp : : B2D2H + Rectangular Cell

• portalEB3.inp : : B3D2H + Rectangular Cell

• portalDB2.inp : B2D2H + Meshed Cell

• portalDB3.inp : B3D2H + Meshed Cell

E.1.3 Space Frame

The three-dimensional beam elements (B3D2H and B3D2HM) and beam element loads are verified. For the space frame shown in the figure, the solutions and natural frequencies were calculated for three applied loading conditions.

Figure E.1.4 Analysis model

Figure E.1.5 Analysis model viewed in hfVisualzier

Table E.1.11 Analysis Result (Case 1)

Result Type	Node	B3D2H						B3D2MH
		X	Y	Z	RX	RY	RZ	X	Y	Z	RX	RY	RZ
Displacement	2	0.514842	-0.10838	0.000479	0.000592	0.030721	0.032236	0.517226	-0.10887	0.000479	0.000598	0.030847	0.03238
	4	0.514247	0.108377	-0.00048	-0.00059	0.030663	0.032198	0.516631	0.108865	-0.00048	-0.0006	0.030788	0.032342
	6	0.017355	-0.10838	3.06E-05	0.000592	0.001284	0.032236	0.01751	-0.10887	3.07E-05	0.000598	0.001294	0.032332
	8	0.017355	0.108377	-3.1E-05	-0.00059	0.001283	0.032198	0.01751	0.108865	-3.1E-05	-0.0006	0.001294	0.032342
Reaction	1	-48.5501	0.885508	-40.2654	-4.46897	-277.158	-2.925	-48.5438	0.889093	-40.2257	-4.48729	-277.268	-2.93804
	3	-48.5096	-0.88551	40.2654	4.46897	-276.89	-2.92152	-48.5034	-0.88909	40.2257	4.48729	-277	-2.93455
	5	-1.4699	0.885508	-2.57215	-4.46897	-8.78713	-2.925	-1.47609	0.889093	-2.58131	-4.48729	-8.83017	-2.93804
	7	-1.47042	-0.88551	2.57215	4.46897	-8.78887	-2.92152	-1.47661	-0.88909	2.58131	4.48729	-8.8319	-2.93455

Table E.1.12 Analysis Result (Case 2)

Result Type	Node	B3D2H						B3D2MH
		X	Y	Z	RX	RY	RZ	X	Y	Z	RX	RY	RZ
Displacement	2	0.266099	3.2E-05	0.000255	-5E-07	0.016003	-0.17642	0.267368	3.2E-05	0.000255	-5E-07	0.016071	-0.17644
	4	0.265801	-3.2E-05	-0.00025	4.98E-07	0.015973	0.048355	0.26707	-3.2E-05	-0.00025	4.98E-07	0.016041	0.048342
	6	0.266099	-3.2E-05	0.000255	4.98E-07	0.016003	0.176423	0.267368	-3.2E-05	0.000255	4.98E-07	0.016071	0.176436
	8	0.265801	3.2E-05	-0.00025	-5E-07	0.015973	-0.04835	0.26707	3.2E-05	-0.00025	-5E-07	0.016041	-0.04834
Reaction	1	-25.01	-0.00025	-21.4188	0.001275	-142.973	16.008	-25.01	-0.00025	-21.4035	0.001274	-143.049	16.0091
	3	-24.99	0.000248	21.4188	-0.00127	-142.839	-4.38753	-24.99	0.000248	21.4035	-0.00127	-142.916	-4.38636
	5	-25.01	0.000248	-21.4188	-0.00127	-142.973	-16.008	-25.01	0.000248	-21.4035	-0.00127	-142.916	-16.0091
	7	-24.99	-0.00025	21.4188	0.001275	-142.839	4.38753	-24.99	-0.00025	21.4035	0.001274	-142.916	4.38636

Table E.1.13 Analysis Result (Case 3)

Result Type	Node	B3D2H						B3D2MH
		X	Y	Z	RX	RY	RZ	X	Y	Z	RX	RY	RZ
Displacement	2	0.266099	4.8E-05	0.000255	-7.5E-07	0.016003	-0.26464	0.267368	4.8E-05	0.000255	-7.5E-07	0.016071	-0.26465
	4	0.265801	-4.8E-05	-0.00025	7.47E-07	0.015973	0.072532	0.26707	-4.8E-05	-0.00025	7.46E-07	0.016041	0.072513
	6	0.266099	-4.8E-05	0.000255	7.47E-07	0.016003	0.264635	0.267368	-4.8E-05	0.000255	7.46E-07	0.016041	0.264654
	8	0.265801	4.8E-05	-0.00025	-7.5E-07	0.015973	-0.07253	0.26707	4.8E-05	-0.00025	-7.5E-07	0.016041	-0.07251
Reaction	1	-25.01	-0.00037	-21.4188	0.001912	-142.973	24.012	-25.01	-0.00037	-21.4037	0.001912	-143.049	24.0137
	3	-24.99	0.000372	21.4188	-0.00191	-142.839	-6.5813	-24.99	0.000372	21.4037	-0.00191	-142.916	-6.57955
	5	-25.01	0.000372	-21.4188	-0.00191	-142.973	-24.012	-25.01	0.000372	-21.4037	-0.00191	-142.916	-24.0137
	7	-24.99	-0.00037	21.4188	0.001912	-142.839	6.5813	-24.99	-0.00037	21.4037	0.001912	-142.916	6.57955

Table E.1.14 Analysis Result – Natural Frequencies

Mode number	B3D2H	B3D2MH
1	0.685552	0.685408
2	1.11262	1.11236
3	2.30255	2.29755
4	2.83181	2.82592
5	2.96413	2.96413
6	6.06135	6.06101
7	6.29894	6.29773
8	8.69497	8.69498
9	9.00954	9.00956
10	13.1401	13.1419

Input File

• SpaceFrame.inp : B3D2H elements are used for the model

• SpaceFrameS.inp : B3D2MH elements are used for the model

E.1.4 Effect of Mass Formulation on Beam Natural Frequencies

As shown in Figure E.1.6, the natural frequencies of a simply supported three-dimensional beam element were analyzed and compared with theoretical solutions. A simply supported beam has four types of vibration modes: bending about the strong axis, bending about the weak axis, axial mode, and torsional mode. The theoretical solutions for each mode are presented in the figure. Analyses were performed by varying the mass matrix formulation (consistent mass or lumped mass) and the number of elements (1, 4, and 10), and the results for each vibration mode were compared with the theoretical solutions. Tables E.1.15 through E.1.18 present the comparison of analysis results for each vibration mode.

Figure E.1.6 Natural frequencies of simply supported beam

Table E.1.15 Natural Frequencies of Strong-Axis Bending Mode (Global XZ Plane, Local XY Plane)

Mode	Analytic solution	1 element		4 elements		10 elements
		Consistent	Lumped	Consistent	Lumped	Consistent	Lumped
1	2.8771071	3.18589	19.0415	2.87122	2.87623	2.87049	2.87709
2	11.508428	14.4918		11.4489	11.4429	11.4046	11.5071
3	25.893964	23.321		21.2859	21.0141	21.1715	21.128
4	46.033713			25.8331	24.2575	25.3847	25.8765
5	71.927677			49.2791	59.8432	44.4925	45.9199
6	103.57585			67.1494	89.5616	64.0379	62.8638
7	140.97825			76.8024	105.645	68.3648	71.4055
8	184.13485			118.392		96.6521	101.654
9	233.04567			121.979		108.483	103.052
10	287.71071			171.617		129.065	134.924

Table E.1.16 Natural Frequencies of Weak-Axis Bending Mode (Global XY Plane, Local XZ Plane)

Mode	Analytic solution	1 element		4 elements		10 elements
		Consistent	Lumped	Consistent	Lumped	Consistent	Lumped
1	0.9590357	1.06417	19.0415	0.959038	0.958743	0.958796	0.959029
2	3.8361428	4.8726		3.84733	3.8083	3.83262	3.83569
3	8.6313212	23.321		8.76871	8.08584	8.61603	8.62552
4	15.344571	16.9607		21.0141	15.3072	15.3066
5	23.975892			21.2859		21.1715	21.128
6	34.525285			26.8951		23.9173	23.8018
7	46.992749			42.3952		34.4816	33.8845
8	61.378284			63.2384		47.0686	44.9746
9	77.681891			67.1494		61.7799	55.7976
10	95.903569			76.7159		64.0379	62.8638

Table E.1.17 Natural Frequencies of Axial Deformation Mode

Mode	Analytic solution	1 element		4 elements		10 elements
		Consistent	Lumped	Consistent	Lumped	Consistent	Lumped
1	21.149764	23.321	19.0415	21.2859	21.0141	21.1715	21.128
2	63.449291			67.1494	59.8432	64.0379	62.8638
3	105.77482			121.979	89.5616	108.483	103.052
4	148.04835			176.392	105.645	155.57	140.702
5	190.34787					206.278	174.888
6	232.6474					261.212	204.768
7	274.94693					319.842	229.605
8	317.24645					378.975	248.789
9	359.54598					430.669	261.847
10	401.84551					462.145	268.457

Table E.1.18 Natural Frequencies of Torsional Mode

Mode	Analytic solution	1 element		4 elements		10 elements
		Consistent	Lumped	Consistent	Lumped	Consistent	Lumped
1	7.6754318	8.4634	6.91034	7.72487	7.62623	7.68335	7.66757
2	23.026295			24.3692	21.7177	23.24	22.8139
3	38.377159			44.2674	32.5028	39.3697	37.3985
4	53.728022			64.0145	38.3397	56.4578	51.0622
5	69.078886					74.8603	63.4686
6	84.429749					94.7963	74.3122
7	99.780613					116.074	83.3259
8	115.13148					137.534	90.288
9	130.48234					156.294	95.0268
10	145.8332					167.717	97.4257

□ Remark : Derivation of natural frequency

In the absence of external forces, the governing equations and boundary conditions for bending, axial deformation, and torsion of a simply supported beam are as follows:

• Bending mode

\[ \small \begin{array}{r} m\frac{\partial^{2}v}{\partial t^{2}} + EI\frac{\partial^{4}v}{\partial x^{4}}0;\ \ v = v(x,t)\tag{E.1.3a} \end{array} \]

\[ \small \begin{array}{r} v(0,t) = 0,\ \ \ \frac{\partial v}{\partial x}(0,t) = 0,\ \ \ v(L,t) = 0,\ \ \ \frac{\partial v}{\partial x}(L,t) = 0\tag{E.1.3b} \end{array} \]

• Axial deformation mode

\[ \small \begin{array}{r} m\frac{\partial^{2}u}{\partial t^{2}} - EA\frac{\partial^{2}u}{\partial x^{2}}0;\ \ u = u(x,t)\tag{E.1.4a} \end{array} \]

\[ \small \begin{array}{r} u(0,t) = 0,\ \ \ \frac{\partial u}{\partial x}(L,t) = 0\tag{E.1.4b} \end{array} \]

• Torsional mode

\[ \small \begin{array}{r} m_{r}\frac{\partial^{2}\theta}{\partial t^{2}} - GJ\frac{\partial^{2}\theta}{\partial x^{2}}0;\ \ \theta = \theta(x,t)\tag{E.1.5a} \end{array} \]

\[ \small \begin{array}{r} \theta(0,t) = 0,\ \ \ \frac{\partial\theta}{\partial x}(L,t) = 0\tag{E.1.5b} \end{array} \]

Here, \(\small m\) is the mass per unit length (\(\small m = \rho A\)), and \(\small m_{r}\) is the rotary mass per unit length (\(\small m_{r} = \rho I_{p}\), where \(\small I_{p}\) is the polar moment of inertia, \(\small I_{p} = I_{x} + I_{y}\)). By applying separation of variables in the form \(\small u(x,t) = \phi(x)q(t)\) and imposing the boundary conditions, the natural frequencies and mode shapes for each case can be derived as follows:

• Bending mode

\[ \small \begin{array}{r} \omega_{n} = \frac{n^{2}\pi^{2}}{L^{2}}\sqrt{\frac{EI}{m}},\ \ \ \phi_{n}(x) = \sin\frac{n\pi x}{L}\tag{E.1.6} \end{array} \]

• Axial deformation mode

\[ \small \begin{array}{r} \omega_{n} = \frac{(2n - 1)}{2}\frac{\pi}{L}\sqrt{\frac{EA}{m}},\ \ \ \phi_{n}(x) = \sin\frac{2(n - 1)\pi x}{2L}\tag{E.1.7} \end{array} \]

• Torsional mode

\[ \small \begin{array}{r} \omega_{n} = \frac{(2n - 1)}{2}\frac{\pi}{L}\sqrt{\frac{GJ}{m_{r}}},\ \ \ \phi_{n}(x) = \sin\frac{2(n - 1)\pi x}{2L}\tag{E.1.8} \end{array} \]

Input file

• frqbeam1c.inp : 1 element, consistent mass

• frqbeam4c.inp : 4 elements, consistent mass

• frqbeam10c.inp : 10 elements, consistent mass

• frqbeam1l.inp : 1 element, lumped mass

• frqbeam4l.inp : 4 elements, lumped mass

• frqbeam10l.inp : 10 elements, lumped mass

E.1.5 Offset of Beam

As shown in Figure E.1.7, a comparison is made between two modeling approaches for a cantilever: one with the nodes located at the centroid and the other with an offset applied based on the bottom of the cross-section. Three loading conditions are considered, with a length of \(L = 10\,\mathrm{m}\) and a rectangular cross-section of \(0.1\,\mathrm{m} \times 0.4\,\mathrm{m}\). When comparing the two models in the figure, points A and B must satisfy the following relationship.

\[ \small \begin{array}{r} u_{A} + \frac{h}{2}\theta_{A} = u_{B}\tag{E.1.9a} \end{array} \]

\[ \small \begin{array}{r} v_{A} = v_{B}\tag{E.1.9b} \end{array} \]

\[ \small \begin{array}{r} \theta_{A} = \theta_{B}\tag{E.1.9c} \end{array} \]

Figure E.1.7 Cantilever with or without offset

Regardless of the offset coefficient, \(\small u_{A} = 0.00119048\), \(\small v_{A} = 0.0892857\), \(\small \theta_{A} = 0.0178571\), \(\small u_{B} = 0.0047619\), \(\small v_{B} = 0.0892857\), and \(\small \theta_{B} = 0.0178571\). These values satisfy the above equation.

Input file

• Offset-cant1c.inp : 1 element, without offset

• Offset-cant1b.inp : 1 element, with offset

• Offset-cant2c.inp : 2 elements, without offset

• Offset-cant2b.inp : 2 elements, with offset

• Offset-cant10c.inp : 10 elements, without offset

• Offset-cant10b.inp : 10 elements, with offset

For the simply supported beam condition shown in Figure E.1.7, unlike Equation E.1.9, \(\small v_{A} = v_{B}\) and \(\small \theta_{A} = \theta_{B}\), but the condition for the axial displacement is not satisfied. Regardless of the number of elements, the results are \(\small v_{A} = v_{B} = 0.186012\) and \(\small \theta_{A} = \theta_{B} = 0\).

Figure E.1.8 Simply suppoted beam with or without offset

Input file

• Offset-B2D2H-simple2c.inp : 2 B2D2H element, without offset

• Offset-B2D2H-simple2b.inp : 2 B2D2H element, with offset

• Offset-B2D2H-simple10c.inp : 10 B2D2H element, with offset

• Offset-B2D2H-simple10b.inp : 10 B2D2H element, with offset

• Offset-B3D2H-simple2c.inp : 2 B3D2H element, without offset

• Offset-B3D2H-simple2b.inp : 2 B3D2H element, with offset

• Offset-B3D2H-simple10c.inp : 10 B3D2H element, without offset

• Offset-B3D2H-simple10b.inp : 10 B3D2H element, with offset

References

1. Friedman, Z., & Kosmatka, J. B. (1993). An improved two-node Timoshenko beam finite element. Computers & structures, 47(3), 473-481.