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Influence of joint flexibility on buckling analysis of free–free beams

  • D. V. Ramana Reddy , K. T. Balaram Padal and Jagadish Babu Gunda EMAIL logo
Published/Copyright: April 3, 2023
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Nonlinear Engineering
From the journal Nonlinear Engineering Volume 12 Issue 1

Abstract

In this work, an application of two noded beam finite element methodology, which is demonstrated in the previous research work for vibration analysis of beam with a flexible joint problem, has been further extended here to investigate the buckling behaviour of free–free beam subjected to an in-plane compressive load. Joint is modelled as rotational spring, where the rotational spring stiffness governs the behaviour of the flexible joint. Variation of first five non-dimensional buckling loads of free–free beam with reference to the joint location as well as joint stiffness parameters are briefly presented. It is understood that looseness of the joint can significantly influence the buckling behaviour of free–free beam and plays an important role in accurately determining the buckling behaviour of jointed beams subjected to an in-plane compressive loads.

Keywords: buckling analysis; spring-hinged beam/column; flexible joints; buckling loads; beam/column with rotational springs; free–free beam; influence of flexible joint

Nomenclature

JRC

joint rotation constant (Rad/N-m)

K S

shear stiffness of the flexible joint (N/m)

K T

rotational (or torsional) spring stiffness of the flexible joint ( 1 JRC ) (N-m/Rad)

[ G ] e

element geometric stiffness matrix

[ G ]

global or assembled geometric stiffness matrix

P

applied in-plane compressive load (N)

P L

linear or critical buckling load (N)

ξ

non-dimensional length of the beam ( x L )

λ L

non-dimensional critical buckling load P L L 2 E I

Γ

non-dimensional rotational spring stiffness of the flexible joint K T L E I or L JRCEI

1 Introduction

Buckling behaviour of jointed connections is an active area of the current research [1–18], and research in this area helps the designer in better understanding of the behaviour of real-life structures with complex joints. Chen and Lui [1] provided an analytical procedure for carrying out the analysis of framed structures by considering the flexibility of steel beam-to-column connections. Dado et al. [2] studied the behaviour of fixed-free two segmented column connected by a linear rotational spring with different flexural stiffness and lengths. This problem has been investigated by using a semi-analytical as well as numerical integration approaches, and both of these approaches shows excellent agreement with each other. Secer [3] studied the stability analysis of frames with flexible beam-to-column connections, and these connections are modelled with rotational spring analogy. It is emphasized that assuming flexible connections in steel columns always leads to reduction of the buckling loads due to increased flexibility.

Rao et al. [4] studied the thermal post-buckling and large amplitude free vibration behaviour of beam with spring-hinged ends by utilizing an energy-based approach (Rayleigh-Ritz method), where spring-hinged end connections can be modelled as either hinged or clamped connections or combination of both as limiting cases of rotational spring-stiffness parameter. Gunda and Krishna [5] investigated the effect of joint flexibility (or looseness) on vibration analysis of free–free beams by using a finite element (FE) formulation where the influence of joint location as well as joint stiffness parameters is discussed on variation of first five vibration modes of free–free beam. Accuracy of the proposed formulation is also verified by using a commercially available FE analysis software. Alkhaldi et al. [6] investigated the large deflection behaviour of spring-hinged-free beam by using an analytical approach based on elliptic integral and demonstrated its efficiency with the available literature results. Cao et al. [7] investigated the buckling behaviour of spring hinge ended columns from first principles and derived the effective length factor equation for these columns while providing effective length factor expression as functions of relevant stiffness ratios of the column. Sun et al. [8] studied the post-buckling behaviour of column with spring-hinged ends by solving an integral governing differential equation by making use of the Galerkin method. Accuracy of analytical expressions is demonstrated with the numerical solutions based on shooting method. Phunpaingam and Chucheepsakul [9] investigated the post-buckling behaviour of elastica subjected to an end loading where rotational spring is considered within the span length of the elastica. Numerical results obtained from elliptic integral method and shooting method are in excellent agreement with each other in this work. Batista [10] investigated the influence of spring behaviour on the large deflections and stability of a spring-hinged cantilever subjected to a conservative force. Ryu and Kim [11] highlighted the necessity of considering Timoshenko Beam theory in the problems, of beam vibrations excited by supported motions, even in cases where the beams are slender one. Umakanth et al. [12] investigated the influence of number of studs on variation of joint rotation compliance as well as joint rotation stiffness by means of 3D FE idealization where the specified pre-tightening torque on fasteners is taken into account in the analysis of intersection fasteners. Amadio and Bedon [13,14], Bedon et al. [15], Bedon [16,17], and Santo et al. [18] extensively investigated the vibration (including dynamic characterization) and buckling (lateral torsional buckling) behaviour of flexible monolithic and layered (or laminated) glass structural elements, where the looseness of the supporting restraints is mathematically modelled by rotational springs and contributed to the detailed experimental validation study by considering flexibility of interfaces as well as edge conditions.

In the present work, buckling analysis of free–free beam with a flexible joint has been investigated by using a two-noded beam FE formulation, which was presented by Gunda and Krishna [5], wherein the joint flexibility was primarily represented by means of rotational spring analogy at each node of the FE idealization. Wide range of joint stiffness parameter values are simulated to demonstrate the case from a completely loose joint to a sufficiently stiffer (or rigid) joint. Influence of joint location as well as joint stiffness parameters on first five considered modes from buckling analysis of free–free beam are briefly summarized.

2 FE formulation with a flexible joint

The total potential energy of a slender beam/column (Figure 1) that is subjected to an axial in-plane compressive load ( P ) with a flexible joint located at any reference distance ( x ) ( 0 < x < L ) or ( 0 < ξ < 1 ) along the length (L) of the beam can be written as follows:

(1) U = E I 2 ∫ 0 L d 2 w d x 2 2 d x + 1 2 K T d w d x ∣ x 2 + 1 2 K S w 2 ∣ x − P 2 ∫ 0 L d w d x 2 d x ,

where E I is the flexural stiffness of the beam and K T and K S are the rotational spring stiffness and the shear stiffness of the flexible joint located at any reference distance x along the length of the beam. Figure 2 represents a typical FE idealization of a flexible joint located in between elements I and J.

Figure 1 Beam/column with a flexible joint subjected to in-plane compressive load.
Figure 1

Beam/column with a flexible joint subjected to in-plane compressive load.

Figure 2 FE idealization of a flexible joint between two FEs.
Figure 2

FE idealization of a flexible joint between two FEs.

Minimization of the total potential energy results in a system of simultaneous equations for an element and are expressed as follows:

(2) [ K e ] { δ e } − P [ G e ] { δ e } = 0 .

FE approach detailed by Gunda and Krishna [5] for vibration problem is employed here subsequently for the considered buckling problem of flexible joint. For more details on complete FE formulation, readers are encouraged to refer to ref. [5], and details of the same are omitted here to avoid further repetition.

Subsequent to assembly of element level matrices, the final governing matrix equation for the buckling analysis with a flexible joint located at any reference distance ( x ) is expressed as follows:

(3) [ K ] { δ } − λ L [ G ] { δ } = 0 ,

where

(4) λ L = P L L 2 E I .

3 Results and discussion

In present work, FE formulation discussed by Gunda and Krishna [5] is employed here to study the buckling behaviour of slender, free–free beam with a flexible joint subjected to an in-plane compressive load (P). In this study, joint location ( ξ ) as well as joint rotation stiffness ( K T ) parameters are assumed to vary along the length of the beam in order to understand the beam buckling behaviour with a flexible joint. Variation of first five non-dimensional buckling loads ( λ L ) of a free–free beam with variation in joint location as well as joint rotation stiffness parameters are shown in Figures 3 and 4 and Tables 1, 2, 3, 4, 5. It is understood from Tables 1–5 that for very small values of joint rotational stiffness ( K T ) , the beam acts as loose (or hinged) joint. Subsequent increase in joint rotation stiffness parameter ( K T ) increases the buckling load gradually and attains a constant value with further increase and becomes insensitive with any further variation in joint rotation stiffness parameter ( K T ) . All the first five considered critical buckling loads have shown considerable variation (shown in Tables 1–5) with reference to variation in joint location ( ξ ) as well as joint rotation stiffness ( K T ) parameters. As the flexible joint locates at any nodal point location on the considered buckled mode, its influence on that critical buckling load variation is minimal with reference to variation in joint rotation stiffness parameter ( K T ) . It is understood that the joint location ( ξ ) as well as joint rotation stiffness ( K T ) parameters play an important role in accurately estimating the critical buckling loads of flexibly connected beams.

Figure 3 Influence of joint location as well as joint stiffness on first three (Modes 1–3) buckling modes of uniform slender free–free beam ( L = 20 m L=20\hspace{0.33em}{\rm{m}} , E I = 5.026629 × 1 0 9 N m 2 EI=5.026629\times 1{0}^{9}\hspace{0.33em}{\rm{N}}\hspace{0.33em}{{\rm{m}}}^{2} ).
Figure 3

Influence of joint location as well as joint stiffness on first three (Modes 1–3) buckling modes of uniform slender free–free beam ( L = 20 m , E I = 5.026629 × 1 0 9 N m 2 ).

Figure 4 Influence of joint Location as well as joint stiffness on buckling modes (Modes 4 and 5) of uniform slender free–free beam ( L = 20 m L=20\hspace{0.33em}{\rm{m}} , E I = 5.026629 × 1 0 9 N m 2 EI=5.026629\times 1{0}^{9}\hspace{0.33em}{\rm{N}}\hspace{0.33em}{{\rm{m}}}^{2} ).
Figure 4

Influence of joint Location as well as joint stiffness on buckling modes (Modes 4 and 5) of uniform slender free–free beam ( L = 20 m , E I = 5.026629 × 1 0 9 N m 2 ).

Table 1

Variation of first non-dimensional buckling load of free–free beam with variation in joint location ( ξ ) as well as spring stiffness ( K T ) of the joint (or JRC or Γ variation)

K T → 1 0 6 1 0 7 1 0 8 1 0 9 1 0 10 1 0 15
JRC → 1 0 − 6 1 0 − 7 1 0 − 8 1 0 − 9 1 0 − 10 1 0 − 15
Γ → 0.0039 0.039 0.39 3.9 39 39.78 × 1 0 5
Joint location ( ξ ) ↓ First non-dimensional buckling load ( λ L ) ↓
0.031 0.1313 1.2956 8.6941 9.8184 9.8648 9.8696
0.063 0.0678 0.6698 5.6053 9.6563 9.8505 9.8696
0.094 0.0468 0.4620 3.9984 9.3857 9.8272 9.8696
0.125 0.0363 0.3589 3.1443 9.0326 9.7958 9.8696
0.156 0.0301 0.2978 2.6250 8.6377 9.7578 9.8696
0.188 0.0261 0.2577 2.2802 8.2404 9.7148 9.8696
0.219 0.0233 0.2297 2.0377 7.8682 9.6687 9.8696
0.250 0.0212 0.2094 1.8605 7.5358 9.6215 9.8696
0.281 0.0197 0.1942 1.7278 7.2487 9.5751 9.8696
0.313 0.0185 0.1828 1.6272 7.0072 9.5311 9.8696
0.344 0.0176 0.1741 1.5508 6.8093 9.4913 9.8696
0.375 0.0170 0.1675 1.4934 6.6522 9.4569 9.8696
0.406 0.0165 0.1628 1.4516 6.5331 9.4290 9.8696
0.438 0.0161 0.1596 1.4232 6.4498 9.4086 9.8696
0.469 0.0160 0.1577 1.4066 6.4005 9.3960 9.8696
0.500 0.0159 0.1571 1.4012 6.3842 9.3918 9.8696
0.531 0.0160 0.1577 1.4066 6.4005 9.3960 9.8696
0.563 0.0161 0.1596 1.4232 6.4498 9.4086 9.8696
0.594 0.0165 0.1628 1.4516 6.5331 9.4290 9.8696
0.625 0.0170 0.1675 1.4934 6.6522 9.4569 9.8696
0.656 0.0176 0.1741 1.5508 6.8093 9.4913 9.8696
0.688 0.0185 0.1828 1.6272 7.0071 9.5311 9.8696
0.719 0.0197 0.1942 1.7278 7.2487 9.5751 9.8696
0.750 0.0212 0.2094 1.8605 7.5358 9.6215 9.8696
0.781 0.0233 0.2297 2.0377 7.8682 9.6687 9.8696
0.813 0.0261 0.2577 2.2802 8.2404 9.7148 9.8696
0.844 0.0301 0.2978 2.6250 8.6377 9.7578 9.8696
0.875 0.0363 0.3589 3.1443 9.0326 9.7958 9.8696
0.906 0.0468 0.4620 3.9984 9.3857 9.8272 9.8696
0.938 0.0678 0.6698 5.6053 9.6563 9.8505 9.8696
0.969 0.1312 1.2956 8.6941 9.8184 9.8648 9.8696
Table 2

Variation of second non-dimensional buckling load of free–free beam with variation in joint location ( ξ ) as well as spring stiffness ( K T ) of the joint (or JRC or Γ variation)

K T → 1 0 6 1 0 7 1 0 8 1 0 9 1 0 10 1 0 15
JRC → 1 0 − 6 1 0 − 7 1 0 − 8 1 0 − 9 1 0 − 10 1 0 − 15
Γ → 0.0039 0.039 0.39 3.9 39 39.78 × 1 0 5
Joint location ( ξ ) ↓ Second non-dimensional buckling load ( λ L ) ↓
0.031 10.5249 10.6097 14.9168 38.4577 39.4009 39.4785
0.063 11.2379 11.3191 12.8116 34.9624 39.1749 39.4785
0.094 12.0260 12.1079 13.2456 30.9517 38.8364 39.4785
0.125 12.9000 12.9838 14.0123 28.2706 38.4477 39.4785
0.156 13.8729 13.9592 14.9462 26.8807 38.0785 39.4785
0.188 14.9602 15.0493 16.0235 26.3999 37.7877 39.4785
0.219 16.1805 16.2729 17.2498 26.5517 37.6135 39.4785
0.250 17.5566 17.6525 18.6419 27.1704 37.5727 39.4784
0.281 19.1159 19.2158 20.2248 28.1620 37.6646 39.4784
0.313 20.8927 20.9969 22.0307 29.4736 37.8737 39.4785
0.344 22.9293 23.0382 24.1009 31.0723 38.1730 39.4785
0.375 25.2789 25.3930 26.4868 32.9248 38.5248 39.4785
0.406 28.0092 28.1290 29.2524 34.9665 38.8819 39.4785
0.438 31.2070 31.3329 32.4735 37.0383 39.1917 39.4785
0.469 34.9855 35.1176 36.2079 38.7625 39.4032 39.4785
0.500 39.4785 39.4785 39.4785 39.4785 39.4785 39.4785
0.531 34.9855 35.1176 36.2079 38.7625 39.4032 39.4785
0.563 31.2070 31.3329 32.4735 37.0383 39.1917 39.4785
0.594 28.0092 28.1290 29.2524 34.9665 38.8819 39.4785
0.625 25.2789 25.3930 26.4868 32.9248 38.5248 39.4785
0.656 22.9293 23.0382 24.1009 31.0723 38.1730 39.4785
0.688 20.8927 20.9969 22.0307 29.4736 37.8737 39.4785
0.719 19.1159 19.2158 20.2248 28.1620 37.6646 39.4785
0.750 17.5566 17.6525 18.6419 27.1704 37.5728 39.4785
0.781 16.1805 16.2729 17.2498 26.5517 37.6135 39.4785
0.813 14.9602 15.0493 16.0235 26.3999 37.7877 39.4785
0.844 13.8729 13.9592 14.9462 26.8807 38.0785 39.4785
0.875 12.9000 12.9838 14.0123 28.2706 38.4477 39.4785
0.906 12.0260 12.1079 13.2456 30.9517 38.8364 39.4785
0.938 11.2379 11.3191 12.8116 34.9624 39.1749 39.4785
0.969 10.5249 10.6097 14.9168 38.4577 39.4009 39.4785
Table 3

Variation of third non-dimensional buckling load of free–free beam with variation in joint location ( ξ ) as well as spring stiffness ( K T ) of the joint (or JRC or Γ variation)

K T → 1 0 6 1 0 7 1 0 8 1 0 9 1 0 10 1 0 15
JRC → 1 0 − 6 1 0 − 7 1 0 − 8 1 0 − 9 1 0 − 10 1 0 − 15
Γ → 0.0039 0.039 0.39 3.9 39 39.78 × 1 0 5
Joint location ( ξ ) ↓ Third non-dimensional buckling load ( λ L ) ↓
0.031 42.0748 42.1512 43.2138 81.1606 88.4278 88.8273
0.063 44.9262 45.0037 45.8871 65.9752 87.3277 88.8273
0.094 48.0777 48.1574 49.0112 61.9544 85.9771 88.8273
0.125 51.5729 51.6551 52.5088 62.6707 84.9411 88.8273
0.156 55.4635 55.5486 56.4136 65.1893 84.5448 88.8273
0.188 59.8117 59.8998 60.7819 68.6899 84.8250 88.8273
0.219 64.6919 64.7835 65.6855 72.9007 85.6426 88.8273
0.250 70.1949 70.2900 71.2117 77.6799 86.7635 88.8273
0.281 76.4311 76.5301 77.4628 82.7990 87.8854 88.8273
0.313 83.5368 83.6393 84.5221 87.4343 88.6605 88.8273
0.344 83.5484 83.7531 85.4502 88.3731 88.7841 88.8273
0.375 70.2055 70.3957 72.2245 82.4229 88.1644 88.8273
0.406 59.8214 59.9977 61.7481 74.5461 87.0261 88.8273
0.438 51.5820 51.7463 53.4345 67.9406 85.7943 88.8273
0.469 44.9347 45.0896 46.7996 63.4879 84.8783 88.8273
0.500 39.5103 39.7961 42.5870 61.8680 84.5433 88.8273
0.531 44.9347 45.0896 46.7996 63.4879 84.8783 88.8273
0.563 51.5820 51.7463 53.4345 67.9406 85.7943 88.8273
0.594 59.8214 59.9977 61.7481 74.5461 87.0261 88.8273
0.625 70.2055 70.3957 72.2245 82.4229 88.1644 88.8273
0.656 83.5484 83.7531 85.4502 88.3731 88.7841 88.8273
0.688 83.5368 83.6393 84.5221 87.4343 88.6605 88.8273
0.719 76.4311 76.5301 77.4628 82.7990 87.8854 88.8273
0.750 70.1949 70.2900 71.2117 77.6799 86.7635 88.8273
0.781 64.6919 64.7835 65.6855 72.9007 85.6426 88.8273
0.813 59.8117 59.8998 60.7819 68.6899 84.8250 88.8273
0.844 55.4635 55.5486 56.4136 65.1893 84.5448 88.8273
0.875 51.5729 51.6551 52.5088 62.6707 84.9411 88.8273
0.906 48.0777 48.1574 49.0112 61.9544 85.9771 88.8273
0.938 44.9263 45.0037 45.8871 65.9752 87.3277 88.8273
0.969 42.0748 42.1512 43.2138 81.1606 88.4278 88.8273
Table 4

Variation of fourth non-dimensional buckling load of free–free beam with variation in joint location ( ξ ) as well as spring stiffness ( K T ) of the joint (or JRC or Γ variation)

K T → 1 0 6 1 0 7 1 0 8 1 0 9 1 0 10 1 0 15
JRC → 1 0 − 6 1 0 − 7 1 0 − 8 1 0 − 9 1 0 − 10 1 0 − 15
Γ → 0.0039 0.039 0.39 3.9 39 39.78 × 1 0 5
Joint location ( ξ ) ↓ Fourth non-dimensional buckling load ( λ L ) ↓
0.031 94.6589 94.7339 95.5908 126.5076 156.6304 157.9187
0.063 101.0745 101.1514 101.9608 114.6079 153.4677 157.9187
0.094 108.1653 108.2445 109.0554 118.4985 150.8819 157.9187
0.125 116.0293 116.1112 116.9349 125.1162 150.4213 157.9187
0.156 124.7832 124.8680 125.7087 133.0828 151.8816 157.9187
0.188 134.5666 134.6545 135.5112 142.0474 154.3874 157.9187
0.219 145.5474 145.6384 146.4953 151.5001 156.8326 157.9187
0.250 157.9187 157.9187 157.9187 157.9187 157.9187 157.9187
0.281 124.8020 125.0564 127.5601 144.7827 156.7074 157.9187
0.313 101.0915 101.3219 103.7131 125.5058 153.8214 157.9187
0.344 91.6818 91.7942 93.2355 115.1450 151.1981 157.9187
0.375 101.0788 101.1940 102.4031 116.9186 150.3379 157.9187
0.406 111.9983 112.1191 113.3412 125.2402 151.5387 157.9187
0.438 124.7879 124.9151 126.1738 136.5309 154.1253 157.9187
0.469 139.9005 140.0347 141.3232 149.3979 156.7741 157.9187
0.500 157.9187 157.9187 157.9187 157.9187 157.9187 157.9187
0.531 139.9005 140.0347 141.3232 149.3979 156.7741 157.9187
0.563 124.7879 124.9151 126.1738 136.5309 154.1253 157.9187
0.594 111.9983 112.1191 113.3412 125.2402 151.5387 157.9187
0.625 101.0788 101.1940 102.4031 116.9186 150.3379 157.9187
0.656 91.6818 91.7943 93.2355 115.1450 151.1981 157.9187
0.688 101.0915 101.3219 103.7131 125.5058 153.8214 157.9187
0.719 124.8021 125.0564 127.5601 144.7827 156.7074 157.9187
0.750 157.9187 157.9187 157.9187 157.9187 157.9187 157.9187
0.781 145.5474 145.6384 146.4953 151.5001 156.8326 157.9187
0.813 134.5666 134.6545 135.5112 142.0474 154.3874 157.9187
0.844 124.7832 124.8680 125.7087 133.0828 151.8816 157.9187
0.875 116.0293 116.1112 116.9349 125.1162 150.4213 157.9187
0.906 108.1653 108.2445 109.0554 118.4985 150.8819 157.9187
0.938 101.0745 101.1514 101.9608 114.6079 153.4677 157.9187
0.969 94.6589 94.7339 95.5908 126.5076 156.6304 157.9187
Table 5

Variation of fifth non-dimensional buckling load of free–free beam with variation in joint location ( ξ ) as well as spring stiffness ( K T ) of the joint (or JRC or Γ variation)

K T → 1 0 6 1 0 7 1 0 8 1 0 9 1 0 10 1 0 15
JRC → 1 0 − 6 1 0 − 7 1 0 − 8 1 0 − 9 1 0 − 10 1 0 − 15
Γ → 0.0039 0.039 0.39 3.9 39 39.78 × 1 0 5
Joint location ( ξ ) ↓ Fifth non-dimensional buckling load ( λ L ) ↓
0.031 168.2803 168.3548 169.1550 186.9389 243.5506 246.7595
0.063 179.6866 179.7632 180.5489 190.2948 237.1667 246.7595
0.094 192.2933 192.3724 193.1684 201.3403 234.9812 246.7595
0.125 206.2750 206.3568 207.1685 214.4934 237.6257 246.7595
0.156 221.8392 221.9238 222.7485 229.1087 242.4403 246.7595
0.188 239.2340 239.3212 240.1084 243.6991 246.3096 246.7596
0.219 206.3023 206.6293 209.8548 232.3513 245.6521 246.7596
0.250 157.9612 158.3429 162.1280 193.4916 240.1238 246.7595
0.281 171.9614 172.0616 173.1115 187.4031 235.4701 246.7595
0.313 187.9505 188.0548 189.1093 200.0221 235.7244 246.7595
0.344 206.2781 206.3871 207.4685 216.9532 239.9943 246.7595
0.375 227.4236 227.5378 228.6365 236.0216 245.0209 246.7596
0.406 239.2438 239.4180 240.9679 245.5290 246.6396 246.7596
0.438 206.2841 206.4477 208.0674 221.5080 242.9242 246.7595
0.469 179.6951 179.8483 181.4144 197.7898 237.4211 246.7595
0.500 157.9506 158.2369 161.0821 185.6115 234.9451 246.7595
0.531 179.6951 179.8483 181.4144 197.7898 237.4211 246.7595
0.563 206.2841 206.4477 208.0674 221.5080 242.9242 246.7595
0.594 239.2438 239.4180 240.9679 245.5290 246.6396 246.7596
0.625 227.4236 227.5378 228.6365 236.0216 245.0209 246.7596
0.656 206.2781 206.3871 207.4685 216.9532 239.9943 246.7595
0.688 187.9505 188.0548 189.1093 200.0221 235.7244 246.7595
0.719 171.9614 172.0616 173.1115 187.4031 235.4701 246.7595
0.750 157.9612 158.3429 162.1280 193.4916 240.1238 246.7595
0.781 206.3023 206.6293 209.8547 232.3513 245.6521 246.7596
0.813 239.2340 239.3212 240.1084 243.6991 246.3096 246.7596
0.844 221.8392 221.9238 222.7485 229.1087 242.4403 246.7595
0.875 206.2750 206.3568 207.1685 214.4934 237.6257 246.7595
0.906 192.2933 192.3724 193.1684 201.3403 234.9812 246.7595
0.938 179.6866 179.7632 180.5489 190.2948 237.1667 246.7595
0.969 168.2803 168.3548 169.1550 186.9389 243.5506 246.7595

Furthermore, the proposed study can be further extended to investigate the buckling behaviour of short beams with flexible joints, where the shear deformation effects also play an important role in addition to joint rotation stiffness ( K T ) as well as joint location ( ξ ) parameters. Subsequently, similar FE procedure can be developed and applied to investigate the buckling behaviour of thin and thick plates with flexible joints. Same will be investigated in detail in future works planned by the authors.

4 Conclusion

In present study, an application of two noded beam FE formulation with flexible joint has been demonstrated for investigating the buckling behaviour of beam with flexible joint. Joint is modelled as rotational spring in FE studies, and its influence on first five considered modes is aptly summarized. It is understood that loose joints as well as their locations can significantly alter the beam buckling behaviour and must be accounted in understanding and evaluating the actual buckling behaviour of these structures.

Acknowledgements

The authors would like to thank the anonymous reviewers and the editor for their valuable and constructive comments and suggestions that helped in improving the manuscript’s quality.

  1. Funding information: The authors state that there is no funding involved.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors declare that there are no conflicts of interest regarding the publication of this article.

  4. Data availability statement: All data generated or analysed during this study are included in this published article.

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Received: 2022-07-15
Revised: 2022-11-12
Accepted: 2022-12-16
Published Online: 2023-04-03

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/nleng-2022-0274
Keywords for this article
buckling analysis; spring-hinged beam/column; flexible joints; buckling loads; beam/column with rotational springs; free–free beam; influence of flexible joint
Creative Commons
BY 4.0

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