Riveted Joints and Linear Buckling in the Steel Load-bearing Structure

Transcription

1 American Journal of Mechanical Engineering, 017, Vol. 5, No. 6, Availale online at Science and Education Pulishing DOI: /ajme Riveted Joints and Linear Buckling in the Steel Load-earing Structure Pavol Lengvarský *, Miroslav Pástor, Jozef Bocko Department of Applied Mechanics and Mechanical Engineering, Faculty of Mechanical Engineering, Technical University of Košice, Košice, Slovakia *Corresponding author: Astract The article deals ith the theoretical asis of riveted joints and linear uckling. The steel roof structure is loaded ith force effects that take into account eight of the roof structure, the eight of the roofing, the sno load and the eight of the air conditioning. For the loaded roof structure, a stress analysis is performed using the finite element method, from hich the stress fields and the size of the axial forces in rods of the roof structure are otained. These axial forces strained the riveted elements and can also cause the uckling of the longest and the most stressed rods. For this reason, a check is carried out on the riveted elements at the most stressed places of the structure, and a check is then carried out on the strut. It is clear from the results that all the riveted elements are suitale, and there is no loss staility of the rods. Keyords: riveted joints, rod, linear uckling strength, truss, roof Cite This Article: Pavol Lengvarský, Miroslav Pástor, and Jozef Bocko, Riveted Joints and Linear Buckling in the Steel Load-earing Structure. American Journal of Mechanical Engineering, vol. 5, no. 6 (017): doi: /ajme Introduction A rivet is a short cylindrical ar ith a head integral to it. The cylindrical portion of the rivet is called shank or ody and loer portion of shank is knon as tail. The rivets are used to make permanent fastening eteen the plates such as in structural ork, ship uilding, ridges, tanks and oiler shells. The riveted joints are idely used for joining light metals. The fastenings (i.e. joints) may e classified into the folloing to groups: permanent fastenings and temporary or detachale fastenings [1]. Column uckling is a curious and unique suject. It is perhaps the only area of structural mechanics in hich failure is not related to the strength of the material. A column uckling analysis consists of determining the maximum load a column can support efore it collapses. But for long columns, the collapse has nothing to do ith material yield. It is instead governed y the column's stiffness, oth material and geometric []. The article deals ith the theoretical introduction for riveted joints and linear uckling in the elastic area. Susequently, the finite element analysis of the roof structure is performed, from hich output data are used to check the riveted elements and check the rods for loss of linear staility. Only static calculations are solved in the article, ut it is important take into account, that residual stresses can also occur in steel supporting structures.. Riveted Joints A relatively large numer of parts of machines are shear stressed, these parts are joining elements (rivets, pins, elds, etc.). Depending on the type of construction of the joints and the method of loading, the joints elements can e stressed y shear load, also y tension and ending load. The normal stresses from tension and ending load are small compared to the shear stresses and therefore are not take into account in the technical calculations. When performing stress calculations, it is assumed that shear stresses across the shear plane are evenly distriuted. This assumption is sometimes inconsistent ith the reality, therefore it is compensated y the loer value of the alloed shear stresses. In case that transfer of forces to an element stressed y shear is mediated y a small surface (eg eteen the rivet and the connected part), the influence of the contact stresses is also applied. The contact stresses are often distriuted unevenly trought the contact area. In computational practice, therefore, the assumption of a uniform distriution of these stresses over an area equal to the projection of the contact surface to the plane perpendicular to the load force is introduced. These deviations from the fact are taken into account hen determining the values of the permissile stresses at the crushing. Deformation from the contact stresses is called crushing. For strength checks, e take into account not only the shear stress, ut also to the tension and crushing resistance, depending on the nature of the element to e controlled [1-6]. Check the eakness part on tension N σ = σall, (1) A here A is the area of eakened cross section, N is the axial force in this cross section and σ all is the alloale stress.

2 330 American Journal of Mechanical Engineering The crushing resistance is defined as F σ = σall, () A here F is the force transmitted y the contact surface and A is the area of the rectangle representing the projection of the riveted on the plate section. For riveted joints loaded axially, it is that the line of action force passes through the center of the cross section of the rivets, e assume the proportional distriution of the loading force on the individual rivets. The strength condition for stress of rivets has form F τ = τ i n A s D, (3) here i is the numer of shearing areas, n is the numer of rivets and A is the shearing area. The aove relationships represent strength conditions for the stressing of the fasteners and the connected parts. The Euler relations for critical force are applicale in elastic area, i.e. for λ λ m. For the common structural steels λm 100. If λ < λ m, i.e. σcr R U, e speak aout uckling in elasto-plastic area, here the Euler relations for critical forces are inapplicale. 3. Linear Buckling The critical force in relation to the linear uckling is called a compression force in hich a eam loaded in the axial direction loses staility. In the area of elastic deformations (to the limit of proportionality), critical force and critical stress can e determined from Euler's relationship Fcr = σ π EJmin, ( ) (4) π E cr =, (5) here E is the Young s modulus, J min is the minimal moment of inertia, λ is the ar slenderness ratio defined as λ λ =, i min (6) here β is the column effective length factor depending on supports of ar ends and i min = J min / A is the minimum radius of gyration of the ar cross section. The effective length factor expresses fact, ho many times e have to increase length l of rod pinned on oth ends, in order to have the same critical force as has the rod of length l under investigated support conditions. In Figure 1 are given some types of supports ith corresponding effective length factors. The relation (4) can e considered valid if the critical stress is ithin the range 0 σ cr R U, here R u is the limit of proportionality, respectively if applicale λ λ m (7) hile the limit magnitude of slenderness ratio λ m is defined as E λm = π. (8) RU The limit magnitude of slenderness ratio λ m creates order eteen uckling in elastic and elasto-plastic area. Figure 1. Column effective length factor depending on supports of rod ends 4. Roof Construction The roof truss structure consists of L profiles laeled L50x50x6, L 70x70x8, L 10x10x1 stored on support pillars. Each rod is made up of a pair of rolled L profiles (Figure 3), hich are riveted to the steel joint plate usually y 3 rivets. The roof structure ill e considered as a truss system here the rods are only loaded y axial forces, it is stressed y tension, respectively y compression. Figure. Roof construction A strength analysis of the roof structure is carried out, in hich the axial forces and the axial stresses in the rods

3 American Journal of Mechanical Engineering 331 are determined. The shape of the rolled L profiles is shon in Figure 4 and their asic dimensions are shon in Tale 1. t r z T r 1 y z T y T Figure 4. Basic dimension of L profiles Figure 3. Rods formed y pair of L profiles Tale 1. Basic dimension of L profiles Parameters Profile L 50x50x6 L 70x70x8 L 10x10x1 (mm) t (mm) r 1 (mm) r (mm) y T=z T (mm) A (mm ) m (kg/m) The Figure 5 shos a model of the roof structure on hich the oundary conditions are applied and is loaded y force effects. In the calculation model, the force F corresponds to the load applied to the area defined eteen the to joint plates adjacent to the vertical rods and the distance eteen the truss. The rod numers are defined automatically y the finite element program and are shon in Figure 6. For the calculation, the cross sectional area of individual rods according to Tale 1 is defined, the material used for the steel roof structure is The tension strength of the material is 360 MPa and the yield strength is 35 MPa. A finite element mesh is created on the model. The used type of the element is a truss element and the element size is equal to the length of the profile. Figure 5. Applied oundary conditions and loads on the construction Figure 6. Fields of axial stresses in axis of truss The load forces corresponding to the eight of a hole construction and the positions of the forces taking into account the skylight in the middle part of the roof structure (Figure ). The load consists from the eight of the steel structure, the eight of the roofing and the eight of the sno or icing (F 1 = 3 kn + 64 kn kn = 56 kn), and from the eight of the air conditioning F = kn acting in the joint plates II, IV and V (Figure 5). Figure 6 shos the stress fields in the individual rods for the mentioned load. 5. Analysis of Loaded Elements 5.1. Riveted Joints In this section the strength analysis of the rivets used in the steel supporting structure is performed. To types of rivets are used for riveted joints (Figure 7): Rivets ith diameter d = 0 mm joint the L 50x50x6 and L 70x70x8 profiles to the joint plate.

4 33 American Journal of Mechanical Engineering The shear area of the rivets ith diameter d = 0mm is A s = 314 mm ; Rivets ith diameter d = mm joint the L 10x10x1 profiles to the joint plate. The shear area of rivets ith diameter d = mm is A s = 380 mm ; In the strength analysis of the rivets, the investigators consider ith the commonly used rivets material, i.e., ith steel The tensile strength of the material is Rm = MPa. The value of the yield strength is equal Re = 05 MPa. Figure 7. The used rivets to joining the L profiles ith joint plate Each rod is joint y 3 rivets, check of rivets has een performed only for the most stressed rods. The most stressed rod is rod no. 35 (Figure 6). To check the rivet joint, the rod no. 35 is selected ith the profile L 10x10x1. The value of the axial and at the same time the shear force in the rod is F s = 349 kn. The shear stress in the rivet loaded y shear force is τ = = 153 MPa (9) We can conclude that sear stress is in elastic limit. In this case, it should e noted that eams no. 35 and no. 36 do not form separate L profiles, i.e., they are made from one rolled L profile connected from oth sides ith the joint plate. For this reason, the stresses in the rivets are s significantly loer. The stress in the eakened part of the profile L 10x10x1 is equal to N σ = = = 157 MPa. A 1111 (10) Checking of rivets in the pressed rod no. 9 - profile L 50x50x6 in hich the size of the axial and shear forces F s = 10.5 kn. The value of the shear stress in rivet loaded y shear forces is equal τ = = 5.57 MPa (11) We can conclude that sear stress is in elastic limit. The tension in the eakened part of the profile L 50x50x8 is equal N σ = = = 4 MPa. A 15 (1) 5.. Linear Buckling For the check the rods on the linear uckling, the compression rod no. 9 (Figure 6) is selected ased on the numerical calculation. The elastic modulus of the used 5 material is E =.1 10 MPa,, the minimal moment of 3 4 inertia is J min = mm and the column-effective length factor depending on supports of rod ends is β = 1 (Figure 1), length of the ar is = 500mm. Checking the condition λ λm for calculating the Euler's critical force. The slenderness ratio of the rod is equal λ = = = = 365. i 3 min J min A 750 (13) The limit of the slenderness ratio of the rod λ m is equal 5 E.1 10 λm = π = π = (14) R 35 U The condition for calculating the Euler's critical force is fulfilled if Euler's critical force has value F cr ( 1 500) λ > λ m. (15) 5 3 π = = 4.78 kn. (16) The axial force in rod no. 9 has a value of 10.5 kn and the safety factor is equal F cr 4.78 k = = = (17) F 10.5 The rod no. 9 ill not loss of staility. 6. Conclusion The introduction to the riveted joints and linear uckling theory as presented in the article. Susequently, a stress analysis of the roof structure as performed. Where the distriution of the axial stresses and the axial forces in the rods as found. Susequently, for the most stressed rods, the stress control as performed for the joining elements. The controlled riveted joints are suitale and ill not e violation. For the rod ith the highest compression stress due to its length, a linear uckling as performed here it as found that the rod is suitale. It should e noted that residual stresses play an important role in the assessment of steel load-earing structures. Acknoledgements This research as supported y a grant from the Slovak Grant Agency VEGA No. 1/0731/16, and KEGA 049TUKE-4/017.

5 American Journal of Mechanical Engineering 333 References [1] Meriam, J.L. and Kraige, L.G., Engineering Mechanics: Statics, Wiley, Texas, 011. [] Cherchuk, R.N., Method of Joints (Easy Way), Amazon Digital Services LLC, 014, [E-ook] Availale: Amozon.com. [3] Bocko, J. and Delyová, I., Statics, Technical University of Košice, Košice, 015. [4] Jančo, R., Numerical and exact solution of uckling load for eam on elastic foundation, in Sorník vědeckých prací Vysoké školy áňské-technické univerzity Ostrava, Technical University West Bohemia, 1-6. [5] Lengvarský, P. and Bocko, The Static Analysis of the Truss, American Journal of Mechanical Engineering, 4 (7) Dec.016. [6] Pierre, F., et al., Mechanics of materials, McGra-Hill, Ne York, 009.