Other advantages and disadvantages of SPR process are reported [23, 24]. This is a preview of subscription content, access via Woodhead Publishing, 2020: Chapter 8, 275-308. Think about grabbing and suspending it at one end with some sort of clamp across the entire 1.5 inch width and then hanging weights on the other end from another clamp. What are the common LTB failure modes and how can you prevent or mitigate them? [109] employed rivets with different hardness levels for joining upper high strength press-hardened steel to lower heat-treated aluminium alloy 6111. Multidisciplinary Digital Publishing Institute Proceedings, Brussels, Belgium, July 1-5, 2018, 2(8): 384-389. Article SPR process parameter optimisation includes optimisation of rivet geometry, riveting direction, riveting force, rivet setting velocity, die, sheet thickness and materials properties of sheets and rivet. Answered: To determine the strength of the | bartleby SHOW THE COMPLETE SOLUTION GIVEN, REQUIRED AND SOLUTION. A conclusion in fatigue properties for riveted joints is that larger squeeze force leads to better fatigue performance of joints [25, 26]. These SPR quality monitoring processes can be classified into destructive and non-destructive testing methods. Since then, a significant progression has been the application of SPR technique to both space frame (Audi A2) [5] and monocoque designs (Jaguar) [6, 7]. G L Song, A Atrens. In this article, you will learn how to do that using some basic formulas and principles. https://doi.org/10.1007/s42154-020-00108-y, Zhao L, He X, Xing B, Lu Y, Gu F, Ball A (2015) Influence of sheet thickness on fatigue behavior and fretting of self-piercing riveted joints in aluminum alloy 5052. The International Journal of Advanced Manufacturing Technology, 2017, 92: 1777-1824. Civil Engineering To determine the strength of the riveted joints in a cylindrical vessel, tensile tests were performed on the 6-in.-wide specimens, as shown in the gure. Material preparation, data collection, and analysis were performed by Yahui Liu, Zhiwang Zhu, Huipeng Yu, and Jun Wang. H Duan, G Han, M Wang, et al. J Manuf Process 69:290302. https://doi.org/10.1016/j.matdes.2015.08.121, Haque R, Durandet Y (2016) Strength prediction of self-pierce riveted joint in cross-tension and lap-shear. Although the application of a PTFE insert at the interface between the sheets managed to lower the frictional force and reduce fretting damage, there was a decrease in the fatigue life of the SPR joint with PTFE insert. Eng. As the rivet pierces into the upper sheet, sheet material flows outwards, creating internal tensile stresses within the material [107]. The next part of the study describes experimental . [9] to effectively join advanced high strength steel. Springer, 2006,108(12): 1086-1090. Journal of Manufacturing Science and Engineering, 2011, 133(3): 031009. EP-SPR joints of aluminium alloy and high strength steel were observed to have less risk of lower sheet penetration, better strength and energy absorption performance as compared to conventional SPR joints [9]. Without sufficient support from the upper sheet, the rivet was rotated clockwise, and the sheets were bent severely, leading to rivet pull-out. If the internal shear force Q of the beam is 50kN, verify the shear strength of the rivets. In a composite-metal structure, composite sheet is always recommended to be placed on top and metal sheet below for a better joint quality [11, 63]. J Mater Process Technol 199:2736. Start your trial now! Hydraulics Pneumatics The term shear strength is often associated with the soil. Thin-Walled Struct 131:115. Black debris resulted from fretting may accumulate around the rivet leg, increasing the roughness of the contact interface, thereby further increasing the friction and resulting in more debris [87]. Fluids Flow Engineering Terms and Conditions, PDF Instructional Objectives: 1. Strength of riveted joint - IDC-Online Calculate and check the strength of the riveted joint. Dynamic strength evaluations for self-piercing rivets and resistance spot welds joining similar and dissimilar metals. Mucha, J., Witkowski, W. Mechanical Behavior and Failure of Riveting Joints in Tensile and Shear Tests. Arrows in a indicate sheet cracking and in b fibre breakage and delamination. In Pkw. Tearing of the plate at an edge Figure: Tearing of the plate at an edge As shown in the above diagram, the tearing may happen at the nearest edge of the plate. Besides high strength steel and aluminium castings, another difficult material to rivet is magnesium alloys. (PDF) The Structure of the Strength of Riveted Joints Determined in the WordPress Download Manager - Best Download Management Plugin. For industry applications, the remaining bottom thickness and undercut must be greater than 0.2 mm [49]. A ductile lower sheet also allows wider spread of rivet leg for good interlocking [54], and this is advantageous to the static and fatigue strengths of the joint [26]. An example of fibre breakage and delamination of CFRP sheet in a composite-aluminium joint is shown in Figure5(b), reproduced from [68]. W. Voelkner, F. Jesche, and L. Lachmann, Joining by forming: newer developments, J. Technol. 3, 386390 (2013). Introduction Self-Pierce Riveting (SPR) is a cold forming process used to fasten two or more sheet materials as shown in Fig. Effect of galvanic corrosion on durability of aluminium/steel self-piercing rivet joints. Google Scholar, Kim K-Y, Sim J, Jannat N-E, Ahmed F, Ameri S (2019) Challenges in riveting quality prediction: a literature survey. In this paper, research and progress of SPR are critically reviewed in different aspects. Self-piercing riveting connections using aluminium rivets. The shear force acting on each rivet is given by: . A solid rivet differs from a semi-tubular rivet because a solid rivet has grooves on its lateral surface. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. q = Q/T = 1000/10 = 100ft2/s Hope you learned something from this. [24], if the upper sheet is thicker than the lower sheet, joint is likely to fail by rivet tail pull-out (Figure9(b)), whilst if the upper sheet is thinner, joint is failed by tearing of upper sheet, leading to rivet head pull-out (Figure9(a)). However, with the strength and hardness improved, ductility is usually compromised as a result of embrittlement. Journal of Materials Processing Technology, 2008, 199(1): 27-36. Mater. Would a 100-pound set of barbells do it? Fatigue properties of clinched, self-piercing riveted and spot welded joints in steel and aluminium alloy sheet. Therefore, SPTR composite joints were observed to have better static strength than the conventional SPR composite joints [139]. G. Di Franco, L. Fratini, and A. F Moroni, A Pirondi, F Kleiner. Wang et al. RSW is not suitable for aluminium joints because aluminium has high thermal conductivity and low melting point; the electrical conductivity of aluminium also makes the electrical current requirement for RSW process high. Plast., 1-2, 517 (2002). All authors contributed to the study conception and design. A Baldan. As seen in a steel-aluminium joint, placing high strength steel on top may cause rivet leg compression due to similar strength between the sheet and rivet while placing high strength steel below may result in sheet penetration and necking [71]. https://doi.org/10.1016/j.promfg.2020.04.322, Casalino G, Rotondo A, Ludovico A (2008) On the numerical modelling of the multiphysics self piercing riveting process based on the finite element technique. Normally, some geometrical features, such as a torx-shaped or hexagonal-shaped hole, are designed in the rivet head to transmit the rotational motion from a spindle or a drilling machine. A gross-weight Grumman Tiger at 2400 pounds? p = pitch of inner row of rivets (in) Copyright Glasair Aircraft Owners Association. 2023, by Engineers Edge, LLC www.engineersedge.com Figure 6 shows the results of these tests. J Mater Eng Perform 29:48884897. Google Scholar. How do you collaborate and learn from other experts and peers in crack growth analysis and machine design? Separation of upper and lower sheets, on the other hand, is caused by insufficient penetration or spreading of the rivet leg to join the sheets. Materials Science Forum, 1997, 242: 231-238. A 500-pound set? https://doi.org/10.1016/j.jma.2020.12.016, Ying L, Gao T, Dai M, Hu P, Dai J (2021) Towards joinability of thermal self-piercing riveting for AA7075-T6 aluminum alloy sheets under quasi-static loading conditions. Therefore, process parameter optimisation is generally preferred over advanced SPR techniques. Int J Mech Sci 202203:106499. https://doi.org/10.1016/j.ijmecsci.2021.106499, Porcaro R, Hanssen AG, Langseth M, Aalberg A (2006) An experimental investigation on the behaviour of self-piercing riveted connections in aluminium alloy AA6060. Jaguar uses X350 car to pioneer use of self-piercing rivets. The different types of joint defects and failure modes under mechanical loading described in this Section are summarised in Figure12. Die cavity may have different sizes; a deeper die cavity will obviously introduce larger plastic deformation in the lower sheet, increasing the likelihood of necking at the joint button [24]. C. C. Menzemer, L. Fei, and T. S. Srivatsan, Failure of bolted connections in an aluminum alloy, J. The presence of a coating not only provides a barrier action to galvanic corrosion phenomena, but, at the same time, reduces crevice corrosion due to the sealing of crevice area. The over driven were squashed nearly flat and the under driven were barely set at all. Sheet cracking. EN 1993-1-8. = Strength of Joint / Strength of Un-riveted plate (%) Article Adhesively-bonded joints and repairs in metallic alloys, polymers and composite materials: adhesives, adhesion theories and surface pretreatment. Journal of Composite Materials, 2015, 49(12): 1493-1502. a, b Cross-section and c, d bottom surface of aluminium-magnesium joints formed by SPR and F-SPR [136]. tc = thickness of cover plates (in) M Lou, Y B Li, Y Wang, et al. T Stepinski, M Engholm. This suggests that the failure mode is also determined by the resistance of the sheet materials to bearing and to rivet being pulled out. Civil Struct. All the calculation errors are lower than 8%. In addition to material properties, sheet thickness and die size, friction between the upper and lower sheet interfaces is another factor to consider. I was interested in the force required to cause the initial failure, as well as the nature and appearance of that initial failure; i.e. This was seen in magnesium-aluminium joints, where magnesium die casting cracked when placed below aluminium, despite the different rivet and die configurations used in the study [61]. Strength of Materials Finally, an industrial internet of things (IIoT) was developed to realize the data processing and the SPR joint strength predication. Method of securing a self-piercing and clinching element to a sheet of metal. Self-piercing riveting (SPR) process was originated around half a century ago [1] and it was mainly used by appliance and packaging industry. Lower sheet materials will undergo large plastic deformation when deforming into the die cavity, and therefore must have sufficient ductility; otherwise severe cracks will be generated at the joint button. 2, 621634 (2011). Journal of Aircraft, 2000, 37(1): 130-137. In weeks 3 and 7, corrosion product increases progressively within the joint (compare locations B, and C in Figure13), thinning the lower aluminium sheet, and eventually leading to joint failure at locations D and E in Figure13. . J Manuf Process 7:8393. The rivets are then hammered or pressed to form heads on both ends, creating a tight and secure connection. J. Mucha, The effect of material properties and joining process parameters on behavior of self-pierce riveting joints made with the solid rivet, Mater. Numerical simulation of hybrid joining processes: self-piercing riveting combined with adhesive bonding. Quality control and non-destructive testing of self-piercing riveted joints in aerospace and other applications. y1= 10ft G. Di Franco, L. Fratini, and A. P = pitch of outer row of rivets (in) https://doi.org/10.1016/j.ijfatigue.2006.02.054, Ma Y, Niu S, Liu H, Li Y, Ma N (2021) Microstructural evolution in friction self-piercing riveted aluminum alloy AA7075-T6 joints. Influence of sheet thickness on fatigue behavior and fretting of self-piercing riveted joints in aluminum alloy 5052. 126 142, September October, 2015. A Kochan. Types of rivet fractures: a compression cracks and b internal cracks [72]. https://doi.org/10.1007/s11223-015-9712-5, access via Frankly, I was surprised when it passed 1000 pounds and still going strong. A: Conjugate beam method Eurocode 3. Characterisation of force-displacement curve in self-pierce riveting. Two days ago I got around to doing something that I had planned last yearactual pull tests on riveted aluminum coupons to see how critical it is to drive rivets to the correct height. Fatigue analyses of self-piercing rivets and clinch joints in lap-shear specimens of aluminum sheets. Learn from the communitys knowledge. A new self-piercing riveting process and strength evaluation. In: Materials, Design and Manufacturing for Lightweight Vehicles. Journal of Materials Engineering and Performance, 2014, 23(5): 1515-1523. Other types of coating include electrophoretic coating (e-coat), and electroplating with zinc and a combination of tin-zinc. The more grooves are filled by the lower sheet material, the higher joint strength [127]. M Jckel, T Grimm, D Landgrebe. D = Diameter of the rivet (in) L = Safe Tensile Load (lbs) Remember this is .032, 2024-T3 sheet 4 inches long and 1.5 inches wide with no holes or rivets in it. The maximum allowable force Ps can be determined as, \[{P_s}=n{\sigma _s}\left({{\pi\over 4}{d^2}} \right)\] for lap joints / single cover butt joints (Figure 26.2a), \[{P_s}=2n{\sigma _s}\left({{\pi\over 4}{d^2}}\right)\] for double cover butt joints (Figure 26.2b), \[{\sigma _s}={\rm{ allowable shear stress of the rivet material}}\]. It is clear that the fretting areas are covered with oxidised wear debris coloured black and the boundary between the fretting and the non-fretting zones can be clearly distinguished. Assessment of rivet coating corrosion effect on strength degradation of CFRP/aluminum self-piercing riveted joints. Friction stir spot welding of advanced high-strength steels-a feasibility study. 1 Crore+ students have signed up on EduRev. International Journal of Advanced Manufacturing Technology, 2009, 43(1-2): 61-66. Fretting may develop at different locations during cyclic loading as observed by Chen et al. The shear force acting on each rivet is given by: The force acting on each rivet due to crushing is: The area of the rivet that is being crushed is: Area of shear surface for the dowel pin A.Area of extrusion surface for the dowel pin Ajy. Pickin et al. - 178.63.98.22. Bearing Strength and Failure Mechanisms of Riveted Woven Carbon - MDPI Materials and Deign, 2007, 28(8): 2361-2365. [68]). Low ductility of composites prohibits the rivet leg from spreading, creating more lateral pressure on the composites. The force-displacement curve is often compared with a reference curve [38]. Performance comparison of spot-welded, adhesive bonded, and self-piercing riveted aluminium joints. Typically, the rivet leg length is about 70%80% of the total rivet length [110] and the total rivet length (including rivet head) is approximately equal to the combined thickness of the sheets to be joined, plus 75% of the rivet shank diameter. Failure of aluminium self-piercing rivets: An experimental and numerical study. L-SPR has effectively joined aluminium and magnesium sheets with a total stack thickness up to 6.6 mm under 5 s [119]. The current review is more focused, discussing the different types of failure mechanisms of SPR joints and their corrosion issues as well as summarising the potential joining optimisation techniques to improve different rivet joints. Sheet bending. In addition to the aforementioned SPR joint defects, sheet cracking may occur when piercing through materials with low room-temperature ductility such as magnesium alloys [8, 58,59,60,61], aluminium die castings [62], and fibre-reinforced composites [11, 55, 63,64,65]. Resistance spot welding of aluminum alloy to steel with transition material-from process to performance-Part I: Experimental study. J Constr Steel Res 144:211220. (PDF) The Structure of the Strength of Riveted Joints Determined in the To overcome this issue, a novel self-piercing-through riveting (SPTR) was recently proposed [139]. Soil stabilization can be achieved by:- The presence of such frictional force is essential for the proper function of the joint; without this frictional force, the joint will be weaker. SAE Technical Paper, 2006-01-0774. Rivet damage: a leg compression, b leg bending, and c leg fracture. Self-piercing riveting of aluminiumlithium alloy sheet materials. This contradicted the findings from Zhao et al. No defect is observed in the upper sheet of aluminium but cracks are observed in the lower sheet of magnesium alloy after riveting which may lead to subsequent failure under loading. By following these guidelines, you can design a riveted joint that can perform well under various conditions and requirements. V. Blanchot and A. Daidie, riveting assembly modelling: Study and numerical characterisation of a riveting process, J. To design a riveted joint that can withstand the applied loads and stresses, you need to calculate its efficiency and strength. Fatigue behaviour of hybrid clinch-bonded and self-piercing rivet bonded joints. This two-step process also increases process complexity and cost, especially when laser heating is used. Self-piercing riveting (SPR) is a cold forming technique used to fasten together two or more sheets of materials with a rivet without the need to predrill a hole. Strength modeling of Al-alloy sheet self-piercing riveting considering However, SPR joining of these advanced light materials remains a challenge as these materials often lack a good combination of high strength and ductility to resist the large plastic deformation induced by the SPR process. D A Dillard (Ed.). During F-SPR, a rivet rotating at high speed will pierce the joining sheets, generating frictional heat between the rivet and sheets. Single-Riveted Lap-Joint Equations and Calculator - Engineers Edge To calculate the strength, you need to know the nominal diameter of the rivet, d; the pitch, p; the gauge, g; the thickness, t; and the width, w of the plate; as well as the number of rows and rivets in each row. Journal of Materials Processing Technology, 2015, 216: 28-36. Zhang and Yang [55] studied the riveting of composites with aluminium alloy, and they observed fibre delamination and folding in the GFRP composite sheet. Fatigue performance of a self-piercing rivet joint between aluminum and glass fiber reinforced thermoplastic composite. Thermally activated plastic flow of metals and ceramics with an electric field or current. A: ASKED: To choose the correct description of given soil based on its composition by weight. What I did was to make up 10 test coupons. M Niinomi (Ed.). AIP Conference Proceedings, France, April 27-29, 2016, 1769(1): 100010. your institution. Figure13 shows sheet thinning in a steel-aluminium joint at different ageing time. The maximum punching force of a punch press is P = 400KN, the allowable compressive stress [] of the punch material is 440MPa, and the shear strength limit of the steel plate is = 360MPa. The gunpowder is first stored in a shell, and when ignited, it impacts on a punch, which hits the rivet head, creating an impact SPR process, which can be completed in less than a millisecond. As a result, there are less tensile stresses created by impact riveting, giving better joint quality and strength than the conventional SPR process. 1 Introduction The Glasair Aircraft Owners Association is the one must-join association for Glasair, GlaStar, Sportsman 2+2, Merlin, and OMF Symphony builders, owners, pilots, and enthusiasts. The feasibility of using SPR to join metals and composites has also been investigated [11, 55, 63, 64, 66,67,68]. Refer to Figure9(c), sheet bending is particularly observed in thin sheets (thickness of less than or equal to 1 mm) on the rivet head side, leading to either tearing or shearing of the upper sheet. SPR is a cold forming process that is used to fasten together two or more sheets of materials by driving a rivet through the upper sheet and upsetting the rivet in the lower sheet without penetration into the lower one. Process. Hence, it is expedient to have more insight on the fracture mechanism of various joints during tensile tests. The interlock distance is another important joint quality to ensure good amount of rivet flaring and no gap between the upper and lower sheets around the rivet leg. These advanced light materials have relatively low ductility at room temperature, and often lead to crack formation both inside the joint and on the top or bottom surfaces of the joint during SPR process. https://doi.org/10.1533/9780857098849.2.171, Wang J, Zhang G, Zheng X, Li J, Li X, Zhu W, Yanagimoto J (2021) A self-piercing riveting method for joining of continuous carbon fiber reinforced composite and aluminum alloy sheets. SPR only rose to prominence during the last decades when it was adopted by Audi to join the structural panels in the all-aluminium Audi A8 [4]. Eng. Two common defects observed in riveted composite sheets are fibre damage and delamination. Han et al. [50] and [100], respectively). Self-piercing riveting of similar and dissimilar titanium sheet materials. This may have just been the luck of the draw for this single sample, so I wouldnt put any real faith in it. Intelligent stannate-based coatings of self-healing functionality for magnesium alloys. Fretting wear under fatigue loading is also discussed. Y Miyashita, Y C J Teow, T Karasawa, et al. A 1200-pound small car? The initiation of such failure mechanism also depends on the fibres orientation of the laminate material [64]. In: Advances in structural adhesive bonding. What do you think of it? https://doi.org/10.1016/j.jmatprotec.2015.04.006, Kim C, Min KM, Choi H, Kim HJ, Lee M-G (2021) Development of analytical strength estimator for self-piercing rivet joints through observation of finite element simulations. Adhesiv., 30, 367379 (2010). Our experts are here to help with a quick response within 24 hours. https://doi.org/10.1016/j.eng.2020.06.015, Tiryakiolu M, Robinson JS, Salazar-Guapuriche MA, Zhao YY, Eason PD (2015) Hardnessstrength relationships in the aluminum alloy 7010. Rivet with higher strength and ductility than the sheets is recommended [9] so that the sheets would be deformed without obvious upsetting, although this may significantly increase the rivet cost. Int J Solids Struct 47:427439. Some adhesives require heat curing after bonding to reach the required strength, and this may take up to weeks to complete. [29] reported 25% lower static failure loads in SPR joints as compared to RSW joints in joining steel sheets, under both lap shear and T-peel testing. High strength aluminium alloys are not as problematic as the improved strength is still considerably lower than that of a steel rivet while maintaining a reasonable ductility. Some researchers reported that crevice corrosion could not occur in magnesium alloys because magnesium corrosion is insensitive to oxygen concentration differences [103, 104]. Process parameter optimisation involves optimisation of riveting parameters, such as rivet and die while advanced SPR techniques often require significant changes to the conventional SPR process. Rivet pull-out can occur from the upper sheet (rivet head pull-out) or from the lower sheet (rivet tail pull-out); the type of rivet pull-out is dependent on sheet thickness and sheet material properties against bearing and tension. In: Welding and Joining of Aerospace Materials. https://doi.org/10.1016/j.jmatprotec.2007.10.071, Hoang NH, Hanssen AG, Langseth M, Porcaro R (2012) Structural behaviour of aluminium self-piercing riveted joints: an experimental and numerical investigation. To avoid excessive deformation of the rivet head and upper sheet, an anvil is placed between the punch and upper sheet to absorb some of the impact forces. In this blog post, well dive into the main types of rivet connections and the formulas used to calculate their strength. I did this to see just how poorly a joint made with this sort of gross error would hold up. This process is known as solid SPR [111, 127, 128]. As the name suggests, crevice corrosion arises due to surface irregularities or crevices formed in SPR joints. In the SPR joint, large cracks were observed in the lower magnesium alloy sheet which propagated to the bottom surface; in the F-SPR joint, no visible cracks were observed when the generated frictional heat was large enough to soften the magnesium alloy sheet [136]. Most commonly used rivet is semi-tubular, and rivet leg length can directly affect the volume of the sheet material filling into the hollow cavity of the rivet [50]. Archiwum Motoryzacji, 2011, 2: 57-67. 2023 BioMed Central Ltd unless otherwise stated. Simplifying the eccentric load P acting on the rivet assembly to the centroid point O, we obtain a force P passing through the point O and a moment m = Pe that rotates around the point O. Narrowband ultrasonic spectroscopy for inspecting multilayered aerospace structures. https://doi.org/10.1533/ijcr.2005.0108, He X, Pearson I, Young K (2008) Self-pierce riveting for sheet materials: state of the art. We all have some of these flying in formation in our airplanes. The author read and approved the final manuscript. J Magnes Alloys. Joining steel with aluminium is a common cause of galvanic corrosion due to the great discrepancy in electric potential [97]. G Han, M Wang, Z Liu, et al. Woodhead Publishing, 2012: 215-234. In case of shear test of conventional riveting joints, their strength is determined by the mechanical properties of the fastener material. How much weight would it take to break this .032 inch thick sheet? Beam Deflections and Stress Assuming that any straight line on the steel plate (such as OA or OB) remains straight after the rotation, the average shear strain of each rivet is proportional to the distance from the center of the rivet cross-section to point O. SPR involves large and highly localised plastic deformation [45]. For steel-aluminium joints, rivets with blade angle of 40 was found to be the most suitable [50]. Mechanical properties and deformation behaviour of high-pressure die-cast magnesium-aluminium based alloys. A defect-free joint is shown in d for comparison [35]. Simulation der verbrauchsverbesserung durch gewichtsreduzierung. A form of corrosion occurs at the crevice of magnesium alloys is called filiform corrosion, and it is not a true crevice corrosion [105]. Some suggestions are to use higher-ductility material as the lower sheet, for example, placing steel sheet on top and aluminium sheet below [35], or use a shorter rivet to avoid full penetration through the lower sheet [55]. I am constantly thinking, reading, and writing about these subjects, constantly striving to stay at the forefront of my field. Even an occasional grossly over driven rivet is probably OK, especially if getting rid of it could cause damage. Experimental analysis and comparison of the strength of simple and hybrid structural joints. L Huang, J Bonnen, J Lasecki, et al. Struct., 94, 33883393 (2012). Effect of heating time in the welding stage on the microhardness and tensile shear strength of the joint (Data taken from Ref. JOM, 2008, 60(11): 57-62. 34, 2 (2021). According to the shear failure condition of the steel plate: Using two steel rails to rivet into a composite beam, the connection situation is shown in figure a and b.
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