1 Introduction

Shear pins are generally used as a mechanical safeguard in assembly operations. They are designed specifically to shear and fracture at a particular load that would otherwise damage the sensitive and critical parts of the machinery [1, 2]. Shear pins are comparatively inexpensive and are thus commonly used as sacrificial members in an assembly. To allow a shear pin to fracture in an orderly manner a groove or notch is cut on the surface of pins at the intended plane of shear. The grooved notch is expected to initiate the shearing thereby decreasing the fracture load, which may lead to shear the pin by decreasing the resisting shear area [3, 4]. Shear pins are only useful for a single operating cycle, after each operation, they have to be replaced. A very simple example is the plastic or wire loop affixed to the handles of common fire extinguishers. Its presence prevents accidental discharge by only allowing the handle to be depressed once a high amount of initial force is applied; by breaking, it allows the handle to subsequently be depressed more easily [5]. A shear pin could potentially be made from any material although metal is the most common. A shear pin is often tempered to make the metal brittle. Thus, it may break or shatter rather than bending when the required force is applied. The material of a shear pin is selected and treated so that it is relatively resistant to fatigue (material). That is, when subjected to small forces, each one insufficient to break the pin, the pin does not retain damage [6, 7].

Shear pins are used in several applications[8, 9]. They are preferred in special purpose applications where underwater vehicles lay underground oceanic network cables with the assistance of a ship [10]. In the event of a fault or a stray oceanic current, the cable snaps free from the underwater vehicle and prevents its cable from getting snagged in the propeller mechanical piston assembly which may endanger the life of the underwater crew or equipment involved. Shear pins are housed in devices such as snow movers and auger drives. In the event of the worm blade hitting a stone that may receive a load rated than a higher load, a coupling link with a shear pin may fracture before the rest of the expensive drive train and involved gears fracture[11]. This increases the longevity of device life whilst ensuring a cheap sustainable way to replace pins. Typically, two shear pins (shear bolts) connect the auger to the auger shaft [12]. They feature specially designed grooves, allowing them to snap if there’s too much torque. This prevents damage to the gear case and auger shaft. In the role of conditional operator, a shear pin may use to prevent a mechanical device from operating before the criteria for operation are met.

Currently, majority of the industrial applications use solid shear pins in assembly joints. A lot of literature is available on solid shear pins design, analysis, and application. Solid pins for sacrificial shear are chosen in a very theoretical manner. The material of the pin, the diameter of the pin at the notch, and the notch depth are the essential parameters required to choose a pin for a rated fracture load [13, 14]. In the case of steel-made solid shear pins, the ferrite grains in the steel do not influence the shearing but distribute the stress among the carbide particles of the steel matrix. It is the spheroidal carbide particles that initiate the shear. The continuous ferrite matrix phase has a lower strength and gets deformed easily along with spheroidal cementite particles and this assists the failure in shear [1]. Bending results in the failure of pins earlier than the intended shear load. Pins above a critical diameter fail in shear whereas those below the critical diameter fail in bending [15]. Another factor that leads to the failure of pins is the shock loads and fatigue. Further, the misalignment of the assembly components is also a critical factor that coupled with fatigue causes premature cracks which would propagate and cause the material to fail at a lower than rated load [3].

On the other hand, minimal literature is available on the use of hollow shear pins. Uses of hollow pins have been studied in applications of heat transfer where the governing partial differential equations are solved numerically using a well-known implicit, iterative and finite-difference method. The numerical solutions are validated against various analytical solutions derived based on different constraints. The hollow pins provide a larger surface area for heat dissipation in situations demanding good thermal conductivity [16,17,18]. Enhancing heat transfer in a permeable hollow pin over that of the solid pin is only reachable when an external suction flow is present at the hollow-pin outer surface. The maximum reported heat transfer rate due to permeable hollow-pin is 362% above that of solid pin.

The aircraft industry uses shear pins excessively in their assembly operations. The landing gear assembly has a set number of these shear pins at certain locations which are designed to fracture in the event of a harsh landing or any other such malfunction. They are the sacrificial members who are designed to protect the rest of the expensive airframe in such events. There exists never-ending demand for lightweight yet strong materials for serval components of several industries including biomedical, marine, automotive, military, and aerospace industries [19,20,21,22]. Especially in the case of aerospace industries weight of the components is a major concern. A decrease in a small amount of weight of any of the components can be a great contribution to decreasing the kerb weight of the aircraft and thereby the fuel savings in aircraft [23]. There is an unabated demand for high-performance materials yet light in weight for use in aircraft, automobile, marine, etc. applications [24,25,26].

Unfortunately, hollow pins as sacrificial members have not been considered vital industrially, as there are design limitations to hollow pins. Manufacturing of hollow pins is expensive and also their fracture characteristics are deemed unreliable. In this regard, hollow pins appear to be a lucrative option for the lightweight-high performance material made components and which needs to be studied. Thus, there exist a need for exploring the industrial applicability of the hollow shear pins and the current work is aimed at exploring the feasibility of the above-mentioned.

The current study encompasses modeling shear pins, analyzing, assessing, and quantifying the fracture characteristics of hollow shear pins, and eventually comparing them with the theoretically estimated values. The fracture characteristics of hollow shear pins of various pin diameters, hollowness factors, and notch depth factors are studied. The hollow pins are tested and analyzed under static structural loading and their effect on fracture load is determined. The study is focused on single shear. A comparative study of solid pins with hollow pins is carried out to estimate the feasibility of replacing solid pins with hollow pins in aerospace applications. Data points obtained are compared against those obtained for solid pins of similar stresses and similar pin geometries. Theoretically calculated values of fracture loads for various cases of variables with a plethora of subcases of variables are compared with the computed values. The result of extensive data is graphically illustrated. A correlation is carried out between theoretical results of solid and hollow pins and is then compared with the corresponding results from modeling results.

2 Materials and methodology

The current study aims to create a robust model which can allow an organization/user in aerospace industries to swap or replace a solid pin used in assembly joints with a hollow pin for an equivalent fracture load. The focus is to study the effects of various parameters of a pin on its fracture load and then derive a relation for the same. Using that relationship, one can use the derived formula to facilitate the swap between a solid pin and a hollow pin. The modeling of the elements for the study is carried out in the CATIA workbench and the analysis of the elements is carried out in the ANSYS workbench.

The dimensions of the pin are chosen as per National Aerospace Standards (NAS). In the present study, the pin of dimensions as per NASM20392 is modeled [27]. The 2D diagram of the pin and its dimensions are shown in Fig. 1. In the study, five different families of φA, the outer diameter of the pin are selected from the table of standards given in Table 1. Each family has a different outer diameter (A), corresponding pin head diameter (B), and an effective length (G). The chosen outer diameters are 0.560″, 0.622″, 0.747″, 0.871″, and 0.996″ inches. Each family is modeled and tested in eight different conditions listed in Table 2.

Fig. 1
figure 1

2D diagram of pin as per MS20392 Standard

Full size image
Table 1 NASM20392 standard and dimensioning
Full size table
Table 2 Various conditions of pin family
Full size table

The four main variants of the pin are solid pin, solid notched pin, hollow pin, and hollow notched pin. In the case of hollow and hollow notched pins, the sub-variants based on the (ID/A) ratio are considered where ID indicates the inner diameter of the hollow pin. The chosen (ID/A) ratios are 0.25, 0.5, and 0.75. Altogether 40 models constituting 5 different families of A and sub-variants of each family are created on a CATIA workbench. The dimension details of all the variants considered are given in Table 3. In the case of notched pins, the depth of the notch of 0.5″ is kept constant across all the families. By this, the effect of the Notch Depth Factor (NDF) can also be studied as the A in each family would remain constant.

Table 3 Dimension details of pins modelled on CATIA
Full size table

2.1 Theory of static fracture load

The cross-sectional area at the shear plane is the area resisting shear. Considering the ultimate shear strength of the material, the outer and inner diameters, and the notch depth the theoretical fracture load for static failure during shear can be calculated using Eq. 1.

$${FL}_{ST}=\left(0.25\pi \right)*\left[({A-2{N}_{D})}^{2}-\left({A*H}_{F}\right)\right]*FSU$$
(1)

where \({FL}_{ST}\) = Fracture Load for Static theoretical analysis, A = Outer diameter of the pin as per NASM20392 standard. \({N}_{D}\) = Notch depth. Kept fixed at 0.5 inches. \({H}_{F}\) = Hollowness Factor = Inner Diameter/Outer diameter. Studied for 0.25, 0.5 and 0.75. FSU = Ultimate shear strength of the material.

Considering FSU = 85kpsi for the material 15-5PH H1150 [28], the variations of FLST with other pin parameters are studied. The variation is plotted against other pin parameters and in this plot identical pin variants but from different pin families are considered for each curve.

2.2 Theory of static stress variation

The shear stress induced in a pin under shear [29] is expressed as Eq. 2

$$ {\text{Stress}} = {\text{Force}}/{\text{Area}} $$
(2)

A constant force of 444.9 kN is applied on all pins with varying hollowness factor and notch depth factor (NDF) and the max shear stress is determined. The variations are assessed graphically to serve as an ideal curve against which all future simulated values should align.

2.3 Swapping of a solid pin with a hollow pin

Theoretically, the shear resisting force that a material can offer is a product of the area at the cross-section and the ultimate shear strength of the material as shown in the Eq. 2. A hollow pin can substitute the solid pin only when the fracture loads of the both are same. Hence the equations for fracture load of the solid pin and hollow pin are equated as shown in Eq. 3.

$$ \begin{aligned} & \left(0.25\pi \right)*{{(A}_{S}-2{N}_{D})}^{2}*FSU= \left(0.25\pi \right) \\ &\quad *[\left({{A}_{H}-2{N}_{D})}^{2}-{\left({{A}_{H}*H}_{F}\right)}^{2}\right]*FSU \end{aligned} $$
(3)

The L.H.S represents the solid pin whereas the R.H.S represents the hollow pin.

Assuming that the material for both pins is the same, after removing constants, Eq. 3 can be expressed as Eq. 4.

$${(A}_{S}-{A}_{H}){(A}_{S}+{A}_{H}-4{N}_{D})+{\left({{A}_{H}*H}_{F}\right)}^{2}=0$$
(4)

Further simplified the Eq. 4 in the quadratic form is expressed in Eq. 5.

$$\left(1-{{H}_{F}}^{2}\right){{A}_{H}}^{2} -4{N}_{D}{A}_{H}-{(4{N}_{D}{A}_{S}+{A}_{S})}^{2}=0$$
(5)

where AS = Original solid pin diameter, AH = New pin diameter needed, ND = Notch depth which is kept fixed for both pins, HF = Intended hollowness factor.

By substituting the values for the original solid pin diameter, notch depth, and intended hollowness factor, there are two values as per the quadratic equation and negative results are discarded.

2.3.1 Substitution using un-notched hollow pin

A solid pin without a notch and its two holders are modelled in ANSYS workbench. For each solid pin of the nominal diameter shown in Table 3, three hollowness profiles 0.25, 0.5, and 0.75 are selected and the corresponding pin diameters are calculated using the quadratic Eq. 5. These values are shown in Table 4.

Table 4 Calculated substitution values for un-notched hollow pins
Full size table

2.3.2 Swapping solid notch pin with hollow pin

Since pin shearing is paramount during application the effect of the notch in the shearing process is also studied. A solid pin with a notch and its two holders are modelled in ANSYS workbench. For each solid pin of the nominal diameter shown in Table 3, three hollowness profiles 0.25, 0.5, and 0.75 are selected and the corresponding pin diameters are calculated using the quadratic Eq. 5.

To verify these values, the above pins and their corresponding pin-holders were modelled and assembled in the CATIA workbench as per Table 5.

Table 5 Calculated substitution values for notched hollow pins
Full size table

2.4 Model of the pin

Although the 2D geometry of the pin is shown in Fig. 1. The computer-aided design model of the pin with the notch and without the notch is shown in Fig. 2. Pins of 40 different configurations were modelled and analysed using the ANSYS workbench. The details of all the configurations are given in Table 3.

Fig. 2
figure 2

CAD models of the notched and unnotched pins

Full size image

2.5 Workbench simulation and verification

To understand and verify the deviations in the fracture load against various parameters such as hollowness factor and pin outer diameter, the modelled pins are simulated against a predetermined load in the ANSYS workbench. The assemblies from CATIA were imported to the ANSYS workbench. A shear pin whose primary purpose is to shear is considered in simulations by eliminating other motions and degrees of freedom whereas the real-time conditions may vary from case to case. The pin assembly is carried out in such a way that the factors like bending and pin rotation are hindered during assembly. Thus it is ensured that the shear pin boundary conditions are commensurate with their industrial use. The pin assembly is shown in Fig. 3. The simulation was carried out using the following boundary conditions and assumptions: (1) the contact between pin and holders was bonded as shown in Fig. 3a, b, (2) the contact between two holder sliding surfaces was frictionless as shown in Fig. 3c, (3) the contact between pin and pin lock was bonded as shown in Fig. 3d, (4) one of the holders had all six surfaces treated as fixed support as shown in Fig. 3e, and (5) the other holder had four surfaces treated as frictionless surfaces to prevent bending and movement in other planes due to the rudimentary nature of the pin holders as shown in Fig. 3f.

Fig. 3
figure 3

a, b Bonded contact between pin and holder, c frictionless contact between two holder surfaces, d bonded contact between pin and pin lock, e holder having all six surfaces as fixed support, and f other holder having four surfaces as frictionless contact

Full size image

The meshing of the assembly was carried out using the auto-meshing of the ANSYS workbench. The meshed assembly is shown in Fig. 4a. A fixed force of 444.9 kN was applied on one surface of the pin such that the pin is subject to shear in the YZ plane along the Z axis as shown in Fig. 4b.

Fig. 4
figure 4

a Auto-generated mesh and b applying fixed load

Full size image

3 Results and discussion

The purpose of the study so far has been to understand the shearing patterns of hollow pins with or without the notch. However, the ulterior motive is to understand their applicability in an industry where weight reduction or functionality is essential. Thus, studying and justifying the industrial applicability of pins is carried out.

3.1 Variation in theoretical static fracture load

3.1.1 Variation with change in pin size

It is paramount to understand the importance of the overall pin size and how its presence affects all other variables in the fracture load Eq. 1. Fracture load values are mathematically calculated and plotted to visually observe the correlation between pin diameter and fracture load on a pin. The variation of static fracture load (FLST) with change in pin size for all the families is shown in Fig. 5.

Fig. 5
figure 5

Variation of static fraction load with change in diameter of the pin

Full size image

It is observed that the FLST increases with A, the outer diameter. The increase is not linear but is parabolic. Further, it is observed that the variation is high in the case of larger pins as compared to the smaller pins.

3.1.2 Variation against changes in hollowness factor

The prime aspect of the current work is to study the substitutability of a hollow pin with a solid pin. Thus, it is essential to determine the effect of hollowness on the fracture load, hence the pins are loaded and the FLST characteristics against variation in the Hollowness Factor (HF) are determined. The variation of FLST with HF is shown in Fig. 6.

Fig. 6
figure 6

Variation in FLST versus HF for un-notched hollow pins

Full size image

It is very straightforward to understand that the pin with a higher Hollowness Factor (HF) will have a lower fracture load as the area supporting shear will be lesser. However, what is also observed is that theoretically, the presence of a notch does not affect the slope of the graph. Additionally, it is again observe that pins of larger diameter have a much larger variation in magnitude of fracture load than smaller pins as shown in Fig. 6.

3.2 Variation in static shear stress

3.2.1 Variation of maximum shear stress

From the above graph and table, it is noticed that, a non-linear increase in theoretical maximum shear stress due to an increase in the hollowness factor (HF). This can be attributed to the change in the cross-section area of the pin supporting the shearing. The stress variations are concurrent with the fracture load variations. The computed values of maximum shear stress are also shown in Fig. 7. It is observed that the increase in the computed maximum shear stress for the hollow un-notched pins is in concordance with theoretical calculations. As observed in Fig. 7, it shows a relatively linear increase in stress until when the pin starts getting too thin at the hollowness factor (HF) of 0.75 where it shows a jump. The jump is more pronounced in the smaller pin 4 as shown in Table 6.

Fig. 7
figure 7

Variation in maximum shear stress with hollowness factor (HF)

Full size image
Table 6 Maximum shear stress variation versus hollowness factor (HF)
Full size table

3.2.2 Verification of simulated static shear stress

To understand and verify the deviations in fracture load against various parameters such as hollowness factor and pin outer diameter, the actual performance of the pins is determined by simulating the assembly in ANSYS Workbench. While real-life conditions may vary from case to case; since the study considers a shear pin whose primary purpose is to shear, other motions and degrees of freedom are eliminated. The simulated values are given in Table 7. The variation is shown in Fig. 8.

Table 7 Theoretical maximum shear stress Variation versus pin diameter (A)
Full size table
Fig. 8
figure 8

Simulated sharp increase in shear stress due to presence of notch

Full size image

The presence of a notch has induced a steep change in the theoretical maximum shear stresses. The effect due to the notch is exacerbated in smaller pins as the notch depth is kept fixed. Since the effective area at the cross-section is reduced, the stress increase is much more. Additionally, it can be made as an astute observation from Fig. 8 that the presence of a notch has a very significant impact on the shear stress induced in the material and the effect of the notch is much more pronounced when the notch is deeper with respect to the outer diameter of the pin. Thus, the theoretically calculated effect of the notch vastly differs from simulation results.

3.3 Verification of pin substitution

To validate the considered equation, the simulated results need to be analysed graphically. Hence, simulated results are tabulated and analysed graphically in the following sections.

3.3.1 Substitution using un-notched hollow pins

Three out of five un-notched pins undergoing simulation are studied for Von-Mises equivalent stress and maximum shear stress. Pins are selected such that the largest, the smallest, and the middle diameters are covered. The Von-Mises equivalent stress and maximum are obtained for the solid pin and then the corresponding values are entered in Table 8. The variation is shown in Fig. 9. The shearing of the un-notched hollow pin during loading as observed during simulation is shown in Fig. 10. The images are captured at different time steps.

Table 8 Von-Mises equivalent and maximum shear stress of the un-notched
Full size table
Fig. 9
figure 9

Comparison of simulated results versus ideal theoretical values for un-notched pins

Full size image
Fig. 10
figure 10

Simulation of un-notched hollow pin

Full size image

From workbench simulations and obtained graph data, it is observed that the theoretically derived quadratic equation has yielded results that are in good agreement with the simulated results in the case of substitution of the solid pin with hollow pins. It is verified from the graph data that shows minor changes in stress with the variation in the hollowness factor HF (between 0 and 0.75). Thus, it is clear that the quadratic relation holds good even for different pin diameters and can be successfully applied to any pin.

3.3.2 Substitution using notched hollow pins

For the application of shearing, it is essential to test the effect of the notch in the shearing process. A solid pin with a notch and its two holders is modelled and simulated in the ANSYS workbench. For each solid pin of the nominal diameter shown, three hollowness profiles 0.25, 0.5, and 0.75 are selected and the corresponding pin diameters are calculated using the quadratic Eq. 4. These values are shown in Table 9.

Table 9 Equivalent stress and maximum shear obtained for notched pins
Full size table

The shearing of the notched hollow pin during loading as observed during simulation is shown in Fig. 11. The images are captured at different time steps. From a large number of workbench simulations, an astute observation is that Eq. 4 does not hold well for pins consisting of a notch. It is imperative to note that the equivalence equation which equates a solid pin to a hollow one assumes that shear is affected by the area at the cross-section. However, even though the area has been kept constant as per the equation, drastic deviations in stresses due to the notch are observed. This can thus be attributed to only one potential cause and that is stress concentration [29]. The significance of stress concentration can be seen and verified from the graphs shown in Fig. 12.

Fig. 11
figure 11

Simulation of notched hollow pin

Full size image
Fig. 12
figure 12

Variation of shear stress with hollowness factor

Full size image

Further, it is seen that the stress increases with the increase in the Notch Depth Factor (NDF = ND/A). As shown in Fig. 13, with the increase in the slope of the Notch Depth Factor, the effects of the Notch become more pronounced and they greatly influence the equivalent pin diameter hence, the pin substitution equation fails. Until the current stage of analysis, the presence of the notch dominates and the shear resisting capability of the pin depends less on the area offering resistance and more on how deep the notch is relative to the body of the pin.

Fig. 13
figure 13

Notch depth factor (NDF) Slopes for Various Pins as per Equivalent Alternatives

Full size image

3.4 Verification of weight savings

The present study pertains to an aircraft industry where weight reduction in any component by 0.1% can be considered as an advantage and appreciable contribution to the total weight savings of an aircraft. This section describes briefly on weight saved by the hollow pins for the equivalent fracture load. Although the attempt to develop a conclusive model for equivalent stress and fracture load for notched pins was not in good agreement, the efforts toward un-notched pins were appreciating. It is found that the un-notched hollow pins may be used for joining parts together without being needed for safety shearing applications. The volumes of such pins can be calculated and the product of volume and density of a commonly used engineering material can be determined. In this regard, an attempt has been to determine the weight saving by considering structural steel with a density of 0.2836 lb/in3 [27]. The weights of solid pins and equivalent hollow counterparts are shown in Table 10.

Table 10 Shows the weight saving parameter
Full size table

It is observed from Table 10 that no reliable savings in weight are achieved for the same amount of shear resistance offered by the pins. This is possibly because a pure shear perspective is considered in the present case whereas in reality, bending during shear is also a large component of the total stress endured by the material. Hollow shafts can withstand larger bending loads than solid ones for the same weight and this property is fruitfully exploited in the hollow axles of the aircraft landing gear. Thus, in a practical scenario, hollow pins will provide greater resistance to shear. Eventually, it is imbibed that the benefit of a hollow un-notched pin will boil down to functionality. There may arise situations, where objects such as wires or cables may have to pass through an interface, and drilling adjacent holes, may not be feasible as it may compromise structural integrity or there may be physical constraints. Thus, the industrial applicability of these hollow pins can be considered as beneficial components primarily towards functionality.

3.5 Impact loading

The static structural simulation assumes that force varies over an infinite period. However, shearing pins with notches are designed to fracture under impact loads. In the majority of industrial applications when pins are used as a safeguard, the pins are designed to protect equipment against mechanical loads higher than the rated capacity. These higher-than-rated loads usually occur as transients during non-routine procedures.

From Fig. 11, it is observed that even for the same cross-sectional area, the notched pins fail to adhere to Eq. 4 due to the notch depth factor (NDF) that plays a large role in localizing stress and increasing stress concentration. Therefore, for the same weight in the material, a hollow notched shear pin may never sustain the same fracture load. Impact loads generally fracture the pins with more ease due to the impulse of the force. The effect of the impact is more pronounced in notches and it is industrially observed that pins rated for a static load may fail at an even smaller impact load. Due to the profound effect of the notch which changes stress concentration, it appears that weight savings using hollow notched pins possibly are not feasible.

From the above discussion, it implies that impact loading analysis is impractical for a notched shear pin, which is designed to fracture because there are no savings in weight for the same fracture load and there is no practical functionality of the hollow cavity from an engineering standpoint.

4 Conclusions

The current study consists of theoretical analysis and part modeling of the solid and hollow pin with a notch and without a notch. The following are the significant inferences from the exhaustive study.

  1. 1.

    The magnitude of variation in theoretical fracture load due to changes in parameters such as hollowness fracture, and notch factor is more in the case of larger diameter pins due to the larger change in the area under shear. However, this is not in good agreement with the simulation results because the stress concentration plays a vital role once the pin size gets too small for the same rated load.

  2. 2.

    The theoretical values greatly differ from the simulated values in the case of pins with a notch. The presence of a notch on a pin, in both hollow and solid pins, can increase the shear stress by a large factor due to stress localization and stress concentration. Such observation is made during the simulation studies while the theoretical approach does not account for the induced stress concentration.

  3. 3.

    The solid to hollow pin substitution equation holds good for all un-notched pins and can be verified by the results of simulation which indicate minor or zero variation in stresses and maximum shear stress. Thus, a suitable substitution equation has been developed which can be used to replace solid pins with hollow pins where functionality is of primary concern.

  4. 4.

    The Notch Depth Factor (NDF) has a profound effect on the stress and maximum shear stress. At higher Notch Depth Factors (NDF), the shearing stress depends more on the depth of the notch relative to the pin than the area at the cross-section.

  5. 5.

    There is no industrial applicability with respect to weight savings for hollow notched pins as they require a much larger increase in the material; assuming material properties to be constant; to be rated for the same fracture load.