Introduction

Grains such as corn and wheat produced from farms and processed in industry are first stored in silo facility to ensure all year round availability of food for human consumption. These grains are usually discharged into the silo by means of a belt bucket elevator system, which is one of the important material handling equipment used in industries to convey bulk materials from the ground level to a required height. The belt bucket elevator system (Fig. 1) consists of a flat flexible belt with cup-shaped buckets attached to it. The belt is firmly joined over a spaced head and boot pulleys and driven by the head pulley. During the rotational drive, the buckets are filled with grains at the elevator boot (bottom), conveyed to the top where the grains are discharged as the bucket turns downward over the head pulley. The head pulley is attached to the head shaft which is rotated by a drive mechanism (Gearbox motor). Generally, a shaft is a component of a mechanical device usually of circular cross section which transmits motion or power from one point to another. It is usually stepped to provide shoulder for other mechanical components. The head shaft of a belt bucket elevator system is an important structural component of the elevator system because it is the drive shaft of the system which supports the weight of the belt, the weight of buckets filled with grains and the head pulley, hence any failure of the head shaft means a failure of the entire elevator system1,2. A broken head shaft can be devastating with consequences such as loss production time with its associated downtime cost, drop in quality of grains and wastage (due to moisture and weevils’ infestation). Also of safety concern, the procedure for replacement of the failed shaft is very difficult, most especially for very tall and heavy elevators, where the belts and cups will have to be suspended to remove the fractured shaft, this poses a significant safety risk to the maintenance engineers. Thus considering the load, the shaft will be subjected to and the consequences of any failure, a proper design of the head shaft of the elevator is critical and a generous safety factor for an infinite life is recommended to ensure safe and reliable operation of a developed elevator system3,4. The head shaft should be designed to sustain both static and cyclic loading in its service life. The cyclic load carried by the shaft causes bending which produces stresses within the shaft and results in fatigue failure when the cyclic loading over a period of time leads to the failure of the shaft even when the magnitude of the stress is below the material yield strength. Since it is on record that majority of engineering shaft failures are caused by fatigue, it is recommended that at the conceptual design stage, shafts should be correctly sized to prevent fatigue failure within the expected period of usage by the industry5.

Figure 1
figure 1

Bucket belt elevator system.

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There are many literature which have dealt with problems relating to elevator shaft system. Goksenli et al.6 investigated the failure of an elevator drive shaft due to torsion-bending fatigue and found out that failure occurred at the keyway of the shaft due to faulty design of the keyway, which resulted to a high notch effect at the keyway section. Brijesh et al. optimized the weight and discharge capacity as well as the stress and deformation of the bucket and shaft of a modeled belt driven elevator for material transport using SolidWorks7. Chavhan et al.8 studied the load carrying capacity of an elevator bucket, the strain in the bucket edges and the clamp bolts using CATIA V5 3D model and ANSYS software and concluded that failure of elevator shaft was the main cause of the bucket failure. Gililland et al. reported that the loads imposed on the boot shaft are very low when compared to the head shaft. This establishes the critical nature of the head shaft of an elevator system9. Snehal et al.10 reviewed that failure on shaft takes place due to high stress on the keyway and areas where there are abrupt changes in cross sectional area. Yin et al.11 investigated the failure of a bucket elevator chain link and reported that the failure was due to propagation of crack embedded within the material during loading. This signifies that the elevator shaft was equally affected by such loading. The drive shaft of a system was examined and reported to have failed under fatigue due to design error of the fillet radius of the shaft12. Ariwibowo et al.13 equally investigated the failure of a shaft and reported that the failure was due to crack propagation initiated from fatigue loading.

Osakue developed a reliable model for a probabilistic design approach for shafts under combined bending and torsional loading, it also reported that for conditions of combined torsional and bending stress, distortion energy (DE) theory of failure was appropriate for the design analysis of the component14. Baig et al.15 investigated a shaft for failure and reported that failure always occurs at the areas of high stress concentration. Pelaseyed et al.16 studied the failure of the shaft of a unit and concluded that the failure was as a result of fatigue loading. Gurudath et al.17 investigated the failure of a bucket elevator shaft and reported that the failure was due to crack initiated at the heat affected zone of tack-welded spot on the shaft which propagated transversely by fatigue due to cyclic loading. Butlovic et al.18 study on design of machine parts such as shaft inferred that the use of CAD softwares provides simple and exact values of stress and deformation distribution on the shaft. It was also reported that the Goodman theory was widely used and adjudged to be a safe design approach in case of combined static and cyclic stress. Adekunle et al.19 developed a software for shaft design that satisfactorily handled design based on strength and safety. Sathishkumar et al.20 created a shaft model using CREO software, performed static analysis of the model to determine the stress and deformation under load using ANSYS software, it was concluded that the result obtained in good agreement and is within the safe limit. Ofolabi et al. investigated the parameters of fatigue life of a shaft and produced a 3D model of the shaft using Inventor software. It was reported that the results of the finite element analysis when compared with calculate values were satisfactory21. Engel et al. studied the failure mode of a shaft, analytic method was carried out to determine the stress and deformation of the shaft under combined torsion and bending loads, the results where then compared with that of finite element analysis method. It was concluded that the shaft needs some surface treatment to increase its fatigue life22. Rasovic et al. analyzed a SolidWorks-modeled drive shaft using finite element analysis and reported unsatisfactory result from the analysis as a result of the shaft geometrical change and low yield strength of the shaft material23. Robothan et al.24 analyzed a shaft and concluded that the behavior of shaft under combined loads can be accurately predicted using finite element analysis.

With the level of advancement in mechanical design in this era, shaft prototypes with optimal performance are developed through modeling of the physical system, as this helps to eliminate failure which comes with costly consequences25,26. With model-based design approach, shaft design engineers can make accurate prediction of the system performance as well as its service life, and make corrections of potential failure points of the shaft at the conceptual design stage. In this study, the limitations to obtaining safe shaft design for infinite life, such as the presence of stress risers and the centrifugal force action were analyzed using standard shaft design calculation procedure. The model of the head shaft was produced and simulated, so as to make accurate prediction of the service life of the designed head shaft.

Hence, this research work provides a comprehensive approach for design and modeling of the head shaft of a belt bucket elevator system which will not fail under fatigue as well as in predicting the service life of the shaft, using DE-Goodman criterion and SolidWorks engineering software.

Methodology

Physical model

The assembly model of the bucket elevator head as shown in Fig. 2 consists of the head pulley coupled to a head shaft supported by two self-aligning pillow bearings. The dimensions for the design analysis of the pulley were chosen at the conceptual design stage while that of the shaft were derived from the design calculations.

Figure 2
figure 2

Mechanical model of the elevator head shaft with pulley and its bearing support.

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Material selection

For the design, AISI 1020 steel was selected for the pulley because of its high ductility, machinability, strength, polished finished property and good weldability. AISI 1018 steel was selected for the elevator cups because of it great ductility, malleability, toughness, ease of welding and cheaper cost. AISI 1045 cold drawn steel was selected for the shaft because of its high tensile strength, hardness, excellent size accuracy, straightness and good surface condition27,28,29.

Design methodology

During operation of the belt bucket elevator system, the head shaft is subjected to torque from the gear drive motor and bending moment from the weight of the belt, cups filled with wheat grains and pulley, these generates torsional shear stress and bending stresses on the shaft, because of these reasons, the shaft was designed for combined stress using distortion energy theory of failure. The shaft design involves the determination of the preliminary diameter of the shaft. The graphical representation of the methodology is shown in Fig. 3.

Figure 3
figure 3

Graphical representation of the methodology.

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Hypothesis

For the model based design, it was considered that there was no resisting friction from the bearing and the weight of the shaft was negligible. It was also considered that both the pulley and the gear drive were mounted to the shaft using shrink disc, hence, the need for welding and keyway which act as stress risers was eliminated. It was also considered that the gear drive was well suspended by external clamp, hence its weight was not on the shaft. The grain product used was wheat.

The design analysis covers some areas which includes:

Pulley

Volume of pulley (Vp)

The head pulley of the elevator is a crowned type which uses flat belt. To determine the volume of the pulley, we split the pulley into two sections as shown in Figs. 4 and 5.

Figure 4
figure 4

Hollow cylinder of the Pulley.

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Figure 5
figure 5

Side cover plates of pulley.

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The hollow cylinder and the two side plates of the pulley were considered. The volume of the pulley was determined using Eq. (1)

$$ V_{p} = \frac{\pi W}{4}\left[ {d_{o}^{2} - d_{1}^{2} } \right] + \frac{{\pi t_{2} }}{2}\left[ {d_{3}^{2} - d_{4}^{2} } \right] $$
(1)

where VP—Volume of pulley, W—Width of pulley, dO—External diameter of pulley(drum), d1—Internal diameter of pulley, d3—External diameter of pulley side cover plate, d4—Internal diameter of pulley side cover plate, t1—Thickness of pulley plate, t2—Thickness of side cover plate.

Weight of head pulley

The weight(N) of the head pulley was determined from Eq. (2),

$$ W_{P} = \rho \; \times \; V_{P} \; \times \; g $$
(2)

where WP—Weight of the head pulley, \(\rho\)—Mass density of pulley material, g—Acceleration due to gravity.

Elevator belt

Belt length

The elevator operates on an open belt drive system as shown in Fig. 6.

Figure 6
figure 6

Sketch diagram of the belt pulley arrangement in belt elevator system.

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The length of the open belt drive was determined from Eq. (3) given as

$$ L_{B} = \frac{\pi }{2}\left[ {D_{d1} + D_{d2} } \right] + \left[ {\frac{{(D_{d1} - D_{d2} )^{2} }}{{4 \; \times \; S_{D} }}} \right] + 2 \; \times \;S_{D} $$
(3)

where LB—length of belt, Dd1—Diameter of the head pulley (Drive pulley), Dd2—Diameter of the bottom pulley (Driven pulley), SD—Centre to centre shaft distance,

θ1, θ2—Angel of contact (or wrap) for the head and bottom pulleys successively30,31.

Mass of belt (MB)

The mass of the elevator belt was determined from Eq. (4) given as

$$ M_{B} = \rho_{B} \; \times \; W_{tB} \; \times \; t_{B} \; \times \; L_{B} $$
(4)

where \(\rho_{B}\)—Density of belt, WtB—Width of the belt, tB—Thickness of belt, LB—Length of Belt.

The density of the belt was also determined from Eq. (5) as:

$$ \rho_{B} = M_{BL} \left( { \frac{1}{{W_{tB} \; \times \; t_{B} }}} \right) $$
(5)

where MBL—mass of belt per metre length.

Furthermore, the weight of the belt (WB) was determined using Eq. (6)

$$ W_{B} = M_{B} x g $$
(6)

Number of cups (TNC)

The total number of cups on the elevator was determined from Eq. (7) given as;

$$ T_{NC} = N_{CL} x L_{B} $$
(7)

where NCL—Number of cups per metre.

Volume of cup (Wc)

The elevator cup was sketched as represented in Fig. 7.

Figure 7
figure 7

Sketch diagram of elevator cup (bucket).

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From the sketched diagram, the volume of the cup was determined mathematically by considering a section of the cup in Fig. 7 and representing it in Fig. 8 as a triangle and semi-ellipse joined as a section.

Figure 8
figure 8

Section of the elevator cup (bucket).

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From Fig. 8, the length of X was determined from the triangular section using Pythagoras theorem and this was given in Eq. (8) as:

$$ X = \sqrt {H_{C}^{2} + W_{C}^{2} } $$
(8)

where X—Hypotenuse of the triangle, HC—Height of the cup, WC—Width of the cup.

Similarly, the radius(r) of the semi-ellipse section of Fig. 8 was determined using the derived Eq. (9) given below;

$$ r = \frac{{4P_{eC} - X\pi }}{2\pi } $$
(9)

where r—radius of semi- ellipse, PC ≈ Pec—Perimeter of semi-ellipse (surface curve length).

Thus, considering the hollow section of the elevator cup, the volume of the cup (VC) was determined as the difference between the volume of external cup dimension and the volume of the internal cup dimension; this is equivalent to the difference of the sum of the area of the triangle and area of the semi-ellipse multiplied by length for external section and that of the sum of the area of the triangle and area of semi-ellipse multiply by length for the internal section of the cup. Thus, the volume of the cup material was determined using derived Eq. (10) as given:

$$ V_{C} = \left[ {\left( {\frac{{W_{C} H_{C} }}{2} + \frac{\pi Xr}{4}} \right)L_{C} } \right] - \left[ {\left( {\frac{{(W_{C} - 2t_{C } )\left( {H_{C} - t_{C} } \right) }}{2} + \frac{\pi Xr}{4}} \right)\left( {L - 2t_{c} } \right)} \right] $$
(10)

where LC—Length of the cup, Wtc—Width (or projection) of cup, tc—Thickness (gauge) of cup material.

Mass of cup (MC)

The mass of each elevator cup was determined using Eq. (11) as shown below,

$$ M_{C} = \rho_{C} \; \times \;V_{C} $$
(11)

where \(\rho_{C}\)—Mass density of elevator cup material, VC—Volume of elevator cup.

Weight of cup (Wc)

The weight of each elevator cup was determined using Eq. (12) given below32,

$$ W_{C} = M_{C} \; \times \; g $$
(12)

Mass of wheat per cup (MW)

It is worthy to note that during operation, only 67% of the designed cup capacity is actually filled with the grain(wheat). Thus, the actual mass of wheat (excluding the cup weight) only transported by each cup was given in Eq. (13) as:

$$ M_{W} = \rho_{g} \; \times \;C_{C} \; \times \; 0.67 $$
(13)

where \({\uprho }_{{\text{g}}}\)—Bulk density of grain (wheat), \(C_{C}\)—Designed capacity of elevator cup.

The designed capacity of the elevator cup is equivalent to the volume of internal cup dimension, hence, the designed capacity of the elevator cup was determined from Eq. (14)33,34

$$ C_{C} = \left( {\frac{{W_{C} - 2t_{C} }}{2}} \right)\left( {H_{C} - t_{C} } \right) + \left( {\frac{\pi Xr}{4}} \right)\left( {L_{C} - 2t_{C} } \right) $$
(14)

Mass of cup and wheat (MCW)

The mass of each elevator cup carrying wheat was determined using Eq. (15)

$$ M_{CW} = M_{C} + M_{W} $$
(15)

Forces on shaft (FS)

The forces acting on the shaft are the weights of the head pulley, elevator cups, grain (wheat) and the belt. This was resolved using the free body diagram shown in Fig. 9

Figure 9
figure 9

Free body diagram (FBD) of top section of belt elevator system.

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Where F1—Tension(force) on tight side of belt, F2—Tension(force) on the slack side of belt, WP—weight of the head pulley, T—Torque.

Tension on tight side of belt (F1)

Taking centrifugal force into consideration, the force/tension on the tight side (F1) of the belt was determined using Eq. (16) below:

$$ F_{1} = F_{P} + F_{ce} + \frac{{F_{C} }}{2} $$
(16)

where FP—Initial (Preload) tension(force), FCE—Tension due to centrifugal force,

FC- Effective/circumferential force.

Effective/circumferential force (FC)

For the elevator belt to rotate, the circumferential force must be equal to the frictional force (FF) of the belt on the pulley interface (i.e. FF ≈ FC).

For this elevator system, the frictional force (FF) was given in Eq. (17) as:

$$ F_{C} \approx F_{F} = \mu .g\left[ {M_{B} + \frac{{M_{CW } \; \times \;T_{NC} }}{2} + \frac{{M_{C } \; \times \;T_{NC} }}{2}} \right] $$
(17)

where \(\mu\)—Coefficient of friction of elevator belt.

The coefficient of friction \(\left( \mu \right)\) of the elevator belt was determined from Eq. (18)

$$ \mu = 0.54 - \left( { \frac{42.6}{{152.6 + V_{eb} }}} \right) $$
(18)

where Veb—Velocity of the belt.

Considering the design capacity of the elevator, the velocity of the belt was determined from Eq. (19)

$$ V_{eb} = \frac{Q}{{M_{W } \; \times \; N_{CL} \; \times \; 3600}} $$
(19)

where Q—Capacity of the bucket elevator.

To ensure an effective throw of the grain(wheat) into the chute at the head pulley, a speed in the range of 1–2 m/s is recommended for centrifugal bucket elevator. Also, it has been reported that rubber coated flat belts running over pulleys in practice displays friction coefficient between 0.3 and 0.8. Thus, these criteria were used as validation values during the analysis35,36,37.

Initial tension (FP)

The Initial tension or preload force of the belt on the shaft was determined using Eq. (20)

$$ F_{P} = F_{C} \left[ {\frac{{\left( {e^{{\mu \theta_{1} }} } \right) + 1}}{{2\left( {(e^{{\mu \theta_{1} }} ) - 1} \right)}}} \right] $$
(20)

The angle of contact for the head (θ1) in radians was determined from Eq. (21) given as38:

$$ \theta_{1} = \left[ {180 + 2 \; \times \; Sin\left( {\frac{{D_{d1} - D_{d2} }}{{2 x S_{D} }}} \right)} \right]\; \times \; \frac{\pi }{180} $$
(21)

Centrifugal force (Fce)

For the elevator system, centrifugal forces are generated at the angle of wrap where the belt rotates around the pulley. The centrifugal force (Fce) was determined from the derived Eq. (22)

$$ F_{ce} = \left[ {\frac{{\left( {M_{CW} \; \times \; {\raise0.7ex\hbox{${N_{CA} }$} \!\mathord{\left/ {\vphantom {{N_{CA} } 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}} \right) + \left( {M_{C} \; \times \; {\raise0.7ex\hbox{${N_{CA} }$} \!\mathord{\left/ {\vphantom {{N_{CA} } 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}} \right)}}{{L_{ce} }}} \right]V_{eb}^{2} $$
(22)

where Lce—Length of belt contact on the pulley with reference to the angle of wrap ≈ Arc length of the pulley, NCA—Number of elevator cups within the Arc length of the pulley.

Furthermore, the elevator was designed such that the diameters of the head and bottom pulleys are equal, for this reason, the length of the belt contact on the pulley with reference to the angle of wrap will be equal to the length of semi-circle of the head pulley, this is shown in Fig. 10.

Figure 10
figure 10

Diagram of head pulley/belt contact.

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Therefore, Length of belt contact on the pulley with reference to the angle of wrap was determined from the derived Eq. (23)

$$ L_{ce} = \left( \pi \right)\; \times \;\frac{{D_{d1} }}{2} $$
(23)

The Number of cups within the arc length (NCA) was determined from the Eq. (24)

$$ N_{CA} = N_{CL} \; \times \;L_{ce} $$
(24)

Tension on slack side of belt (F2)

The tension on the slack side of the elevator belt was determined using Euler model in Eq. (25), which is the simplest theoretical model for belt drives39,40.

$$ \frac{{F_{1} }}{{F_{2} }} = e^{{\mu \theta_{1} }} $$
(25)

Total force on shaft (FS)

The total force acting of the shaft was determined from Eq. (26) stated below:

$$ F_{s} = F_{1} + F_{2} + W_{P} $$
(26)

Torque (T)

The total torque(T) required to operate the elevator system was determined from Eq. (27) given as T = Torque from belt (TB) + Torque to rotate the head pulley (THP) + Torque to rotate the bottom pulley (TBP).

This is represented thus:

$$ T = \frac{{\left( {F_{1} - F_{2} } \right)D_{d1} }}{2} + J_{1} \alpha_{1} + J_{2} \alpha_{2} $$
(27)

where J1—Mass moment of Inertia of the head pulley, J2—Mass moment of Inertia of the bottom pulley, \(\alpha_{1} - \user2{ }\) Angular acceleration of the head pulley, \(\alpha_{2}\)- Angular acceleration of the bottom pulley.

$$ {\text{Generally}},\quad J_{1} = \frac{{\pi \rho W\left( {r_{0}^{4} - r_{1}^{4} } \right)}}{2} + n_{sp} \left[ {\frac{{\rho V_{{2\left( { r_{3}^{2} + r_{4}^{2} } \right)}} }}{2}} \right] $$

Thus, the mass moment of Inertia of the head pulley (J1) was derived from Eq. (28) below:

$$ J_{1} = \frac{{\pi \rho W\left( {d_{0}^{4} - d_{1}^{4} } \right)}}{32} + n_{sp} \left[ {\frac{{\rho V_{2} \left( {d_{3}^{2} + d_{4}^{2} } \right)}}{8}} \right] $$
(28)

where nsp—Number of side plates of the head pulley.

Furthermore, the angular acceleration of the head pulley (\({\upalpha }_{1}\)) was determine from Eq. (29)

$$ \alpha_{1} = \frac{{2 \cdot V_{eb}^{2} }}{{d_{0} }} $$
(29)

Since the dimensions of the head and bottom pulleys are equal and of the same material, it therefore means: J1 \(\alpha_{1}\) ≈ J2 \(\alpha_{2}\)41,42

Power (P)

The power required to operate this elevator was determined from Eq. (30) given as:

$$ P = \frac{{C \times Q \times S_{D} }}{367} $$
(30)

where C—Coefficient factor (for grain: 1.2)43.

In summary, the principal dimensions and mechanical parameters for the elevator system with respect to the pulley and belt are listed in Tables 1 and 2.

Table 1 Parameters for the shaft load (force) design.
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Table 2 Calculated parameter values based on Table 1 definition.
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The parameters from Table 1 were chosen at the conceptual design stage for the elevator system, the parameters in Table 2 were calculated using parameters in Table 1.

Shaft diameters

Considering the space on the elevator where the shaft will be mounted, a schematic diagram with length dimensions (mm) was developed as shown in Fig. 11.

Figure 11
figure 11

Schematic diagram of the shaft.

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Where d5, d6, d3, d7—diameter of each shoulder section of shaft in mm. Also, the length of the shaft cross sections is in mm.

The shaft design was based on determining the safe design diameters for each section of the shaft to carry loads efficiently without failure. Thus, the areas of the design analysis covered are:

Support reactions

This was determined from bending moment diagram. The free body diagram was first developed as from the schematic diagram in Fig. 11 by considering the shaft as a beam as shown in Fig. 12.

Figure 12
figure 12

Free body diagram (FBD) of the shaft in beam representation.

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From the design, the two sections of the pulley side plates are fixed to the shaft. For this reason,

FS1 is equal to FS2 (i.e. FS1–FS2). Thus, FS1 is determined using Eq. (31)

$$ F_{S1} = \frac{{F_{S} }}{2} $$
(31)

According to Newton’s law for linear mechanical system, the sum of external forces acting on a rigid body is equal to the mass times the acceleration, as shown in Eq. (32)

$$ \Sigma F_{External} = Ma $$
(32)

D’ Alembert’s law also relates the sum of all forces acting on a rigid body as shown in Eq. (33)44,45,46:

$$ \Sigma F_{All} = 0 $$
(33)

Applying these two laws, the dynamic equation of motion for the shaft system are as follows in Eq. (34):

$$ \Sigma F_{All} = 0 $$
$$ \begin{gathered} R_{A} + R_{B} - F_{S1} - F_{S2} = 0 \hfill \\ R_{A} = F_{S1} + F_{S2} - R_{B} \hfill \\ \end{gathered} $$
(34)

Also, summation of all moments about Point A is equal to zero, as shown in Eqs. (35) and (36)

$$ \Sigma M_{A} = 0 $$
(35)
$$ R_{B} x \left( {L_{AE} + L_{EF} + L_{FB} } \right) - F_{S2} x \left( {L_{EF} + L_{AE} } \right) - F_{S1} \left( {L_{AE} } \right) = 0 $$
$$ R_{B} = \frac{{F_{S2} \times \left( {L_{EF} + L_{AE} } \right) + F_{S1} \left( {L_{AE} } \right) }}{{L_{AE} + L_{EF} + L_{FB} }} $$
(36)

where RA, RB are the reactions from support bearings 1 and 2.

Bending moment

Taking sections of the beam,

The bending moment at E from Fig. 13 was determined from Eq. (37),

Figure 13
figure 13

FBD for analysis of section AE.

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Therefore, bending moment at E (ME), we have Eq. (37),

$$ M_{E} = R_{A} \; \times \; X_{AE} $$
(37)

Thus, the bending moment at A (i.e. XAE = 0), is MA = 0.

The bending moment at F from Fig. 14 was determine from Eq. (38)

Figure 14
figure 14

FBD for analysis of section EF.

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Bending moment at F (MF)

$$ M_{F} = R_{A} \left( {L_{AE} + X_{EF} } \right) - F_{S1} X_{EF} $$
(38)

Bending moment at G (MG) was determined from Eq. (39):

$$ M_{G} = R_{A} \left( {L_{AE} + L_{EF} + X_{FG} } \right) - F_{S1} \left( {L_{EF} + X_{FG} } \right) - F_{S2} X_{FG} $$
(39)

Bending moment at H (MH) was determined from Eq. (40):

$$ M_{H} = R_{A} \left( {L_{AE} + L_{EF} + X_{GH} } \right) - F_{S1} \left( {L_{EF} + L_{FH} } \right) - F_{S2} L_{FH} $$
(40)

Bending moment at B (MB) was determined from Eq. (41):

$$ M_{B} = R_{A} \left( {L_{AE} + L_{EF} + X_{FB} } \right) - F_{S1} \left( {L_{EF} + X_{FB} } \right) - F_{S2} X_{FB} $$
(41)

Bending moment at J (MJ) was determined from Eq. (42):

$$ M_{X} = R_{A} \left( {L_{AE} + L_{EF} + L_{FB} + X_{BJ} } \right) + R_{A} X_{BJ} - F_{S1} \left( {L_{EF} + L_{FB} + X_{BJ} } \right) - F_{S2} \left( {L_{FB} + X_{BJ} } \right) $$
(42)

In summary, the principal and mechanical parameters of the elevator shaft system with respect to bending moment analysis are listed in Tables 3 and 4

Table 3 Parameters for bending moment analysis.
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Table 4 Calculated parameter values based on Table 3 definitions.
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The parameters from Table 3 were chosen at the conceptual design stage for the elevator shaft, the parameters in Table 4 were calculated using parameters in Table 3.

The bending moment diagram is shown in Fig. 15 below47.

Figure 15
figure 15

Bending moment diagram for the shaft analysis.

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Because stresses are usually higher at the points along the surface of the shaft where there are shoulders which invariably are stress risers, fatigue cracks are most likely to originate from these points and progress to fatigue failure of the shaft, hence for a safe design, the diameter of the shaft will be determined by considering this sections as follows:

For small diameter of the shoulder at H (d5)

At this section, torque and bending moments are present. This is also the section of the shaft where the bearings are installed, hence the section was designed with sharp shoulder (step). Assuming generous fillet radius, the standard 1st estimated recommended for geometric stress concentration for shaft with shoulder fillet radius are:

$$ {\text{r}}/{\text{d }} = \, 0.0{2},{\text{ K}}_{{{\text{tB}}}} = { 2}.{7},{\text{ K}}_{{{\text{tT}}}} = {2}.{2} $$

where r—fillet radius, d—smaller diameter, KtB—theoretical stress concentration factor for bending, KtT—theoretical stress concentration factor for torsion.

Also, for quick conservative 1st pass, we assume KtT = KFATT1 = 2.7, KtB = KFATB1 = 2.2

Where KFATT—Fatigue stress concentration factor for torsion in 1st estimate, KFATB—Fatigue stress concentration factor for bending in 1st estimate.

The diameter (d5) at the shoulder at H was determined using DE-Goodman criterion as shown in Eq. (43), considering alternating torque and midrange bending moment for rotating shaft was equal to zero.

$$ d_{5} = \left[ {\frac{{16n_{1} }}{\pi }\left\{ {\frac{{[4(K_{FATB1 } \; \times \;M_{H} )^{2} ]^{1/2} }}{{Se_{1} }} + \frac{{[3(K_{FATT1} \; \times \; Tm)^{2} ]^{1/2} }}{Sut}} \right\}} \right]^{{^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}}} }} $$
(43)

where Se1—Endurance limit correction factor for 1st pass, Sut—Tensile strength of material (AISI 1045 steel, cold drawn), n1—Safety factor for 1st pass, MH—Alternating bending moment at H, Tm—midrange torque.

The endurance limit correction factor (Se1) for 1st pass was determined from Eq. (44) given as:

$$ Se_{1} = K_{a} \; \times \; K_{b1} \; \times \; K_{C} \; \times \; k_{d} \; \times \; K_{e} \; \times \; K_{F} \; \times \; Se^{\prime } $$
(44)

where Ka—Surface factor, Kb1—Size factor for 1st pass, KC—Load factor, Kd—Temperature factor, Ke—Reliability factor, KF—Miscellaneous factor, \(Se^{\prime }\)—Uncorrected endurance limit.

The surface factor (Ka) was determined from Eq. (45) given as:

$$ K_{a} = aSut^{b} $$
(45)

where a and b are constant (Values of a = 4.51 and b = − 0.265 for machined/cold drawn shaft).

The size factor (Kb1) = 1 (for 1st estimate), Load factor (Kc) = 1(for combined loading), temperature factor (Kd) = 1(Assumed room temperature), Reliability factor (Ke) = 1, Miscellaneous factor (Kf) = 1.

The uncorrected endurance limit was determined from Eq. (46)

$$ Se^{\prime } = 0.5 Sut $$
(46)

Furthermore, considering that the stress type and material condition is uncertain, a safety factor (n1) of 5 (for the 1st pass) was used48

From Eq. (43), the standard shaft size (i.e. d5 = 90 mm) was analyzed to determine its fatigue and yielding factors of safety.

From DE-Goodman equation, the fatigue factor safety was determined from Eq. (47) given as:

$$ n_{5} = \frac{{Se_{5} \; \times \; Sut}}{{\left( {Sut \; \times \; \delta a_{5}^{\prime } + Se \; \times \; \delta m_{5}^{\prime } } \right)}} $$
(47)

where Se5—endurance limit correction factor for d5, \(\delta a_{5}^{1} \)—Alternating von misses stress for d5, \(\delta m_{5}^{1}\)—midrange von misses stress for d5.

The endurance limit correction factor for d5 (Se5) was determined from Eq. (48) given as:

$$ Se_{5} = K_{a} \; \times \;K_{b5} \; \times \; K_{C} \; \times \; k_{d} \; \times \; K_{e} \; \times \; K_{F} \; \times \; Se^{\prime } $$
(48)

The size factor for d5 (Kb5) was calculated from Eq. (49) given as:

$$ K_{b5} = 1.51\; \times \; d_{5}^{ - 0.157} $$
(49)

Also, the alternating von misses stress (\(\delta a_{5}^{1}\)) was determined from Eq. (50):

$$ \delta a_{5}^{1} = \frac{{K_{FATBH} \; \times \; 32M_{H} }}{{\pi d_{5}^{3} }} $$
(50)

KFATBH is the fatigue stress concentration factor for bending at section(point) H and was determined using Eq. (51)

$$ K_{FATBH} = 1 + \frac{{\left( {K_{tBH} - 1} \right)\sqrt {r_{H} } }}{{\left( {\sqrt {r_{H} } + \sqrt {a_{B} } } \right)}} $$
(51)

where \(\sqrt {a_{B} }\)—is the Neuber constant for bending given in Eq. (52) as:

$$ \sqrt {a_{B} } = 0.246 - 3.08 \; \times \; 10^{ - 3} Sut + 1.51 \; \times \; 10^{ - 5} Sut^{2} - 2.67\; \times \; 10^{ - 8} Sut^{3} $$
(52)

\(r_{H}\) is the fillet radius at H, KtBH is the theoretical stress concentration factor for bending at point H and its value was determine from chart of theoretical stress concentration for shaft with shoulder fillet in bending for \(\frac{{d_{6} }}{{d_{5} }}\) against \(\frac{{r_{H} }}{{d_{5} }}\)49.

d6 was determined from Eq. (53):

$$ d_{6} = d_{5} + 2 x h_{SH} $$
(53)

where \(h_{SH}\) is the shoulder height at section H.

Considering the bearing(self-aligning bearing with adaptive sleeve) placement at section H(d5) and the standard shaft size (d5 = 90 mm), the fillet radius and shoulder height for the shaft from bearing catalogue are \(r_{H} = 2\;{\text{mm}}\; and\; h_{SH} = 4\;{\text{mm}}\)50.

Similarly, the midrange von misses stress for d5 was determined from Eq. (54)

$$ \delta m_{5}^{^{\prime}} = \left[ {3\left( {\frac{{K_{FATTH } \times \;16Tm}}{{\pi d_{5}^{3} }}} \right)^{2} } \right]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} $$
(54)

KFATTH is the fatigue stress concentration factor for torsion at section H(d5) and was determined from Eq. (55):

$$ K_{FATTH} = 1 + \frac{{\left( {K_{tTH} - 1} \right)\sqrt {r_{H} } }}{{(\sqrt {r_{H} } + \sqrt {a_{T} } }} $$
(55)

where \(\sqrt {a_{T} }\)—is the Neuber constant for torsion given in Eq. (56) as:

$$ \sqrt {a_{T} } = 0.190 - 2.57\; \times \; 10^{ - 3} Sut\; + \;1.35\; \times \; 10^{ - 5} Sut^{2} \; - \;2.62\; \times \; 10^{ - 8} Sut^{3} $$
(56)

KtTH is the theoretical stress concentration factor for torsion at point H and its value was determine from chart of theoretical stress concentration for shaft with shoulder fillet in torsion for \(\frac{{d_{6} }}{{d_{5} }}\) against \(\frac{{r_{H} }}{{d_{5} }}\)49.

Furthermore, the yielding factor of safety for section H (d5) was determined from Eq. (57):

$$ n_{y5} = \frac{Sy}{{\delta max_{5}^{^{\prime}} }} $$
(57)

where Sy—yield strength of AISI steel material, \(n_{y5}\)—yielding factor of safety for section H, \(\delta max_{5}^{^{\prime}}\)—Von misses Maximum stress at section H (d5) determined from Eq. (58)

$$ \delta max_{5}^{^{\prime}} = \left[ {\left( {\frac{{K_{FATBH } x32M_{H} }}{{\pi d_{5}^{3} }}} \right)^{2} + 3\left( {\frac{{K_{FATTH } x 16Tm}}{{\pi d_{5}^{3} }}} \right)^{2} } \right]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} $$
(58)

Generally, for safe design, \(\delta max_{5}^{^{\prime}} \) ≤ Sy51.

For small diameter of the shoulder at G (d6)

At this section, there was presence of bending and torsion. The fatigue factor of safety was determined from Eq. (59)

$$ n_{6} = \frac{{Se_{6} x Sut}}{{\left( {Sut x \delta a_{6}^{^{\prime}} + Se x \delta m_{6}^{^{\prime}} } \right)}} $$
(59)

where Se6—endurance limit correction factor for d6, \(\delta a_{6}^{1} \)–Alternating von misses stress for d6, \(\delta m_{6}^{1}\)- midrange von misses stress for d6.

The endurance limit correction factor for d6 (Se6) was determined from Eq. (60) given as:

$$ Se_{6} = K_{a} \; \times \;K_{b6} \; \times \; K_{C} \; \times \; k_{d} \; \times \;K_{e} \; \times \; K_{F} \; \times \; Se^{^{\prime}} $$
(60)

The size factor for d6 (Kb6) was calculated from Eq. (61) given as:

$$ K_{b6} = 1.51 \; \times \;d_{6}^{ - 0.157} $$
(61)

Also, the alternating von misses stress (\(\delta a_{6}^{1}\)) was determined from Eq. (62):

$$ \delta a_{6}^{1} = \frac{{K_{FATBG} \; \times \; 32M_{G} }}{{\pi d_{6}^{3} }} $$
(62)

KFATBG is the fatigue stress concentration factor for bending at section(point) G and was determined using Eq. (63):

$$ K_{FATBG} = 1 + \frac{{\left( {K_{tBG} - 1} \right)\sqrt {r_{G} } }}{{(\sqrt {r_{G} } + \sqrt {a_{B} } }} $$
(63)

where \(r_{G}\) is the fillet radius at G, KtBG is the theoretical stress concentration factor for bending at point G and its value was determined from chart of theoretical stress concentration for shaft with shoulder fillet in bending for \(\frac{{d_{3} }}{{d_{6} }}\) against \(\frac{{r_{G} }}{{d_{6} }}\)49

In design of machine shaft where bending and torsion loading are present, small value of D/d near 1 is recommended. For this, d3 was determined from Eq. (64)

$$ d_{3} = 1.1 \; \times \; d_{6} $$
(64)

To obtain least stress concentration factor for shaft shoulder, the fillet height was assumed to be equal to the fillet radius52 Thus, rG was obtained from Eq. (65):

$$ r_{G} = \frac{{d_{3} - d_{6} }}{2} $$
(65)

Similarly, the midrange von misses stress for d6 was determined from Eq. (66)

$$ \delta m_{6}^{\prime } = \left[ {3\left( {\frac{{K_{FATTG } \; \times \; 16Tm}}{{\pi d_{6}^{3} }}} \right)^{2} } \right]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} $$
(66)

KFATTG is the fatigue stress concentration factor for torsion at section G(d6) and was determined from Eq. (67):

$$ K_{FATTG} = 1 + \frac{{\left( {K_{tTG} - 1} \right)\sqrt {r_{G} } }}{{\left( {\sqrt {r_{G} } + \sqrt {a_{T} } } \right)}} $$
(67)

KtTG is the theoretical stress concentration factor for torsion at point G and its value was determined from chart of theoretical stress concentration for shaft with shoulder fillet in torsion for \(\frac{{d_{3} }}{{d_{6} }}\) against \(\frac{{r_{G} }}{{d_{6} }}\)49.

Furthermore, the yielding factor of safety for section G (d6) was determined from Eq. (68):

$$ n_{y6} = \frac{Sy}{{\delta max_{6}^{\prime } }} $$
(68)

where \(n_{y6}\)—yielding factor of safety for section G, \(\delta max_{6}^{\prime }\)—Von Misses Maximum stress at section G (d6), it was determined from Eq. (69)

$$ \delta max_{6}^{\prime } = \left[ {\left( {\frac{{K_{FATBG } \times 32M_{G} }}{{\pi d_{6}^{3} }}} \right)^{2} + 3\left( {\frac{{K_{FATTG } \times 16Tm}}{{\pi d_{6}^{3} }}} \right)^{2} } \right]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} $$
(69)

For safe design, \(\delta max_{6}^{\prime } \) ≤ Sy.

For small diameter of the shoulder at I (d7)

At this section, only torque is present. The load is entirely shear stress; hence the yielding factor of safety was determined from Eq. (70):

$$ n_{y7} = \frac{Sys}{{\tau max_{7}^{\prime } }} = \frac{0.577 \; \times \; Sy}{{\tau max_{7}^{\prime } }} $$
(70)

where Sys—yield shear strength, \(\tau max_{7}\)—maximum shear stress at I (d7) and \(\delta max_{7}^{\prime }\)—Von misses Maximum stress at section I (d7) where determined from Eqs. (71) and (72):

$$ \tau max_{7} = \frac{{\delta max_{7}^{\prime } }}{\sqrt 3 } $$
(71)
$$ \delta max_{7}^{\prime } = \sqrt 3 x \left( {\frac{{K_{FATTI} \; \times \; 16Tm}}{{\pi d_{7}^{3} }}} \right) $$
(72)

where KFATTI is the fatigue stress concentration factor for torsion at section I (d7) and was determined from Eq. (73):

$$ K_{FATTI} = 1 + \frac{{\left( {K_{tTI} - 1} \right)\sqrt {r_{I} } }}{{\left( {\sqrt {r_{I} } + \sqrt {a_{T} } } \right)}} $$
(73)

KtTI is the theoretical stress concentration factor for torsion at point I and its value was determined from chart of theoretical stress concentration for shaft with shoulder fillet in torsion for \(\frac{{d_{5} }}{{d_{7} }}\) against \(\frac{{r_{I} }}{{d_{7} }}\)49.

Also, rI is the fillet radius at I and was determined from Eq. (74):

$$ r_{I} = \frac{{d_{5} - d_{7} }}{2} $$
(74)

The value of d7 was determined from Eq. (75):

$$ d_{7} = \frac{{d_{5} }}{1.1} $$
(75)

Thus, for safe design, \(\tau max_{7}^{\prime }\) \(<\) \(Sys\).

In summary, the principal parameters for determining the safe diameters of the sections of the elevator shaft are listed in Tables 5 and 6

Table 5 Parameter for shaft diameter.
Full size table
Table 6 Calculated parameter values based on Table 5 definition.
Full size table

The parameters from Table 5 were chosen at the conceptual design stage for the elevator shaft, the parameters in Table 6 was calculated using parameters in Table 5.

Model development

The model of the shaft was produced using Solidworks, which is a solid modelling computer-aided design and computer aided engineering programme. The 2D line sketch of the half section of the shaft with reference to a center axis line was first created. The dimensions from the design calculations were then added to the sketch so as to define its sizes. The revolved boss/base tool of the feature manager was then used to rotate the contour of the line sketch about the axis line thereby creating a round shape object which is the shaft model as shown in Fig. 16.

Figure 16
figure 16

Shaft model.

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Simulation and boundary conditions

Because the shaft will be subjected to static and cyclic loading over its service life, most commonly, the shaft will tend to fail under fatigue. For proper design of the shaft, a stress-life (S–N) curve which is one of the fatigue life methods to predict the number of cycles to failure for specific load level the shaft will be subjected to was constructed. A well-defined stress-life (S–N) characteristic of the shaft material will aid in obtaining precise fatigue life prediction for the shaft. The simulation analysis of the shaft model for both static and fatigue loading was done using Solidworks engineering software. The software uses finite element analysis (FEA) in predicting real world physical behavior of modeled mechanical components53. For this analysis, the shaft model was first created from the calculated shaft sizes, followed by static analysis using the following calculated parameters in Table 7.

Table 7 Parameters for static analysis.
Full size table

The step-by-step procedure for the simulation on solidworks are as follows:

Material

AISI 1045 cold drawn steel was selected as the shaft material. The material properties of the shaft material are given in Table 8.

Table 8 Properties of the shaft material.
Full size table

Fixtures

The shaft has a bearing on the left and right hand of the shaft, for this reason, bearing fixture was applied at the left and right hand end of the shaft as shown in Fig. 17.

Figure 17
figure 17

Fixtures and external loads on the shaft model.

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External loads

Equal forces of 8157.5 N were applied at the slit sections of the shaft where the pulley is fixed. A torque of 2316 Nm was also applied to the shaft end at a position the gear drive motor will be coupled, as also shown in Fig. 17.

Mesh

This is a crucial step in design analysis which involves subdividing the model into small pieces. A good mesh increases the accuracy of the simulation result. Mesh setting was first carried out before the meshed model of the shaft was produced. The mesh setting was done in steps as shown in Fig. 18.

Figure 18
figure 18

Mesh setting sequence.

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The meshed model and a complete mesh information as shown in Fig. 19 and Table 9 were subsequently generated after the application of the parameter setting. The simulation was then put to run after the setting process.

Figure 19
figure 19

Meshed model of the head shaft.

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Table 9 Mesh generated details.
Full size table

The parameter setting used for fatigue are as follows.

  • Criterion MAX Von Misses Stress

  • Computing alternating stress Equivalent stress (Von misses)

  • Mean stress correction Goodman

The simulation results for the maximum von misses stress and that of the yielding factor of safety were determined from the static analysis. Following that, fatigue analysis was carried out with the same software and the value for the fatigue life, fatigue load and S–N curve were equally obtained from the simulation results.

Results and discussion

At this conceptual design stage, the performance of the head shaft was tested and compared with value obtained from the analytic design calculations. The results (Figs. 20, 21, 22, 23, 24, 25, 26, 27, 28) were obtained from the simulation analysis of the model. The results from static analysis are Figs. 20, 21, 22 and 23 while that of fatigue analysis are Figs. 24, 25, 26, 27 and 28.

Figure 20
figure 20

Plots of the stress.

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Figure 21
figure 21

Simulation graph for stress.

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Figure 22
figure 22

Plots of factor of safety.

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Figure 23
figure 23

Simulation graph for factor of safety.

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Figure 24
figure 24

Fatigue load plot.

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Figure 25
figure 25

Simulation graph for fatigue load.

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Figure 26
figure 26

Fatigue life plot.

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Figure 27
figure 27

Simulation graph for fatigue life.

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Figure 28
figure 28

Fatigue stress-life (S–N) curve.

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Plots from static analysis

See Figs. 20, 21, 22 and 23.

Plots from fatigue analysis

From the plots (Figs. 20 and 21), the maximum Von misses stress occurred at the small diameter of section I and the value was 4.817 × 107 N/m2 which was equivalent to the value (4.852 × 107 ≈ 48.52 N/mm2) obtained from the design calculation. The value of the maximum von misses stress was less than the yield strength value of the shaft material, this signifies that the shaft will operate in the elastic region, this means that the designed diameters calculated for the shaft are safe for the elevator system operation. From the plots in Figs. 22 and 23, the yielding factor of safety at the small diameter section of I was 11 and was equivalent to the calculate value (\(n_{y7}\) = 11), this shows that a generous design factor was considered for the head shaft design. The plots in Figs. 24 and 25 show that the load factor was 11.55, this was higher than 1 (i.e. 11.55 > 1), hence this signifies the shaft will not go into failure mode within the 1.0 × 106 cycles. The plots of Figs. 26 and 27 show that the value of the fatigue life was 1.0 × 106 cycles, this result signifies infinite life for the shaft. This was validated from the result of the S–N curve of Fig. 28 which shows a fatigue strength of 3.62 × 108 N/m2, this value was higher than the value of the maximum von misses stress (4.817 × 107 N/m2), thus signifying that the shaft will not fail but will survive indefinitely in operation. The closeness of the calculated values with the values from the analysis show that FEA is a very useful and effective tool for design of mechanical components that will be subjected to combined loading as well as for predicting the service life of the component.

Also, this research used the DE-Goodman criterion because of the need for accurate prediction of failure for the ductile shaft material and also due to the combined load of bending and torsion on the head shaft. The analysis also showed the difficulty of getting optimized numerical values for some design parameters such as the material variation and shaft weight, this point may surely be future research work for a more accurate and optimized head shaft model analysis. Also, fatigue test of the developed model should be performed in future research to confirm the theoretical data.

Conclusion

This study presented a detailed approach for the design, modeling and simulation analysis of the head shaft of a belt bucket elevator for transporting grains (wheat). As a critical component of the elevator system, the forces acting on the shaft were first determined, the load type and the stresses at potential critical locations were considered in the design calculations. To ensure a proper design, modeling and simulation analysis were performed on the designed shaft so as to determine the fatigue strength and to make accurate prediction of the fatigue life of the head shaft. The result from the simulation analysis shows that the designed diameters for the shaft were safe for operation considering the 200 tons/h design capacity of this bucket elevator system.

This study can be used in industry as a reference for predicting the service life of shaft designed for different capacity of elevator systems. It will also help industrial engineers to have better understanding of the behavior of elevator shaft as well as to gain advanced knowledge on how to improve the fatigue life of shafts in manufacturing machines using FEA tools; this will directly reduce cost associated to downtime.

Note also that the complexity of the model is such that authors where not able to show and discuss all the details of their work. The authors stay away with pleasure to the disposal of the interested readers for any further discussion on the approach followed here.