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Stability analysis of trailed tankers on slopes
J. agric. Engng Res. (1985) 32,3 1l-320
Stability
Analysis of Trailed Tankers on Slopes A. G. M. HUNTER*
The stability analysis of trailed tankers is a new area, which has not previously been covered in the study of farm machines on slopes. It is an important area because tankers have increased in size over recent years and the massive quantity of fluid carried has a dominating effect. The overturning accidents which occur, arise from a previously unexplored cause, namely that the overall centre of gravity of the tanker moves markedly with movement of the fluid inside, unlike the immobile centre of gravity of a trailer filled with a solid load. The stability analysis covers tankers with circular and rectangular cross-section tanks. An unexpected result is that the stability of a rectangular cross-section tanker can be least when the tank is only 30% full. The analysis has been confirmed experimentally. 1.
Introduction
The stability of farm vehicles has been studied extensively because tractor overturning accidents may result in severe injury to the driver or even in his death. Often, an accident is caused by a trailer, which is less stable than the towing tractor, overturning and pulling the tractor with it.’ One group of farm trailers which has not previously been studied comprises farm tankers normally used either for slurry spreading or for crop spraying. When a tanker is driven on a slope, its overall centre of gravity shifts due to the fluid inside finding its own level. The tanker may be inclined at a compound angle and its stability is therefore a function of the slope and of the direction in which the tanker is facing on the slope. The stability is also a function of the fraction of tanker capacity filled with fluid, and this fraction decreases continuously as the tanker contents are discharged onto the field. The need for stability analysis of tankers became apparent after a few accidents had been investigated, where trailed tankers had overturned on slopes. In one example, a 4000 1 capacity slurry tanker of nearly rectangular cross-section overturned in a field of slope 24” while partly full. Using existing methods of stability analysis for a trailer with an immobile centre of gravity,’ the stability limit (i.e. overturning slope) for the empty tanker was calculated as 37”, and for the full tanker 28”, both values greater than the actual overturning value of 24”. The displaced centre of gravity of the fluid in the tanker on a slope was then roughly estimated and the corresponding stability limit was calculated as 23”, which was less than the stability limit for the tanker empty or full. The first stage in improving the centre of gravity estimate for this and other tankers was to develop a centre of gravity analysis of fluid in inclined tanks .j The second and final stage, the subject of this paper, was to combine the new centre of gravity analysis with the existing stability analysis for trailers. The method followed is described in the next section. The succeeding sections describe how tanker stability can best be presented graphically, and how the analysis was experimentally confirmed. It was confined to tanks of circular and rectangular cross-sections, because these shapes approximately fitted most farm tankers. The result described above, where the static stability on a slope is least when the tanker is only partly full, is unexpected and has not previously been recognized. It may also apply to off-road vehicles outside agriculture, for example military ones. Although there has been considerable work on the dynamic effects of liquids in tanks fitted to road, rail, sea, air and space vehicles, it has all been related to level or upright tanks.’ There has been no attempt here to cover dynamic of Agricultural Engineering,
*Scottish
Institute
Received
28 September
1983; accepted
Bush Estate,
Penicuik,
t” revised form 17 November
Midlothm
EH26 OPH, Scotland
1984
311 0021-8634r85/080311+
10 $03,M):O
a
1985 The British Society
for Research
in Agricultural
Engineering
312
STABILITY
ANALYSIS
OF TRAILED
TANKERS
NOTATION G j
k 2t m
OXYZ oxyz O’uvw x V
! 2 P v, b ; S
centre of gravity hitch distance forward from axle hitch height track width mass tank-fixed axis system moveable axis system trailer axis system centre of gravity co-ordinate (typical) volume heading angle slope slope of fluid surface relative to ox fraction of tank capacity filled by fluid density of fluid rotation angle Yoy Subscripts
body of trailer container (tank) fluid stability
effects, because static stability on compound slopes is considered to be the dominant problem. The question of the optimum shape of tanker for use on slopes is a closely related subject which will be covered in a later paper. 2.
Method of analysis
The analysis follows from that for the centre of gravity of fluid in tanks. A trailed tanker is shown schematically in Fig. 1. The tank is of circular cross-section, partially filled with fluid and mounted transversely on a trailer frame. The trailer rests on a place surface of slope /I, at a heading angle a to the slope line. The set of axes 0’ uvw is trailer-fixed with origin at the wheel-ground contact point, 0’ shown. The set of axes OXYZ is tank-fixed, parallel to 0’ uvw, with origin, 0, at the tank centre. The third set of axes, oxyz, is rotated through the angle v, about the -0X direction so that oy lies parallel to the fluid surface. As explained in the centre of gravity paper, the origin of this third set may move for the convenience of the analysis. The fluid surface lies at an angle, 6, to the axis ox. The relationship between the angles is found to be a, = tan- ‘(cosa tan /3),
. . .(1)
S = sin- ‘(sina sin 8).
. . .(2)
and
The transverse tank shown is representative of the arrangement on some types of crop sprayer. On other types, the tank is mounted in the fore-and-aft direction, i.e. the tank is rotated 90” about the 0’ w direction so that OX lies parallel with 0’ v. The set of axes oxyz also rotates so that ox remains parallel with OX. In this case the relationship between the angles is
313
A. G. M. HUNTER
Fig. 1. Circular cross-section tanker partly filled with Jluid facing along heading angle, a, on slope. /I. Three sets of coordinate axes shown
W
Z t
t
I--+--+j Y
1
0'
0'
I
I
J
u
I
2t
i
Fig. 2. Geometry o/rectangular cross-section tanker
withfiwed co-ordinate axis systems
p=tan-‘(-sinatanj?),
. . 43)
and 6=&t-’
(cosasinB).
. . .(4)
Tanks with rectangular cross-section are also commonly found, as in the slurry tanker accident example mentioned in the Introduction. Such tanks are considered as belonging to the transverse case, with Eqns (1) and (2) applying, but care must be taken that the correct tank dimensions are associated with the tank axes OX, 0 Y and OZ (Fig. 2). Tanker stability at a known heading angle a, with a known fraction A of the tank capacity filled
314
STABILITY
ANALYSIS
OF TRAILED
TANKERS
with fluid, is determined iteratively. First, on an assumed slope j3, the centre of gravity of the fluid G, is found relative to tank-fixed axes OXYZ using the methods and equations given in the earlier paper. The co-ordinates are . . .(5)
X, = X, (a, P, n),
and Y, and Z, which are similarly dependent on a, j3 and 2. The calculated co-ordinates transformed to co-ordinates in the trailer-fixed axes 0’ uvw as follows:
. . .(6)
Ur=uo+X,, with similar equations for vf and wf. If the tank is mounted in the fore-and-aft position, the transformation $=u,-
are
becomes
r,,
. . (7)
with similar equations for vf and wf. The overall centre of gravity is found from _ Zd=
mf”f
_ + mbUb
mf+mb
’
. . (8)
where the mass of fluid . . .(9)
m,=pVf =pAV,.
The equations for the V and Wco-ordinates are similar. The stability limit & is the slope on which the gravity vector passing through the overall centre of gravity of the tanker intersects one side of AO’JK. The equations for & may be derived using vector analysis as (ty-Jq
tan j3,=
sina - (ku- tE) cosa
@-j$
. . .(10)
for tipping about 0’ K, or tan&=
(kv
_
(2$-J%-jiF)
t$
sina + (2kt - hi-- tF) cosa
. . .(11)
for tipping about KJ, or tan&=
V -1 Wcosa
. . .(12)
for tipping about 0’ J. The lowest positive-valued solution for B, is found from Eqns (lo), (11) and (12). This value for B, is taken as a new value for the assumed slope in an iterative fashion until ,!?, and the assumed slope converge. The stability analysis of the tanker is completed by calculating the stability limit /I, for all values of heading angle a and for all fractions ;1 from 0.0 (empty) to 1.O (full). An additional stage to the analysis is to analyse the stability of the tanker when filled with an imaginary series of solid loads, of the same mass and volume as the fluid ones, but occupying the lower parts of the tank which the fluid loads would occupy when the tank was level. These imaginary solid loads are not free to move, and the changes which are introduced by them to the stability of the tanker, represent the contributions made by the mass but not the movement of the fluid loads. Therefore, the differences in stability value between the two stages of the analysis show the effect on tanker stability of fluid movement within the tank. 3.
Presentation of analytical results
The calculated values of stability limit were difficult to present graphically because stability
315
A. G. M. HUNTER
(up) 0 0
0.2
04
0.6
0.8
I.0
Fill, X
Fig. 3. Contours of tanker stability limit, &, plotted against tanker heading angle, a. and tanker$ll. section tanker
1. Rectangular
cross-
limit /3, had to be plotted against both heading angle a and fractional tanker fill 1. In addition, the results calculated for both fluid and imaginary solid loads had to be presented. The accepted method of presenting stability results using polar diagrams, which will be described later, was unsatisfactory because a new polar diagram w’as required for each 2 value. This gave a number of polar diagrams which were difficult to compare visually and from which it was difficult to visualize trends in stability limit. One possibility was to use an isometric surface view of j?, plotted against a and 1. Several computer-generated plots were made, but it proved difficult to provide an absolute scale of stability values which could be superimposed on these plots, with the result that only qualitative and not quantitative deductions could be made from the plots. Even the qualitative information was imperfect, since relative values of stability limit were difficult to assess, due to the perspectives from which the plots were viewed.
316
STABILITY
0
02
04
0.6
ANALYSIS
0.8
OF TRAILED
TANKERS
1.
FIII, X
Fig. 4. Plots of tanker stability limit, /I,, against tanker Jill at one tanker heading angle of 100”. (i.e. facing almost directly across the slope and slightly downhill). Rectangular cross-section tanker
A contour plot proved much more satisfactory and had the merit that quantitative values of stability limit could be read from it (Fig. 3). A proportion of the calculated information was lost in transferring to contours at fixed intervals, but there was a corresponding gain in being able to see trends in stability limit. In the example shown, the lowest stability value, about 14”, is at A when the tank is about 45% full (A= 0.45) and the tanker is facing directly uphill. Under these circumstances the tanker hitch will lift, although the tanker is unlikely to overturn backwards because the weight of the towing tractor will hold the hitch down. The practical situation, where the tanker is likely to overturn, is when travelling across the slope. The stability values between B and C, where the tanker is travelling directly across the slope, are slightly higher than those between D and E, where the tanker is facing slightly downhill. The lowest stability value between D and E is not apparent from the contour plot, but it is clear that it lies somewhere between the empty and full values. There is a ridge FF of higher stability values on the contour plot, between the values for travel directly uphill, when the hitch will lift, and those for travel across the slope, when one wheel will lift. The highest stability values on the plot are between G and H, where the tanker is facing directly downhill. The close contour spacings indicate that the stability values rise very rapidly towards G and H from those at D and E. The detailed variation of stability values between D and E, where the tanker is facing across the slope and slightly downhill, is best shown by plotting a graph of these values against fractional tank fill (Fig. 4). The “X’S on the curve marked “fluid” correspond to those in Fig. 3. The minimum stability value of 24.5” at I is seen to occur when the tank is between 30% and 40% full. The stability value when the tank is 100% full at E is 28.5”, and the stability value when the
317
A. G. M. HUNTER
tank is empty at D is 38.5”. The large effect on tanker stability of the fluid movement within the tank is shown by plotting the “solid” curve, calculated for imaginary solid loads in the tank, as well as the “fluid” one. The difference in values between these two curves is shown as the “difference” curve in the lower part of the figure. The largest difference is 10”at J, where the tank is only 20% full. 4.
Experimental confirmation
Experiments were used to confirm that both the stability analysis and the computer programming used to implement it were accurate. These experiments followed a series of numerical tests which were also completed satisfactorily. The approach was to carry out extensive stability measurements with physical models, where slope and heading angles were easy to adjust and there were no safety problems, and to carry out limited stability measurements with a full-scale tanker. The models of circular and rectangular cross-section tankers rested on an adjustable sloping board and were hitched to a post at its centre. A protractor scale was laid out on the board to indicate heading angle. The transparent tanks were marked with a scale on the side to indicate 1. The circular tank could be rotated from the fore-and-aft to the transverse position as required. The stability limit for each tanker was measured by inclining the board until the tanker started to tip. Either a wheel lifted from the board or the hitch lifted. This was done for 19 heading angles (0” to 180” in 10” steps) and 11 fractional fills (0.0 to 1.0 in 0.1 steps) for each tank. A smaller number of measurements were repeated with the circular tanker in the alternative position. The dimensions of the tankers were measured and the centre of gravity of each was estimated from knife-edge weighings, made when empty. The computed values of stability limit were plotted as continuous lines on polar diagrams and the experimental values were plotted as points on the
Heading, a 0”
x
lup)
x
Fig. 5. Computed polar stability boundary (full line) plotted with experimental points measured on scale model circular cross-section tanker. Fractional tankfill 1= 0.3
318
STABILITY Heading,
I
OF TRAILED
TANKERS
CI
180” (dawn)
:
ANALYSIS
I
: x
x
Fig. 6. Computedpolar stability boundary (fill line) plotted with experimental points measured on scale model rectangular cross-section tanker. Fractional tankjill 2 = 0.3
same diagrams. In each diagram, heading angle is represented by polar heading and slope is represented by radial distance from the diagram centre. Slope is plotted as percentage value: slope % = tan/? x 100. Typical examples are given in Figs 5 and 6 for circular and rectangular tankers, respectively. The agreement between the computed and experimental values is seen to be very close. The full-scale measurements were made with a small farm slurry-tanker of circular crosssection (Fig. 7). The tanker was hitched behind a tractor and both machines were fitted sideways on three beams placed under the two sets of tractor wheels and the tanker wheels, respectively. The load under the upper tanker wheel was measured with an electronic weighpad fitted into the one beam. Clearly, if the load under the tanker wheel approached zero as the tanker was tipped, then the tanker was approaching the point of overturn. It would have been preferable to use the precise method of estimating the stability limit of farm machines, which avoids the need for actually approaching the point of overturn, by using instead the formula relating wheel load reduction to slope.5 However, the formula does not take account of machines with a centre of gravity which shifts due to fluid movement and so the formula was unsuitable. In fact, the reduction of load under the slurry tanker wheel varied almost linearly with slope and good estimates of the stability limits were obtained from linear regressions on the results (Fig. 8). The measurements were made with four different levels of fill in the tank, from empty to completely full. The experimental values of stability limit were compared with the computed ones. The set of input data for the computed values consisted of the measured dimensions of the tanker, the fractional fill of the tank, and the empty tanker centre of gravity co-ordinates. Of these co-ordinates, the centre of gravity height was first estimated by eye, as no other method was readily available. This estimate was improved by a process of iteration, making comparisons of the experimentally
319
A. G. M. HUNTER
measured stability limit for the empty tanker with a series of computed ones, incorporating successive estimates of centre of gravity height in the input data. The best estimate of centre of gravity height was then used in computing the curves for stability limit against I for both the fluid-filled and imaginary solid-filled cases (Fig. 9). It can be seen that the curves have been made
Fig. 7. Full-scale stability measurement on circular cross-section slurry tanker
Slope, o Fig. 8. Experimental data points of wheel loadplotted against slope angle for full-scale slurry tanker. Fractional tankjll values: A, 1= 0.0 (empty) ; B, 1= 0.35; C, I = 0.72: D. 1= I.0 (full)
320
STABILITY
ANALYSIS
OF TRAILED
TANKERS
Fill. A
Fig. 9. Experimental
values of stability limit at B, C and D plotted against predicted curve for full-scale slurry tanker. -2 calculated; x , experimental
to coincide with the measured value at A. The three other measured stability limits, at B, C and D, agree closely with the computed curve for the fluid-filled tanker. A brief comment may be made here about the differences between the curves shown in Fig. 4 and Fig. 9. In Fig. 4, the “fluid” curve shows a distinct minimum at I and a drop in stability value of 10” between D and E, while in Fig. 9 there is no minimum and the overall drop is only 6.5”. These two curves were calculated for two different tankers: the curve in Fig. 4 was for the tanker of rectangular cross-section cited in the Introduction, and the curve in Fig. 9 was for the circular section tanker shown in Fig. 7. It is clear that the differences in tanker shape result in markedly different stability characteristics. Conclusions
6.
The stability analysis of a trailed farm tanker is considerably more complex than that of a simple farm trailer because the fluid inside finds its own level when the tanker is towed on a slope and causes an overall centre of gravity shift. The analysis and associated computer programming can be used confidently, now that these have been experimentally confirmed using models and a full-size tanker. Tanks of both circular and rectangular cross-section have been covered. One effect of a centre of gravity shift can be that the tanker is less stable when partly full than when completely full, with the result that the tanker becomes progressively less stable as its contents are discharged during work. The implications of this should be considered both in advisory practices for working on slopes and in designing tankers for greater stability. This paper is not, in itself, a design study, although the examples given show that there are marked differences in the stability characteristics of rectangular and circular cross-section tankers. REFERENCES
Hunter, A. G. M.; Owen, G. MD Tractor overturning accidents on slopes. J. occup. Accid., 1983 5, 195-210 Spencer, H. B. Stability and control of two wheel drive tractors and machinery on sloping ground. J. agric.
Engng Res., 1978 23 (2) 169-188 Hunter, A. G. M. Centre of gravity analysis offluid in inclined tanks. J. agric. Engng Res. (to be published) Bauer, H. F. On the destabilising efict of liquids in various vehicles. Vehic. Syst. Dynamics 1972, 1
227-260 Spencer, H. B.; Owen, G. M.; Glasbey, C. A. On-site measurements of the stability of agricultural
machines. J. agric. Engng Res. 1985 31,81-91
Stability
Analysis of Trailed Tankers on Slopes A. G. M. HUNTER*
The stability analysis of trailed tankers is a new area, which has not previously been covered in the study of farm machines on slopes. It is an important area because tankers have increased in size over recent years and the massive quantity of fluid carried has a dominating effect. The overturning accidents which occur, arise from a previously unexplored cause, namely that the overall centre of gravity of the tanker moves markedly with movement of the fluid inside, unlike the immobile centre of gravity of a trailer filled with a solid load. The stability analysis covers tankers with circular and rectangular cross-section tanks. An unexpected result is that the stability of a rectangular cross-section tanker can be least when the tank is only 30% full. The analysis has been confirmed experimentally. 1.
Introduction
The stability of farm vehicles has been studied extensively because tractor overturning accidents may result in severe injury to the driver or even in his death. Often, an accident is caused by a trailer, which is less stable than the towing tractor, overturning and pulling the tractor with it.’ One group of farm trailers which has not previously been studied comprises farm tankers normally used either for slurry spreading or for crop spraying. When a tanker is driven on a slope, its overall centre of gravity shifts due to the fluid inside finding its own level. The tanker may be inclined at a compound angle and its stability is therefore a function of the slope and of the direction in which the tanker is facing on the slope. The stability is also a function of the fraction of tanker capacity filled with fluid, and this fraction decreases continuously as the tanker contents are discharged onto the field. The need for stability analysis of tankers became apparent after a few accidents had been investigated, where trailed tankers had overturned on slopes. In one example, a 4000 1 capacity slurry tanker of nearly rectangular cross-section overturned in a field of slope 24” while partly full. Using existing methods of stability analysis for a trailer with an immobile centre of gravity,’ the stability limit (i.e. overturning slope) for the empty tanker was calculated as 37”, and for the full tanker 28”, both values greater than the actual overturning value of 24”. The displaced centre of gravity of the fluid in the tanker on a slope was then roughly estimated and the corresponding stability limit was calculated as 23”, which was less than the stability limit for the tanker empty or full. The first stage in improving the centre of gravity estimate for this and other tankers was to develop a centre of gravity analysis of fluid in inclined tanks .j The second and final stage, the subject of this paper, was to combine the new centre of gravity analysis with the existing stability analysis for trailers. The method followed is described in the next section. The succeeding sections describe how tanker stability can best be presented graphically, and how the analysis was experimentally confirmed. It was confined to tanks of circular and rectangular cross-sections, because these shapes approximately fitted most farm tankers. The result described above, where the static stability on a slope is least when the tanker is only partly full, is unexpected and has not previously been recognized. It may also apply to off-road vehicles outside agriculture, for example military ones. Although there has been considerable work on the dynamic effects of liquids in tanks fitted to road, rail, sea, air and space vehicles, it has all been related to level or upright tanks.’ There has been no attempt here to cover dynamic of Agricultural Engineering,
*Scottish
Institute
Received
28 September
1983; accepted
Bush Estate,
Penicuik,
t” revised form 17 November
Midlothm
EH26 OPH, Scotland
1984
311 0021-8634r85/080311+
10 $03,M):O
a
1985 The British Society
for Research
in Agricultural
Engineering
312
STABILITY
ANALYSIS
OF TRAILED
TANKERS
NOTATION G j
k 2t m
OXYZ oxyz O’uvw x V
! 2 P v, b ; S
centre of gravity hitch distance forward from axle hitch height track width mass tank-fixed axis system moveable axis system trailer axis system centre of gravity co-ordinate (typical) volume heading angle slope slope of fluid surface relative to ox fraction of tank capacity filled by fluid density of fluid rotation angle Yoy Subscripts
body of trailer container (tank) fluid stability
effects, because static stability on compound slopes is considered to be the dominant problem. The question of the optimum shape of tanker for use on slopes is a closely related subject which will be covered in a later paper. 2.
Method of analysis
The analysis follows from that for the centre of gravity of fluid in tanks. A trailed tanker is shown schematically in Fig. 1. The tank is of circular cross-section, partially filled with fluid and mounted transversely on a trailer frame. The trailer rests on a place surface of slope /I, at a heading angle a to the slope line. The set of axes 0’ uvw is trailer-fixed with origin at the wheel-ground contact point, 0’ shown. The set of axes OXYZ is tank-fixed, parallel to 0’ uvw, with origin, 0, at the tank centre. The third set of axes, oxyz, is rotated through the angle v, about the -0X direction so that oy lies parallel to the fluid surface. As explained in the centre of gravity paper, the origin of this third set may move for the convenience of the analysis. The fluid surface lies at an angle, 6, to the axis ox. The relationship between the angles is found to be a, = tan- ‘(cosa tan /3),
. . .(1)
S = sin- ‘(sina sin 8).
. . .(2)
and
The transverse tank shown is representative of the arrangement on some types of crop sprayer. On other types, the tank is mounted in the fore-and-aft direction, i.e. the tank is rotated 90” about the 0’ w direction so that OX lies parallel with 0’ v. The set of axes oxyz also rotates so that ox remains parallel with OX. In this case the relationship between the angles is
313
A. G. M. HUNTER
Fig. 1. Circular cross-section tanker partly filled with Jluid facing along heading angle, a, on slope. /I. Three sets of coordinate axes shown
W
Z t
t
I--+--+j Y
1
0'
0'
I
I
J
u
I
2t
i
Fig. 2. Geometry o/rectangular cross-section tanker
withfiwed co-ordinate axis systems
p=tan-‘(-sinatanj?),
. . 43)
and 6=&t-’
(cosasinB).
. . .(4)
Tanks with rectangular cross-section are also commonly found, as in the slurry tanker accident example mentioned in the Introduction. Such tanks are considered as belonging to the transverse case, with Eqns (1) and (2) applying, but care must be taken that the correct tank dimensions are associated with the tank axes OX, 0 Y and OZ (Fig. 2). Tanker stability at a known heading angle a, with a known fraction A of the tank capacity filled
314
STABILITY
ANALYSIS
OF TRAILED
TANKERS
with fluid, is determined iteratively. First, on an assumed slope j3, the centre of gravity of the fluid G, is found relative to tank-fixed axes OXYZ using the methods and equations given in the earlier paper. The co-ordinates are . . .(5)
X, = X, (a, P, n),
and Y, and Z, which are similarly dependent on a, j3 and 2. The calculated co-ordinates transformed to co-ordinates in the trailer-fixed axes 0’ uvw as follows:
. . .(6)
Ur=uo+X,, with similar equations for vf and wf. If the tank is mounted in the fore-and-aft position, the transformation $=u,-
are
becomes
r,,
. . (7)
with similar equations for vf and wf. The overall centre of gravity is found from _ Zd=
mf”f
_ + mbUb
mf+mb
’
. . (8)
where the mass of fluid . . .(9)
m,=pVf =pAV,.
The equations for the V and Wco-ordinates are similar. The stability limit & is the slope on which the gravity vector passing through the overall centre of gravity of the tanker intersects one side of AO’JK. The equations for & may be derived using vector analysis as (ty-Jq
tan j3,=
sina - (ku- tE) cosa
@-j$
. . .(10)
for tipping about 0’ K, or tan&=
(kv
_
(2$-J%-jiF)
t$
sina + (2kt - hi-- tF) cosa
. . .(11)
for tipping about KJ, or tan&=
V -1 Wcosa
. . .(12)
for tipping about 0’ J. The lowest positive-valued solution for B, is found from Eqns (lo), (11) and (12). This value for B, is taken as a new value for the assumed slope in an iterative fashion until ,!?, and the assumed slope converge. The stability analysis of the tanker is completed by calculating the stability limit /I, for all values of heading angle a and for all fractions ;1 from 0.0 (empty) to 1.O (full). An additional stage to the analysis is to analyse the stability of the tanker when filled with an imaginary series of solid loads, of the same mass and volume as the fluid ones, but occupying the lower parts of the tank which the fluid loads would occupy when the tank was level. These imaginary solid loads are not free to move, and the changes which are introduced by them to the stability of the tanker, represent the contributions made by the mass but not the movement of the fluid loads. Therefore, the differences in stability value between the two stages of the analysis show the effect on tanker stability of fluid movement within the tank. 3.
Presentation of analytical results
The calculated values of stability limit were difficult to present graphically because stability
315
A. G. M. HUNTER
(up) 0 0
0.2
04
0.6
0.8
I.0
Fill, X
Fig. 3. Contours of tanker stability limit, &, plotted against tanker heading angle, a. and tanker$ll. section tanker
1. Rectangular
cross-
limit /3, had to be plotted against both heading angle a and fractional tanker fill 1. In addition, the results calculated for both fluid and imaginary solid loads had to be presented. The accepted method of presenting stability results using polar diagrams, which will be described later, was unsatisfactory because a new polar diagram w’as required for each 2 value. This gave a number of polar diagrams which were difficult to compare visually and from which it was difficult to visualize trends in stability limit. One possibility was to use an isometric surface view of j?, plotted against a and 1. Several computer-generated plots were made, but it proved difficult to provide an absolute scale of stability values which could be superimposed on these plots, with the result that only qualitative and not quantitative deductions could be made from the plots. Even the qualitative information was imperfect, since relative values of stability limit were difficult to assess, due to the perspectives from which the plots were viewed.
316
STABILITY
0
02
04
0.6
ANALYSIS
0.8
OF TRAILED
TANKERS
1.
FIII, X
Fig. 4. Plots of tanker stability limit, /I,, against tanker Jill at one tanker heading angle of 100”. (i.e. facing almost directly across the slope and slightly downhill). Rectangular cross-section tanker
A contour plot proved much more satisfactory and had the merit that quantitative values of stability limit could be read from it (Fig. 3). A proportion of the calculated information was lost in transferring to contours at fixed intervals, but there was a corresponding gain in being able to see trends in stability limit. In the example shown, the lowest stability value, about 14”, is at A when the tank is about 45% full (A= 0.45) and the tanker is facing directly uphill. Under these circumstances the tanker hitch will lift, although the tanker is unlikely to overturn backwards because the weight of the towing tractor will hold the hitch down. The practical situation, where the tanker is likely to overturn, is when travelling across the slope. The stability values between B and C, where the tanker is travelling directly across the slope, are slightly higher than those between D and E, where the tanker is facing slightly downhill. The lowest stability value between D and E is not apparent from the contour plot, but it is clear that it lies somewhere between the empty and full values. There is a ridge FF of higher stability values on the contour plot, between the values for travel directly uphill, when the hitch will lift, and those for travel across the slope, when one wheel will lift. The highest stability values on the plot are between G and H, where the tanker is facing directly downhill. The close contour spacings indicate that the stability values rise very rapidly towards G and H from those at D and E. The detailed variation of stability values between D and E, where the tanker is facing across the slope and slightly downhill, is best shown by plotting a graph of these values against fractional tank fill (Fig. 4). The “X’S on the curve marked “fluid” correspond to those in Fig. 3. The minimum stability value of 24.5” at I is seen to occur when the tank is between 30% and 40% full. The stability value when the tank is 100% full at E is 28.5”, and the stability value when the
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tank is empty at D is 38.5”. The large effect on tanker stability of the fluid movement within the tank is shown by plotting the “solid” curve, calculated for imaginary solid loads in the tank, as well as the “fluid” one. The difference in values between these two curves is shown as the “difference” curve in the lower part of the figure. The largest difference is 10”at J, where the tank is only 20% full. 4.
Experimental confirmation
Experiments were used to confirm that both the stability analysis and the computer programming used to implement it were accurate. These experiments followed a series of numerical tests which were also completed satisfactorily. The approach was to carry out extensive stability measurements with physical models, where slope and heading angles were easy to adjust and there were no safety problems, and to carry out limited stability measurements with a full-scale tanker. The models of circular and rectangular cross-section tankers rested on an adjustable sloping board and were hitched to a post at its centre. A protractor scale was laid out on the board to indicate heading angle. The transparent tanks were marked with a scale on the side to indicate 1. The circular tank could be rotated from the fore-and-aft to the transverse position as required. The stability limit for each tanker was measured by inclining the board until the tanker started to tip. Either a wheel lifted from the board or the hitch lifted. This was done for 19 heading angles (0” to 180” in 10” steps) and 11 fractional fills (0.0 to 1.0 in 0.1 steps) for each tank. A smaller number of measurements were repeated with the circular tanker in the alternative position. The dimensions of the tankers were measured and the centre of gravity of each was estimated from knife-edge weighings, made when empty. The computed values of stability limit were plotted as continuous lines on polar diagrams and the experimental values were plotted as points on the
Heading, a 0”
x
lup)
x
Fig. 5. Computed polar stability boundary (full line) plotted with experimental points measured on scale model circular cross-section tanker. Fractional tankfill 1= 0.3
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STABILITY Heading,
I
OF TRAILED
TANKERS
CI
180” (dawn)
:
ANALYSIS
I
: x
x
Fig. 6. Computedpolar stability boundary (fill line) plotted with experimental points measured on scale model rectangular cross-section tanker. Fractional tankjill 2 = 0.3
same diagrams. In each diagram, heading angle is represented by polar heading and slope is represented by radial distance from the diagram centre. Slope is plotted as percentage value: slope % = tan/? x 100. Typical examples are given in Figs 5 and 6 for circular and rectangular tankers, respectively. The agreement between the computed and experimental values is seen to be very close. The full-scale measurements were made with a small farm slurry-tanker of circular crosssection (Fig. 7). The tanker was hitched behind a tractor and both machines were fitted sideways on three beams placed under the two sets of tractor wheels and the tanker wheels, respectively. The load under the upper tanker wheel was measured with an electronic weighpad fitted into the one beam. Clearly, if the load under the tanker wheel approached zero as the tanker was tipped, then the tanker was approaching the point of overturn. It would have been preferable to use the precise method of estimating the stability limit of farm machines, which avoids the need for actually approaching the point of overturn, by using instead the formula relating wheel load reduction to slope.5 However, the formula does not take account of machines with a centre of gravity which shifts due to fluid movement and so the formula was unsuitable. In fact, the reduction of load under the slurry tanker wheel varied almost linearly with slope and good estimates of the stability limits were obtained from linear regressions on the results (Fig. 8). The measurements were made with four different levels of fill in the tank, from empty to completely full. The experimental values of stability limit were compared with the computed ones. The set of input data for the computed values consisted of the measured dimensions of the tanker, the fractional fill of the tank, and the empty tanker centre of gravity co-ordinates. Of these co-ordinates, the centre of gravity height was first estimated by eye, as no other method was readily available. This estimate was improved by a process of iteration, making comparisons of the experimentally
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measured stability limit for the empty tanker with a series of computed ones, incorporating successive estimates of centre of gravity height in the input data. The best estimate of centre of gravity height was then used in computing the curves for stability limit against I for both the fluid-filled and imaginary solid-filled cases (Fig. 9). It can be seen that the curves have been made
Fig. 7. Full-scale stability measurement on circular cross-section slurry tanker
Slope, o Fig. 8. Experimental data points of wheel loadplotted against slope angle for full-scale slurry tanker. Fractional tankjll values: A, 1= 0.0 (empty) ; B, 1= 0.35; C, I = 0.72: D. 1= I.0 (full)
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STABILITY
ANALYSIS
OF TRAILED
TANKERS
Fill. A
Fig. 9. Experimental
values of stability limit at B, C and D plotted against predicted curve for full-scale slurry tanker. -2 calculated; x , experimental
to coincide with the measured value at A. The three other measured stability limits, at B, C and D, agree closely with the computed curve for the fluid-filled tanker. A brief comment may be made here about the differences between the curves shown in Fig. 4 and Fig. 9. In Fig. 4, the “fluid” curve shows a distinct minimum at I and a drop in stability value of 10” between D and E, while in Fig. 9 there is no minimum and the overall drop is only 6.5”. These two curves were calculated for two different tankers: the curve in Fig. 4 was for the tanker of rectangular cross-section cited in the Introduction, and the curve in Fig. 9 was for the circular section tanker shown in Fig. 7. It is clear that the differences in tanker shape result in markedly different stability characteristics. Conclusions
6.
The stability analysis of a trailed farm tanker is considerably more complex than that of a simple farm trailer because the fluid inside finds its own level when the tanker is towed on a slope and causes an overall centre of gravity shift. The analysis and associated computer programming can be used confidently, now that these have been experimentally confirmed using models and a full-size tanker. Tanks of both circular and rectangular cross-section have been covered. One effect of a centre of gravity shift can be that the tanker is less stable when partly full than when completely full, with the result that the tanker becomes progressively less stable as its contents are discharged during work. The implications of this should be considered both in advisory practices for working on slopes and in designing tankers for greater stability. This paper is not, in itself, a design study, although the examples given show that there are marked differences in the stability characteristics of rectangular and circular cross-section tankers. REFERENCES
Hunter, A. G. M.; Owen, G. MD Tractor overturning accidents on slopes. J. occup. Accid., 1983 5, 195-210 Spencer, H. B. Stability and control of two wheel drive tractors and machinery on sloping ground. J. agric.
Engng Res., 1978 23 (2) 169-188 Hunter, A. G. M. Centre of gravity analysis offluid in inclined tanks. J. agric. Engng Res. (to be published) Bauer, H. F. On the destabilising efict of liquids in various vehicles. Vehic. Syst. Dynamics 1972, 1
227-260 Spencer, H. B.; Owen, G. M.; Glasbey, C. A. On-site measurements of the stability of agricultural
machines. J. agric. Engng Res. 1985 31,81-91