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An approximate method of predicting the bending of thermoplastic beams
An approximate method of predicting the bending of thermoplastic beams R. M. Ogorkiewicz Department of Mechanical Engineering, Imperial College of Science and Technology, Exhibition Road, London SW7 2BX, UK
and A. A. M. Sayigh College of Engineering, University of Riyadh, Riyadh, Saudi Arabia (Received 31 May 1973) Availability of an increasing amount of data on the deformational characteristics of thermoplastics makes it possible to predict more accurately the deflection under load of plastics articles. This is shown by the close agreement between the calculated and observed deflections of PVC beams under bending loads of short and long duration.
INTRODUCTION The effort devoted in recent years to the study of the mechanical properties of thermoplastics has resulted in the publication of an increasing amount of data about their deformational characteristics 1, 2. The generation of these data was inspired very largely by the hope that they would lead not only to a a better understanding of the behaviour of plastics in general but also to the selection of thermoplastics more appropriate to particular applications and, ultimately, to a more rational design of thermoplastic articles. So far, however, relatively little has been published about the use of such data for the design of plastics components within limits of deflection prescribed by functional or aesthetic considerations. Most of the data published so far about the deformational characteristics of thermoplastics has been based on their response under simple, uniaxial loading systems and in particular on tensile creep tests z-3. This type of data is generally considered to be best for characterizing deformational behaviour but it is not directly applicable to the design of plastics articles because in most cases they are subjected to more complex stress systems than uniaxial tension. There is a need, therefore, to devise or establish methods that would enable the basic data obtained under uniaxial loading systems to be used under other conditions which arise in engineering design with thermoplastics. In particular, this calls for methods of predicting the deflection of articles under loads of given magnitude and duration. METHODS OF PREDICTING DEFLECTIONS General approach
Since thermoplastics behave in general as non-linear viscoelastic materials rigorous solutions to the deformation of articles made of them are bound to be compficated and prohibitively so for general design purposes. It is
584 POLYMER, 1973, Vol 14, November
necessary, therefore, to resort to simpler, approximate methods 4. The approximate methods which have been put forward so far are based on adaptations of the theory of linear elastic materials established for many years with metals. One of the earliest adaptations of this is the procedure put forward by Baer et al. 5 which amounts to the substitution of a time-dependent modulus into the standard linear elastic solutions for deflections, the modulus being considered strain independent provided that the procedure is applied within prescribed strain limitsS, 6. Similar procedures have been put forward, among others, by Alfrey and Gurnee 7, as applicable to linear viscoelastic materials or, at least, the quasilinear viscoelastic range of behaviour of polymeric materials. This approach has already been applied with some success to beams by McCammonds who has found, however, that it predicted deflection less accurately than a method which took into account the variation of the modulus not only with time but also with strain. The second approach is obviously more realistic, particularly at other than small strains, and deserves further consideration, especially as it can be applied in an alternative and simpler form than McCammond's, which involves integration of areas under stress-strain curves. Moreover, the authors have had the opportunity of putting it to test in a bending apparatus more accurate and capable of taking more readings than those available to earlier workers. Method used
The method which has been followed is similar to that already applied to the deflection of plastics sandwich beams 9. It is based on constant load isochronous stressstrain curves, which are now a common form of presenting basic data on thermoplastics z and from which corresponding values of stress and strain can be obtained
Bending of thermoplastic beams: R. M. Ogorkiewicz and A. A. M. Sayigh at different times from the commencement of the application of a constant load. Thus, selection of a particular strain E at a time t leads to the corresponding stress o and the secant creep modulus E = o/e. Knowing the stress it is also possible to calculate the corresponding load W on a particular beam from the standard equation10: (1) o = M] y
and at the points of load application"
where M =bending moment,
To examine the validity of the theoretical approach, slender rectangular section beams of ICI Etd Darvic 110 PVC were considered and their deflections were calculated from equations (3) to (7) for a series of values of load at 100 and 1000 sec from the applications of the loads using isochronous tensile stress-strain curves given for Darvic 110 in ref. 2, and additional tensile data obtained by the authors. Corresponding Darvic 110 beams were then tested using an interrupted step-loading procedure, similar to that commonly used to obtain isochronous stressstrain curves in tension 11, which gave 100 and 1000sec isochronous load-deflection curves. The beams were nominally 6 mm thick and 24mm wide and they were supported on small, fixed rollers, set 250mm apart, which were adopted as a result of an earlier investigation into the effects of different supports on the results of bending tests 12. The beams were loaded at their centre or at two symmetrically disposed points in a specially developed lever-loading machine and their deflections were recorded with capacitive transducers accurate to within + 0 . 0 0 5 ~ 13 A set of results for a beam in three-point bending 100sec from the application of the load is shown in Figure 1. The correlation between the observed deflections and those calculated from equations (3), (4) and (5), respectively, at the centre of the beam and at 0.7 and 0.9 of the span is evidently close. Similarly, close agreement was obtained at 1000 sec from the application of the load.
I =second moment of area of beam about its neutral axis, 2y=thickness of the beam. If ~ is the skin stress at the mid-span of centrally loaded rectangular section beam equation (1) reduces to: 2bd 2
W=
31 cr
(2)
where b = w i d t h of beam, d =thickness of beam, l =span of beam supports. The theory of bending of beams also gives solutions to the deflection of beams in terms of W, E and the geometry of the beam and loading system. For the particular case of the central deflection 3 of the centrally loaded rectangular section beam it is1°: 3
W13 = 4Ebd 3
(3)
Repeating the process for a series of values of e makes it possible to plot a curve of W vs. 8, which can then be compared with the observed relationships between deflections and loads. Although in the study of the bending of simply loaded plastics beams attention has tended to be focused on their central deflections, it is also of interest to consider deflection at other points along the beam. The authors have therefore done this at two additional points which are at a distance x = 7 1 / l O and 91/10, measured from one of the supports. The equations to the deflection at these points obtained from classical beam theory 10 are:
3
-
WI 3 8 Ebd 3
(7)
where W is the sum of the two equal loads acting on the beam. COMPARISON
WITH EXPERIMENTAL
RESULTS
t2C
at x = 90l
80
37 WI 3 3 - 5 x 108 E b d 3
(4) "10 C3
and at x = 7 l 99 WI 3 3 - 5 x 108 Ebd 3
(5)
From some points of view the bending of beams simply supported at both ends and acted on by two identical, symmetrically disposed loads is of even greater interest, because of the constancy of the bending moment acting on the section of the beam between the loads. The authors have therefore considered the bending of beams under this condition also, with each load acting at a distance equal to 1/4 from the adjacent support. In this case the deflection at mid-span is: 11 WI 3 3 = 64 Ebd 3 (6)
40
Ir
O
I
i
8
I
I
I
16 Deflection (ram)
I
24
Figure I
Comparison of calculated load-deflection curves and the observed deflections of a PVC beam in three-point bending at 100sec after the application of loads at: C], mid-span; A, 0.7 of span; ©, 0-9 of span
POLYMER, 1973, Vol 14, November
585
Bending of thermoplastic beams: R. M. Ogorkiewicz and A. A. M. Sayigh Close agreement was also obtained in four-point bending. This is illustrated in Figure 2, which shows curves obtained from equations (6) and (7) for the deflection of the beam at mid-span and under the loads 100sec after the application of the loads and the corresponding observed deflections. Similarly, close agreement was obtained at 1000 sec. The agreement between the calculated and the observed results is brought out even more clearly by considering the deflected shape of the beams. This is shown in Figure 3 for a beam in three-point bending at 100sec
18
14 E
o
I0
u
E3
6
I
IO
i
IO 2
240
I
IO 3 Time (sec)
I
IO4
I
IO s
IO6
Figure 4
Calculated and observed creep deflections at the centre of a PVC beam in three-point bending. O, 9.8N; x , 19.6N; [], 29.4N; A, 3g.2N; e , 49.0N; V, 58.8N
D
after the application of each of a series of loads. The vertical scale in Figure 3 has been considerably magnified but, nevertheless, the deflections of different points o f the beam come very close to its calculated shape.
160 A Z
PREDICTION OF CREEP
"cD O _J
80
•
i
0
I
8
I
16
24
D~l¢ction (ram)
Figure 2 Comparison of calculated load-deflection curves and the observed deflection of a PVC beam in four-point bending at 100sec after the application of the loads at: O, mid-span; U, quarter span
0
25
75
Beam length (ram) 125 175
225 2 5 0
~5 15
g 2o Figure 3 Comparison of calculated and observed deflected shapes of a PVC beam in three-point bending 100sec after the application of different loads. ©, 19.6N; A, 39.2N; I-1, 58.8N; ×, 78.4N
586
P O L Y M E R , 1973, V o l 14, N o v e m b e r
The agreement obtained at 100 and 1000see from the application of the load, led to an extension of the work to the prediction of creep deflections at times of up to 106sec. This required the derivation of additional isochronous stress-strain curves from ref. 2 but otherwise followed the same procedure for working out series of values of load and deflection as before, except that each series was worked out for a constant load. A typical set of results, for the central deflection of a beam in three-point bending, is shown in Figure 4. Evidently the agreement between the calculated and observed deflections is close. Similarly close agreement was obtained in four-point bending. REFERENCES 1 'Shell Polyolefms Engineering Design Data', Shell, London, 1966 2 'Engineering Properties of Thermoplastics', (Ed. Ogorkiewicz, R. M.), Wiley, London, 1970 3 Turner, S. Br. Plast. 1964, 37, 440 4 Turner, S. Trans. J. Plast. Inst. 1966, 34, 127 5 Baer, E., Knox, J. R., Linton, T. J. and Maier, R. E. SPE Jl. 1960, 16, 396 6 'Designing with Du Pont Plastics', Du Pont, Wilmington, Delaware, 1961 7 Alfrey, T. and Gumee, E. F. 'Organic polymers', Prentice-Hall, Englewood Cliffs, 1967 8 McCammond. D. PhD Thesis Queen's University, Belfast, 1968 9 Sayigh, A. A. M. PhD Thesis University of London, 1966 10 Timoshenko, S. 'Strength of Materials', Van Nostrand, New York, Vol 1, 1957 11 Thomas, D. A. and Turner, S. in 'Testing of Polymers', (Ed. W. E. Brown), Wiley, New York, Vol 4, 1969 12 Ogorkiewicz, R. M. and Mucci, P. E. R. Composites 1971, 2, 139 13 Mucci, P. E. R. and Ogorkiewicz, R. M. to be published
and A. A. M. Sayigh College of Engineering, University of Riyadh, Riyadh, Saudi Arabia (Received 31 May 1973) Availability of an increasing amount of data on the deformational characteristics of thermoplastics makes it possible to predict more accurately the deflection under load of plastics articles. This is shown by the close agreement between the calculated and observed deflections of PVC beams under bending loads of short and long duration.
INTRODUCTION The effort devoted in recent years to the study of the mechanical properties of thermoplastics has resulted in the publication of an increasing amount of data about their deformational characteristics 1, 2. The generation of these data was inspired very largely by the hope that they would lead not only to a a better understanding of the behaviour of plastics in general but also to the selection of thermoplastics more appropriate to particular applications and, ultimately, to a more rational design of thermoplastic articles. So far, however, relatively little has been published about the use of such data for the design of plastics components within limits of deflection prescribed by functional or aesthetic considerations. Most of the data published so far about the deformational characteristics of thermoplastics has been based on their response under simple, uniaxial loading systems and in particular on tensile creep tests z-3. This type of data is generally considered to be best for characterizing deformational behaviour but it is not directly applicable to the design of plastics articles because in most cases they are subjected to more complex stress systems than uniaxial tension. There is a need, therefore, to devise or establish methods that would enable the basic data obtained under uniaxial loading systems to be used under other conditions which arise in engineering design with thermoplastics. In particular, this calls for methods of predicting the deflection of articles under loads of given magnitude and duration. METHODS OF PREDICTING DEFLECTIONS General approach
Since thermoplastics behave in general as non-linear viscoelastic materials rigorous solutions to the deformation of articles made of them are bound to be compficated and prohibitively so for general design purposes. It is
584 POLYMER, 1973, Vol 14, November
necessary, therefore, to resort to simpler, approximate methods 4. The approximate methods which have been put forward so far are based on adaptations of the theory of linear elastic materials established for many years with metals. One of the earliest adaptations of this is the procedure put forward by Baer et al. 5 which amounts to the substitution of a time-dependent modulus into the standard linear elastic solutions for deflections, the modulus being considered strain independent provided that the procedure is applied within prescribed strain limitsS, 6. Similar procedures have been put forward, among others, by Alfrey and Gurnee 7, as applicable to linear viscoelastic materials or, at least, the quasilinear viscoelastic range of behaviour of polymeric materials. This approach has already been applied with some success to beams by McCammonds who has found, however, that it predicted deflection less accurately than a method which took into account the variation of the modulus not only with time but also with strain. The second approach is obviously more realistic, particularly at other than small strains, and deserves further consideration, especially as it can be applied in an alternative and simpler form than McCammond's, which involves integration of areas under stress-strain curves. Moreover, the authors have had the opportunity of putting it to test in a bending apparatus more accurate and capable of taking more readings than those available to earlier workers. Method used
The method which has been followed is similar to that already applied to the deflection of plastics sandwich beams 9. It is based on constant load isochronous stressstrain curves, which are now a common form of presenting basic data on thermoplastics z and from which corresponding values of stress and strain can be obtained
Bending of thermoplastic beams: R. M. Ogorkiewicz and A. A. M. Sayigh at different times from the commencement of the application of a constant load. Thus, selection of a particular strain E at a time t leads to the corresponding stress o and the secant creep modulus E = o/e. Knowing the stress it is also possible to calculate the corresponding load W on a particular beam from the standard equation10: (1) o = M] y
and at the points of load application"
where M =bending moment,
To examine the validity of the theoretical approach, slender rectangular section beams of ICI Etd Darvic 110 PVC were considered and their deflections were calculated from equations (3) to (7) for a series of values of load at 100 and 1000 sec from the applications of the loads using isochronous tensile stress-strain curves given for Darvic 110 in ref. 2, and additional tensile data obtained by the authors. Corresponding Darvic 110 beams were then tested using an interrupted step-loading procedure, similar to that commonly used to obtain isochronous stressstrain curves in tension 11, which gave 100 and 1000sec isochronous load-deflection curves. The beams were nominally 6 mm thick and 24mm wide and they were supported on small, fixed rollers, set 250mm apart, which were adopted as a result of an earlier investigation into the effects of different supports on the results of bending tests 12. The beams were loaded at their centre or at two symmetrically disposed points in a specially developed lever-loading machine and their deflections were recorded with capacitive transducers accurate to within + 0 . 0 0 5 ~ 13 A set of results for a beam in three-point bending 100sec from the application of the load is shown in Figure 1. The correlation between the observed deflections and those calculated from equations (3), (4) and (5), respectively, at the centre of the beam and at 0.7 and 0.9 of the span is evidently close. Similarly, close agreement was obtained at 1000 sec from the application of the load.
I =second moment of area of beam about its neutral axis, 2y=thickness of the beam. If ~ is the skin stress at the mid-span of centrally loaded rectangular section beam equation (1) reduces to: 2bd 2
W=
31 cr
(2)
where b = w i d t h of beam, d =thickness of beam, l =span of beam supports. The theory of bending of beams also gives solutions to the deflection of beams in terms of W, E and the geometry of the beam and loading system. For the particular case of the central deflection 3 of the centrally loaded rectangular section beam it is1°: 3
W13 = 4Ebd 3
(3)
Repeating the process for a series of values of e makes it possible to plot a curve of W vs. 8, which can then be compared with the observed relationships between deflections and loads. Although in the study of the bending of simply loaded plastics beams attention has tended to be focused on their central deflections, it is also of interest to consider deflection at other points along the beam. The authors have therefore done this at two additional points which are at a distance x = 7 1 / l O and 91/10, measured from one of the supports. The equations to the deflection at these points obtained from classical beam theory 10 are:
3
-
WI 3 8 Ebd 3
(7)
where W is the sum of the two equal loads acting on the beam. COMPARISON
WITH EXPERIMENTAL
RESULTS
t2C
at x = 90l
80
37 WI 3 3 - 5 x 108 E b d 3
(4) "10 C3
and at x = 7 l 99 WI 3 3 - 5 x 108 Ebd 3
(5)
From some points of view the bending of beams simply supported at both ends and acted on by two identical, symmetrically disposed loads is of even greater interest, because of the constancy of the bending moment acting on the section of the beam between the loads. The authors have therefore considered the bending of beams under this condition also, with each load acting at a distance equal to 1/4 from the adjacent support. In this case the deflection at mid-span is: 11 WI 3 3 = 64 Ebd 3 (6)
40
Ir
O
I
i
8
I
I
I
16 Deflection (ram)
I
24
Figure I
Comparison of calculated load-deflection curves and the observed deflections of a PVC beam in three-point bending at 100sec after the application of loads at: C], mid-span; A, 0.7 of span; ©, 0-9 of span
POLYMER, 1973, Vol 14, November
585
Bending of thermoplastic beams: R. M. Ogorkiewicz and A. A. M. Sayigh Close agreement was also obtained in four-point bending. This is illustrated in Figure 2, which shows curves obtained from equations (6) and (7) for the deflection of the beam at mid-span and under the loads 100sec after the application of the loads and the corresponding observed deflections. Similarly, close agreement was obtained at 1000 sec. The agreement between the calculated and the observed results is brought out even more clearly by considering the deflected shape of the beams. This is shown in Figure 3 for a beam in three-point bending at 100sec
18
14 E
o
I0
u
E3
6
I
IO
i
IO 2
240
I
IO 3 Time (sec)
I
IO4
I
IO s
IO6
Figure 4
Calculated and observed creep deflections at the centre of a PVC beam in three-point bending. O, 9.8N; x , 19.6N; [], 29.4N; A, 3g.2N; e , 49.0N; V, 58.8N
D
after the application of each of a series of loads. The vertical scale in Figure 3 has been considerably magnified but, nevertheless, the deflections of different points o f the beam come very close to its calculated shape.
160 A Z
PREDICTION OF CREEP
"cD O _J
80
•
i
0
I
8
I
16
24
D~l¢ction (ram)
Figure 2 Comparison of calculated load-deflection curves and the observed deflection of a PVC beam in four-point bending at 100sec after the application of the loads at: O, mid-span; U, quarter span
0
25
75
Beam length (ram) 125 175
225 2 5 0
~5 15
g 2o Figure 3 Comparison of calculated and observed deflected shapes of a PVC beam in three-point bending 100sec after the application of different loads. ©, 19.6N; A, 39.2N; I-1, 58.8N; ×, 78.4N
586
P O L Y M E R , 1973, V o l 14, N o v e m b e r
The agreement obtained at 100 and 1000see from the application of the load, led to an extension of the work to the prediction of creep deflections at times of up to 106sec. This required the derivation of additional isochronous stress-strain curves from ref. 2 but otherwise followed the same procedure for working out series of values of load and deflection as before, except that each series was worked out for a constant load. A typical set of results, for the central deflection of a beam in three-point bending, is shown in Figure 4. Evidently the agreement between the calculated and observed deflections is close. Similarly close agreement was obtained in four-point bending. REFERENCES 1 'Shell Polyolefms Engineering Design Data', Shell, London, 1966 2 'Engineering Properties of Thermoplastics', (Ed. Ogorkiewicz, R. M.), Wiley, London, 1970 3 Turner, S. Br. Plast. 1964, 37, 440 4 Turner, S. Trans. J. Plast. Inst. 1966, 34, 127 5 Baer, E., Knox, J. R., Linton, T. J. and Maier, R. E. SPE Jl. 1960, 16, 396 6 'Designing with Du Pont Plastics', Du Pont, Wilmington, Delaware, 1961 7 Alfrey, T. and Gumee, E. F. 'Organic polymers', Prentice-Hall, Englewood Cliffs, 1967 8 McCammond. D. PhD Thesis Queen's University, Belfast, 1968 9 Sayigh, A. A. M. PhD Thesis University of London, 1966 10 Timoshenko, S. 'Strength of Materials', Van Nostrand, New York, Vol 1, 1957 11 Thomas, D. A. and Turner, S. in 'Testing of Polymers', (Ed. W. E. Brown), Wiley, New York, Vol 4, 1969 12 Ogorkiewicz, R. M. and Mucci, P. E. R. Composites 1971, 2, 139 13 Mucci, P. E. R. and Ogorkiewicz, R. M. to be published