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Non-linear bending of beams with uniformly distributed loads
hr. 1. ?fm-m
Mackaks.
Vol. 4, pp. 38%395. PargmoB
Pmm 1969. Printed in orclt
B&in.
NON-LINEAR BENDING OF BEAMS WITH UNIFORMLY DISTRIBUTED LOADS T. M. WANG Department of Civil Engineering, University of New Hampshire, Durham,
N.H.
Abatrac&The non-linear bending of both cantilever and simply supported beams subjected to a uniformly distributed load has been studied. The exact solutions for the s1opc.shave been obtained and the solution for the maximum deflection and the horizontal projection of the beam length for the cantilever case are compared with a known approximate solution.
fNTRODUCTION
THE subject of the non-linear bending of thin beams has received renewed attention during the past two decades. Barton [l] and Bisshopp and Drucker [2] provided solutions for the cantilever beam with a vertical load at the free end. The solutions for a simply supported beam with a central concentrated load have been obtained by Conway [33. In all these cases, the dilferential equations are integrated directly which leads to solutions in terms of elliptic integrals. For beams subjected to ~~0~1~ distributed load, however, the equations apparently do not lead to any direct solutions, and only approx~ate solutions are possible. Rohde [4] has expanded the slope in a power series of the arc length to analyze the case of a cantilever with uniformly distributed load. This problem has later been solved by Seames and Conway [S] by using a numerical procedure. Applying the same technique as Rhode’s, Iyengar [6] presented solutions for a simply supported beam under the action of a uniformly distributed load. The present paper is concerned with the non-linear bending of beams of constant cross-section carrying a uniformly distributed load. Both a cantilever and a simply supported beam are considered and the equations are integrated leading to the solutions for slope in terms of the horizontal projection of the arc length. Numerical methods are then used to obtain the deflections.
CANTILEVER BEAM WTI’H UNIFORMLY DiSTRIBuTED
LOAD
Consider the beam loaded as shown in Fig. 1. With the origin at the free end of the beam, the Euler-Bernoulli equation of bending M
d8 ai=
(1)
-5
leads to Dd’B dJV --=:ds2 = ds 389
wscose
(21
T. M. WAUG
390
FIG. 1. The 1oad~I cantilever.
where I) is the flexural rigidity of the beam, s, the arc length at any point measured from the free end, 0, the slope, and w, the load intensity. Substituting cos 8 = dx/ds into equation (2) yields d28 ds=
w -Tis
(3)
Differentiating equation (3) with respect to s and then integrating with respect to x, the horizontal projection of s, one obtains d20 s=
-;x+c,
(44)
At X=O,
$=O
and
C,=O
Equation (4) becomes d28 dS2=
--X
w
D
Now, d28
d’(sin e)
d
ds2=ds
=-iii-G*
(6)
Using equation (6) in equation (5) and integrating with respect to x, it follows that
d(sh 0) = ds
c0se
0dcf -$xz+c, ds =
At
de
x= 0 , &=O
and
C,=O.
(7)
Non-lbrearbend@
of&mm with una$wnsIydistributed loa&
391
Hence, =
--_x w
2
(8)
2D Substituting di3/ds = cos 6 (de/dx) and integrating, 8 sin 28 2+4=
(9)
-$x”+c,
At x=h,
and
@=O
C3=g
where h is the horizontal projection of the beam length Substituting C3 in equation (9) we finally obtain
For small values of 8, equation (10) yields the linear bending theory
With the 8’s computed from equation (lo), the values of h and y_, the maximum deflection, can be obtained from the following relationships L = ‘secedx
(11)
d
h
Ymax
=
tan6dx
(12)
In the present problem, h is divided into eight equal intervals. The calculations are carried out as follows : Consider, for example, w~~/D = 3. By using the ~e~o~-Rap~n method [7], the values of 8 at all points can be obtained from equation (10) and are given in Table 1. TABLE 1. Vn~ues x/L - 1
-.. .-
0=0
3 9” 33
a
t 17” 3
22” 49
OF 8 RX
f 26” 58
wh’/D = 3
3 29” 39
f 31”4
t 31”37
Using Simpson’s rule, numerical integration of equations (11) and (12) yields h=0904L
and
y_=O*397L
0 31”41’
392
T, M. WANG
Substituting h = 0904 L into wh3/D = 3 yields wL?/D = 4Q6
RG. 2 Variation of y~~~~ and h/L with WC/D for the cantilever. -----obtained fmm [4]
The numerical results of ymax and h for different values of wL?/D between 1 to 10 arc shown in Fig. 2 and are compared with those obtained by an approximate solution [4]. SIMPLY SWPPORIED BEAM Wll% BRAY
Dads
Ph
FIG.3. The loaded simply supported beam
LOAD
393
Non-linear bending of beams with uniformly distributed loa&
The second case to be considered is shown in Fig. 3 in which the beam is carrying a uniformly distributed load. It is assumed that the reactions at the ends of the beam are vertical The Euler-~rnoul~ equation gives
&f!
d’s =
- w(L - s)cos8
(13)
In this problem, the boundary conditions are d% ds2
-=0
da ds=O
at
x=h
at
x=0
(14) t
8=0
at
x=h
J
Following the same procedure as given in the previous case, equation (13) will yield the tokens equation 2e+sh2e=D
2x3 Wh3 4 2*2+57;” j-
(
(15)
h2
>
Again for small values of 8, equation (15) yields
I36
0.4
0.3
@2
01
0
I
2
FIG. 4. Variation ofy,&L
3
4
5 *LS D
6
7
8
9
IO
and h/L with wL?/Dfor the simply supported beam.
394
which agrees with the linear bending theory. The results for y,, and h for WC/D = 1 to 10 are shown in Fig. 4.
CONCLUSION The expressions for slopes in terms of the horizontal projection of the arc length for beams carrying a uniformly distributed load have been derived. By using numerical methods, the slopes at all points, the maximum deflection, and the horizontal projection of the beam length are obtained. The question may arise as to the accuracy of the proposed solutions with regard to the numerical methods used. It has been pointed out by Scarborough [7) that the results obtained by the Newton-Raphson method are correct to their last figures. This can be proved by substitution of the tf values from Table 1 into equation (10). Thus, the correctness of the solutions may depend on the numerical integration of equations (11) and (12). Wang [S] has shown that the results obtained by Simpson’s rule with eight equal intervals for h for the cantilever beam subjected to a concentrated load at the free end are in excellent agreement with those obtained by an exact solution. Therefore, the results obtained in this paper are closer to the actual values. It can be concluded therefore that the proposed technique offers a more accurate solution.
ftJcFERENcEs
[I] H. J. BATON, On the deflection of a cantilever beam. Q. uppl. Math. 2, 168-171 (1944) atxd Q. appl. Muzfi. 3, 275-276 (I 945). [2] K. E. BIWOPP and D. C. DRUCKER,Large deflections of cantilever beams. Q. appi. Math. 3,272-275 (1945). [3] H. D. CONWAY,The large defleotion of simply supported beams. Phil. Mug. 38,905-911 (1947). [4] F. V. ROHDE,Large deflections of a cantilever beam with uniformly distributed load. Q. appl. Math. 11, 337-338 (1953). [5] A. E. S~hhtxs and H. D. COMKAY,A numerical procedure for calculating the large deflections of straight and curved beams. .I. appl. Mech. 24,289-294 (1957). [6] R. FRNH-FAY, Flexible Bars. Butterworths (1962). [7] J. B. SCARBOROUGH, Numericof Mafhemaficul Anaiysis. John Hopkins Press (1955). [8] T. M. WANG,Nonlinear bending of beams with concentrated loads. J. Frnnklin Inst. 285,386390 (1968). (Received 21 May 1968)
R6samb-h
flexion non-li&aire d’tm encorbelkment et de poutres simplement supportceS soumises a tme charge uniformtment distribuee, a ttt CtudiC J.aasolutions cxactes pour lea pentes ont ttt obtenuea d la solution pour le lkchissement maximum et la projection horizontale de la longueur de la poutre dam le cas de I’encorbellement sont compares & une solution approximative connuc
Zwammenfassaag-Die nichtlineare Biegung von Ausleger- und einfach unterstiitxten Balken unter txleicbm&ssin verteilter Last w&de. tmtersucht Die e&t& tcisung& fiir die Neigung wurden erhalten, und dk Ergebnii fftr die maximak Durchbicgung und waagrechte Projektion der Balkenlgnge for den Fall dcs Aualegerbalkens werden mit einer bckamten N~~~l~~g vergleichen.
Non-hear
bending of beams with unifomdy distributed
Ioads
395
Henmetinblit cm6 KaK KOHCOJtbHOfi, TaH ha npoc~o no~epwmaelvofi Pe*epm-PaccntaTpmann BGKH c paenorveptfo pacnpefieneenofi narpyaKo#. Halunw npaBHJIbKbCfi OTBeT ~JI~IHaKnonoB. Paapeluam Mawmanbnoe 0TKnoneme m &umiy rOpHaonTanbHOr0 earmyna 6iwmf ma Kopnyca K~HC~HZ~,YTO cpaBnnsanK C HaBecTnm4 Kp~6nn~eHn~m pemesneu.
Mackaks.
Vol. 4, pp. 38%395. PargmoB
Pmm 1969. Printed in orclt
B&in.
NON-LINEAR BENDING OF BEAMS WITH UNIFORMLY DISTRIBUTED LOADS T. M. WANG Department of Civil Engineering, University of New Hampshire, Durham,
N.H.
Abatrac&The non-linear bending of both cantilever and simply supported beams subjected to a uniformly distributed load has been studied. The exact solutions for the s1opc.shave been obtained and the solution for the maximum deflection and the horizontal projection of the beam length for the cantilever case are compared with a known approximate solution.
fNTRODUCTION
THE subject of the non-linear bending of thin beams has received renewed attention during the past two decades. Barton [l] and Bisshopp and Drucker [2] provided solutions for the cantilever beam with a vertical load at the free end. The solutions for a simply supported beam with a central concentrated load have been obtained by Conway [33. In all these cases, the dilferential equations are integrated directly which leads to solutions in terms of elliptic integrals. For beams subjected to ~~0~1~ distributed load, however, the equations apparently do not lead to any direct solutions, and only approx~ate solutions are possible. Rohde [4] has expanded the slope in a power series of the arc length to analyze the case of a cantilever with uniformly distributed load. This problem has later been solved by Seames and Conway [S] by using a numerical procedure. Applying the same technique as Rhode’s, Iyengar [6] presented solutions for a simply supported beam under the action of a uniformly distributed load. The present paper is concerned with the non-linear bending of beams of constant cross-section carrying a uniformly distributed load. Both a cantilever and a simply supported beam are considered and the equations are integrated leading to the solutions for slope in terms of the horizontal projection of the arc length. Numerical methods are then used to obtain the deflections.
CANTILEVER BEAM WTI’H UNIFORMLY DiSTRIBuTED
LOAD
Consider the beam loaded as shown in Fig. 1. With the origin at the free end of the beam, the Euler-Bernoulli equation of bending M
d8 ai=
(1)
-5
leads to Dd’B dJV --=:ds2 = ds 389
wscose
(21
T. M. WAUG
390
FIG. 1. The 1oad~I cantilever.
where I) is the flexural rigidity of the beam, s, the arc length at any point measured from the free end, 0, the slope, and w, the load intensity. Substituting cos 8 = dx/ds into equation (2) yields d28 ds=
w -Tis
(3)
Differentiating equation (3) with respect to s and then integrating with respect to x, the horizontal projection of s, one obtains d20 s=
-;x+c,
(44)
At X=O,
$=O
and
C,=O
Equation (4) becomes d28 dS2=
--X
w
D
Now, d28
d’(sin e)
d
ds2=ds
=-iii-G*
(6)
Using equation (6) in equation (5) and integrating with respect to x, it follows that
d(sh 0) = ds
c0se
0dcf -$xz+c, ds =
At
de
x= 0 , &=O
and
C,=O.
(7)
Non-lbrearbend@
of&mm with una$wnsIydistributed loa&
391
Hence, =
--_x w
2
(8)
2D Substituting di3/ds = cos 6 (de/dx) and integrating, 8 sin 28 2+4=
(9)
-$x”+c,
At x=h,
and
@=O
C3=g
where h is the horizontal projection of the beam length Substituting C3 in equation (9) we finally obtain
For small values of 8, equation (10) yields the linear bending theory
With the 8’s computed from equation (lo), the values of h and y_, the maximum deflection, can be obtained from the following relationships L = ‘secedx
(11)
d
h
Ymax
=
tan6dx
(12)
In the present problem, h is divided into eight equal intervals. The calculations are carried out as follows : Consider, for example, w~~/D = 3. By using the ~e~o~-Rap~n method [7], the values of 8 at all points can be obtained from equation (10) and are given in Table 1. TABLE 1. Vn~ues x/L - 1
-.. .-
0=0
3 9” 33
a
t 17” 3
22” 49
OF 8 RX
f 26” 58
wh’/D = 3
3 29” 39
f 31”4
t 31”37
Using Simpson’s rule, numerical integration of equations (11) and (12) yields h=0904L
and
y_=O*397L
0 31”41’
392
T, M. WANG
Substituting h = 0904 L into wh3/D = 3 yields wL?/D = 4Q6
RG. 2 Variation of y~~~~ and h/L with WC/D for the cantilever. -----obtained fmm [4]
The numerical results of ymax and h for different values of wL?/D between 1 to 10 arc shown in Fig. 2 and are compared with those obtained by an approximate solution [4]. SIMPLY SWPPORIED BEAM Wll% BRAY
Dads
Ph
FIG.3. The loaded simply supported beam
LOAD
393
Non-linear bending of beams with uniformly distributed loa&
The second case to be considered is shown in Fig. 3 in which the beam is carrying a uniformly distributed load. It is assumed that the reactions at the ends of the beam are vertical The Euler-~rnoul~ equation gives
&f!
d’s =
- w(L - s)cos8
(13)
In this problem, the boundary conditions are d% ds2
-=0
da ds=O
at
x=h
at
x=0
(14) t
8=0
at
x=h
J
Following the same procedure as given in the previous case, equation (13) will yield the tokens equation 2e+sh2e=D
2x3 Wh3 4 2*2+57;” j-
(
(15)
h2
>
Again for small values of 8, equation (15) yields
I36
0.4
0.3
@2
01
0
I
2
FIG. 4. Variation ofy,&L
3
4
5 *LS D
6
7
8
9
IO
and h/L with wL?/Dfor the simply supported beam.
394
which agrees with the linear bending theory. The results for y,, and h for WC/D = 1 to 10 are shown in Fig. 4.
CONCLUSION The expressions for slopes in terms of the horizontal projection of the arc length for beams carrying a uniformly distributed load have been derived. By using numerical methods, the slopes at all points, the maximum deflection, and the horizontal projection of the beam length are obtained. The question may arise as to the accuracy of the proposed solutions with regard to the numerical methods used. It has been pointed out by Scarborough [7) that the results obtained by the Newton-Raphson method are correct to their last figures. This can be proved by substitution of the tf values from Table 1 into equation (10). Thus, the correctness of the solutions may depend on the numerical integration of equations (11) and (12). Wang [S] has shown that the results obtained by Simpson’s rule with eight equal intervals for h for the cantilever beam subjected to a concentrated load at the free end are in excellent agreement with those obtained by an exact solution. Therefore, the results obtained in this paper are closer to the actual values. It can be concluded therefore that the proposed technique offers a more accurate solution.
ftJcFERENcEs
[I] H. J. BATON, On the deflection of a cantilever beam. Q. uppl. Math. 2, 168-171 (1944) atxd Q. appl. Muzfi. 3, 275-276 (I 945). [2] K. E. BIWOPP and D. C. DRUCKER,Large deflections of cantilever beams. Q. appi. Math. 3,272-275 (1945). [3] H. D. CONWAY,The large defleotion of simply supported beams. Phil. Mug. 38,905-911 (1947). [4] F. V. ROHDE,Large deflections of a cantilever beam with uniformly distributed load. Q. appl. Math. 11, 337-338 (1953). [5] A. E. S~hhtxs and H. D. COMKAY,A numerical procedure for calculating the large deflections of straight and curved beams. .I. appl. Mech. 24,289-294 (1957). [6] R. FRNH-FAY, Flexible Bars. Butterworths (1962). [7] J. B. SCARBOROUGH, Numericof Mafhemaficul Anaiysis. John Hopkins Press (1955). [8] T. M. WANG,Nonlinear bending of beams with concentrated loads. J. Frnnklin Inst. 285,386390 (1968). (Received 21 May 1968)
R6samb-h
flexion non-li&aire d’tm encorbelkment et de poutres simplement supportceS soumises a tme charge uniformtment distribuee, a ttt CtudiC J.aasolutions cxactes pour lea pentes ont ttt obtenuea d la solution pour le lkchissement maximum et la projection horizontale de la longueur de la poutre dam le cas de I’encorbellement sont compares & une solution approximative connuc
Zwammenfassaag-Die nichtlineare Biegung von Ausleger- und einfach unterstiitxten Balken unter txleicbm&ssin verteilter Last w&de. tmtersucht Die e&t& tcisung& fiir die Neigung wurden erhalten, und dk Ergebnii fftr die maximak Durchbicgung und waagrechte Projektion der Balkenlgnge for den Fall dcs Aualegerbalkens werden mit einer bckamten N~~~l~~g vergleichen.
Non-hear
bending of beams with unifomdy distributed
Ioads
395
Henmetinblit cm6 KaK KOHCOJtbHOfi, TaH ha npoc~o no~epwmaelvofi Pe*epm-PaccntaTpmann BGKH c paenorveptfo pacnpefieneenofi narpyaKo#. Halunw npaBHJIbKbCfi OTBeT ~JI~IHaKnonoB. Paapeluam Mawmanbnoe 0TKnoneme m &umiy rOpHaonTanbHOr0 earmyna 6iwmf ma Kopnyca K~HC~HZ~,YTO cpaBnnsanK C HaBecTnm4 Kp~6nn~eHn~m pemesneu.