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Post-yield behaviour of a rigid-plastic beam with partial axial and rotational end fixities
Int. J. Mech. Sci. Vol. 32, No. 7, pp. 623 630, 1990 Printed in Great Britain.
0020 7403/90 $3.00+.00 g, 1990Pergamon Pressplc
POST-YIELD BEHAVIOUR OF A RIGID-PLASTIC BEAM WITH PARTIAL AXIAL AND ROTATIONAL END FIXITIES F. TIN-Lol School of Civil Engineering, The University of New South Wales, Kensington, N S W 2033, Australia (Received 25 Auoust 1989; and in revised form 10 January 1990)
Abstra¢~Complete load~leflection relationships of a r i g i d - p l a s t i c b e a m subjected to a nons y m m e t r i c a l l y placed point load are derived. In addition to the inclusion of the effects of geometry change, the analysis considers the supports as being capable Of variable combined axial and rotational restraints. As expected, the increase in load-carrying capacity with deflection can be quite large for certain support conditions.
NOTATION a, b cl, c z
K k L M M0 m N NO n P
Po U W w A
rigid lengths constants defined by equations (22a) and (22b) support stiffness dimensionless support stiffness half-span of beam bending moment full plastic m o m e n t dimensionless bending moment axial force full axial capacity d i m e n s i o n l e s s axial force
applied load yield-point load support displacement m a x i m u m deflection d i m e n s i o n l e s s m a x i m u m deflection lateral m o v e m e n t of load elongation
dimensionless load hinge rotation rotational constraint factor ( ' ) denotes differentiation with respect to w # 0
INTRODUCTION
The study of the effects of change of geometry on the load-carrying capacity of structures has received considerable attention since Haythornthwaite's early work [1] on the behaviour of a fixed-ended rigid-plastic rectangular beam under a point load. Since then a number of papers have been published on the plastic behaviour of transversely loaded beams with various end support conditions, under changes in geometry. Notable results have been achieved by Jones [2] and Hodge [3] who studied the influence of small in-plane boundary displacements on the response of a rigid-plastic rectangular beam loaded by a concentrated load at midspan. While Jones assumed these displacements to be proportional to the square of the maximum transverse deflection, Hodge adopted a physically more meaningful representation for which the axial support movement was considered to be proportional to the induced axial force until the membrane or string phase was reached when the axial force remained constant. Several researchers [4-7] have also shown that the exact analysis for the distributed loading case is very difficult, essentially because of the complex mode changes which can occur. In their analyses, Giirk6k and Hopkins [7] accounted for the additional complexity of combined rotational and axial support restraints. Approximate solutions which violate the yield 623
624
F. Tly-Lol
condition have been suggested by Jones [2] and Schubak et al. [8], among others. The effect of elasticity [-9] and of loading through a rigid circular indenter [10] have also been investigated. Applications of some of the results achieved have been made, for instance, to study the post-yield behaviour of tubular bracing members of offshore platforms I1 I] and the dynamic response of beams of various cross-sections [12]. In this paper, Hodge's work I-3] on the post-yield behaviour of a simply supported rigid-plastic beam under a central point load is extended to the non-trivial case of an asymmetrically located point load. The inclusion of rotational restraints at the support, in addition to axial constraints, is also carried out in the fashion proposed by Giirk6k and Hopkins [7]. The results are given in the form of general and simple expressions for the load~deflection response. These, it will be shown, can be used to regain, for instance, the specific results of Haythornthwaite [1] and Hodge [3]. PRELIMINARY CONSIDERATIONS Consider a rigid-plastic beam of length 2L under a point load P located as shown in Fig. l(a). The end supports are assumed to be capable of both axial and rotational restraints, a feature which will be clarified later on in this section. As the load is increased, the beam will remain motionless until the yield-point load, given by conventional rigid plastic analysis, is reached. At that stage, hinges will have formed in the beam at the supports and at the load point position. Thereafter, the deformation is assumed to follow the mode shown in Fig. l(b); for the sake of clarity, two coincident hinges are assumed to be formed within the span of the beam under the load. The deformation mode adopted herein is consistent with the assumption of the load being attached to a fixed point on the beam and not otherwise constrained. In such a case, the load P will move laterally as well as vertically (Fig. 1b) with the corresponding static and kinematic relations for the rigid length AB of the beam being shown in Fig. l(c). Moreover, as pointed out by Hodge [13], two other possibilities exist. Firstly, the load can still be attached to a fixed point on the beam but is constrained to move only vertically. Secondly, the load can be assumed to move vertically but the beam is not constrained as it is free to slide under the load. The implications of both of these models will not, however, be discussed in this paper. The problem to be solved is that of obtaining the complete load-deflection relationship of the rigid-plastic beam in the presence of geometry changes. It is also assumed that a small strain approximation holds. This implies that the maximum deflection W is sufficiently small (W/a and W/b being of the order of the beam depth to span ratio) compared to the rigid lengths a and b of the beam (Fig. l a} so that the induced axial force N may be considered horizontal and constant in the beam.
p
(b)
(c)
FIG. 1. Rigid-plastic beam with end-constraints. (a) Structure, (b) mechanism at finite deflection, (c) static and kinematic relations.
Post-yield behaviour of a rigid-plastic beam
625
The nature of the variable end-constraints will now be clarified. In the first instance, the axial restraints at the supports are provided by two linear elastic springs each of stiffness K, a model first used by Hodge I-3-]. In view of the assumption that the axial force remains constant and horizontal in the beam, the displacements U at ends A and C must be equal (Fig. lb), and can be calculated from: N = KU.
(1)
Clearly, K = 0 represents complete axial freedom while K = ~ corresponds to complete axial restraint at the supports. In addition to this specified axial constraint, it is also assumed that the supports have variable rotational rigidity. The consequent rotation which can occur implies that the full plastic moment cannot be achieved at the end sections of the beam. A simple and suitable model, essentially proposed by Gfirk6k and Hopkins [7], is to specify that yielding at the supports occurs when the bending moment reaches ~M (Fig. lc), where M is the moment at the span hinges 2 and 3 (Fig. lb) and ~ is a rotational constraint factor. The value of ~ = 0 corresponds to complete rotational freedom (pinned-end), ~ = 1 to complete rotational constraint (fixed-end), while values in the range 0 < ~ < 1 represent intermediate degrees of rotational constraint. This model also implies that the equation to the yield locus of support hinges is given, within a possible sign difference, in non-dimensional form by m = ~f(n);
0 ~< ~ ~< 1
(2)
where M N m = Moo' n - No
(3a,b)
with M o, N o and m = f ( n ) representing, respectively, the full plastic moment capacity, the full axial capacity and the equation to the yield curve of a span hinge. Figure 2 clarifies this situation. Thus, if hinge 2 of Fig. l(b) is at point Q, then the corresponding location for hinge 1 is at Q' since n is the same at both hinges and the moments are of opposite signs. As deformation proceeds, n increases so that both stress points Q and Q' migrate simultaneously along their respective yield curves towards the common pure axial point R. RIGID-PLASTIC ANALYSIS The analysis can be carried out by considering either section AB or BC of the beam at finite deflection. This statement is not immediately obvious since the beam is not symmetrical about the point load so that it is not clear from the beginning why the behaviour of portion BC should be the same as AB. That the kinematics of BC leads to the same results as AB will be done by first deriving the necessary relations for AB and
tn
u.0
5/
-I
FIG. 2. Yield curves for support and span hinges.
626
F. TIN-LOl
corresponding hinges 1 and 2 (Fig. 1), and then showing that identical results will be obtained when BC is considered. With the assumption that U and A are both much less than length a, the ratios U/a and A/a being of the order of the square of the beam depth to span ratio or less, moment equilibrium of AB in Fig. l(c) leads to
Pab
[1 + ~)M + W N -
2L
(4)
Clearly, the yield point load Po for the beam occurs when N = 0 and M = M o, or Po -
2MoL (1 + ¢) ab
(5)
assuming that there is no pretension at zero deflection and that the beam is straight at zero load. Equation (4) can also be written in conventional dimensionless form as
2nw
p = m + -1+~
{6)
P p = Po
~7~
where
WNo w-
2Mo"
(8)
The general load-deflection response given by (6) can be further simplified to furnish explicit expressions for p in terms of w, provided both m and n can be expressed in terms of w. This will be carried out in the following by using information regarding the yield curves of the hinges and their corresponding flow rules. In the first instance, the total elongation CABof bar AB can be calculated by application of Pythagoras' theorem to the configuration shown in Fig. l(c). That is, (a -- U -- A) 2 +
W 2 =
(a
+ tC;AB)2.
(9)
Since A is given by (refer to Appendix)
A = U(~-~b-a)_
(10)
and terms involving the squares of U and CABcan be neglected, then (9) can be simplified to give
W2 ~AB- 2a
Ub
(11)
L
The quantity CABrepresents the sum of the elongations of hinges 1 and 2 located at the ends of the rigid bar AB. The individual contributions of these hinges can be calculated from the normality rule of the assumed flow theory plasticity. As shown in Fig. 2, the requirement that the strain rate vector, with components (Not, MOO), is normal to the yield locus gives dm MoO dn No~
1
(12)
where ( ) represents differentiation with respect to any monotonically increasing parameter. In this work, the dimensionless maximum defection w will be used as that parameter. Assume further that the equation of the yield locus for hinge 2 is given by m = f(n) and that for support hinge 1 by m = - ~f(n); sagging moments and corresponding rotations, and tensions and corresponding extensions are taken to be positive. Then, app.lication.of condition (12) to hinges 1 and 2, together with the kinematic condition that 01 . . . . 02,
Post-yield behaviour of a rigid-plastic beam
627
~ez = kl
(13)
gives
where the subscripts indicate the appropriate hinges. The stress point movement for hinge 2 will now be traced in order to develop the necessary relations between n and w. From (13), the extension rate for hinge 2 can be calculated as /;AS ~2 - 1 + ~"
(14)
Differentiating (11) with respect to w and substituting the answer into (14) leads to k2 = 1 + ~
--
L
Finally, (15) and the obvious geometrical relation 0 2 = the following ratio
"
W/a can now be combined to give
i2 1 ( Uab~ 0~=1+¢ w - ~:/~/.
(16)
The governing equations (4) and (16), derived through considerations of portion AB, can be used with any explicitly specified yield curve relation to generate the load~:leflection history for the beam. However, before this is carried out, it will be briefly explained why the kinematics of section BC would lead to the same results. In the first instance, it should be noted that lengths a and b appear as a product in both key relations (4) and (16), indicating that if one considered BC, the same equation (4) would be obtained, and the ratio of strain rate to rotation rate for hinge 3 would also be given by (16). This can be explicitly checked by deriving, in an identical manner as above, the corresponding relations for BC; this straightforward exercise will not be carried out in this paper. As an example of how the load-deflection response can be obtained, consider the case of a beam with a solid rectangular section for which the yield equation for positive moment, as for hinge 2, is defined by m = 1 - n Z ; m>~0. (17) Equation (17) applies to span hinges 2 and 3 while the yield curve equation for support hinges 1 and 4 follows from (2). The evolution for hinge 2 (or, equivalently hinge 3) is chosen to trace the deformation history of the structure. As mentioned earlier, the stress point movement for this hinge (n 1> 0, m/> 0), with increase of load, is along yield curve (17). Initially, the stress point is at (0, 1) corresponding to the pure bending situation at the yield-point load. As the load is increased, the stress point migrates towards the pure axial or membrane situation. The deflection behaviour corresponding to the chosen yield condition is then clearly in two stages, namely, a first stage during which n < 1 and the strain rate vector must be normal to the yield curve, and a second stage when n = 1 and the stress point is at a corner of the yield locus. Hence, the normality condition (12) is required in the first stage only. These two stages will now be considered.
Staoe l : n <
1 For hinge 2, (12) and (17) can be used to give
E2
2nMo
02
No
(18)
which when substituted into (16) produces the following relation:
n=2Mo(l+~)
W
~,'L
"
(19)
628
F. T I N - L O I
Now, (1) and (3b) can be combined to calculate the derivative of U while (8) leads to the derivative of W as follows: 0 - fiN° K '
W - 2M° NO
(20a,b)
Further, Wdefined by (8) and the derivatives given by (20) can be substituted into (19) to give, after some simplification, the following first order linear differential equation in n and w: r/ =
(21)
C l W - - C2fi
where 1
ab
cl - 1 + ~'
c2
(1 + ~ ) L Z k
(22a,b)
and k is a dimensionless spring constant, defined in an identical manner by Hodge [3], through the expression k -
4M2°K NaoL
(22c)
Again, it should be noted that c 2 involves the product ab indicating that consideration of either AB or BC would lead to identical results, ensuring continuity of axial force and moment across the middle hinges. Equation (21) can now be easily solved for the initial condition w = 0, n = 0 to give the following explicit expression for n in terms of w: n = clc2exp
(w) ---
+ clw -
clc2.
(23)
c2
In turn, the general deflection history given by (6) can be expressed in terms of n through substitution of (17) as 2nw
p = ( 1 - n 2) + -1 -+ ~;
n<
1
(24)
which together with (23), fully defines the first stage load-deflection behaviour of the structure. Stage 2:n=
1 This stage or membrane condition for the structure occurs when the stress point reaches (1, 0), in which case the load-deflection response can be obtained directly from (6). It is given by 2w / ~ - 1 + ~;
n=
1.
(251
The entire load-deflection behaviour of the beam considered is therefore defined by relations (24) and (25). It is easy to verify that both Haythornthwaite's [1] and Hodge's [3] solutions can be recovered from these expressions. In the former case, the obvious substitutions of ~ - 1 and k = ~ are made, while in the latter case ( = 0 and a = b = L. It should also be noted that the present solutions, although founded on an upper bound approach, are in fact exact, or complete, within rigid-plastic theory and the assumptions made on the magnitudes of the deflections. The deformation mode is kinematically admissible so that equilibrium at each deflection stage is ensured. Static admissibility requirements are also satisfied since the yield condition for the whole beam is never violated, with the possibility of plastic hinges forming only at the assumed locations 1 to 4 where the (n, m) stress combinations are likely to be critical. This latter observation is due to the fact that the axial force throughout the beam is constant, and the total bending moment, with maximum values at the hinge locations, is made up of two linear moment diagrams caused by the point load and the axial force. However, adoption of this deformation mode when a distributed loading is applied would lead to violation of yield in the beam [4].
Post-yield behaviour of a rigid-plastic beam
629
5.0
/ 4.5
--
4.0--
+ v
o =
0.8L,
b =
- -
3.5
:::
3.0
--
/ /
t/
°
. . . .
fff
--
:¢
1.2L
///I:
2.5 2.0
.. 1.5
0o,
_
1.0
k -- 0 " " 0.5 0.0
I
I
I
I
I
0.4
0.8
1.2
I .ll
2,0
2,.4
W
FIG. 3. Load~leflection curves for rigid-plastic beam with different degrees of axial and rotational support constraints.
Finally, to complete this section, a specific example with a = 0.8 L and b = 1.2 L has been worked out to illustrate the effects of varying the end restraint conditions. Figure 3 shows the load-deflection responses; two sets of curves corresponding to ~ = 0 (pinned-end) and k = ~ (complete axial restraint) are given. Clearly, for some conditions, the increase in loadcarrying capacity of the beam with deflection can be quite large. CONCLUSION
An analysis which includes the effects of finite geometry changes and variable endconstraints has been developed to predict the response of a rigid-plastic beam subjected to an asymmetrically applied point load. Fairly general closed-form expressions relating the load to the maximum transverse displacement have been derived. These theoretical solutions are exact since both static and kinematic admissibility requirements are satisfied. Once again, the importance of considering both induced axial forces and end-constraints has been demonstrated. Whilst the method presented herein is eminently suitable for simple structures and loading conditions, a computer analysis would be necessary for more complex problems. A worthwhile extension would be to develop a general rigid-plastic computer-oriented formulation to enable such occurrences as travelling and splitting hinges to be considered. Acknowled#ements--The author would like to thank Professor P. G. Hodge Jr for his interest and useful comments on this problem, and the reviewers for their constructive and detailed remarks on an earlier version of this paper. REFERENCES 1. R. M. HAYTHORNTHWAITE,Beams with full end fixity. En#ineerin# 183, 110 (1957). 2. N. JONES, Influence of in-plane displacements at the boundaries of rigid-plastic beams and plates. Int. J. Mech. Sci. 15, 547 (1973). 3. P. G. HODGE, Post-yield behaviour of a beam with partial end fixity. Int. J. Mech. Sci. 16, 385 (1974). 4. R. M. HAYTHORNTHWAITE,Mode change during the plastic collapse of beams and plates. In Developments in Mechanics (Proc. 7th Midwestern Mech. Conf., Michigan State University, 6-8 September 1961) (edited by J. E. LAY and L. E. MALVERN),Vol. 1, p. 203. Plenum Press, New York (1961). 5. S. S. GILL, Effect of deflexion on the plastic collapse of beams with distributed load. Int. J. Mech. Sci. 15, 465 (1973). 6. A. Gt~RKOK and H. G. HOPKINS, The effect of geometry changes on the load carrying capacity of beams under transverse load. S I A M J. appl. Math. 25, 500 (1973). 7. A. GIJRKOK and H. G. HOPKINS,Plastic beams at finite deflection under transverse load with variable endconstraints. J. Mech. Phys. Solids 29, 447 (1981). 8. R. B. SCHUBAK,M. D. OLSON and D. L. ANDERSON, Simplified rigid-plastic beam analysis. J. appl. Mech. 54, 720 (1987).
630
F. Tm-Lol
9. T. I. CAMPBELL and T. M. CHARLTON, Finite deformation of a fully fixed beam comprised of a non-linear material. Int. J. Mech. Sci. 15, 415 (1973). 10. H. Y. Low, Behaviour of a rigid-plastic beam loaded to finite deflections by a rigid circular indenter. Int. J. Mech. Sci. 23, 387 (1981). 11. C.G. SOARESand T. H. SOREIDE, Plastic analysis of laterally loaded circular tubes. J. Struct. Engng ASCE 109, 451 (1983). 12. R. VAZIRI, M. D. OLSON and D. L. ANDERSON, Dynamic response of axially constrained plastic beams to blast loads. Int. J. Solids Struct. 23, 153 (1987). 13. P. G. HODGE, Private communication (June 1989).
APPENDIX In this appendix, the relation given by (10) will be derived. The ratio of the elongations of portions AB and BC of the beam is given by ~AB
qg2
(Al)
where (14) has been used in conjunction with the corresponding expressions for BC. Also, as noted in the text after (16), the ratios of strain rates to rotation rates for hinges 2 and 3 are equal so that L~2
02
g3
= -03
(A2)
which, when combined with (AI), 02 = W/a and 03 = W/b, leads to b eAB=
a
eac.
(A3)
Now, (19) can be simplified by neglecting terms involving the squares of U, A, e and their products to give
-2a(A + U) + W 2 = 2aeAB.
(A4)
Similarly, the corresponding expression for BC is 2 b ( A - U) + W 2 = 2beBc.
(A5}
Finally, (A3), (A4) and (A5) can be used to solve for A in terms of a, b and U to give A =
u(b - a \a+b
thus proving the expression given in (10).
. /
(A6)
0020 7403/90 $3.00+.00 g, 1990Pergamon Pressplc
POST-YIELD BEHAVIOUR OF A RIGID-PLASTIC BEAM WITH PARTIAL AXIAL AND ROTATIONAL END FIXITIES F. TIN-Lol School of Civil Engineering, The University of New South Wales, Kensington, N S W 2033, Australia (Received 25 Auoust 1989; and in revised form 10 January 1990)
Abstra¢~Complete load~leflection relationships of a r i g i d - p l a s t i c b e a m subjected to a nons y m m e t r i c a l l y placed point load are derived. In addition to the inclusion of the effects of geometry change, the analysis considers the supports as being capable Of variable combined axial and rotational restraints. As expected, the increase in load-carrying capacity with deflection can be quite large for certain support conditions.
NOTATION a, b cl, c z
K k L M M0 m N NO n P
Po U W w A
rigid lengths constants defined by equations (22a) and (22b) support stiffness dimensionless support stiffness half-span of beam bending moment full plastic m o m e n t dimensionless bending moment axial force full axial capacity d i m e n s i o n l e s s axial force
applied load yield-point load support displacement m a x i m u m deflection d i m e n s i o n l e s s m a x i m u m deflection lateral m o v e m e n t of load elongation
dimensionless load hinge rotation rotational constraint factor ( ' ) denotes differentiation with respect to w # 0
INTRODUCTION
The study of the effects of change of geometry on the load-carrying capacity of structures has received considerable attention since Haythornthwaite's early work [1] on the behaviour of a fixed-ended rigid-plastic rectangular beam under a point load. Since then a number of papers have been published on the plastic behaviour of transversely loaded beams with various end support conditions, under changes in geometry. Notable results have been achieved by Jones [2] and Hodge [3] who studied the influence of small in-plane boundary displacements on the response of a rigid-plastic rectangular beam loaded by a concentrated load at midspan. While Jones assumed these displacements to be proportional to the square of the maximum transverse deflection, Hodge adopted a physically more meaningful representation for which the axial support movement was considered to be proportional to the induced axial force until the membrane or string phase was reached when the axial force remained constant. Several researchers [4-7] have also shown that the exact analysis for the distributed loading case is very difficult, essentially because of the complex mode changes which can occur. In their analyses, Giirk6k and Hopkins [7] accounted for the additional complexity of combined rotational and axial support restraints. Approximate solutions which violate the yield 623
624
F. Tly-Lol
condition have been suggested by Jones [2] and Schubak et al. [8], among others. The effect of elasticity [-9] and of loading through a rigid circular indenter [10] have also been investigated. Applications of some of the results achieved have been made, for instance, to study the post-yield behaviour of tubular bracing members of offshore platforms I1 I] and the dynamic response of beams of various cross-sections [12]. In this paper, Hodge's work I-3] on the post-yield behaviour of a simply supported rigid-plastic beam under a central point load is extended to the non-trivial case of an asymmetrically located point load. The inclusion of rotational restraints at the support, in addition to axial constraints, is also carried out in the fashion proposed by Giirk6k and Hopkins [7]. The results are given in the form of general and simple expressions for the load~deflection response. These, it will be shown, can be used to regain, for instance, the specific results of Haythornthwaite [1] and Hodge [3]. PRELIMINARY CONSIDERATIONS Consider a rigid-plastic beam of length 2L under a point load P located as shown in Fig. l(a). The end supports are assumed to be capable of both axial and rotational restraints, a feature which will be clarified later on in this section. As the load is increased, the beam will remain motionless until the yield-point load, given by conventional rigid plastic analysis, is reached. At that stage, hinges will have formed in the beam at the supports and at the load point position. Thereafter, the deformation is assumed to follow the mode shown in Fig. l(b); for the sake of clarity, two coincident hinges are assumed to be formed within the span of the beam under the load. The deformation mode adopted herein is consistent with the assumption of the load being attached to a fixed point on the beam and not otherwise constrained. In such a case, the load P will move laterally as well as vertically (Fig. 1b) with the corresponding static and kinematic relations for the rigid length AB of the beam being shown in Fig. l(c). Moreover, as pointed out by Hodge [13], two other possibilities exist. Firstly, the load can still be attached to a fixed point on the beam but is constrained to move only vertically. Secondly, the load can be assumed to move vertically but the beam is not constrained as it is free to slide under the load. The implications of both of these models will not, however, be discussed in this paper. The problem to be solved is that of obtaining the complete load-deflection relationship of the rigid-plastic beam in the presence of geometry changes. It is also assumed that a small strain approximation holds. This implies that the maximum deflection W is sufficiently small (W/a and W/b being of the order of the beam depth to span ratio) compared to the rigid lengths a and b of the beam (Fig. l a} so that the induced axial force N may be considered horizontal and constant in the beam.
p
(b)
(c)
FIG. 1. Rigid-plastic beam with end-constraints. (a) Structure, (b) mechanism at finite deflection, (c) static and kinematic relations.
Post-yield behaviour of a rigid-plastic beam
625
The nature of the variable end-constraints will now be clarified. In the first instance, the axial restraints at the supports are provided by two linear elastic springs each of stiffness K, a model first used by Hodge I-3-]. In view of the assumption that the axial force remains constant and horizontal in the beam, the displacements U at ends A and C must be equal (Fig. lb), and can be calculated from: N = KU.
(1)
Clearly, K = 0 represents complete axial freedom while K = ~ corresponds to complete axial restraint at the supports. In addition to this specified axial constraint, it is also assumed that the supports have variable rotational rigidity. The consequent rotation which can occur implies that the full plastic moment cannot be achieved at the end sections of the beam. A simple and suitable model, essentially proposed by Gfirk6k and Hopkins [7], is to specify that yielding at the supports occurs when the bending moment reaches ~M (Fig. lc), where M is the moment at the span hinges 2 and 3 (Fig. lb) and ~ is a rotational constraint factor. The value of ~ = 0 corresponds to complete rotational freedom (pinned-end), ~ = 1 to complete rotational constraint (fixed-end), while values in the range 0 < ~ < 1 represent intermediate degrees of rotational constraint. This model also implies that the equation to the yield locus of support hinges is given, within a possible sign difference, in non-dimensional form by m = ~f(n);
0 ~< ~ ~< 1
(2)
where M N m = Moo' n - No
(3a,b)
with M o, N o and m = f ( n ) representing, respectively, the full plastic moment capacity, the full axial capacity and the equation to the yield curve of a span hinge. Figure 2 clarifies this situation. Thus, if hinge 2 of Fig. l(b) is at point Q, then the corresponding location for hinge 1 is at Q' since n is the same at both hinges and the moments are of opposite signs. As deformation proceeds, n increases so that both stress points Q and Q' migrate simultaneously along their respective yield curves towards the common pure axial point R. RIGID-PLASTIC ANALYSIS The analysis can be carried out by considering either section AB or BC of the beam at finite deflection. This statement is not immediately obvious since the beam is not symmetrical about the point load so that it is not clear from the beginning why the behaviour of portion BC should be the same as AB. That the kinematics of BC leads to the same results as AB will be done by first deriving the necessary relations for AB and
tn
u.0
5/
-I
FIG. 2. Yield curves for support and span hinges.
626
F. TIN-LOl
corresponding hinges 1 and 2 (Fig. 1), and then showing that identical results will be obtained when BC is considered. With the assumption that U and A are both much less than length a, the ratios U/a and A/a being of the order of the square of the beam depth to span ratio or less, moment equilibrium of AB in Fig. l(c) leads to
Pab
[1 + ~)M + W N -
2L
(4)
Clearly, the yield point load Po for the beam occurs when N = 0 and M = M o, or Po -
2MoL (1 + ¢) ab
(5)
assuming that there is no pretension at zero deflection and that the beam is straight at zero load. Equation (4) can also be written in conventional dimensionless form as
2nw
p = m + -1+~
{6)
P p = Po
~7~
where
WNo w-
2Mo"
(8)
The general load-deflection response given by (6) can be further simplified to furnish explicit expressions for p in terms of w, provided both m and n can be expressed in terms of w. This will be carried out in the following by using information regarding the yield curves of the hinges and their corresponding flow rules. In the first instance, the total elongation CABof bar AB can be calculated by application of Pythagoras' theorem to the configuration shown in Fig. l(c). That is, (a -- U -- A) 2 +
W 2 =
(a
+ tC;AB)2.
(9)
Since A is given by (refer to Appendix)
A = U(~-~b-a)_
(10)
and terms involving the squares of U and CABcan be neglected, then (9) can be simplified to give
W2 ~AB- 2a
Ub
(11)
L
The quantity CABrepresents the sum of the elongations of hinges 1 and 2 located at the ends of the rigid bar AB. The individual contributions of these hinges can be calculated from the normality rule of the assumed flow theory plasticity. As shown in Fig. 2, the requirement that the strain rate vector, with components (Not, MOO), is normal to the yield locus gives dm MoO dn No~
1
(12)
where ( ) represents differentiation with respect to any monotonically increasing parameter. In this work, the dimensionless maximum defection w will be used as that parameter. Assume further that the equation of the yield locus for hinge 2 is given by m = f(n) and that for support hinge 1 by m = - ~f(n); sagging moments and corresponding rotations, and tensions and corresponding extensions are taken to be positive. Then, app.lication.of condition (12) to hinges 1 and 2, together with the kinematic condition that 01 . . . . 02,
Post-yield behaviour of a rigid-plastic beam
627
~ez = kl
(13)
gives
where the subscripts indicate the appropriate hinges. The stress point movement for hinge 2 will now be traced in order to develop the necessary relations between n and w. From (13), the extension rate for hinge 2 can be calculated as /;AS ~2 - 1 + ~"
(14)
Differentiating (11) with respect to w and substituting the answer into (14) leads to k2 = 1 + ~
--
L
Finally, (15) and the obvious geometrical relation 0 2 = the following ratio
"
W/a can now be combined to give
i2 1 ( Uab~ 0~=1+¢ w - ~:/~/.
(16)
The governing equations (4) and (16), derived through considerations of portion AB, can be used with any explicitly specified yield curve relation to generate the load~:leflection history for the beam. However, before this is carried out, it will be briefly explained why the kinematics of section BC would lead to the same results. In the first instance, it should be noted that lengths a and b appear as a product in both key relations (4) and (16), indicating that if one considered BC, the same equation (4) would be obtained, and the ratio of strain rate to rotation rate for hinge 3 would also be given by (16). This can be explicitly checked by deriving, in an identical manner as above, the corresponding relations for BC; this straightforward exercise will not be carried out in this paper. As an example of how the load-deflection response can be obtained, consider the case of a beam with a solid rectangular section for which the yield equation for positive moment, as for hinge 2, is defined by m = 1 - n Z ; m>~0. (17) Equation (17) applies to span hinges 2 and 3 while the yield curve equation for support hinges 1 and 4 follows from (2). The evolution for hinge 2 (or, equivalently hinge 3) is chosen to trace the deformation history of the structure. As mentioned earlier, the stress point movement for this hinge (n 1> 0, m/> 0), with increase of load, is along yield curve (17). Initially, the stress point is at (0, 1) corresponding to the pure bending situation at the yield-point load. As the load is increased, the stress point migrates towards the pure axial or membrane situation. The deflection behaviour corresponding to the chosen yield condition is then clearly in two stages, namely, a first stage during which n < 1 and the strain rate vector must be normal to the yield curve, and a second stage when n = 1 and the stress point is at a corner of the yield locus. Hence, the normality condition (12) is required in the first stage only. These two stages will now be considered.
Staoe l : n <
1 For hinge 2, (12) and (17) can be used to give
E2
2nMo
02
No
(18)
which when substituted into (16) produces the following relation:
n=2Mo(l+~)
W
~,'L
"
(19)
628
F. T I N - L O I
Now, (1) and (3b) can be combined to calculate the derivative of U while (8) leads to the derivative of W as follows: 0 - fiN° K '
W - 2M° NO
(20a,b)
Further, Wdefined by (8) and the derivatives given by (20) can be substituted into (19) to give, after some simplification, the following first order linear differential equation in n and w: r/ =
(21)
C l W - - C2fi
where 1
ab
cl - 1 + ~'
c2
(1 + ~ ) L Z k
(22a,b)
and k is a dimensionless spring constant, defined in an identical manner by Hodge [3], through the expression k -
4M2°K NaoL
(22c)
Again, it should be noted that c 2 involves the product ab indicating that consideration of either AB or BC would lead to identical results, ensuring continuity of axial force and moment across the middle hinges. Equation (21) can now be easily solved for the initial condition w = 0, n = 0 to give the following explicit expression for n in terms of w: n = clc2exp
(w) ---
+ clw -
clc2.
(23)
c2
In turn, the general deflection history given by (6) can be expressed in terms of n through substitution of (17) as 2nw
p = ( 1 - n 2) + -1 -+ ~;
n<
1
(24)
which together with (23), fully defines the first stage load-deflection behaviour of the structure. Stage 2:n=
1 This stage or membrane condition for the structure occurs when the stress point reaches (1, 0), in which case the load-deflection response can be obtained directly from (6). It is given by 2w / ~ - 1 + ~;
n=
1.
(251
The entire load-deflection behaviour of the beam considered is therefore defined by relations (24) and (25). It is easy to verify that both Haythornthwaite's [1] and Hodge's [3] solutions can be recovered from these expressions. In the former case, the obvious substitutions of ~ - 1 and k = ~ are made, while in the latter case ( = 0 and a = b = L. It should also be noted that the present solutions, although founded on an upper bound approach, are in fact exact, or complete, within rigid-plastic theory and the assumptions made on the magnitudes of the deflections. The deformation mode is kinematically admissible so that equilibrium at each deflection stage is ensured. Static admissibility requirements are also satisfied since the yield condition for the whole beam is never violated, with the possibility of plastic hinges forming only at the assumed locations 1 to 4 where the (n, m) stress combinations are likely to be critical. This latter observation is due to the fact that the axial force throughout the beam is constant, and the total bending moment, with maximum values at the hinge locations, is made up of two linear moment diagrams caused by the point load and the axial force. However, adoption of this deformation mode when a distributed loading is applied would lead to violation of yield in the beam [4].
Post-yield behaviour of a rigid-plastic beam
629
5.0
/ 4.5
--
4.0--
+ v
o =
0.8L,
b =
- -
3.5
:::
3.0
--
/ /
t/
°
. . . .
fff
--
:¢
1.2L
///I:
2.5 2.0
.. 1.5
0o,
_
1.0
k -- 0 " " 0.5 0.0
I
I
I
I
I
0.4
0.8
1.2
I .ll
2,0
2,.4
W
FIG. 3. Load~leflection curves for rigid-plastic beam with different degrees of axial and rotational support constraints.
Finally, to complete this section, a specific example with a = 0.8 L and b = 1.2 L has been worked out to illustrate the effects of varying the end restraint conditions. Figure 3 shows the load-deflection responses; two sets of curves corresponding to ~ = 0 (pinned-end) and k = ~ (complete axial restraint) are given. Clearly, for some conditions, the increase in loadcarrying capacity of the beam with deflection can be quite large. CONCLUSION
An analysis which includes the effects of finite geometry changes and variable endconstraints has been developed to predict the response of a rigid-plastic beam subjected to an asymmetrically applied point load. Fairly general closed-form expressions relating the load to the maximum transverse displacement have been derived. These theoretical solutions are exact since both static and kinematic admissibility requirements are satisfied. Once again, the importance of considering both induced axial forces and end-constraints has been demonstrated. Whilst the method presented herein is eminently suitable for simple structures and loading conditions, a computer analysis would be necessary for more complex problems. A worthwhile extension would be to develop a general rigid-plastic computer-oriented formulation to enable such occurrences as travelling and splitting hinges to be considered. Acknowled#ements--The author would like to thank Professor P. G. Hodge Jr for his interest and useful comments on this problem, and the reviewers for their constructive and detailed remarks on an earlier version of this paper. REFERENCES 1. R. M. HAYTHORNTHWAITE,Beams with full end fixity. En#ineerin# 183, 110 (1957). 2. N. JONES, Influence of in-plane displacements at the boundaries of rigid-plastic beams and plates. Int. J. Mech. Sci. 15, 547 (1973). 3. P. G. HODGE, Post-yield behaviour of a beam with partial end fixity. Int. J. Mech. Sci. 16, 385 (1974). 4. R. M. HAYTHORNTHWAITE,Mode change during the plastic collapse of beams and plates. In Developments in Mechanics (Proc. 7th Midwestern Mech. Conf., Michigan State University, 6-8 September 1961) (edited by J. E. LAY and L. E. MALVERN),Vol. 1, p. 203. Plenum Press, New York (1961). 5. S. S. GILL, Effect of deflexion on the plastic collapse of beams with distributed load. Int. J. Mech. Sci. 15, 465 (1973). 6. A. Gt~RKOK and H. G. HOPKINS, The effect of geometry changes on the load carrying capacity of beams under transverse load. S I A M J. appl. Math. 25, 500 (1973). 7. A. GIJRKOK and H. G. HOPKINS,Plastic beams at finite deflection under transverse load with variable endconstraints. J. Mech. Phys. Solids 29, 447 (1981). 8. R. B. SCHUBAK,M. D. OLSON and D. L. ANDERSON, Simplified rigid-plastic beam analysis. J. appl. Mech. 54, 720 (1987).
630
F. Tm-Lol
9. T. I. CAMPBELL and T. M. CHARLTON, Finite deformation of a fully fixed beam comprised of a non-linear material. Int. J. Mech. Sci. 15, 415 (1973). 10. H. Y. Low, Behaviour of a rigid-plastic beam loaded to finite deflections by a rigid circular indenter. Int. J. Mech. Sci. 23, 387 (1981). 11. C.G. SOARESand T. H. SOREIDE, Plastic analysis of laterally loaded circular tubes. J. Struct. Engng ASCE 109, 451 (1983). 12. R. VAZIRI, M. D. OLSON and D. L. ANDERSON, Dynamic response of axially constrained plastic beams to blast loads. Int. J. Solids Struct. 23, 153 (1987). 13. P. G. HODGE, Private communication (June 1989).
APPENDIX In this appendix, the relation given by (10) will be derived. The ratio of the elongations of portions AB and BC of the beam is given by ~AB
qg2
(Al)
where (14) has been used in conjunction with the corresponding expressions for BC. Also, as noted in the text after (16), the ratios of strain rates to rotation rates for hinges 2 and 3 are equal so that L~2
02
g3
= -03
(A2)
which, when combined with (AI), 02 = W/a and 03 = W/b, leads to b eAB=
a
eac.
(A3)
Now, (19) can be simplified by neglecting terms involving the squares of U, A, e and their products to give
-2a(A + U) + W 2 = 2aeAB.
(A4)
Similarly, the corresponding expression for BC is 2 b ( A - U) + W 2 = 2beBc.
(A5}
Finally, (A3), (A4) and (A5) can be used to solve for A in terms of a, b and U to give A =
u(b - a \a+b
thus proving the expression given in (10).
. /
(A6)