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Reinforcement minimization of beams with axial forces
Int..I. Mech. Sci. Vol. 23, No. 10, pp. 607-617, 1981 Printed in Great Britain.
0020-7403/81/100607-11502.00/0 Pergamon Press Ltd.
REINFORCEMENT MINIMIZATION OF BEAMS WITH AXIAL FORCES R. E. MELCHERS Department of Civil Engineering, Monash University, Australia (Received 10 June 1980; in revised [orm 24 November 1980)
Summary--Within the confines of absolute minimum weight design of plastic structures, it has generally been assumed that the relationship between material (cost) and the vector of internal actions is linear (or piecewise linear) that is, sandwich approximations, or essentially constant lever-arm assumptions for reinforced materials, have been employed. Where both axial load and moment act on the one cross-section, or moments are relatively large, such idealization of cost function is no longer consistent with actual behaviour. A nonlinear cost function model must be substituted. The cost-function for an idealized reinforced section under bending moment and axial load (tensile or compressive) is derived herein. Two examples of encastre reinforced beams with axial forces are considered. When the applied load consists of a transverse central point load the optimum design is found to accord with earlier results employing a simplified cost function. When the applied lateral load is uniformly distributed earlier results are only approached in the limit as the lateral load approaches infinity.
A~, A2 a b c D _G H k~ L Mx mx Nx nx P _Q _q s s' Sr wa
NOTATION tensile and compressive cross-sectional areas of reinforcement distance to inflexion points from beam end beam width specific cost structural domain cost gradient vector beam section depth curvature of associated structure beam length moment 2MJ(trcH2b) axial tension Nx/(~ycHb) applied point load vector of generalised stresses vector of generalised strains (al~ro)/(o'cHb) (A2~ro)/(~cHb) total specific cost (reinforcement content) applied uniformly distributed loading
strain in associated structure u proportion of beam depth in compression q5 total cost ~b specific cost function, strain angle generic point ,~ linear multiplier tr~ concrete matrix compressive strength tr0 steel yield strength 1. INTRODUCTION T h e c l a s s o f p r o b l e m c o n s i d e r e d h e r e i n c o n c e r n s r e i n f o r c e d s t r u c t u r e s f o r w h i c h the c o s t ( w e i g h t , v o l u m e ) o f r e i n f o r c i n g m a t e r i a l p r e d o m i n a t e s o v e r the c o s t o f the m a t r i x m a t e r i a l , a n d f o r w h i c h a d e s i g n is s o u g h t s u c h t h a t the t o t a l c o s t ( w e i g h t , v o l u m e ) o r r e i n f o r c i n g m a t e r i a l s is r e d u c e d to a n a b s o l u t e m i n i m u m . T h e u s u a l a s s u m p t i o n s f o r d e s c r i b i n g b e h a v i o u r a r e m a d e . T h u s it is a s s u m e d t h a t s i m p l e ( " e n g i n e e r s " ) b e n d i n g t h e o r y is v a l i d , t h a t the influence o f s h e a r c a n be i g n o r e d o r a s s u m e d c a r r i e d e n t i r e l y b y the m a t r i x m a t e r i a l a n d t h a t l o a d s a r e q u a s i - s t a t i c . It is a l s o a s s u m e d t h a t i n s t a b i l i t y e f f e c t s m a y b e i g n o r e d . T h e r e i n f o r c i n g m a t e r i a l a n d the m a t r i x a r e a s s u m e d to h a v e r i g i d - p l a s t i c m a t e r i a l p r o p e r t i e s . This is a first a p p r o x i m a t i o n to the b e h a v i o u r o f r e i n f o r c e d c o n c r e t e in the p l a s t i c r a n g e . 607
608
R.E. MELCHERS Piece
- wise
lineor
,C
,
V \
Nonlineor
,c : s
: t-(l-
,0.5 mxJ
\ \
J
\
/.i to
~ ~
Ideal,zotlon for reinforced concrete
~y O ~
Im
: ~
slobs
H=b)
FIO. 1. Specific cost function approximations.
Previous investigations into the absolute optimal arrangement of reinforcing materials [1-3] have usually assumed that the relationship between the unit cost (cost per unit area, per unit length, etc.) and the vector of internal actions is linear (see Fig. 1). This follows naturally in the case of reinforced concrete slabs since the percentage of reinforcing material is usually low. The linearity assumption has also been applied to reinforced concrete shells [4, 5], steel/-beams [6] and beam grillages [7]. Apart from the work of Mroz[5], the work of Morley[l] on one-way and circular slabs and a limited amount of work using piecewise linear cost functions[3,8], nonlinear cost functions do not appear to have been applied in optimization problems. A fundamental difficulty has been finding a way of applying an appropriate optimality criterion such as that of Prager-Shield[8] to the particular problem under consideration. 2. OPTIMALITY CRITERION Various techniques exist to find a solution to an optimization problem. The approach used here is due to Prager and Shield[8]. In their terminology, the local state of stress at a generic point ~ in the structural domain D is defined by generalized stresses Q~(O (i = 1,2 . . . . . n) corresponding to generalized strains q~(~) (i = 1, 2 . . . . . n). Let the total quantity to be minimized be termed the total cost (q~) and the cost per unit area or unit length of the structure be termed specific cost (~b). The relationship between the specific cost ~h and the generalized stresses _Q is given by the specific cost function ~b(Q). The cost gradient vector _G(40 is defined as
[ am am aq, ] G(~b) = (grad ~), = t-~' ~'~2 ..... O--~.J
(1)
for all states of generalized stress Q* for which 4'(_Q) is differentiable. If ~b(_Q) is discontinuous at some generalized stress state Q_d,then _G fs defined as any convex combination of the gradient vectors G i for the adjacent regions (in each of which 4~(_Q)is differentiable): _G = ~ A,_G, 9=9, = ~ At(grad 4~), 9=9'
(2)
which for all i: ,~ > 0 and E A~= 1. PraiSer and Shield[8] have shown that on the structural domain D, the condition for minimum cost is a kinematic condition expressed by[3]: q(¢) = G[_Q(¢)]
(3)
where q is a kinematically admissible generalized strain field corresponding to the statically admissible generalized stress field _Q.
3. COST FUNCTION FOR COMBINED BENDING AND AXIAL FORCE Consider the idealized reinforced concrete beam section shown in Fig. 2, in which concrete cover to the reinforcement has been ignored, both tensile (A~ 3 0 ) and compressive (A2~>0) reinforcement may be present and in which the reinforcing bars have been idealized as a uniform flat sheet on each beam face. Let the concrete matrix have ideal rigid-plastic material properties with compressive strength + ~c and zero tensile strength. The reinforcement is assumed ideally rigid-plastic with tensile strength+ ~0 and compressive strength-tr0. Each identical reinforcement layer has negligible thickness compared to the beam section depth, H t> 0.
Reinforcement minimization of beams with axial forces
609
M
]:--N ~
A l f fo
4:at
A1
FIG. 2. Material idealization, strain state and stress state. 3.1 Case 1. N o compression reinforcement Under the action of the stress vector {Mx, N~} the equations of equilibrium for the cross-section about its centre line are (see Fig. 2): n M~ = Ato'0-~- + vHtrc ~
Hb
(4)
and
Nx = A~,ro - vHtrcb.
(5)
The reduced notation (see Mroz[5]): 2Mx mx = - - - ~ - ~
nx
= Nx trcI-Ib
Ato'c s = tT~nb
will be adopted. Combining equations (4) and (5) to eliminate the parameter v: mx = 2s - nx - (s - n~)2.
(6)
For a unit length of beam, the cross-sectional area, AI of reinforcing material is proportional the specific cost so that it is sufficient to optimize 's'. For the simplest case, nx = 0; equation (6) leads, for s/> 0, to: s = 1 - (1 - m~) 1/2
(7)
which is plotted in Fig. 1. It is evident that only at low values of m~ is the linear approximation used in earlier optimization work adequate. 3.2 Case 2. Compression reinforcement The total specific cost is now sT = Is[ + Is'l
(8)
where s' = A:,7o/((rcHb). In reduced notation, equilibrium can be expressed as n~ = s + s ' - v m~ = s - s ' +
v(1 - v)
(9) (10)
which, for the special case nx = 0, reduces to m~ = 2s - (s + s') 2.
(11)
When both layers A~ and A2 are present the distribution of the minimal reinforcement sr = [s[ + Is'l, between the two reinforcement layers in a beam section, to transmit a given vector {m~, nx} of applied stress resultants, can be obtained by applying the optimality condition, equation (3) to (8), where Sr is the specific cost, and s and s' are treated as stress resultants. (This is valid since the applied stress resultants, {M~, N~} may always be transformed to any other equivalent system.) Hence et = -~-ST= + 1
BST e2 = "~'TS,= --1
(12)
where the values of the strains el and 42 are in reduced notation and the signs correspond to the sign of s and s' respectively. Equations (12) express the requirement that when both reinforcements A~ and A2 are
610
R . E . MELCHERS
present in the structure, they must be equally (but oppositely) strained in the (associated) structure for the actual structure to be optimal. When this is the case, then v = 1/2, i.e. the neutral axis corresponds with the centre line, and for n~ = O, equations (9) and (10) give !
= s + s'
(9a) 1
mx = s - s ' + ~
(lOa)
so that mx
1
mx
3
s =Y+i; s=--T+i sr = m x - ~ for m,
(13a, b,c)
The relation between sT and m, for n, = 0 is sketched in Fig. 3(a). Evidently, for m,/> 3/4, compression reinforcement produces a lower specific cost c = st. The relationship between s and S' could also have been obtained by a trial and error approach to minimizing sT. More generally, either A] or A2 or both may be preseat. To obtain the requirements for optimality under these circumstances, equation (3) is applied to the specific cost: kx =
#c
(14)
where k~ is the curvature in the associated structure, corresponding to m~, and c = ST.
ST
CT:
k~
~,~ /,~
i s
.~
~'~/c,/ /
©
sr'nx
is
state (c) - - ST" m x _ 0 2 5
,At
ST=(l+nx)-
/Z / 0.7.5-
5
/
/>I"-.-
~ ,m~l ,m~_ . ~/ - x v..,~-, ~,, ~ . . , , , , f /
~
""-zero
+nx~mx
S,o,e compression reinforcement
/ nx
-025(-075)
% =-0.5
kx
•
I
Oc T amx
n nx =-0.25
,o
(-0.75)
)/
/
o 75
025
I
0"25
0 5~
075
~o
mx
© 1875
FIG. 3. Specific cost function for various values of n, and corresponding curvatures for associated beam. (The notation n, = I(-2), etc. indicates that n, may take the values of +1 or - 2 respectively.)
Reinforcement minimization of beams with axial forces
611
Using equation (7) for 0 ~< m, ~< 3/4 and equation (13c) for mx/> 3/4, equation (14) becomes
O<~ m~ <~~: k~ =l [l - mx(x)]-'n 3 ~ < m~: k~ = 1.
(15a)
(15b)
This relation is shown in Fig. 3(b) for n~ = 0. 3.3 C a s e 3. G e n e r a l c a s e In the most general case n~ > 0 so that, depending on the precise value of n~, either or both At and A, may be in tension or compression, or be absent. The optimality criterion then produces (see equation 12) Osr=+-l; el = -~s
-Osr-+-I e, - ~s----7-
(16)
where (+) corresponds to tension and ( - ) to compression. This means that only the following stress-strain states can exist at a beam cross-section in the optimal design: (a) (b) (c) (d)
s>0, s >0, s > 0, s =0,
s'>0;el=+l,¢2=+l s ' = 0 ; ~1 = + l , - l ~ < e 2 ~< +1 s' < 0; el = +1, ~2 = - 1 s ' < 0 ; - 1 ~<~l~<+l,e2 = - 1
(e) s
e I =-1,~2
(17)
= -1.
Note that states (d) and (e) are similar to states (b) and (a) respectively. An equivalent optimization criterion has been given by Morley[l]. The various stress-strain states will now be considered in more detail: For stress state (a) which denotes large tensile n~, v = 0 in equations (9, 10), so that nx = s + s ' = s t .
(18)
This state remains valid provided s' > 0. But s' = (n~ - m D / 2 from equations (9) and (10) so that a condition on the validity of this state is that n~ ~> m~. The appropriate cost function is sketched in Fig. 3(a). State (b) has zero "compression" reinforcement and is a generalization of Case (1). Equations (9) and (10) with s' = 0 produce s r = s = (1 + n~) - [1 + nx - rex] In
(19)
which is sketched in Fig. 3(a). Equation (19) is valid in the range n~ ~< mx ~< nx + 3/4. For State (c), s > 0, s' < 0 and upon combining equations (9, 10) to eliminate v there is obtained: m~ = 2s - nx - (s + s' - n~)2
(20)
(see equation I1). Using the same approach as for Case 2, s = (m~ + n D I 2 + 1/8 s' = (n~ - m x ) / 2 + 3/8
(21a, b)
and Sr : Isl + Is'l = m~ - 1/4
(21c)
since s' < 0, and provided ms ~> n~ + 3/4. This cost function is also shown on Fig. 3(a). It is exactly the result obtained for Case 2 with only the limit of validity modified to account for n,. The requirements for an optimal design can most easily be given in terms of the curvature kx corresponding to the moment m~ for a given value of nx. Applying equation (14) to the curves shown in Fig. 3(a) produces the requirements on the curvature given in Fig. 3(b). It is evident that these agree with the requirements on reinforcement strain (equations 17). For state (b), formal derivation of equation (19) results in the curve shown in Fig. 4(b): k ~ f x ) = ~Osr -_ ~
(I -
m~
+
nx)-In.
(22)
State (b) remains valid until nx = - 1 / 2 at which point states (b) and (c) reach their limiting values at mx = +1/4. The cost function is then bilinear: ST = 0 for 0 ~< m x ~< 1/4 and sT = m x - 1/4 for m, ~> 1/4. For - 1 / 2 ~< nx ~<0 the cost function includes part of state (b), equation (19) and state (c). A typical example is n, = - 1 / 4 shown in Figs. 3(a) and (b). State (d) denotes s = 0, s ' < 0, indicating compressive nx with zero net tension forces on cross-section. This state has certain similarities to state (b). Again from equations (9) and (10) it follows now that Sr = Is'l = - s ' = -n~ - (-nx - mx) 'n
OJ)MS Vol. 23, No. 10--C
(lOb)
612
R . E . MELCHErS
and OST _
1
I/'~
(23)
k~(x)='~m - ~(-n~ - m~) -.
Equations (10b) and (23) can be recast into the form of equations (19) and (22) for State (b) by replacing nx in State (b) by -1 - n~. The symmetry of equations about n~ = -1/2 can be seen in Fig. 4. Similarly, State (e) can be obtained by replacing n~ by -1 - n~ in state (a). 3.4 Cost contours An alternative form of presenting the results of Fig. 3(a) is given in Fig. 4, which shows cost contours for given values of ST. It is clear that the bilinear cost function n~ = -1/2 in Fig. 3(a) is obtained in Fig. 4 by taking a section at n~ = - 1 / 2 . Taking of sections for other values of nx will produce appropriate cost functions. Figure 4 is symmetric about ms = 0 and nx = - 1 / 2 , the latter corresponding to the observation above for cases (d) and (e). Also shown are the stress states relevant to each region of the cost contour diagram. 4. APPLICATION In applying the cost function information derived above to particular problems of structural optimization, the nonlinear relationship between k~ and mx implies that in general the optimal design will be a function of both the distribution mx(x) and its intensity as well as the intensity of nx. In the following, two examples of an encastre beam will be considered. 4.1 Example 1 An encastre beam under central point load P is shown in Fig. 5(a). The bending moment diagram and slope continuity diagram are shown in Figs. 5(b) and (c). Let nx be arbitrary. If a = 0.25 L, the bending moments in each half of the beam are antisymmetric about the inflexion points x = a, x = L - a . To ensure slope continuity consistent with the absolute values of the bending moments (i.e. invoking Fig. 4b), it is evident that a = 0.25L irrespective of the precise value of P or n~. In this case, then, the locations of the various region boundaries in the optimal design are loading-independent. This accords entirely with the earlier results of Heyman[6] using a simpler (linear) cost function. 4.2 Example 2 An encastre beam under uniformly distributed load wa is shown in Fig. 6(a), together with the bending moment diagram Fig. 6(b), and a slope continuity diagram, Fig. 6(c). Owing to the lack of symmetry of the bending moment diagram for each half of the beam in this case, it follows that the optimal design will be a function of the applied loading. This will now be explored. 4.2.1 (a) High WA, nx >I 0 (i.e. tensile axial force) [Case I]. Consider first the situation for which the load w is sufficiently high to develop all three types of curvature states shown in Fig. 3(b); viz.: k~ = 0 , k~ = f(nx, rex) and k~ = 1. As evident from Fig. 6(c) there are 5 parameters describing the deformation field and there are 4 conditions on the value of mx (see Fig. 6b). Both moment equilibrium and slope continuity must be satisfied. For moment equilibrium:
M(x)
s'(o
wa(L
=
s'
• s'(O
-
x)x/2
-
~
>>-
0
s~ s'~
s':o mx
s>O
I
/ /
r,
VIII ,,o V
I.,
I
-2.o
E..c231
\\ .
\
/ /
I", I
-,.5
0
/
\\ //°T
Y\
/"
E,o.9
A
I
s)O
"° nx
-,.o
o
-
o5
~.o
.0
t o
FIG. 4. Cost contours and corresponding stress states.
,5
n~
R e i n f o r c e m e n t minimization of b e a m s with axial forces
613
F
(a)
(b)
~
(c)
~
B.MD.
~ m(x) Curvoture
x
FIG. 5. Encastre beam under central point load (a), bending m o m e n t diagram (b) and curvature field for associated beam (c). d
(
nl~ (o)
I. r"
wA
nx
L
J.-]
~
Mc
--Am
. . . . x(l÷nx) m(x) (b) r
L 0 L'L'L"
1 ql.
1 "1
iq 7mnX(t+nx}
L' L
1
L/2
(c)
FIG. 6. Encastre b e a m under uniformly distributed load (a), bending m o m e n t diagram (b) and curvature field for associated beam (c).
or
r e x ( x ) = 2 [ ( L - x ) x - ( L - a )a ]
(24)
where w = (2Wa)/(o'cH2b). U s i n g the t r a n s f o r m e d variable y = x - a - e,
m~(y) = 2 [(Ly - y2 _ 2ay - 2ey - 2ae + L e - e2].
(25)
At the b o u n d a r y b e t w e e n regions g and e, m x ( x ) = n,, (see Fig. 6b) so that e = ( L / 2 - a ) - [ ( L / 2 - a) 2 - 2nx/w] 1/2.
(26)
The negative root has been selected on the basis that if nx = 0, region e would not be expected to exist.
614
R . E . MELCHERS
Also across region g, the greatest change in m o m e n t cannot exceed 3]4, i.e. Am~ g = 3/4 (see Fig. 3b for state b); thus mx
y=g = mx ly=O+ ~
(27)
from which it follows that g = (L/2 - a - e) -
[(L/2 - a - e) 2 - 3/2w] t/2
(28)
with the negative root selected. The parameter e can be eliminated using equation (26). In an exactly analogous manner, using the transformed variable z = a - f - x, it m a y be s h o w n that f = -(L/2
- a) +
[(L/2 - a ) z + 2 n f f w ] v2
(29)
[(L/2 - a + f ) 2 + 3 / 2 w ] l n .
(30)
and h = - (L/2 - a + f) +
At this stage, relationships for e, f, g and h have been obtained in terms of a and given constants. The values r and s follow if the others are known. An expression for a is obtained from slope continuity as follows. In regions r and s, the curvature in the (associated) structure is kx = 1 (see Fig. 3b). Hence, at the boundaries b e t w e e n regions r and g and s and h, the slopes are r and s respectively. In region g, the curvature is given by kx(x)=~(l-m~+nD
-I/2
~1< ~ k ~
~< 1.
(31)
The slope change is obtained by integrating the curvature. The I.h.s. of the half b e a m s h o w n in Fig. 6(b) will be considered first. U s i n g the transformed variable y, equation (25) for rex(y), and introducing a negative sign to a c c o u n t for the reduction in slope with increasing y (see Fig. 6c): slope = - f k ~ ( y ) dy + c~ w (Ly - 2ay - 2ey - y 2 2 a e - L e - e 2) + n~ j 1 - ~-
= -
dy + cl.
(32)
This solution remains valid as long as the square root term/> 0. At y = g, the slope = r, hence c~ can be evaluated. Using equations (26) and (28), the slope at y = 0 can be s h o w n to reduce to slope ,=0 = - ( 2 w ) 1/2In [2(2/w) tn - 2[(L/2 - a ) ~ - 2 n d w ] In + r + (2w) i/2 In 2(2w) t/2 _ 2[(L/2 - a ) 2 - 2 n d w ] "2 - 3 / 2 w where r = [(L/2 - a) 2 - 2 n d w - 3/2w] m
:xxT~oresslve l
tensile n n >-0 tensde
k=l
O
r
1
compressive
g
k=O
e÷f
-I< k
h
s tensile nn < 0
teniile
FIG. 7. Locations of reinforcement in the various zones of the optimal design.
(33)
Reinforcement minimization of beams with axial forces
615
By an analogous approach for the r.h.s, of the half beam of Fig. 6(b), now using the transformed variable z, equation (29) for m~(z) and noting that the slope decreases with z, it may be shown that the slope at z = 0 is given by [
(L/2_a)2+2ndw
slope z=0 = (2w)-m arsin - L(L/2 t2w ~-1/2arsin -'
'
-
]1t2
a) 2 + 2(1 + n,.)wJ
[ ( L / 2 - a) 2 + 2 n d w + 3/2w t/' - t ~2---=$772-~-~
+ L/2 - [(L/2 - a) 2 + 2 n d w + 3/2w] m.
(34)
Since there is no change of slope across regions e and f (as Imxl ~< n~ and hence k~ = 0), equations (33) and (34) may be equated to allow solution for a by trial and error. 1"(2/w)l/z - (A + 1.5/w) m ] _(2wA)l/2 + arsin - ( B - 1.5/w)"2/C m 0 = In [ (2w)_j/z All2 -
-
- arsin - ( B / C ) m + (2w)m(L[2 - Bin).
(35a)
where A = (L/2 - a) 2 - 2[n~ + 3/4]w
(35b)
B = (L/2 - a) 2 + 2[n~ + 3/41/w
(35c)
C = ( L / 2 - a)2 + 2[G + l]/w.
(35d)
Equations (35) were solved for a using a computer. The solution remains valid provided r = A/> 0. The results are presented in Fig. 8 which also shows the region of validity of the present results. (b) High wA, - 1 / 2 ~< nx~<0 (i.e. compressive axial [orce) [Case H]. From Fig. 4 it is evident that for -1/2 ~< nx ~<0, it is possible, within the shaded region, to have some moment capacity mx without incurring any cost. This region corresponds to the intersection of state (b) in Fig. 3(a) with the m~ axis. The boundary to the shaded area in Fig. 4 is given (for -1 ~< n~ ~<0) by Sr = 0 in equation (19) or nx(l + nD = -rex.
(36)
Equation (36) applies at the junctions g - e and [ - h in Fig. 6(b) [with appropriate sign for mx]. As before regions e, [ have zero curvature and zero cost st. The range Am~ for regions g and h is now no longer given by Amx = 314 (except at G = 0) and can be found as follows (see Fig. 3b). The left hand end is given by equation (36) above. The right hand end corresponds to the point where k, = 1 and using equation (21) for k~ gives m~ - nx = 3/4.
(37)
The difference in moment values given by equations (36) and (37) produces the desired result: (38)
Amx = ~ + 2nx + n~z.
Proceeding now in an analogous manner to that for Case/, produces modified expressions for e, g and h which may then be used with slope continuity to produce eventually: 0
In
[(1 + nx)(2/w) llz - [(L/2 - a) 2 + 2nx(l + n D / w l m ] (2w)-m - Al/2 J - (2wA)'/2 [ - [ ( L / a - a) ~ - 2nx(l + n ~/w ll/zl CI a ~' J ] - a r s i n - ( B / C )
+arsin[
I/z
(39)
+ (2w)l12(L[2 - B m)
where A, B, C are as defined previously. Equation (39) is also plotted in Fig. 8. Equation (39) remains valid provided r >/0. (c) L o w e r values of wA [Cases I l l - W ] . When the limit r = 0 is reached, the above equations must be further modified. The region g is then given by g = L/2 - a - e. For even lower values of wA (but depending on nx), region s may also disappear, leading to s = 0 , h = a - [ . The resulting slope continuity equations can again be derived in a manner analogous to that given above, and are as follows: Case I I I ; 0 << - n~ <~ i; r = 0, s 1>0
, f
(21w)I/2-[A+3/2w] m
].
= m [[21w(1 + nx) - (L/2 - a)2]'aJ + arsin - {[B - 31(2w)]/C} 'lz - arsin { - ( B / C ) m} + ( 2 w ) m ( L / 2 - BI/:).
(40)
616
R . E . MELCHERS
~3 e= E
~
x
U
,o
o.~ I .-~ ..,-,
L
II
~2
--I 0
o
o
0
0
0
o
0
0
0
o
o ~_ o
Reinforcement minimization of beams with axial forces
617
Case I V ; -1/2-< n~; r = 0 , s>~0 0 = In (1 +
I/2 - [(L/2 - a) 2 + 2nx(1 + nx)/w] I/2 [2(1 + nx)/w - (L]2 - a)2] 1/2
nx)(2/w)
+ arsin - {[B - 31(2w)]/C} In - arsin [ - ( B / C ) t/2] + (2w)l/2(L/2 - BI/2).
(41)
Case V; 0~
(2/w) I/2 - [A + 3/(2w)] In 0 = In 2(1 + nx)/w - (L/2 - a)2] In + arsin - {[(L/2 - a) 2 + 2nx/w]/C} In - arsin - L/(2C1/2).
(42)
Case VI; -1/2~< nx; r = s = 0
0 = In (1 + nx)(21w) I/2 - [(L/2 - a) + 2nx(1 + nx)lw[ I/2 [2(1 + nx)lw - ( L I 2 - a)2] 1/2 + arsin - {[(L/2 - a) 2 - 2nx(1 + nx)/w]/C} In - arsin - L/(2Clt2).
(43)
The boundary between regions I and 1V is given by r = x / A = [ ( L / 2 - a ) 2 - 2 ( n ~ + 314)/w] i n = O. The boundary between regions 111 and V I is given by s = (L]2 - x/B) = [L]2(L]2 - a) 2 + 2(n~ + 3/4)1w] 1/2 = O. Both limits have been plotted on Fig. 8. A further limit to the solutions exists since it is required in general that g, h/> 0. When g = h = 0, it is easily shown that a / L = 1/2 - X/(1/8) = 0.1464 for all w, nx(-1/2 ~< n~ ~< 1). 5. D I S C U S S I O N Example 2 is by far the more interesting of the two examples considered. Unlike example 1, which because of the special symmetry of its bending moment diagram corresponds with the results originally obtained by Heyman[6], example 2 shows quite marked variation from those elementary results, particularly at low loads. For the simple case n~ = 0, for example, the inflexion point in the optimum design can lie anywhere between 1 / 2 - X / ( 1 / 8 ) ( = 0.1464) and 1/4. These two limits represent, respectively, the cases where maximum positive and negative moments have the same value (as for an encastre beam yielding according to simple plastic theory), and the case given by Heyman[6]. The former case arises naturally when consideration is given to the effect of a small axial load and small lateral load w. To obtain an appropriate deformation for the associated structure non-zero curvature must exist at the beam centre and its ends, even if the rest of the beam has bending moments too small to correspond to non-zero curvature. This indeterminacy is of theoretical interest only, however, since it disappears under non-zero loading w. For n~ = 0, it can be seen that a can be reduced by up to 10% from the value obtained by Heyman[6]. When n x # 0 , the location of the inflexion point is very strongly dependent on the applied load w, particularly for low values of w. As might be expected, at high values of w, the results converge to a / L = 0.25 as obtained by Heyman[6]. 6. C O N C L U S I O N S
It has been shown that when a realistic cost function is used for the optimization of encastre beams under axial load, the parameter a, which defines the regions in the optimal design is, in general, dependent on the applied loading. This is quite unlike a wide range of problem solutions obtained for purely flexural systems[2, 3]. For the particular case of a beam under uniformly distributed loading, it was found that the optimal design parameter a was highly susceptible to the value of both the applied lateral load and the axial load. In particular, it was found that the simple optimal result given by Heyman[6] for the location of the inflexion points of a beam loaded uniformly and transversely, measured from the encastre supports, forms an upper limit on the more exact result given herein. A c k n o w l e d g e m e n t - - T h e author is indebted to the referees for their helpful comments including the suggestion to include Fig. 7.
7. R E F E R E N C E S C. T. MORLEY, Int. J. Mech. Sci. 8, 305 (1966). P. G. LOWE and R. E. MELCHERS, Int. J. Mech. Sci. 14, 311 (1972); 15, 151 (1973); 15, 711 (1973). G. I. N. ROZVANY, Optimal Design o f Flexural Systems. Pergamon Press, Oxford (1976). R. E. MELCHERS and G. I. N. ROZVA~Y, J. Engng Mech. Div., A S C E . 96, 1093 (1970). Z. Mnoz, Optimum design of reinforced shells of revolutions. In Nonclassical Shell Problems (Edited by W. Olszak and Z. Sawczuk), p. 732. North-Holland, Amsterdam (1964). 6. J. HERMAN, Quart. J. Mech. Appl. Math. 12, 314, (1959). 7. W. PRACER and G. I. N. ROZVANY, J. Struct. Mech. 5, 1 (1977). 8. W. PRA6ER and R. T. SHIELD, J. Appl. Mech. 344, 184 (1967).
1. 2. 3. 4. 5.
0020-7403/81/100607-11502.00/0 Pergamon Press Ltd.
REINFORCEMENT MINIMIZATION OF BEAMS WITH AXIAL FORCES R. E. MELCHERS Department of Civil Engineering, Monash University, Australia (Received 10 June 1980; in revised [orm 24 November 1980)
Summary--Within the confines of absolute minimum weight design of plastic structures, it has generally been assumed that the relationship between material (cost) and the vector of internal actions is linear (or piecewise linear) that is, sandwich approximations, or essentially constant lever-arm assumptions for reinforced materials, have been employed. Where both axial load and moment act on the one cross-section, or moments are relatively large, such idealization of cost function is no longer consistent with actual behaviour. A nonlinear cost function model must be substituted. The cost-function for an idealized reinforced section under bending moment and axial load (tensile or compressive) is derived herein. Two examples of encastre reinforced beams with axial forces are considered. When the applied load consists of a transverse central point load the optimum design is found to accord with earlier results employing a simplified cost function. When the applied lateral load is uniformly distributed earlier results are only approached in the limit as the lateral load approaches infinity.
A~, A2 a b c D _G H k~ L Mx mx Nx nx P _Q _q s s' Sr wa
NOTATION tensile and compressive cross-sectional areas of reinforcement distance to inflexion points from beam end beam width specific cost structural domain cost gradient vector beam section depth curvature of associated structure beam length moment 2MJ(trcH2b) axial tension Nx/(~ycHb) applied point load vector of generalised stresses vector of generalised strains (al~ro)/(o'cHb) (A2~ro)/(~cHb) total specific cost (reinforcement content) applied uniformly distributed loading
strain in associated structure u proportion of beam depth in compression q5 total cost ~b specific cost function, strain angle generic point ,~ linear multiplier tr~ concrete matrix compressive strength tr0 steel yield strength 1. INTRODUCTION T h e c l a s s o f p r o b l e m c o n s i d e r e d h e r e i n c o n c e r n s r e i n f o r c e d s t r u c t u r e s f o r w h i c h the c o s t ( w e i g h t , v o l u m e ) o f r e i n f o r c i n g m a t e r i a l p r e d o m i n a t e s o v e r the c o s t o f the m a t r i x m a t e r i a l , a n d f o r w h i c h a d e s i g n is s o u g h t s u c h t h a t the t o t a l c o s t ( w e i g h t , v o l u m e ) o r r e i n f o r c i n g m a t e r i a l s is r e d u c e d to a n a b s o l u t e m i n i m u m . T h e u s u a l a s s u m p t i o n s f o r d e s c r i b i n g b e h a v i o u r a r e m a d e . T h u s it is a s s u m e d t h a t s i m p l e ( " e n g i n e e r s " ) b e n d i n g t h e o r y is v a l i d , t h a t the influence o f s h e a r c a n be i g n o r e d o r a s s u m e d c a r r i e d e n t i r e l y b y the m a t r i x m a t e r i a l a n d t h a t l o a d s a r e q u a s i - s t a t i c . It is a l s o a s s u m e d t h a t i n s t a b i l i t y e f f e c t s m a y b e i g n o r e d . T h e r e i n f o r c i n g m a t e r i a l a n d the m a t r i x a r e a s s u m e d to h a v e r i g i d - p l a s t i c m a t e r i a l p r o p e r t i e s . This is a first a p p r o x i m a t i o n to the b e h a v i o u r o f r e i n f o r c e d c o n c r e t e in the p l a s t i c r a n g e . 607
608
R.E. MELCHERS Piece
- wise
lineor
,C
,
V \
Nonlineor
,c : s
: t-(l-
,0.5 mxJ
\ \
J
\
/.i to
~ ~
Ideal,zotlon for reinforced concrete
~y O ~
Im
: ~
slobs
H=b)
FIO. 1. Specific cost function approximations.
Previous investigations into the absolute optimal arrangement of reinforcing materials [1-3] have usually assumed that the relationship between the unit cost (cost per unit area, per unit length, etc.) and the vector of internal actions is linear (see Fig. 1). This follows naturally in the case of reinforced concrete slabs since the percentage of reinforcing material is usually low. The linearity assumption has also been applied to reinforced concrete shells [4, 5], steel/-beams [6] and beam grillages [7]. Apart from the work of Mroz[5], the work of Morley[l] on one-way and circular slabs and a limited amount of work using piecewise linear cost functions[3,8], nonlinear cost functions do not appear to have been applied in optimization problems. A fundamental difficulty has been finding a way of applying an appropriate optimality criterion such as that of Prager-Shield[8] to the particular problem under consideration. 2. OPTIMALITY CRITERION Various techniques exist to find a solution to an optimization problem. The approach used here is due to Prager and Shield[8]. In their terminology, the local state of stress at a generic point ~ in the structural domain D is defined by generalized stresses Q~(O (i = 1,2 . . . . . n) corresponding to generalized strains q~(~) (i = 1, 2 . . . . . n). Let the total quantity to be minimized be termed the total cost (q~) and the cost per unit area or unit length of the structure be termed specific cost (~b). The relationship between the specific cost ~h and the generalized stresses _Q is given by the specific cost function ~b(Q). The cost gradient vector _G(40 is defined as
[ am am aq, ] G(~b) = (grad ~), = t-~' ~'~2 ..... O--~.J
(1)
for all states of generalized stress Q* for which 4'(_Q) is differentiable. If ~b(_Q) is discontinuous at some generalized stress state Q_d,then _G fs defined as any convex combination of the gradient vectors G i for the adjacent regions (in each of which 4~(_Q)is differentiable): _G = ~ A,_G, 9=9, = ~ At(grad 4~), 9=9'
(2)
which for all i: ,~ > 0 and E A~= 1. PraiSer and Shield[8] have shown that on the structural domain D, the condition for minimum cost is a kinematic condition expressed by[3]: q(¢) = G[_Q(¢)]
(3)
where q is a kinematically admissible generalized strain field corresponding to the statically admissible generalized stress field _Q.
3. COST FUNCTION FOR COMBINED BENDING AND AXIAL FORCE Consider the idealized reinforced concrete beam section shown in Fig. 2, in which concrete cover to the reinforcement has been ignored, both tensile (A~ 3 0 ) and compressive (A2~>0) reinforcement may be present and in which the reinforcing bars have been idealized as a uniform flat sheet on each beam face. Let the concrete matrix have ideal rigid-plastic material properties with compressive strength + ~c and zero tensile strength. The reinforcement is assumed ideally rigid-plastic with tensile strength+ ~0 and compressive strength-tr0. Each identical reinforcement layer has negligible thickness compared to the beam section depth, H t> 0.
Reinforcement minimization of beams with axial forces
609
M
]:--N ~
A l f fo
4:at
A1
FIG. 2. Material idealization, strain state and stress state. 3.1 Case 1. N o compression reinforcement Under the action of the stress vector {Mx, N~} the equations of equilibrium for the cross-section about its centre line are (see Fig. 2): n M~ = Ato'0-~- + vHtrc ~
Hb
(4)
and
Nx = A~,ro - vHtrcb.
(5)
The reduced notation (see Mroz[5]): 2Mx mx = - - - ~ - ~
nx
= Nx trcI-Ib
Ato'c s = tT~nb
will be adopted. Combining equations (4) and (5) to eliminate the parameter v: mx = 2s - nx - (s - n~)2.
(6)
For a unit length of beam, the cross-sectional area, AI of reinforcing material is proportional the specific cost so that it is sufficient to optimize 's'. For the simplest case, nx = 0; equation (6) leads, for s/> 0, to: s = 1 - (1 - m~) 1/2
(7)
which is plotted in Fig. 1. It is evident that only at low values of m~ is the linear approximation used in earlier optimization work adequate. 3.2 Case 2. Compression reinforcement The total specific cost is now sT = Is[ + Is'l
(8)
where s' = A:,7o/((rcHb). In reduced notation, equilibrium can be expressed as n~ = s + s ' - v m~ = s - s ' +
v(1 - v)
(9) (10)
which, for the special case nx = 0, reduces to m~ = 2s - (s + s') 2.
(11)
When both layers A~ and A2 are present the distribution of the minimal reinforcement sr = [s[ + Is'l, between the two reinforcement layers in a beam section, to transmit a given vector {m~, nx} of applied stress resultants, can be obtained by applying the optimality condition, equation (3) to (8), where Sr is the specific cost, and s and s' are treated as stress resultants. (This is valid since the applied stress resultants, {M~, N~} may always be transformed to any other equivalent system.) Hence et = -~-ST= + 1
BST e2 = "~'TS,= --1
(12)
where the values of the strains el and 42 are in reduced notation and the signs correspond to the sign of s and s' respectively. Equations (12) express the requirement that when both reinforcements A~ and A2 are
610
R . E . MELCHERS
present in the structure, they must be equally (but oppositely) strained in the (associated) structure for the actual structure to be optimal. When this is the case, then v = 1/2, i.e. the neutral axis corresponds with the centre line, and for n~ = O, equations (9) and (10) give !
= s + s'
(9a) 1
mx = s - s ' + ~
(lOa)
so that mx
1
mx
3
s =Y+i; s=--T+i sr = m x - ~ for m,
(13a, b,c)
The relation between sT and m, for n, = 0 is sketched in Fig. 3(a). Evidently, for m,/> 3/4, compression reinforcement produces a lower specific cost c = st. The relationship between s and S' could also have been obtained by a trial and error approach to minimizing sT. More generally, either A] or A2 or both may be preseat. To obtain the requirements for optimality under these circumstances, equation (3) is applied to the specific cost: kx =
#c
(14)
where k~ is the curvature in the associated structure, corresponding to m~, and c = ST.
ST
CT:
k~
~,~ /,~
i s
.~
~'~/c,/ /
©
sr'nx
is
state (c) - - ST" m x _ 0 2 5
,At
ST=(l+nx)-
/Z / 0.7.5-
5
/
/>I"-.-
~ ,m~l ,m~_ . ~/ - x v..,~-, ~,, ~ . . , , , , f /
~
""-zero
+nx~mx
S,o,e compression reinforcement
/ nx
-025(-075)
% =-0.5
kx
•
I
Oc T amx
n nx =-0.25
,o
(-0.75)
)/
/
o 75
025
I
0"25
0 5~
075
~o
mx
© 1875
FIG. 3. Specific cost function for various values of n, and corresponding curvatures for associated beam. (The notation n, = I(-2), etc. indicates that n, may take the values of +1 or - 2 respectively.)
Reinforcement minimization of beams with axial forces
611
Using equation (7) for 0 ~< m, ~< 3/4 and equation (13c) for mx/> 3/4, equation (14) becomes
O<~ m~ <~~: k~ =l [l - mx(x)]-'n 3 ~ < m~: k~ = 1.
(15a)
(15b)
This relation is shown in Fig. 3(b) for n~ = 0. 3.3 C a s e 3. G e n e r a l c a s e In the most general case n~ > 0 so that, depending on the precise value of n~, either or both At and A, may be in tension or compression, or be absent. The optimality criterion then produces (see equation 12) Osr=+-l; el = -~s
-Osr-+-I e, - ~s----7-
(16)
where (+) corresponds to tension and ( - ) to compression. This means that only the following stress-strain states can exist at a beam cross-section in the optimal design: (a) (b) (c) (d)
s>0, s >0, s > 0, s =0,
s'>0;el=+l,¢2=+l s ' = 0 ; ~1 = + l , - l ~ < e 2 ~< +1 s' < 0; el = +1, ~2 = - 1 s ' < 0 ; - 1 ~<~l~<+l,e2 = - 1
(e) s
e I =-1,~2
(17)
= -1.
Note that states (d) and (e) are similar to states (b) and (a) respectively. An equivalent optimization criterion has been given by Morley[l]. The various stress-strain states will now be considered in more detail: For stress state (a) which denotes large tensile n~, v = 0 in equations (9, 10), so that nx = s + s ' = s t .
(18)
This state remains valid provided s' > 0. But s' = (n~ - m D / 2 from equations (9) and (10) so that a condition on the validity of this state is that n~ ~> m~. The appropriate cost function is sketched in Fig. 3(a). State (b) has zero "compression" reinforcement and is a generalization of Case (1). Equations (9) and (10) with s' = 0 produce s r = s = (1 + n~) - [1 + nx - rex] In
(19)
which is sketched in Fig. 3(a). Equation (19) is valid in the range n~ ~< mx ~< nx + 3/4. For State (c), s > 0, s' < 0 and upon combining equations (9, 10) to eliminate v there is obtained: m~ = 2s - nx - (s + s' - n~)2
(20)
(see equation I1). Using the same approach as for Case 2, s = (m~ + n D I 2 + 1/8 s' = (n~ - m x ) / 2 + 3/8
(21a, b)
and Sr : Isl + Is'l = m~ - 1/4
(21c)
since s' < 0, and provided ms ~> n~ + 3/4. This cost function is also shown on Fig. 3(a). It is exactly the result obtained for Case 2 with only the limit of validity modified to account for n,. The requirements for an optimal design can most easily be given in terms of the curvature kx corresponding to the moment m~ for a given value of nx. Applying equation (14) to the curves shown in Fig. 3(a) produces the requirements on the curvature given in Fig. 3(b). It is evident that these agree with the requirements on reinforcement strain (equations 17). For state (b), formal derivation of equation (19) results in the curve shown in Fig. 4(b): k ~ f x ) = ~Osr -_ ~
(I -
m~
+
nx)-In.
(22)
State (b) remains valid until nx = - 1 / 2 at which point states (b) and (c) reach their limiting values at mx = +1/4. The cost function is then bilinear: ST = 0 for 0 ~< m x ~< 1/4 and sT = m x - 1/4 for m, ~> 1/4. For - 1 / 2 ~< nx ~<0 the cost function includes part of state (b), equation (19) and state (c). A typical example is n, = - 1 / 4 shown in Figs. 3(a) and (b). State (d) denotes s = 0, s ' < 0, indicating compressive nx with zero net tension forces on cross-section. This state has certain similarities to state (b). Again from equations (9) and (10) it follows now that Sr = Is'l = - s ' = -n~ - (-nx - mx) 'n
OJ)MS Vol. 23, No. 10--C
(lOb)
612
R . E . MELCHErS
and OST _
1
I/'~
(23)
k~(x)='~m - ~(-n~ - m~) -.
Equations (10b) and (23) can be recast into the form of equations (19) and (22) for State (b) by replacing nx in State (b) by -1 - n~. The symmetry of equations about n~ = -1/2 can be seen in Fig. 4. Similarly, State (e) can be obtained by replacing n~ by -1 - n~ in state (a). 3.4 Cost contours An alternative form of presenting the results of Fig. 3(a) is given in Fig. 4, which shows cost contours for given values of ST. It is clear that the bilinear cost function n~ = -1/2 in Fig. 3(a) is obtained in Fig. 4 by taking a section at n~ = - 1 / 2 . Taking of sections for other values of nx will produce appropriate cost functions. Figure 4 is symmetric about ms = 0 and nx = - 1 / 2 , the latter corresponding to the observation above for cases (d) and (e). Also shown are the stress states relevant to each region of the cost contour diagram. 4. APPLICATION In applying the cost function information derived above to particular problems of structural optimization, the nonlinear relationship between k~ and mx implies that in general the optimal design will be a function of both the distribution mx(x) and its intensity as well as the intensity of nx. In the following, two examples of an encastre beam will be considered. 4.1 Example 1 An encastre beam under central point load P is shown in Fig. 5(a). The bending moment diagram and slope continuity diagram are shown in Figs. 5(b) and (c). Let nx be arbitrary. If a = 0.25 L, the bending moments in each half of the beam are antisymmetric about the inflexion points x = a, x = L - a . To ensure slope continuity consistent with the absolute values of the bending moments (i.e. invoking Fig. 4b), it is evident that a = 0.25L irrespective of the precise value of P or n~. In this case, then, the locations of the various region boundaries in the optimal design are loading-independent. This accords entirely with the earlier results of Heyman[6] using a simpler (linear) cost function. 4.2 Example 2 An encastre beam under uniformly distributed load wa is shown in Fig. 6(a), together with the bending moment diagram Fig. 6(b), and a slope continuity diagram, Fig. 6(c). Owing to the lack of symmetry of the bending moment diagram for each half of the beam in this case, it follows that the optimal design will be a function of the applied loading. This will now be explored. 4.2.1 (a) High WA, nx >I 0 (i.e. tensile axial force) [Case I]. Consider first the situation for which the load w is sufficiently high to develop all three types of curvature states shown in Fig. 3(b); viz.: k~ = 0 , k~ = f(nx, rex) and k~ = 1. As evident from Fig. 6(c) there are 5 parameters describing the deformation field and there are 4 conditions on the value of mx (see Fig. 6b). Both moment equilibrium and slope continuity must be satisfied. For moment equilibrium:
M(x)
s'(o
wa(L
=
s'
• s'(O
-
x)x/2
-
~
>>-
0
s~ s'~
s':o mx
s>O
I
/ /
r,
VIII ,,o V
I.,
I
-2.o
E..c231
\\ .
\
/ /
I", I
-,.5
0
/
\\ //°T
Y\
/"
E,o.9
A
I
s)O
"° nx
-,.o
o
-
o5
~.o
.0
t o
FIG. 4. Cost contours and corresponding stress states.
,5
n~
R e i n f o r c e m e n t minimization of b e a m s with axial forces
613
F
(a)
(b)
~
(c)
~
B.MD.
~ m(x) Curvoture
x
FIG. 5. Encastre beam under central point load (a), bending m o m e n t diagram (b) and curvature field for associated beam (c). d
(
nl~ (o)
I. r"
wA
nx
L
J.-]
~
Mc
--Am
. . . . x(l÷nx) m(x) (b) r
L 0 L'L'L"
1 ql.
1 "1
iq 7mnX(t+nx}
L' L
1
L/2
(c)
FIG. 6. Encastre b e a m under uniformly distributed load (a), bending m o m e n t diagram (b) and curvature field for associated beam (c).
or
r e x ( x ) = 2 [ ( L - x ) x - ( L - a )a ]
(24)
where w = (2Wa)/(o'cH2b). U s i n g the t r a n s f o r m e d variable y = x - a - e,
m~(y) = 2 [(Ly - y2 _ 2ay - 2ey - 2ae + L e - e2].
(25)
At the b o u n d a r y b e t w e e n regions g and e, m x ( x ) = n,, (see Fig. 6b) so that e = ( L / 2 - a ) - [ ( L / 2 - a) 2 - 2nx/w] 1/2.
(26)
The negative root has been selected on the basis that if nx = 0, region e would not be expected to exist.
614
R . E . MELCHERS
Also across region g, the greatest change in m o m e n t cannot exceed 3]4, i.e. Am~ g = 3/4 (see Fig. 3b for state b); thus mx
y=g = mx ly=O+ ~
(27)
from which it follows that g = (L/2 - a - e) -
[(L/2 - a - e) 2 - 3/2w] t/2
(28)
with the negative root selected. The parameter e can be eliminated using equation (26). In an exactly analogous manner, using the transformed variable z = a - f - x, it m a y be s h o w n that f = -(L/2
- a) +
[(L/2 - a ) z + 2 n f f w ] v2
(29)
[(L/2 - a + f ) 2 + 3 / 2 w ] l n .
(30)
and h = - (L/2 - a + f) +
At this stage, relationships for e, f, g and h have been obtained in terms of a and given constants. The values r and s follow if the others are known. An expression for a is obtained from slope continuity as follows. In regions r and s, the curvature in the (associated) structure is kx = 1 (see Fig. 3b). Hence, at the boundaries b e t w e e n regions r and g and s and h, the slopes are r and s respectively. In region g, the curvature is given by kx(x)=~(l-m~+nD
-I/2
~1< ~ k ~
~< 1.
(31)
The slope change is obtained by integrating the curvature. The I.h.s. of the half b e a m s h o w n in Fig. 6(b) will be considered first. U s i n g the transformed variable y, equation (25) for rex(y), and introducing a negative sign to a c c o u n t for the reduction in slope with increasing y (see Fig. 6c): slope = - f k ~ ( y ) dy + c~ w (Ly - 2ay - 2ey - y 2 2 a e - L e - e 2) + n~ j 1 - ~-
= -
dy + cl.
(32)
This solution remains valid as long as the square root term/> 0. At y = g, the slope = r, hence c~ can be evaluated. Using equations (26) and (28), the slope at y = 0 can be s h o w n to reduce to slope ,=0 = - ( 2 w ) 1/2In [2(2/w) tn - 2[(L/2 - a ) ~ - 2 n d w ] In + r + (2w) i/2 In 2(2w) t/2 _ 2[(L/2 - a ) 2 - 2 n d w ] "2 - 3 / 2 w where r = [(L/2 - a) 2 - 2 n d w - 3/2w] m
:xxT~oresslve l
tensile n n >-0 tensde
k=l
O
r
1
compressive
g
k=O
e÷f
-I< k
h
s tensile nn < 0
teniile
FIG. 7. Locations of reinforcement in the various zones of the optimal design.
(33)
Reinforcement minimization of beams with axial forces
615
By an analogous approach for the r.h.s, of the half beam of Fig. 6(b), now using the transformed variable z, equation (29) for m~(z) and noting that the slope decreases with z, it may be shown that the slope at z = 0 is given by [
(L/2_a)2+2ndw
slope z=0 = (2w)-m arsin - L(L/2 t2w ~-1/2arsin -'
'
-
]1t2
a) 2 + 2(1 + n,.)wJ
[ ( L / 2 - a) 2 + 2 n d w + 3/2w t/' - t ~2---=$772-~-~
+ L/2 - [(L/2 - a) 2 + 2 n d w + 3/2w] m.
(34)
Since there is no change of slope across regions e and f (as Imxl ~< n~ and hence k~ = 0), equations (33) and (34) may be equated to allow solution for a by trial and error. 1"(2/w)l/z - (A + 1.5/w) m ] _(2wA)l/2 + arsin - ( B - 1.5/w)"2/C m 0 = In [ (2w)_j/z All2 -
-
- arsin - ( B / C ) m + (2w)m(L[2 - Bin).
(35a)
where A = (L/2 - a) 2 - 2[n~ + 3/4]w
(35b)
B = (L/2 - a) 2 + 2[n~ + 3/41/w
(35c)
C = ( L / 2 - a)2 + 2[G + l]/w.
(35d)
Equations (35) were solved for a using a computer. The solution remains valid provided r = A/> 0. The results are presented in Fig. 8 which also shows the region of validity of the present results. (b) High wA, - 1 / 2 ~< nx~<0 (i.e. compressive axial [orce) [Case H]. From Fig. 4 it is evident that for -1/2 ~< nx ~<0, it is possible, within the shaded region, to have some moment capacity mx without incurring any cost. This region corresponds to the intersection of state (b) in Fig. 3(a) with the m~ axis. The boundary to the shaded area in Fig. 4 is given (for -1 ~< n~ ~<0) by Sr = 0 in equation (19) or nx(l + nD = -rex.
(36)
Equation (36) applies at the junctions g - e and [ - h in Fig. 6(b) [with appropriate sign for mx]. As before regions e, [ have zero curvature and zero cost st. The range Am~ for regions g and h is now no longer given by Amx = 314 (except at G = 0) and can be found as follows (see Fig. 3b). The left hand end is given by equation (36) above. The right hand end corresponds to the point where k, = 1 and using equation (21) for k~ gives m~ - nx = 3/4.
(37)
The difference in moment values given by equations (36) and (37) produces the desired result: (38)
Amx = ~ + 2nx + n~z.
Proceeding now in an analogous manner to that for Case/, produces modified expressions for e, g and h which may then be used with slope continuity to produce eventually: 0
In
[(1 + nx)(2/w) llz - [(L/2 - a) 2 + 2nx(l + n D / w l m ] (2w)-m - Al/2 J - (2wA)'/2 [ - [ ( L / a - a) ~ - 2nx(l + n ~/w ll/zl CI a ~' J ] - a r s i n - ( B / C )
+arsin[
I/z
(39)
+ (2w)l12(L[2 - B m)
where A, B, C are as defined previously. Equation (39) is also plotted in Fig. 8. Equation (39) remains valid provided r >/0. (c) L o w e r values of wA [Cases I l l - W ] . When the limit r = 0 is reached, the above equations must be further modified. The region g is then given by g = L/2 - a - e. For even lower values of wA (but depending on nx), region s may also disappear, leading to s = 0 , h = a - [ . The resulting slope continuity equations can again be derived in a manner analogous to that given above, and are as follows: Case I I I ; 0 << - n~ <~ i; r = 0, s 1>0
, f
(21w)I/2-[A+3/2w] m
].
= m [[21w(1 + nx) - (L/2 - a)2]'aJ + arsin - {[B - 31(2w)]/C} 'lz - arsin { - ( B / C ) m} + ( 2 w ) m ( L / 2 - BI/:).
(40)
616
R . E . MELCHERS
~3 e= E
~
x
U
,o
o.~ I .-~ ..,-,
L
II
~2
--I 0
o
o
0
0
0
o
0
0
0
o
o ~_ o
Reinforcement minimization of beams with axial forces
617
Case I V ; -1/2-< n~; r = 0 , s>~0 0 = In (1 +
I/2 - [(L/2 - a) 2 + 2nx(1 + nx)/w] I/2 [2(1 + nx)/w - (L]2 - a)2] 1/2
nx)(2/w)
+ arsin - {[B - 31(2w)]/C} In - arsin [ - ( B / C ) t/2] + (2w)l/2(L/2 - BI/2).
(41)
Case V; 0~
(2/w) I/2 - [A + 3/(2w)] In 0 = In 2(1 + nx)/w - (L/2 - a)2] In + arsin - {[(L/2 - a) 2 + 2nx/w]/C} In - arsin - L/(2C1/2).
(42)
Case VI; -1/2~< nx; r = s = 0
0 = In (1 + nx)(21w) I/2 - [(L/2 - a) + 2nx(1 + nx)lw[ I/2 [2(1 + nx)lw - ( L I 2 - a)2] 1/2 + arsin - {[(L/2 - a) 2 - 2nx(1 + nx)/w]/C} In - arsin - L/(2Clt2).
(43)
The boundary between regions I and 1V is given by r = x / A = [ ( L / 2 - a ) 2 - 2 ( n ~ + 314)/w] i n = O. The boundary between regions 111 and V I is given by s = (L]2 - x/B) = [L]2(L]2 - a) 2 + 2(n~ + 3/4)1w] 1/2 = O. Both limits have been plotted on Fig. 8. A further limit to the solutions exists since it is required in general that g, h/> 0. When g = h = 0, it is easily shown that a / L = 1/2 - X/(1/8) = 0.1464 for all w, nx(-1/2 ~< n~ ~< 1). 5. D I S C U S S I O N Example 2 is by far the more interesting of the two examples considered. Unlike example 1, which because of the special symmetry of its bending moment diagram corresponds with the results originally obtained by Heyman[6], example 2 shows quite marked variation from those elementary results, particularly at low loads. For the simple case n~ = 0, for example, the inflexion point in the optimum design can lie anywhere between 1 / 2 - X / ( 1 / 8 ) ( = 0.1464) and 1/4. These two limits represent, respectively, the cases where maximum positive and negative moments have the same value (as for an encastre beam yielding according to simple plastic theory), and the case given by Heyman[6]. The former case arises naturally when consideration is given to the effect of a small axial load and small lateral load w. To obtain an appropriate deformation for the associated structure non-zero curvature must exist at the beam centre and its ends, even if the rest of the beam has bending moments too small to correspond to non-zero curvature. This indeterminacy is of theoretical interest only, however, since it disappears under non-zero loading w. For n~ = 0, it can be seen that a can be reduced by up to 10% from the value obtained by Heyman[6]. When n x # 0 , the location of the inflexion point is very strongly dependent on the applied load w, particularly for low values of w. As might be expected, at high values of w, the results converge to a / L = 0.25 as obtained by Heyman[6]. 6. C O N C L U S I O N S
It has been shown that when a realistic cost function is used for the optimization of encastre beams under axial load, the parameter a, which defines the regions in the optimal design is, in general, dependent on the applied loading. This is quite unlike a wide range of problem solutions obtained for purely flexural systems[2, 3]. For the particular case of a beam under uniformly distributed loading, it was found that the optimal design parameter a was highly susceptible to the value of both the applied lateral load and the axial load. In particular, it was found that the simple optimal result given by Heyman[6] for the location of the inflexion points of a beam loaded uniformly and transversely, measured from the encastre supports, forms an upper limit on the more exact result given herein. A c k n o w l e d g e m e n t - - T h e author is indebted to the referees for their helpful comments including the suggestion to include Fig. 7.
7. R E F E R E N C E S C. T. MORLEY, Int. J. Mech. Sci. 8, 305 (1966). P. G. LOWE and R. E. MELCHERS, Int. J. Mech. Sci. 14, 311 (1972); 15, 151 (1973); 15, 711 (1973). G. I. N. ROZVANY, Optimal Design o f Flexural Systems. Pergamon Press, Oxford (1976). R. E. MELCHERS and G. I. N. ROZVA~Y, J. Engng Mech. Div., A S C E . 96, 1093 (1970). Z. Mnoz, Optimum design of reinforced shells of revolutions. In Nonclassical Shell Problems (Edited by W. Olszak and Z. Sawczuk), p. 732. North-Holland, Amsterdam (1964). 6. J. HERMAN, Quart. J. Mech. Appl. Math. 12, 314, (1959). 7. W. PRACER and G. I. N. ROZVANY, J. Struct. Mech. 5, 1 (1977). 8. W. PRA6ER and R. T. SHIELD, J. Appl. Mech. 344, 184 (1967).
1. 2. 3. 4. 5.