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Deflection of statically indeterminate structures
Int..k Mech. Sci, Vol. 24, No. 6. pp. 341-347. 1982 Printed in Great Britain.
DEFLECTION
OF
(X)20-74031821V60341-07503.0010 Pergamon Press l a d
STATICALLY STRUCTURES
INDETERMINATE
R. E. MELCHERS Dept. of Civil Engineering, Monash University, Clayton, Victoria, Australia (Received 15 June 1981 ; in revised form I0 November 1981 ) Summary--The traditional procedure for employing the virtual work equation to calculate structural deflections of statically indeterminate structures requires the solution of the internal action state of the structure to be available or to be calculable. This is not always convenient, for example, in design problems for which strength proportioning is based on plasticity theory, or for thermo elastic problems. The present paper demonstrates an alternative method for calculating the linear elastic deflection of statically indeterminate structures using the virtual work approach. However, in contrast to conventional applications, the internal action state under applied loading is not required to be known.
f~ h I L m~ qi Q~ P~ T ui uii U V W a /3 8 A (')
NOTATION force in member a of primary structure due to the ( P load system beam depth moment of inertia length moment at i internal strain (generalised strain) component internal stress (generalised stress resultant) component body force (surface traction) component temperature field body displacement component displacement in i direction due to unit load at j internal work volume (surface) external work coefficient of thermal expansion elastic (and section properties) constant incremental operator deflection denotes( ) may be elastic incompatible.
INTRODUCTION
The usual procedure for calculating the deflections of statically indeterminate linear elastic structures under load is to firstly determine the forces (or stresses) in the structure, and then to use these to calculate strains and hence structural deflections. The first of these steps involves the solution of a statically indeterminate structure using a method of frame analysis (such as the flexibility method for elastic frames) to find the redundant actions. Action-generalised strain relationships then provide the generalised strains in the members. The deflection of the structure may then be found using the virtual work equation in the form of the "unit dummy load method". That is, the generalised strain set is taken as the actual, compatible strain set, and is subject to an internal action system in equilibrium with a "unit dummy" (virtual) load, placed at the point where the deflection is desired to be known. Provided displacements are small, this approach is valid for inelastic as well as elastic structures. For some types of structural analysis or design, it is not always convenient to determine the internal action solution of the statically indeterminate frame. A typical case is in the plastic design of ductile frames using the upper or lower bound theorems of plasticity [1]. Another example is in thermo elastic problems, two cases of which will be given later. 341
342
R . E . MELCHERS
For such situations it is possible to determine the elastic deflection of the redundant structure without reference to the internal action solution. This approach also employs the concept of virtual work, and permits use of the unit dummy load method, but in the special form that the complete elastic solution is required for the unit dummy load, rather than just an equilibrium system (as in the conventional application). THEORY OF VIRTUAL WORK FOR DEFLECTIONS Under the common assumption that the elastic properties of the material of which the structure is composed depend only on the instantaneous state of elastic straining (including temperature effects), the principle of (complementary) virtual work for an isotropic body states [2]:
fv(u,SP,)dV=fvq,6OidV.
(I)
The ui system (i = 1,2,3) denotes the actual displacements in a body subject to forces P~ incremented to Pi + 6P~. P~ may be body forces, in which case the integral in equation (1) is a volume integral, or surface tractions, in which case the integration is over the body surface. As usual, higher order terms (of form 1/26u~ 6P~) have been neglected. BQ~(i = 1..... 9) denotes the incremental internal stress- (generalised stress resultant-) state resulting from the load iincrement 6P~. The strain- (generalized strain-) state q~ is assumed compatible with the displacements u~. In the above equation, the incremental system is the force system 6P~, 6Q~. In terms of the conventional nomenclature, the virtual system is therefore the incremental force system. Attention will be restricted only to linear elasticity, so that the incremental prescripts on the load and stress terms may be dropped, although such terms can still be regarded as a "virtual" system. Expression (1) is independent of any (incremental) change that may occur in u, and correspondingly in q~, provided that the second order terms such as 8uv,SPi are ignored. This implies that the u~- q~ system may be assumed to remain constant under the variation 6P~-6Q~. It should also be noted that whereas the u~-q~ system represents the deformation state of the elastic body under some (as yet undefined) load system, and must therefore be elastic compatible (i.e. obeys compatibility relationships), no such requirement has been placed on the 6Pi-6Qi system. Hence this latter system need be only in equilibrium [2]. Heuristic arguments [2] and a demonstration 14] have been given in support of this conclusion for isothermal problems. Let the notation (^) indicate that the associated stress-state need not be elastic compatible (but is in equilibrium with the applied loads). Further, let ( )u and ( )~ denote the original and the virtual systems respectively:
Equation (2) is the well-known complementary virtual work equation for elastic systems undergoing small deflections. In the special form P~ = I for i = a, = 0 otherwise, it is commonly known as the "unit (dummy) load method", considered particularly suitable for determining the deflection of statically determinate elastic structures for which the stress field QO, and hence the elastic strain field q0 is known, or may be easily obtained. However, for statically indeterminate structures the calculation of q0 may be a demanding problem in itself. EXTENSION OF COMPLEMENTARY VIRTUAL WORK FORMULATION An alternative formulation of equation (2) has recently been developed. Let /3 represent the elastic material (and member) properties (e.g. E if Q~ is a stress, or EA, El, etc. if Q~ is a stress resultant) for the members of the structure being considered. Then at any point in the structure,/3 can be associated with the internal virtual work of either the ( )0 or the ( )~ system:
=
-~ .~dV fv Q° tS'
=fvO°,O',dV.
(4)
The transposition of Q and q from equations (3) to (4) is permitted since both the ( )0 and the ( )~ system refer to an elastic system; the fact that the ( )~ system also refers to a (') stress state, has been retained in the notation in equation (4).
Deflection of statically indeterminate structures
343
The right hand term in equation (4) may also be interpreted in the sense of equation (3); so that
i q i d V -- f v Pi° u ~ d V fv Qo.,
(5)
Since ( )o and ( )~ is arbitrarily assigned nomenclature, expression (5) can be rewritten as:
fv u~0i P iId V = fV q"0i Q iId V
(6)
where now the original structure being analysed, ( )o, may have a (') stress state, and the virtual stress-state ( )~ must be elastic compatible. For finding deflections, the virtual load system Pl is taken to be a unit (dummy) load applied in the direction for which the deflection is sought; so that
a- fv qiQi ~o_ ,o i dV
(7)
which is the well-known expression for the unit (dummy) load method of deflection analysis, but now in the special form that the stress field associated with the virtual load must be elastic compatible, while the stress field associated with the actual load system need not be elastic compatible. As a result the following theorem may be stated:
Theorem The value of the deflection calculated at any point e in a structure is not dependent on the value(s) of the redundant(s) in the statically admissible (but not necessarily elastic compatible) stress-state 5 , provided that the deflection is calculated using the unit dummy load method with a statically admissible and elastic compatible stress-state Q~ corresponding to the unit dummy load P~, = I [1]. The proof of this theorem follows directly from equation (7) when it is noted that it is always possible to convert the elastic incompatible strain field ~0 into an elastic incompatible stress field (~ simply by multiplying by the appropriate elastic constant (and local section properties, if Q~ is a stress resultant). The above result does not appear to have been recognised until recently [I]. However, it appears that Mohr in his 1874 derivation of the "unit" (dummy) load" version of equation (2) for statically determinate frameworks may, according to Argyris [2] have actually used it; "The formula (see equation 2) given is due essentially to Maxwell (1864) and Mohr (1874) who applied it to statically determinate frameworks. Actually Mohr derived this type of equation by using the principle of virtual displacements with the actual elongations taken as the virtual ones and the unit load system as the actual one..." However, it is perhaps not surprising that no attention was given to situations in which the ( )o system was (^), since the need for solution of such problems did not arise until the development of the plastic theory of structures led eventually also to questions about service load behaviour [5]. COROLLARIES TO THEOREM
Corollary 1 Bettrs Reciprocal Theorem, expressed in terms of concentrated surface tractions (i.e. forces) p~ u o = po u~,(summation convention)
(8)
also holds where either the ( )o or the ( )J system is ('). Proof. The proof follows directly from equations (3)--(5):
fvu°f' dV=fvo°
fvP°, dV.
(9)
Retaining only the extreme terms, and surface tractions as concentrated forces; (9a) which is identical to equation (8) since the order of multiplication is immaterial here, but with the proviso that either the ( )0 or the ( )t system may be (').
Corollary 2 Maxwelrs Reciprocal Theorem
u,~ = ub°
(! O)
also holds when either the ( )0 or the ( )t system is ('). Here Uob represents the deflection at point a due to a unit load applied at b.
Proof. The proof follows directly from equation (9a). Let there be only one value of the index i in each summation, say a and b, and let these points correspond to the ( )0 and the ( )J system of (now) single point
344
R . E . MELCHERS
loading. Further let po = p~ = 1, then u ° = U~aor (10a)
Uba = Uab.
It should be noted that in both corollaries the notation C) refers to the stress-state corresponding to the loading or deformation being considered. Hence for the deformation li, the corresponding internal stress state may be an elastic incompatible one, but, according to the above theorem, the deflection ti still corresponds to that for an elastic compatible structure. Hence, it is of no direct relevance to either Betti's theorem or Maxwell's theorem that a statically admissible, but elastic incompatible stress state may be associated with ~ (or /~); in each case ti is an elastic compatible deformation as the result of the theorem. The above corollaries do not change the traditional meanings of either Betti's or Maxwell's theorem, but merely indicate that it is not necessary for both stress states associated with the ( )0 and ( )' systems to be elastic compatible, as is commonly assumed. Corollary I has particular relevance for deflection calculations in thermo-elastic structures, as will now be briefly discussed. DEFLECTIONS IN THERMO-ELASTIC STRUCTURES The applied thermo-elastic strains due to temperature differentials acting on a body may be replaced by an equivalent system of body forces and a surface normal tension force, given respectively by [6]: Pi = - a g T , i
(1 la)
(P~), = a 3 T
(lib)
and
where P~ is the body force at a generic point in the body in the x~ direction (i = 1,2.3), ( ) , denotes partial differentiation with respect to x~, /3 = e(1 - 2v) -t, an elastic constant ( E = Young's modulus, v = Poisson's ratio), a is the coefficient of thermal expansion, and T = T ( x d is the temperature field. Since neither P~ nor (PD, are elastic compatible, they should be denoted ('); they are also the ( )0 system. Substituting in equation (9) with the u °, po as ('), there is obtained:
and applying the divergence theorem as in Ref. [6]
f v ~° P ~ d V = f v ( d T ) °
(9c)
O' d V
where 0 = ~r. + ~r,_2 + w33, which is the sum of principal stresses at any generic point in the body. Equation (9b) corresponds with that given in [6], except that the ( a / p ) term is noted in equation (9b) specifically as ( ) , and that the 0 *term. which corresponds to the internal stress state due to the auxiliary (or unit dummy load) system ( )*, must be elastic compatible. It is evident that the result in [6] is a special case of the results given in the present paper. In conventional thermo-elastic work. the thermo-elastic equivalent to equation (7) is known as Maisel's Integral Solution [6]. EXAMPLE I - - C L A M P E D BEAM Consider the clamped beam subject to two point loads shown in Fig. I. The deflection at c is required. The bending moment diagram M with arbitrary redundant clamping moments X and Y ( X > Y ) is shown. For a unit (dummy) load applied at c, the elastic compatible bending moment diagram is as shown. Assume E l = constant. Using standard integration formulae ([41, p. 172) the deflection calculations are as follows: EI.At = - ~ [ - L L/4
L
{ - 2 Y + ( 5 PL +7
X +3Y)}]4 + - ' - f f ' L I 4 F L f [ 3 p L - - ~ ) 2 + 5 p L - X 4 3 ~ Y } ] [ 8 [ [ 8 PL - Y
- - pL 3
i.e.
Ac = 128EI"
(12)
It is readily verified from standard results that the above value is the correct elastic deflection for a beam loaded and supported as shown. Indeed, the central deflection of a clamped beam loaded at one quarter point by a point load P is given by ( P L 3 / 3 8 4 E I ) , so that L3
pL 3
A¢~.,~ = 384EI (p + 2P) = 128EI"
Deflection of statically indeterminate structures
345
The above result has significance in the plastic design of (rigid-plastic) redundant structures ([1]. Thus, the clamping moments X and Y must be selected so as to be in equilibrium with the applied loading. As this requirement on X and Y is not definitive in setting their values, they may further be chosen so as to allow use of the smallest beam section; a possible design criterion might then be IMdJ = X ( > Y). With the maximum beam moment fixed, the value of I would be known and the elastic deflection A~ calculable directly from equation (12). This would then be the value of the central deflection as a function of the applied loading (P, 2P) while the beam remained elastic. Since it is normally assumed that plastically designed structures remain elastic under service loads, the service load deflection Ac is immediately available. EXAMPLE 2--THERMO-ELASTIC BEAM Consider the uniform, isotropic two-span continuous beam, shown in Fig. 2(a), which is subject to a temperature gradient such that the upper surface is at temperature T and its lower surface at T = 0. It is required to find the central span deflection A. A possible ~0 system of applied strains is shown in Fig. 2(b), in which the beam is taken as simply supported at A and C and ~0= ¢~0= ctTIh where ~b is the curvature, assuming a linear variation of temperature 7", a is the coefficient of thermal expansion and h is the depth of the beam. To find the midspan deflection A, a unit load system P~ = I is applied as shown in Fig. 2(c). It is readily found that the corresponding elastic compatible bending moment distribution is as shown. Application of equation (7) then produces, using standard integration formulae ([4], p. 172): A=
t~° Q~ ds
L2
= -~T3- ~. Hence, the required deflection is upward, as might be expected. It is possible, but tedious, to verify this result by more conventional methods.
EXAMPLE 3--PORTAL FRAME The simple rectangular portal frame shown in Fig. 3(a) is subject to a uniform temperature increase T in member B C only. The resultant deflection A~ is sought. A suitably simple staticallydeterminate substructure for determining ~0 is shown in Fig. 3(b); member B C may expand freely, without inducing actions in any member. From standard texts, or Ref. I, the bending moment distribution due to a unit point load placed at E, is
P
,_.
I
2P
L
I
@
M
©
m
(elostic)
@
m
FIG. 1. Beam held by arbitrary clamping moments. M S Vol. 24, No. 6--B
346
R . E . MELCHERS
ir
,•LA
@
®
B
~t,.T, 0
• '1'~'? - L
J
[ ,~P', 1
©
L
3_~zL.
O'
FiG. 2. Two span beam subject to temperature defferential.
B
E
C
L
..d
@ L.
CC'
B
® ~'~-)~.~ m, I ~ ©
I
~' .Zlmc m rllrlt" O ~ D
""
-'----.~C
®
,, D if • ql
LI.
•1", ~TL
L r
2
~T~ I_
FIG. 3. Clamped portal frame with member
as given in Fig. 3(c), with m• = m o = L I ( 8 N O ma = mc = - 2mA
where NI =
(12h)l(llL) + 2
and [~c =/'/,~ = HD =
(mc + mo)lh
3L
= 8--h [(12h)l(ltL)
= 3LI(gNIh)
+
21-'
~1
BC
2
raised in temperature
Deflection of statically indeterminate structures
347
Hence, using equation (7). we have: ¢
=
,//
ql Q~ds
~. ~0,,i, ds + ~ (AL)Of I
(7) (7a)
but since all curvature (bending) terms d~" are zero in Fig. 2(b), the integral equals zero. The summation extends only over member BC:
A = ~ [(12h)(ItL) + 21-~ ctTL. An alternative approach, rather less drastic in selecting an appropriate d~° system, follows from selecting the system of Fig. 2(d) for ~0. I! can be verified that
aTL h 2 IVl°c= 1(4° = - ' ~ ( ' ~ + --~2 ]-' and that, upon substituting into equation (7a), the integral term vanishes. Hence it is evident that the precise system selected for the applied load strain field does not affect the value of the deflection calculated, but can markedly affect the amount of labour involved in the calculation. CONCLUSION
It has been shown that it is possible to calculate the exact deflection of an elastic structure without knowing its internal stress distribution or the value of any external redundants. The deflection may be obtained by applying the virtual work concept in the form of the unit (dummy) load method but with the special requirement that the stress distribution corresponding to the unit dummy load must be elastic compatible. It is further shown that both Betti's and Maxwell's reciprocal theorems have somewhat wider meaning than is usually recognized and that in the particular case of thermo-elastic problems, the estimate of deflections obtained by Maisel's Integral Solution is a special case of Betti's theorem in its wider context. REFERENCES 1. R. E. MELCHERS, Service load deflections in plastic structural design. Proc. Inst. Civil Engrs. Part 2, 69, 157-174 (1980). 2. J. H. ARGVRISand S. KELSEV, Energy Theorems and Structural Analysis. Butterworths, London (1960). 3. C. H. NORRIS, J. B. WILBUR and S. UTKU, Elementary Structural Analysis, 3rd Edn. McGraw-Hill, New York (1976). 4. J. A. L. MATHESON, Hyperstatic Structures, Vol. I. Butterworths, London (1959). 5. J. HEYMAN, Plastic design and limit state design. The Structural Engineer 51, 127-131 (1973). 6. S. P. TIMOSHENKOand J. N. GOODmR, Theory of Elasticity, 3rd Edn. McGraw-Hill, New York (1970).
DEFLECTION
OF
(X)20-74031821V60341-07503.0010 Pergamon Press l a d
STATICALLY STRUCTURES
INDETERMINATE
R. E. MELCHERS Dept. of Civil Engineering, Monash University, Clayton, Victoria, Australia (Received 15 June 1981 ; in revised form I0 November 1981 ) Summary--The traditional procedure for employing the virtual work equation to calculate structural deflections of statically indeterminate structures requires the solution of the internal action state of the structure to be available or to be calculable. This is not always convenient, for example, in design problems for which strength proportioning is based on plasticity theory, or for thermo elastic problems. The present paper demonstrates an alternative method for calculating the linear elastic deflection of statically indeterminate structures using the virtual work approach. However, in contrast to conventional applications, the internal action state under applied loading is not required to be known.
f~ h I L m~ qi Q~ P~ T ui uii U V W a /3 8 A (')
NOTATION force in member a of primary structure due to the ( P load system beam depth moment of inertia length moment at i internal strain (generalised strain) component internal stress (generalised stress resultant) component body force (surface traction) component temperature field body displacement component displacement in i direction due to unit load at j internal work volume (surface) external work coefficient of thermal expansion elastic (and section properties) constant incremental operator deflection denotes( ) may be elastic incompatible.
INTRODUCTION
The usual procedure for calculating the deflections of statically indeterminate linear elastic structures under load is to firstly determine the forces (or stresses) in the structure, and then to use these to calculate strains and hence structural deflections. The first of these steps involves the solution of a statically indeterminate structure using a method of frame analysis (such as the flexibility method for elastic frames) to find the redundant actions. Action-generalised strain relationships then provide the generalised strains in the members. The deflection of the structure may then be found using the virtual work equation in the form of the "unit dummy load method". That is, the generalised strain set is taken as the actual, compatible strain set, and is subject to an internal action system in equilibrium with a "unit dummy" (virtual) load, placed at the point where the deflection is desired to be known. Provided displacements are small, this approach is valid for inelastic as well as elastic structures. For some types of structural analysis or design, it is not always convenient to determine the internal action solution of the statically indeterminate frame. A typical case is in the plastic design of ductile frames using the upper or lower bound theorems of plasticity [1]. Another example is in thermo elastic problems, two cases of which will be given later. 341
342
R . E . MELCHERS
For such situations it is possible to determine the elastic deflection of the redundant structure without reference to the internal action solution. This approach also employs the concept of virtual work, and permits use of the unit dummy load method, but in the special form that the complete elastic solution is required for the unit dummy load, rather than just an equilibrium system (as in the conventional application). THEORY OF VIRTUAL WORK FOR DEFLECTIONS Under the common assumption that the elastic properties of the material of which the structure is composed depend only on the instantaneous state of elastic straining (including temperature effects), the principle of (complementary) virtual work for an isotropic body states [2]:
fv(u,SP,)dV=fvq,6OidV.
(I)
The ui system (i = 1,2,3) denotes the actual displacements in a body subject to forces P~ incremented to Pi + 6P~. P~ may be body forces, in which case the integral in equation (1) is a volume integral, or surface tractions, in which case the integration is over the body surface. As usual, higher order terms (of form 1/26u~ 6P~) have been neglected. BQ~(i = 1..... 9) denotes the incremental internal stress- (generalised stress resultant-) state resulting from the load iincrement 6P~. The strain- (generalized strain-) state q~ is assumed compatible with the displacements u~. In the above equation, the incremental system is the force system 6P~, 6Q~. In terms of the conventional nomenclature, the virtual system is therefore the incremental force system. Attention will be restricted only to linear elasticity, so that the incremental prescripts on the load and stress terms may be dropped, although such terms can still be regarded as a "virtual" system. Expression (1) is independent of any (incremental) change that may occur in u, and correspondingly in q~, provided that the second order terms such as 8uv,SPi are ignored. This implies that the u~- q~ system may be assumed to remain constant under the variation 6P~-6Q~. It should also be noted that whereas the u~-q~ system represents the deformation state of the elastic body under some (as yet undefined) load system, and must therefore be elastic compatible (i.e. obeys compatibility relationships), no such requirement has been placed on the 6Pi-6Qi system. Hence this latter system need be only in equilibrium [2]. Heuristic arguments [2] and a demonstration 14] have been given in support of this conclusion for isothermal problems. Let the notation (^) indicate that the associated stress-state need not be elastic compatible (but is in equilibrium with the applied loads). Further, let ( )u and ( )~ denote the original and the virtual systems respectively:
Equation (2) is the well-known complementary virtual work equation for elastic systems undergoing small deflections. In the special form P~ = I for i = a, = 0 otherwise, it is commonly known as the "unit (dummy) load method", considered particularly suitable for determining the deflection of statically determinate elastic structures for which the stress field QO, and hence the elastic strain field q0 is known, or may be easily obtained. However, for statically indeterminate structures the calculation of q0 may be a demanding problem in itself. EXTENSION OF COMPLEMENTARY VIRTUAL WORK FORMULATION An alternative formulation of equation (2) has recently been developed. Let /3 represent the elastic material (and member) properties (e.g. E if Q~ is a stress, or EA, El, etc. if Q~ is a stress resultant) for the members of the structure being considered. Then at any point in the structure,/3 can be associated with the internal virtual work of either the ( )0 or the ( )~ system:
=
-~ .~dV fv Q° tS'
=fvO°,O',dV.
(4)
The transposition of Q and q from equations (3) to (4) is permitted since both the ( )0 and the ( )~ system refer to an elastic system; the fact that the ( )~ system also refers to a (') stress state, has been retained in the notation in equation (4).
Deflection of statically indeterminate structures
343
The right hand term in equation (4) may also be interpreted in the sense of equation (3); so that
i q i d V -- f v Pi° u ~ d V fv Qo.,
(5)
Since ( )o and ( )~ is arbitrarily assigned nomenclature, expression (5) can be rewritten as:
fv u~0i P iId V = fV q"0i Q iId V
(6)
where now the original structure being analysed, ( )o, may have a (') stress state, and the virtual stress-state ( )~ must be elastic compatible. For finding deflections, the virtual load system Pl is taken to be a unit (dummy) load applied in the direction for which the deflection is sought; so that
a- fv qiQi ~o_ ,o i dV
(7)
which is the well-known expression for the unit (dummy) load method of deflection analysis, but now in the special form that the stress field associated with the virtual load must be elastic compatible, while the stress field associated with the actual load system need not be elastic compatible. As a result the following theorem may be stated:
Theorem The value of the deflection calculated at any point e in a structure is not dependent on the value(s) of the redundant(s) in the statically admissible (but not necessarily elastic compatible) stress-state 5 , provided that the deflection is calculated using the unit dummy load method with a statically admissible and elastic compatible stress-state Q~ corresponding to the unit dummy load P~, = I [1]. The proof of this theorem follows directly from equation (7) when it is noted that it is always possible to convert the elastic incompatible strain field ~0 into an elastic incompatible stress field (~ simply by multiplying by the appropriate elastic constant (and local section properties, if Q~ is a stress resultant). The above result does not appear to have been recognised until recently [I]. However, it appears that Mohr in his 1874 derivation of the "unit" (dummy) load" version of equation (2) for statically determinate frameworks may, according to Argyris [2] have actually used it; "The formula (see equation 2) given is due essentially to Maxwell (1864) and Mohr (1874) who applied it to statically determinate frameworks. Actually Mohr derived this type of equation by using the principle of virtual displacements with the actual elongations taken as the virtual ones and the unit load system as the actual one..." However, it is perhaps not surprising that no attention was given to situations in which the ( )o system was (^), since the need for solution of such problems did not arise until the development of the plastic theory of structures led eventually also to questions about service load behaviour [5]. COROLLARIES TO THEOREM
Corollary 1 Bettrs Reciprocal Theorem, expressed in terms of concentrated surface tractions (i.e. forces) p~ u o = po u~,(summation convention)
(8)
also holds where either the ( )o or the ( )J system is ('). Proof. The proof follows directly from equations (3)--(5):
fvu°f' dV=fvo°
fvP°, dV.
(9)
Retaining only the extreme terms, and surface tractions as concentrated forces; (9a) which is identical to equation (8) since the order of multiplication is immaterial here, but with the proviso that either the ( )0 or the ( )t system may be (').
Corollary 2 Maxwelrs Reciprocal Theorem
u,~ = ub°
(! O)
also holds when either the ( )0 or the ( )t system is ('). Here Uob represents the deflection at point a due to a unit load applied at b.
Proof. The proof follows directly from equation (9a). Let there be only one value of the index i in each summation, say a and b, and let these points correspond to the ( )0 and the ( )J system of (now) single point
344
R . E . MELCHERS
loading. Further let po = p~ = 1, then u ° = U~aor (10a)
Uba = Uab.
It should be noted that in both corollaries the notation C) refers to the stress-state corresponding to the loading or deformation being considered. Hence for the deformation li, the corresponding internal stress state may be an elastic incompatible one, but, according to the above theorem, the deflection ti still corresponds to that for an elastic compatible structure. Hence, it is of no direct relevance to either Betti's theorem or Maxwell's theorem that a statically admissible, but elastic incompatible stress state may be associated with ~ (or /~); in each case ti is an elastic compatible deformation as the result of the theorem. The above corollaries do not change the traditional meanings of either Betti's or Maxwell's theorem, but merely indicate that it is not necessary for both stress states associated with the ( )0 and ( )' systems to be elastic compatible, as is commonly assumed. Corollary I has particular relevance for deflection calculations in thermo-elastic structures, as will now be briefly discussed. DEFLECTIONS IN THERMO-ELASTIC STRUCTURES The applied thermo-elastic strains due to temperature differentials acting on a body may be replaced by an equivalent system of body forces and a surface normal tension force, given respectively by [6]: Pi = - a g T , i
(1 la)
(P~), = a 3 T
(lib)
and
where P~ is the body force at a generic point in the body in the x~ direction (i = 1,2.3), ( ) , denotes partial differentiation with respect to x~, /3 = e(1 - 2v) -t, an elastic constant ( E = Young's modulus, v = Poisson's ratio), a is the coefficient of thermal expansion, and T = T ( x d is the temperature field. Since neither P~ nor (PD, are elastic compatible, they should be denoted ('); they are also the ( )0 system. Substituting in equation (9) with the u °, po as ('), there is obtained:
and applying the divergence theorem as in Ref. [6]
f v ~° P ~ d V = f v ( d T ) °
(9c)
O' d V
where 0 = ~r. + ~r,_2 + w33, which is the sum of principal stresses at any generic point in the body. Equation (9b) corresponds with that given in [6], except that the ( a / p ) term is noted in equation (9b) specifically as ( ) , and that the 0 *term. which corresponds to the internal stress state due to the auxiliary (or unit dummy load) system ( )*, must be elastic compatible. It is evident that the result in [6] is a special case of the results given in the present paper. In conventional thermo-elastic work. the thermo-elastic equivalent to equation (7) is known as Maisel's Integral Solution [6]. EXAMPLE I - - C L A M P E D BEAM Consider the clamped beam subject to two point loads shown in Fig. I. The deflection at c is required. The bending moment diagram M with arbitrary redundant clamping moments X and Y ( X > Y ) is shown. For a unit (dummy) load applied at c, the elastic compatible bending moment diagram is as shown. Assume E l = constant. Using standard integration formulae ([41, p. 172) the deflection calculations are as follows: EI.At = - ~ [ - L L/4
L
{ - 2 Y + ( 5 PL +7
X +3Y)}]4 + - ' - f f ' L I 4 F L f [ 3 p L - - ~ ) 2 + 5 p L - X 4 3 ~ Y } ] [ 8 [ [ 8 PL - Y
- - pL 3
i.e.
Ac = 128EI"
(12)
It is readily verified from standard results that the above value is the correct elastic deflection for a beam loaded and supported as shown. Indeed, the central deflection of a clamped beam loaded at one quarter point by a point load P is given by ( P L 3 / 3 8 4 E I ) , so that L3
pL 3
A¢~.,~ = 384EI (p + 2P) = 128EI"
Deflection of statically indeterminate structures
345
The above result has significance in the plastic design of (rigid-plastic) redundant structures ([1]. Thus, the clamping moments X and Y must be selected so as to be in equilibrium with the applied loading. As this requirement on X and Y is not definitive in setting their values, they may further be chosen so as to allow use of the smallest beam section; a possible design criterion might then be IMdJ = X ( > Y). With the maximum beam moment fixed, the value of I would be known and the elastic deflection A~ calculable directly from equation (12). This would then be the value of the central deflection as a function of the applied loading (P, 2P) while the beam remained elastic. Since it is normally assumed that plastically designed structures remain elastic under service loads, the service load deflection Ac is immediately available. EXAMPLE 2--THERMO-ELASTIC BEAM Consider the uniform, isotropic two-span continuous beam, shown in Fig. 2(a), which is subject to a temperature gradient such that the upper surface is at temperature T and its lower surface at T = 0. It is required to find the central span deflection A. A possible ~0 system of applied strains is shown in Fig. 2(b), in which the beam is taken as simply supported at A and C and ~0= ¢~0= ctTIh where ~b is the curvature, assuming a linear variation of temperature 7", a is the coefficient of thermal expansion and h is the depth of the beam. To find the midspan deflection A, a unit load system P~ = I is applied as shown in Fig. 2(c). It is readily found that the corresponding elastic compatible bending moment distribution is as shown. Application of equation (7) then produces, using standard integration formulae ([4], p. 172): A=
t~° Q~ ds
L2
= -~T3- ~. Hence, the required deflection is upward, as might be expected. It is possible, but tedious, to verify this result by more conventional methods.
EXAMPLE 3--PORTAL FRAME The simple rectangular portal frame shown in Fig. 3(a) is subject to a uniform temperature increase T in member B C only. The resultant deflection A~ is sought. A suitably simple staticallydeterminate substructure for determining ~0 is shown in Fig. 3(b); member B C may expand freely, without inducing actions in any member. From standard texts, or Ref. I, the bending moment distribution due to a unit point load placed at E, is
P
,_.
I
2P
L
I
@
M
©
m
(elostic)
@
m
FIG. 1. Beam held by arbitrary clamping moments. M S Vol. 24, No. 6--B
346
R . E . MELCHERS
ir
,•LA
@
®
B
~t,.T, 0
• '1'~'? - L
J
[ ,~P', 1
©
L
3_~zL.
O'
FiG. 2. Two span beam subject to temperature defferential.
B
E
C
L
..d
@ L.
CC'
B
® ~'~-)~.~ m, I ~ ©
I
~' .Zlmc m rllrlt" O ~ D
""
-'----.~C
®
,, D if • ql
LI.
•1", ~TL
L r
2
~T~ I_
FIG. 3. Clamped portal frame with member
as given in Fig. 3(c), with m• = m o = L I ( 8 N O ma = mc = - 2mA
where NI =
(12h)l(llL) + 2
and [~c =/'/,~ = HD =
(mc + mo)lh
3L
= 8--h [(12h)l(ltL)
= 3LI(gNIh)
+
21-'
~1
BC
2
raised in temperature
Deflection of statically indeterminate structures
347
Hence, using equation (7). we have: ¢
=
,//
ql Q~ds
~. ~0,,i, ds + ~ (AL)Of I
(7) (7a)
but since all curvature (bending) terms d~" are zero in Fig. 2(b), the integral equals zero. The summation extends only over member BC:
A = ~ [(12h)(ItL) + 21-~ ctTL. An alternative approach, rather less drastic in selecting an appropriate d~° system, follows from selecting the system of Fig. 2(d) for ~0. I! can be verified that
aTL h 2 IVl°c= 1(4° = - ' ~ ( ' ~ + --~2 ]-' and that, upon substituting into equation (7a), the integral term vanishes. Hence it is evident that the precise system selected for the applied load strain field does not affect the value of the deflection calculated, but can markedly affect the amount of labour involved in the calculation. CONCLUSION
It has been shown that it is possible to calculate the exact deflection of an elastic structure without knowing its internal stress distribution or the value of any external redundants. The deflection may be obtained by applying the virtual work concept in the form of the unit (dummy) load method but with the special requirement that the stress distribution corresponding to the unit dummy load must be elastic compatible. It is further shown that both Betti's and Maxwell's reciprocal theorems have somewhat wider meaning than is usually recognized and that in the particular case of thermo-elastic problems, the estimate of deflections obtained by Maisel's Integral Solution is a special case of Betti's theorem in its wider context. REFERENCES 1. R. E. MELCHERS, Service load deflections in plastic structural design. Proc. Inst. Civil Engrs. Part 2, 69, 157-174 (1980). 2. J. H. ARGVRISand S. KELSEV, Energy Theorems and Structural Analysis. Butterworths, London (1960). 3. C. H. NORRIS, J. B. WILBUR and S. UTKU, Elementary Structural Analysis, 3rd Edn. McGraw-Hill, New York (1976). 4. J. A. L. MATHESON, Hyperstatic Structures, Vol. I. Butterworths, London (1959). 5. J. HEYMAN, Plastic design and limit state design. The Structural Engineer 51, 127-131 (1973). 6. S. P. TIMOSHENKOand J. N. GOODmR, Theory of Elasticity, 3rd Edn. McGraw-Hill, New York (1970).