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Creep and temperature effects in concrete structures: reality and prediction
Creep and temperature
effects in concrete structures - reality and prediction: G. L. England
Creep and temperature effects in concrete structures: reality and prediction G. L. England Department of Civil Engineering, London WC2R ZLS, UK (Received August 19 79)
Introduction The creep of concrete covers large areas of research at the microscopic and sub-microscopic material level and at an engineering level, where the behaviour of structural members and indeed whole structures becomes important from the viewpoint of serviceability and safety. Because creep is temperature dependent, structures subjected to spatial variations of temperature will be particularly sensitive to creep. The creep behaviour of concrete at the macroscopic level, its influence on structural behaviour, and its inclusion into theorems of applied mechanics and numerical analyses, form the basis of this paper. In particular, attention is focused on a continuing theme of research being carried out at King’s College into the development of predictive methods of creep analysis which are simple to apply to any problem whether it be small or large, and economically viable for even the largest of problems. Supporting experimental data are presented and some philosophical aspects of the creep theories are mentioned.
Creep as a material
property
At normal working temperatures creep can be represented satisfactorily by two separate strain components:
King’s College, University of London,
The Strand,
strain becomes fully developed a few days after the concrete reaches its new raised temperature state. Figures I and 2 display the form of the creep strains observed in experiments. Such strain behaviour may be described by compound visco-elastic models of the types shown, provided suitably chosen time- and temperaturedependent spring and dashpot parameters can be assigned to the models. In numerical step-by-step analyses little difficulty is experienced by this type of representation, although the costs of solution may be high. In adopting either analytical or semi-analytical procedures, it becomes convenient as described later, to work in terms of a modified time variable, called pseudo-time, which is linked directly to the irreversible creep strain component.
Creep and the structure In describing the influence of creep on structural behaviour it is convenient to consider the two classes: (i), statically determinate and (ii), statically indeterminate. It is also useful to separate the effects of uniform temperature from those associated with temperature gradients and temperature changes in time.
(i) An irreversible strain of the viscous flow type, for which both age and time under load influence the strain rate under constant applied stress. (ii) A reversible strain component, frequently referred to as delayed elastic strain. The rate of development and recovery are age-dependent, and the magnitude of this component when fully developed is approximately 25% to 30% of the immediate elastic strain. At elevated temperatures,
additional
features are present:
(iii) The viscous flow strain is temperature sensitive in an approximately linear way with respect to temperature changes. (iv) The recoverable strain is fairly insensitive to temperature. (v) An additional irreversible strain component is observed when stressed concrete is heated to a previously unattained value for the first time. This temperature-change creep 0307-904X/80/040261-11/$02.00 0 1980 IPC Business Press
Recoverable
I Figure 7 at constant
I
strains
t
Influence stress
Appl.
t Time of temperature on creep and recovery strains
Math.
Modelling,
1980,
Vol
4, August
261
Creep and temperature effects in concrete structures - reality and prediction: G. L. England prestress are very marked in this example.2 The initial thermal deflection is reduced by approximately 60% after a period of 105 days in the heated condition. Figure 5 illustrates the influence of creep on transverse deflections in a simply supported prestressed beam, in relation to time for which the designed live load is actually applied. Some roadway structures fall into this category. For such structures it may be impossible to predict the timedependent deflection behaviour unless the value of k in Figure 5 is known, at least approximately. The longterm effect will be for the beam to tend to hog under the influence of dead load only, and to sag when the live loads approach their designed values. The actual long-term behaviour will be intermediate between the two limits corresponding to dead load only and dead plus live load for the entire life of the structure. A time-dependent increase of the hogging deflections on undertrafficked prestressed structures of this kind can lead to unpleasant vertical motions for motorists travelling at motorway speeds. The cyclic heating of the upper surface of road beams caused by solar heating acts in combination with creep to partially offset the hogging problem. An experimental programme of testing is currently being carried out at King’s College to determine the long-term influence of cyclically varying temperatures on the behaviour of continuous prestressed bridge structures. This work is being carried out for the Department of Transport. Another example where the ambient environment has been observed to influence structural behaviour through creep, is depicted in Figure 6. A vertical bridge pier has been observed to exhibit a time-dependent motion at its
T
B E(T)
Ratchet
Elastic
I
I
strarn
t
b
Time Figure 2 stress
Influence
of change of temperature
on creep while under
Uniform LoodIng
Stress varlatlons t=O to t-m v
V V L7 0
\
t \
Time
\
‘\ ‘. ~--i._
Figure 3 deflection
d L-1
----___
Influence of temperature and temperature behaviour of simply supported beam
r3 >
rr
crossfall on
Statically determinate structures With the exception of possibly small perturbations to the internal states of stress in statically determinate structures the influence of creep is seen primarily as an increase of the time-dependent deformations of the structure. A uniform increase of temperature increases the rate of these deformations. Temperature gradients further influence these rates and introduce a feature not seen in the analysis of beam and column structures by conventional elastic methods. This feature is a coupling between bending moments and extensions to the centroidal axis, and between axial forces and curvature changes. Figure 3 shows the effect of temperature and temperature gradients on the central deflection of a simply supported prestressed beam. Similar enhanced deflection behaviour has been observed in reinforced concrete beams of gravel and limestone concrete, subjected to temperature crossfalls through their depth.’ Figure 4 shows the effect on deflection behaviour of a temperature crossfall applied through the depth of a uniformly prestressed beam which is free from any superimposed loading. The curvature changes caused by the axial
262
Appl.
Math.
Modelling,
1980,
Vol 4, August
Days after
heating
Figure 4 Influence of axial prestress on deflection behaviour of simply supported beam with temperature crossfall (Taken from Ross et a/. *)
1. k
stresses
Trme
1
I
c
Time k, Increasing Dead+hve
Load, contrnuousx
Figure 5 Influence of cyclic load on deflection ported prestressed beam
of simply sup-
Creep and temperature effects in concrete structures - reality and prediction: G. L. England
T,OSun
_
Long term drift to=th
Brldqe _ pier
bending moments at some sections undergo a change of sign with time. This changing internal stress behaviour has been monitored experimentally in a three-span continuous prestressed beam,22 using a series of elastic inclusions similar to those used by Stanculescu and Ionescu” for ultimate load stress measurements on unreinforced concrete prisms in bending. External redundancies and their variation with time have been monitored in heated prestressed portal frames by Krishnamoorthy’O and for flexurally restrained simulated wall elements of oil storage structures subjected to cyclic variations of temperature by Clarke.” Figures 7-9 show examples of structural behaviour as influenced by spatially varying temperatures and the attendant nonhomogeneous creep.
Time
Theoretical behaviour Figure 6 pierz3
Solar heating effect on deflection
behaviour
of bridge
upper end. The motion was predominantly in a southerly direction, and can be described in terms of temperaturedependent creep and cyclic heating caused by the sun’s radiation. Shrinkage may also play some role in this case. However, based on temperature-dependent creep alone the motion at the upper end of the pier can be described as a cyclically varying displacement pattern caused by thermal and thermo-elastic curvature changes, superimposed on a long-term drift displacement towards the most dominantly heated face of the pier. The effect of creep in this situation is thus to cause a ratcheting behaviour which generates a long-term displacement in a preferred direction, even though the motion in the short term appears to be cyclic and repetitive. This motion is shown diagrammatically in Figure 6. Statically indeterminate
considerations
and prediction
of creep
The manner in which the separate components of creep strain are incorporated into creep analyses can depend upon the problem being solved, since justifiable simplifications are frequently possible. For example, in problems for which the temperatures are sustained - and in some cases of cyclically varying temperatures also - the flow and temperature-change components can be treated together in the analysis. At raised temperatures the recoverable
structures
Creep in statically indeterminate structures is frequently thought to be beneficial in that the additional stressing caused to a structure by in-service settlement of foundations is alleviated by subsequent creep. This tends to restore the structure to a state of stress comparable to the result of a conventional elastic calculation for no foundation displacements. Such behaviour may be observed experimentally for structures which are homogeneous in their elastic and creep properties. Theoretically this behaviour is seen to be a special case of a more general understanding of statically indeterminate structures, for which their behaviour appears to move continually towards a state of lower power dissipation in creep. 3,4 In other cases the initial elastic solution may bear no relation to the long-term creep solution. An interesting example where creep can cause a loss of serviceability is that of a multi-span beam structure, designed initially for perhaps continuous or alternate span loading. If this structure is supported on a creep-elastic foundation medium, it can suffer loss of serviceability and possible damage in the long term if there is a partial unloading, even though no damage is caused in the short term.5 It is possible for warehouse floor slabs to be affected in this way. Experiments conducted at King’s College on statically indeterminate beam structures,2p738 have revealed that the existence of temperature gradients causes severe departures, from elastic theory predictions, of measured external support actions. These variations to external actions infer major stress redistribution within the structure, to the extent that
Days after
heating
I!
2000
II
4000 MlowedGheatlng
Unloading
foliowed
Figure 7 Influence of temperature crossfall on support in two-span prestressed beim (Ross eta/.‘). Comparison steady-state stress theory prediction (England14) Elastic i
6d
mcluslons i
/
Ed
_+_ IR
by coozg
reactions with
t_ IR
9ooc
6d
f 6
state
*‘Preferred’ t
5 IF\ 10
Co-Llnear Aports_
30 50 Days after
)_
Support displacement Imposed
200300 100 ioadlng, Log scale
Figure 8 Influence of nonuniform temperatures and support displacements on three-span prestressed beam (Taken from England et al. ‘7
Appl.
Math.
Modelling,
1980,
Vol
4, August
263
Creep and temperature effects in concrete structures - reality and prediction: L
Thermeelastlc
SE Y
,I---__
6 is
r_
stress representation --I_
-.-q
/
-l__,
1-,’ I ’
?
sol&on
G. L. England
is:
i=N U=
(To + 2
afUi
I
I
20
40
60
80
Time since heating,
100
120
days
_--
3-
Steady
state
predictjan
4-
5-
6; Figure 9 Behaviour of prestressed portal frame subjected uniform heating of beam”
where (~0 and Ui represent a set of stresses which satisfies the statical loading of the structure and a set of self-equilibrating stresses, respectively. ai are time-dependent weighting parameters which provide an appropriate combination of the individual ui stresses in the representation of the total stresses, u. The solution procedure is similar to the flexibility method of elastic analysis and results in a set of first order differential equations in ai. Evaluation of the aj coefficients for selected times allows stress variations with time to be deduced from a direct calculation. When a functional representation is adopted and minimization of this functional is conducted with respect to the stress parameters to determine the aj functions, a physical interpretation may be placed on the quantity being minimized. It is a combination of the rate of change of complementary strain energy in the body plus one-half of the rate at which energy is dissipated from the body in creep deformation. An alternative interpretation is simply a statement of the principle of virtual power, viz:
to non-
P6u dV= 0 I
creep becomes a small and insignificant proportion of the total creep, and being insensitive to temperature, it may without much loss in accuracy be included along with the normal elastic strain. This simplification gives rise to small errors in the short-term immediately following a change of stress, and to insignificant errors at large times. It is therefore reasonable to base an initial theory on the simplified series spring and dashpot combination of the Maxwell model. Experimental creep data may be normalized readily with respect to stress and temperature and thus the spring and dashpot parameters may be defined. Analyses are simplified considerably when the ageing characteristics of concrete creep are removed from the analysis. This is done by adopting the use of a pseudo-time parameter (England and Jordaanlz) which is the normalized creep itself. Conversion to real time is then achieved by referring to the single-valued normalized creep/real time curve for the concrete. Direct solutions are available for problems with the following characteristics: (i) Sustained or cyclically repeating applied loading. (ii) Sustained or cyclically varying temperatures. It is convenient groups.
to divide the creep analyses into two
(i) The transient or time-varying stress portion. (ii) The long-term limiting, or steady-state stress solution, for large times. Time-varying stress solution Using the Maxwell creep formulation in conjunction with a Ritz type of stress representation, equation (l), has enabled a complementary power theory to be developed.3 This permits exact solutions to be derived for problems containing a finite number of redundancies and for approximate solutions to be sought in continuum problems. The
264
Appl.
(1)
i=l
Math.
Modelling,
1980,
Vol
4, August
(2)
when the support displacement rates are zero.* In equation (2) P represents a set of internal strain rates which are everywhere compatible. They may be formulated in terms of the stresses and stress rates from equation (l), via the constitutive creep relationship for the Maxwell material. 60 is any one of the sets of self-equilibrating stresses Ui. Thus, as i ranges from 1 to N, a set 0fN simultaneous differentia1 equations emerges for the evaluation of the ai functions. Figure IO shows an example where this method of analysis has been used to analyse a prestressed concrete reactor containment vessel. Comparisons are made with a time-step method of solution in order to assess the accuracy of this approximate solution, for which N = 4 in equation (I). The same approach may be used to solve for the timevarying stresses in a cyclic temperature problem where temperatures vary between two or more repeating states in time. For problems in which the temperature transients, after each temperature change, are of little significance compared to the hold times under steady-state heat flow conditions, and when the true thermal stresses are included among the ui distributions, the solution proceeds in a similar manner to the sustained temperature case, with known changes to the ai functions at each temperature change. An alternative solution may be developed by formulating the power functional in terms of the specific stress and temperature states in the different parts of the temperature cycles. A weighted average power functional is minimized with respect to the free stress parameters.13 This approach allows the general form of the time-varying stresses - over many cycles - to be predicted with fair accuracy, but precision is lost within each part of the temperature cycle, * In the absence of specified displacement rates this situation can always be achieved simply by redefining the boundary to the body or structure
Creep and temperature effects in concrete structures - reality and prediction: G. L. England since only constant-in-time for these intervals.
averaged values are predicted
Limiting long-temz or steady-state stresses Steady-state stresses can exist whenever loads and temperatures are long sustained and the concrete creep capacity is sufficiently high to not impede the stress redistribution process. A quasi-steady state of stress may also be reached when a structure is subjected to cyclically varying loads or temperatures for long periods of time. When the the structure remains serviceable under these conditions a shakedown limit may be defined, for which stresses repeat at corresponding times in successive cycles of load or temperature. During the periods for which the loads or temperatures are constant in time, the stresses exhibit time-dependent variation. Any change in the cyclic pattern of behaviour will result in stress changes towards new and definable quasi-steady-state limits. Although it is possible to evaluate the steady-state (sustained) and shake-down (cyclic) stresses from a special case of the preceding theory relating to time-varying stresses it is often advantageous to take note of another special feature which relates only to these limiting states of stress. Their calculation is dependent on the power dissipation component only, of the power functional, and thus a choice of solution technique is possible. Either a ‘flexibility’ or ‘stiffness’ type of solution may be employed. The latter approach has certain advantages in large continuum problems. However, whichever method is employed it is possible to use standard elastic analyses for the evaluation of the limiting stresses. This is because useful analogies are
Outer
face IS X-l
Temperature
states
Figure 11 Cylindrical concrete oil containment subjected to repeated filling and emptying of hot oil and cold sea water. Cyclic temperature behaviour varies with depth, z (England et al. 24)
Steady
state
IOOOOdays 8
Inner face 0.
seen to exist between certain quantities in elastic theory and the steady-state stress theories. For the sustained temperature and load cases, an elastic calculation will reveal the steady-state stresses when the normal elastic modulus is replaced by 1/H(T), where N(T) is the normalizing creep-temperature functions.14 For the cyclic temperature problem it is necessary to replace the elastic modulus by the reciprocal of a weighted function of the temperatures for the individual parts of the cycle. When 4(T) = T, this quantity is identified as l/T,, where T,, is the weighted mean temperature (Andrews et al. 15) In addition to this substitution an analogous ‘initial strain’ component is also identified and must be incorporated into the modified elastic analysis in order to evaluate the steady-state-cyclic stresses. Examples of such steady-state stress calculations are shown in Figures 3, 7, 9 and 10, for cases of sustained temperature distributions. Figure I2 shows some results for cyclic variations of temperature, as experienced in a thin-walled oil storage structure subjected to repeated filling and emptying (Figure 11).
Discussion
Figure 10 Comparison of creep predictions for stresses in prestressed nuclear containment vessel. Time-step solution and approximate analysis using four self-equilibrating stress distributions?
The previous sections have indicated that creep and temperature can significantly influence structural behaviour, to the extent that elastic theory calculations may be rendered worthless. The creep theories discussed allow solutions for several classes of problem to be achieved without difficulty from semi-analytical calculations. These will usually be
Appl.
Math.
Modelling,
1980,
Vol
4, August
265
Creep and temperature effects in concrete structures - reality and prediction:
0 Circumferential figure
72
Shakedown
and
0
0017
longltudlnal
of composite
coupled with numerical difference type, for the where spatial subdivision transient creep solution states of approximation
structures
containing
elastic
procedures of the finite element or solution of extensive problems is needed. The accuracy of any will depend therefore upon two in the analysis.
(i) The spatial idealization, both in geometry and functional representation of the stress and displacement fields. (ii) The number and suitability of the self-equilibrating stress distributions of equation (1). The structural idealization of(i) governs the overall accuracy of the solution, and therefore the creep solution will be no more precise than the corresponding initial elastic solution. The number of self-equilibrating stress distributions introduces a second approximation into the creep solution, even though each of these may be generated by use of the same elastic procedures and structural idealization used in (i). Experience has indicated that when suitable self-equilibrating stresses can be generated, a creep solution usually requires fewer than ten such distributions to give good time-dependent solutions. Figure 10 shows that when only four distributions are used, the errors are tolerably small even in a zone of the structure where maximum errors are to be expected.6
266
Appl.
Math.
Modelling,
1980,
Vol
4, August
5
10
I5
20
MN/m2
Scale of stress
stresses
stresses due to creep and cyclic temperatures,
Time-dependent
Figure 13 Examples only components
G. L. England
for containment
of Figure
71
Techniques for deriving suitable self-equilibrating distributions automaticalky have been developed and included in a recently completed three-dimensional creep analysis for the Health and Safety Executive. Refinements to all the previously described analyses, to include the delayed elastic strain component of concrete, are readily possible. l6 The complexity of the problem is increased slightly and a solution is obtained from a set of linear second order differential equations in ai, of equation (l), England and Allen.” Standard eigenvalue analyses may be employed to determine these functions. For this material representation it is important to allow freedom of variation of the two creep strain components, with respect to environmental influences such as temperature and humidity.‘83 l9 The recently revised CEC-FIP CodeZo allows the designer to exploit this freedom in his analysis. Hopefully, other Codes will follow this lead when they are next updated. For structures containing discrete elastic components in addition to a creep phase, there is frequently a transfer of stress to the non-creeping component. The simplest example of this is the reinforced concrete column carrying an axial compressive force. For this structure there is a continual transfer of stress from the concrete section to the steel reinforcement, until theoretically, either all load is transferred to the steel, or the steel yields and thereafter load transfer ceases, even though strains continue. In other cases the long-term solution may be less clear. Figure 13 shows two examples, an elastic inclusion completely surrounded by a creeping matrix and a hollow circular cylinder containing an internal pressure and subjected to some restraint at its outer surface due to a close fitting elastic ring. In the first case load is transferred to the elastic inclusion until a definable limit is reached. In the cylinder problem, however, it is possible - depending upon the geometry, temperature and loading parameters of the problem ~ for the outer ring to undergo a time-dependent motion in the radial direction, which may be either outward or inward.*r
Creep and temperature
effects
A special case exists for which there is no surface displacement rate even though there is creep occurring in the the central zone. The implications are of special interest since it is possible to design a pipe section, carrying externally applied prestress of the wire-winding type, such that there is no loss of prestress with time. These and other special features make it impossible to generalize the creep behaviour of statically indeterminate and composite structures.
in concrete 2 3
4
5 6
Conclusions Creep and temperature can give rise to significant redistribution of stress in statically indeterminate structures. These stress changes can affect the long-term serviceability of a structure and should be allowed for in design. Cyclic variations of temperature will normally produce short-term variations to the stresses and displacements, coupled with a long-term ratcheting effect whereby stresses undergo redistribution and displacements accumulate in a preferred direction. As structural designs become more slender creep effects will take on greater dominance in design calculations. Simple creep theories, as described here, are available to predict structural behaviour, for classes of problem where temperatures and loads are either long sustained or vary in time in a repeating manner. The long-term states of stress, caused by creep, may be determined from single step calculations based on analogous elastic analyses; for problems of unifonn and nonuniform temperatures which may be sustained or vary cyclically in time.
References 1
Hannah,
I. W. Nucl. Eng. February
1961
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
structures
- reality
and prediction:
G. L. England
Ross, A. D., England, G. L. and Suan, R. H. Concrete Rex 1965, 17, (52), 117 England, G. L. JBCSA Co12f::Recent advances in stress analysis - new concepts and their practical application. London, March 1968 England, G. L. Some aspects of creep in concrete. Discussion contribution to meeting for discussion: Creep of Engineering Materials and of the Earth. The Royal Society, London, February 1977. England, G. L. and Macleod, J. S. In: Proc. Conf Appl. Numer. Modelling, Madrid, September 1978 England, G. L. and Macleod, J. S. In: Proc. ConfI Strucr. Anal., Design Cons@. Nucl. Power Plants. Porto Alegrc, Brazil, April 1978 (vol. 2) England, G. L. and Ross, A. D. Msg. Concrete Res. 1962, 14, (40), 5 Suan, R. H. PhD thesis, King’s College, Univ. London, 1964 Stanculescu, G. and Ionescu, M. Review Roumaine des Sciences Techniques 1968,9, (7), 161 Krishnamoorthy, S. PhD thesis, Imperial College, Univ. London, 1969 Clarke, J. L. Oceanol. International ‘78 O.ffshore Strucrures Conference, Brighton, March 1978 England, G. L. and Jordaan, I. J. Msg. Concrete Rex 1975,27, (92), 131 Moharram, A. PhD thesis, King’s College, Univ. London, 1979 England, G. L. Nucl. Eng. Design 1966, 3,54 and 1966, 3, 246 Andrews, K. R. F. et al. Conf.: Environmental Forces Eng. Struct., London, July 1979 Allen, S. J. PhD thesis, King’s College, University of London, 1971 England, G. L. and Allen, S. J. Proc. SMiRTI Conf: Berlin 1971, Paper H 3/8 Jordaan, I. J. In IABSE Seminar: Concrete Structures Subjected to Triaxial Stresses. Bergamo, Italy, May 1974 Jordaan, I. J. et al. Proc. ASCE, 1977, 103 (ST3), 475 CEB-FIP Code. Model code for concrete structures, 1978 Phok, M. MSc thesis, King’s College, Univ. London, 1968 England, G. L. et al. Research Seminar of the Cement and Concrete Association, September 1977, Report No. 48 Lebek, D. E. (Private communication), 1979 England, G. L. et al. BOSS ‘79 Con5 London, August 1979
Appl. Math. Modelling, 1980, Vol 4, August
267
effects in concrete structures - reality and prediction: G. L. England
Creep and temperature effects in concrete structures: reality and prediction G. L. England Department of Civil Engineering, London WC2R ZLS, UK (Received August 19 79)
Introduction The creep of concrete covers large areas of research at the microscopic and sub-microscopic material level and at an engineering level, where the behaviour of structural members and indeed whole structures becomes important from the viewpoint of serviceability and safety. Because creep is temperature dependent, structures subjected to spatial variations of temperature will be particularly sensitive to creep. The creep behaviour of concrete at the macroscopic level, its influence on structural behaviour, and its inclusion into theorems of applied mechanics and numerical analyses, form the basis of this paper. In particular, attention is focused on a continuing theme of research being carried out at King’s College into the development of predictive methods of creep analysis which are simple to apply to any problem whether it be small or large, and economically viable for even the largest of problems. Supporting experimental data are presented and some philosophical aspects of the creep theories are mentioned.
Creep as a material
property
At normal working temperatures creep can be represented satisfactorily by two separate strain components:
King’s College, University of London,
The Strand,
strain becomes fully developed a few days after the concrete reaches its new raised temperature state. Figures I and 2 display the form of the creep strains observed in experiments. Such strain behaviour may be described by compound visco-elastic models of the types shown, provided suitably chosen time- and temperaturedependent spring and dashpot parameters can be assigned to the models. In numerical step-by-step analyses little difficulty is experienced by this type of representation, although the costs of solution may be high. In adopting either analytical or semi-analytical procedures, it becomes convenient as described later, to work in terms of a modified time variable, called pseudo-time, which is linked directly to the irreversible creep strain component.
Creep and the structure In describing the influence of creep on structural behaviour it is convenient to consider the two classes: (i), statically determinate and (ii), statically indeterminate. It is also useful to separate the effects of uniform temperature from those associated with temperature gradients and temperature changes in time.
(i) An irreversible strain of the viscous flow type, for which both age and time under load influence the strain rate under constant applied stress. (ii) A reversible strain component, frequently referred to as delayed elastic strain. The rate of development and recovery are age-dependent, and the magnitude of this component when fully developed is approximately 25% to 30% of the immediate elastic strain. At elevated temperatures,
additional
features are present:
(iii) The viscous flow strain is temperature sensitive in an approximately linear way with respect to temperature changes. (iv) The recoverable strain is fairly insensitive to temperature. (v) An additional irreversible strain component is observed when stressed concrete is heated to a previously unattained value for the first time. This temperature-change creep 0307-904X/80/040261-11/$02.00 0 1980 IPC Business Press
Recoverable
I Figure 7 at constant
I
strains
t
Influence stress
Appl.
t Time of temperature on creep and recovery strains
Math.
Modelling,
1980,
Vol
4, August
261
Creep and temperature effects in concrete structures - reality and prediction: G. L. England prestress are very marked in this example.2 The initial thermal deflection is reduced by approximately 60% after a period of 105 days in the heated condition. Figure 5 illustrates the influence of creep on transverse deflections in a simply supported prestressed beam, in relation to time for which the designed live load is actually applied. Some roadway structures fall into this category. For such structures it may be impossible to predict the timedependent deflection behaviour unless the value of k in Figure 5 is known, at least approximately. The longterm effect will be for the beam to tend to hog under the influence of dead load only, and to sag when the live loads approach their designed values. The actual long-term behaviour will be intermediate between the two limits corresponding to dead load only and dead plus live load for the entire life of the structure. A time-dependent increase of the hogging deflections on undertrafficked prestressed structures of this kind can lead to unpleasant vertical motions for motorists travelling at motorway speeds. The cyclic heating of the upper surface of road beams caused by solar heating acts in combination with creep to partially offset the hogging problem. An experimental programme of testing is currently being carried out at King’s College to determine the long-term influence of cyclically varying temperatures on the behaviour of continuous prestressed bridge structures. This work is being carried out for the Department of Transport. Another example where the ambient environment has been observed to influence structural behaviour through creep, is depicted in Figure 6. A vertical bridge pier has been observed to exhibit a time-dependent motion at its
T
B E(T)
Ratchet
Elastic
I
I
strarn
t
b
Time Figure 2 stress
Influence
of change of temperature
on creep while under
Uniform LoodIng
Stress varlatlons t=O to t-m v
V V L7 0
\
t \
Time
\
‘\ ‘. ~--i._
Figure 3 deflection
d L-1
----___
Influence of temperature and temperature behaviour of simply supported beam
r3 >
rr
crossfall on
Statically determinate structures With the exception of possibly small perturbations to the internal states of stress in statically determinate structures the influence of creep is seen primarily as an increase of the time-dependent deformations of the structure. A uniform increase of temperature increases the rate of these deformations. Temperature gradients further influence these rates and introduce a feature not seen in the analysis of beam and column structures by conventional elastic methods. This feature is a coupling between bending moments and extensions to the centroidal axis, and between axial forces and curvature changes. Figure 3 shows the effect of temperature and temperature gradients on the central deflection of a simply supported prestressed beam. Similar enhanced deflection behaviour has been observed in reinforced concrete beams of gravel and limestone concrete, subjected to temperature crossfalls through their depth.’ Figure 4 shows the effect on deflection behaviour of a temperature crossfall applied through the depth of a uniformly prestressed beam which is free from any superimposed loading. The curvature changes caused by the axial
262
Appl.
Math.
Modelling,
1980,
Vol 4, August
Days after
heating
Figure 4 Influence of axial prestress on deflection behaviour of simply supported beam with temperature crossfall (Taken from Ross et a/. *)
1. k
stresses
Trme
1
I
c
Time k, Increasing Dead+hve
Load, contrnuousx
Figure 5 Influence of cyclic load on deflection ported prestressed beam
of simply sup-
Creep and temperature effects in concrete structures - reality and prediction: G. L. England
T,OSun
_
Long term drift to=th
Brldqe _ pier
bending moments at some sections undergo a change of sign with time. This changing internal stress behaviour has been monitored experimentally in a three-span continuous prestressed beam,22 using a series of elastic inclusions similar to those used by Stanculescu and Ionescu” for ultimate load stress measurements on unreinforced concrete prisms in bending. External redundancies and their variation with time have been monitored in heated prestressed portal frames by Krishnamoorthy’O and for flexurally restrained simulated wall elements of oil storage structures subjected to cyclic variations of temperature by Clarke.” Figures 7-9 show examples of structural behaviour as influenced by spatially varying temperatures and the attendant nonhomogeneous creep.
Time
Theoretical behaviour Figure 6 pierz3
Solar heating effect on deflection
behaviour
of bridge
upper end. The motion was predominantly in a southerly direction, and can be described in terms of temperaturedependent creep and cyclic heating caused by the sun’s radiation. Shrinkage may also play some role in this case. However, based on temperature-dependent creep alone the motion at the upper end of the pier can be described as a cyclically varying displacement pattern caused by thermal and thermo-elastic curvature changes, superimposed on a long-term drift displacement towards the most dominantly heated face of the pier. The effect of creep in this situation is thus to cause a ratcheting behaviour which generates a long-term displacement in a preferred direction, even though the motion in the short term appears to be cyclic and repetitive. This motion is shown diagrammatically in Figure 6. Statically indeterminate
considerations
and prediction
of creep
The manner in which the separate components of creep strain are incorporated into creep analyses can depend upon the problem being solved, since justifiable simplifications are frequently possible. For example, in problems for which the temperatures are sustained - and in some cases of cyclically varying temperatures also - the flow and temperature-change components can be treated together in the analysis. At raised temperatures the recoverable
structures
Creep in statically indeterminate structures is frequently thought to be beneficial in that the additional stressing caused to a structure by in-service settlement of foundations is alleviated by subsequent creep. This tends to restore the structure to a state of stress comparable to the result of a conventional elastic calculation for no foundation displacements. Such behaviour may be observed experimentally for structures which are homogeneous in their elastic and creep properties. Theoretically this behaviour is seen to be a special case of a more general understanding of statically indeterminate structures, for which their behaviour appears to move continually towards a state of lower power dissipation in creep. 3,4 In other cases the initial elastic solution may bear no relation to the long-term creep solution. An interesting example where creep can cause a loss of serviceability is that of a multi-span beam structure, designed initially for perhaps continuous or alternate span loading. If this structure is supported on a creep-elastic foundation medium, it can suffer loss of serviceability and possible damage in the long term if there is a partial unloading, even though no damage is caused in the short term.5 It is possible for warehouse floor slabs to be affected in this way. Experiments conducted at King’s College on statically indeterminate beam structures,2p738 have revealed that the existence of temperature gradients causes severe departures, from elastic theory predictions, of measured external support actions. These variations to external actions infer major stress redistribution within the structure, to the extent that
Days after
heating
I!
2000
II
4000 MlowedGheatlng
Unloading
foliowed
Figure 7 Influence of temperature crossfall on support in two-span prestressed beim (Ross eta/.‘). Comparison steady-state stress theory prediction (England14) Elastic i
6d
mcluslons i
/
Ed
_+_ IR
by coozg
reactions with
t_ IR
9ooc
6d
f 6
state
*‘Preferred’ t
5 IF\ 10
Co-Llnear Aports_
30 50 Days after
)_
Support displacement Imposed
200300 100 ioadlng, Log scale
Figure 8 Influence of nonuniform temperatures and support displacements on three-span prestressed beam (Taken from England et al. ‘7
Appl.
Math.
Modelling,
1980,
Vol
4, August
263
Creep and temperature effects in concrete structures - reality and prediction: L
Thermeelastlc
SE Y
,I---__
6 is
r_
stress representation --I_
-.-q
/
-l__,
1-,’ I ’
?
sol&on
G. L. England
is:
i=N U=
(To + 2
afUi
I
I
20
40
60
80
Time since heating,
100
120
days
_--
3-
Steady
state
predictjan
4-
5-
6; Figure 9 Behaviour of prestressed portal frame subjected uniform heating of beam”
where (~0 and Ui represent a set of stresses which satisfies the statical loading of the structure and a set of self-equilibrating stresses, respectively. ai are time-dependent weighting parameters which provide an appropriate combination of the individual ui stresses in the representation of the total stresses, u. The solution procedure is similar to the flexibility method of elastic analysis and results in a set of first order differential equations in ai. Evaluation of the aj coefficients for selected times allows stress variations with time to be deduced from a direct calculation. When a functional representation is adopted and minimization of this functional is conducted with respect to the stress parameters to determine the aj functions, a physical interpretation may be placed on the quantity being minimized. It is a combination of the rate of change of complementary strain energy in the body plus one-half of the rate at which energy is dissipated from the body in creep deformation. An alternative interpretation is simply a statement of the principle of virtual power, viz:
to non-
P6u dV= 0 I
creep becomes a small and insignificant proportion of the total creep, and being insensitive to temperature, it may without much loss in accuracy be included along with the normal elastic strain. This simplification gives rise to small errors in the short-term immediately following a change of stress, and to insignificant errors at large times. It is therefore reasonable to base an initial theory on the simplified series spring and dashpot combination of the Maxwell model. Experimental creep data may be normalized readily with respect to stress and temperature and thus the spring and dashpot parameters may be defined. Analyses are simplified considerably when the ageing characteristics of concrete creep are removed from the analysis. This is done by adopting the use of a pseudo-time parameter (England and Jordaanlz) which is the normalized creep itself. Conversion to real time is then achieved by referring to the single-valued normalized creep/real time curve for the concrete. Direct solutions are available for problems with the following characteristics: (i) Sustained or cyclically repeating applied loading. (ii) Sustained or cyclically varying temperatures. It is convenient groups.
to divide the creep analyses into two
(i) The transient or time-varying stress portion. (ii) The long-term limiting, or steady-state stress solution, for large times. Time-varying stress solution Using the Maxwell creep formulation in conjunction with a Ritz type of stress representation, equation (l), has enabled a complementary power theory to be developed.3 This permits exact solutions to be derived for problems containing a finite number of redundancies and for approximate solutions to be sought in continuum problems. The
264
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i=l
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Modelling,
1980,
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(2)
when the support displacement rates are zero.* In equation (2) P represents a set of internal strain rates which are everywhere compatible. They may be formulated in terms of the stresses and stress rates from equation (l), via the constitutive creep relationship for the Maxwell material. 60 is any one of the sets of self-equilibrating stresses Ui. Thus, as i ranges from 1 to N, a set 0fN simultaneous differentia1 equations emerges for the evaluation of the ai functions. Figure IO shows an example where this method of analysis has been used to analyse a prestressed concrete reactor containment vessel. Comparisons are made with a time-step method of solution in order to assess the accuracy of this approximate solution, for which N = 4 in equation (I). The same approach may be used to solve for the timevarying stresses in a cyclic temperature problem where temperatures vary between two or more repeating states in time. For problems in which the temperature transients, after each temperature change, are of little significance compared to the hold times under steady-state heat flow conditions, and when the true thermal stresses are included among the ui distributions, the solution proceeds in a similar manner to the sustained temperature case, with known changes to the ai functions at each temperature change. An alternative solution may be developed by formulating the power functional in terms of the specific stress and temperature states in the different parts of the temperature cycles. A weighted average power functional is minimized with respect to the free stress parameters.13 This approach allows the general form of the time-varying stresses - over many cycles - to be predicted with fair accuracy, but precision is lost within each part of the temperature cycle, * In the absence of specified displacement rates this situation can always be achieved simply by redefining the boundary to the body or structure
Creep and temperature effects in concrete structures - reality and prediction: G. L. England since only constant-in-time for these intervals.
averaged values are predicted
Limiting long-temz or steady-state stresses Steady-state stresses can exist whenever loads and temperatures are long sustained and the concrete creep capacity is sufficiently high to not impede the stress redistribution process. A quasi-steady state of stress may also be reached when a structure is subjected to cyclically varying loads or temperatures for long periods of time. When the the structure remains serviceable under these conditions a shakedown limit may be defined, for which stresses repeat at corresponding times in successive cycles of load or temperature. During the periods for which the loads or temperatures are constant in time, the stresses exhibit time-dependent variation. Any change in the cyclic pattern of behaviour will result in stress changes towards new and definable quasi-steady-state limits. Although it is possible to evaluate the steady-state (sustained) and shake-down (cyclic) stresses from a special case of the preceding theory relating to time-varying stresses it is often advantageous to take note of another special feature which relates only to these limiting states of stress. Their calculation is dependent on the power dissipation component only, of the power functional, and thus a choice of solution technique is possible. Either a ‘flexibility’ or ‘stiffness’ type of solution may be employed. The latter approach has certain advantages in large continuum problems. However, whichever method is employed it is possible to use standard elastic analyses for the evaluation of the limiting stresses. This is because useful analogies are
Outer
face IS X-l
Temperature
states
Figure 11 Cylindrical concrete oil containment subjected to repeated filling and emptying of hot oil and cold sea water. Cyclic temperature behaviour varies with depth, z (England et al. 24)
Steady
state
IOOOOdays 8
Inner face 0.
seen to exist between certain quantities in elastic theory and the steady-state stress theories. For the sustained temperature and load cases, an elastic calculation will reveal the steady-state stresses when the normal elastic modulus is replaced by 1/H(T), where N(T) is the normalizing creep-temperature functions.14 For the cyclic temperature problem it is necessary to replace the elastic modulus by the reciprocal of a weighted function of the temperatures for the individual parts of the cycle. When 4(T) = T, this quantity is identified as l/T,, where T,, is the weighted mean temperature (Andrews et al. 15) In addition to this substitution an analogous ‘initial strain’ component is also identified and must be incorporated into the modified elastic analysis in order to evaluate the steady-state-cyclic stresses. Examples of such steady-state stress calculations are shown in Figures 3, 7, 9 and 10, for cases of sustained temperature distributions. Figure I2 shows some results for cyclic variations of temperature, as experienced in a thin-walled oil storage structure subjected to repeated filling and emptying (Figure 11).
Discussion
Figure 10 Comparison of creep predictions for stresses in prestressed nuclear containment vessel. Time-step solution and approximate analysis using four self-equilibrating stress distributions?
The previous sections have indicated that creep and temperature can significantly influence structural behaviour, to the extent that elastic theory calculations may be rendered worthless. The creep theories discussed allow solutions for several classes of problem to be achieved without difficulty from semi-analytical calculations. These will usually be
Appl.
Math.
Modelling,
1980,
Vol
4, August
265
Creep and temperature effects in concrete structures - reality and prediction:
0 Circumferential figure
72
Shakedown
and
0
0017
longltudlnal
of composite
coupled with numerical difference type, for the where spatial subdivision transient creep solution states of approximation
structures
containing
elastic
procedures of the finite element or solution of extensive problems is needed. The accuracy of any will depend therefore upon two in the analysis.
(i) The spatial idealization, both in geometry and functional representation of the stress and displacement fields. (ii) The number and suitability of the self-equilibrating stress distributions of equation (1). The structural idealization of(i) governs the overall accuracy of the solution, and therefore the creep solution will be no more precise than the corresponding initial elastic solution. The number of self-equilibrating stress distributions introduces a second approximation into the creep solution, even though each of these may be generated by use of the same elastic procedures and structural idealization used in (i). Experience has indicated that when suitable self-equilibrating stresses can be generated, a creep solution usually requires fewer than ten such distributions to give good time-dependent solutions. Figure 10 shows that when only four distributions are used, the errors are tolerably small even in a zone of the structure where maximum errors are to be expected.6
266
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5
10
I5
20
MN/m2
Scale of stress
stresses
stresses due to creep and cyclic temperatures,
Time-dependent
Figure 13 Examples only components
G. L. England
for containment
of Figure
71
Techniques for deriving suitable self-equilibrating distributions automaticalky have been developed and included in a recently completed three-dimensional creep analysis for the Health and Safety Executive. Refinements to all the previously described analyses, to include the delayed elastic strain component of concrete, are readily possible. l6 The complexity of the problem is increased slightly and a solution is obtained from a set of linear second order differential equations in ai, of equation (l), England and Allen.” Standard eigenvalue analyses may be employed to determine these functions. For this material representation it is important to allow freedom of variation of the two creep strain components, with respect to environmental influences such as temperature and humidity.‘83 l9 The recently revised CEC-FIP CodeZo allows the designer to exploit this freedom in his analysis. Hopefully, other Codes will follow this lead when they are next updated. For structures containing discrete elastic components in addition to a creep phase, there is frequently a transfer of stress to the non-creeping component. The simplest example of this is the reinforced concrete column carrying an axial compressive force. For this structure there is a continual transfer of stress from the concrete section to the steel reinforcement, until theoretically, either all load is transferred to the steel, or the steel yields and thereafter load transfer ceases, even though strains continue. In other cases the long-term solution may be less clear. Figure 13 shows two examples, an elastic inclusion completely surrounded by a creeping matrix and a hollow circular cylinder containing an internal pressure and subjected to some restraint at its outer surface due to a close fitting elastic ring. In the first case load is transferred to the elastic inclusion until a definable limit is reached. In the cylinder problem, however, it is possible - depending upon the geometry, temperature and loading parameters of the problem ~ for the outer ring to undergo a time-dependent motion in the radial direction, which may be either outward or inward.*r
Creep and temperature
effects
A special case exists for which there is no surface displacement rate even though there is creep occurring in the the central zone. The implications are of special interest since it is possible to design a pipe section, carrying externally applied prestress of the wire-winding type, such that there is no loss of prestress with time. These and other special features make it impossible to generalize the creep behaviour of statically indeterminate and composite structures.
in concrete 2 3
4
5 6
Conclusions Creep and temperature can give rise to significant redistribution of stress in statically indeterminate structures. These stress changes can affect the long-term serviceability of a structure and should be allowed for in design. Cyclic variations of temperature will normally produce short-term variations to the stresses and displacements, coupled with a long-term ratcheting effect whereby stresses undergo redistribution and displacements accumulate in a preferred direction. As structural designs become more slender creep effects will take on greater dominance in design calculations. Simple creep theories, as described here, are available to predict structural behaviour, for classes of problem where temperatures and loads are either long sustained or vary in time in a repeating manner. The long-term states of stress, caused by creep, may be determined from single step calculations based on analogous elastic analyses; for problems of unifonn and nonuniform temperatures which may be sustained or vary cyclically in time.
References 1
Hannah,
I. W. Nucl. Eng. February
1961
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
structures
- reality
and prediction:
G. L. England
Ross, A. D., England, G. L. and Suan, R. H. Concrete Rex 1965, 17, (52), 117 England, G. L. JBCSA Co12f::Recent advances in stress analysis - new concepts and their practical application. London, March 1968 England, G. L. Some aspects of creep in concrete. Discussion contribution to meeting for discussion: Creep of Engineering Materials and of the Earth. The Royal Society, London, February 1977. England, G. L. and Macleod, J. S. In: Proc. Conf Appl. Numer. Modelling, Madrid, September 1978 England, G. L. and Macleod, J. S. In: Proc. ConfI Strucr. Anal., Design Cons@. Nucl. Power Plants. Porto Alegrc, Brazil, April 1978 (vol. 2) England, G. L. and Ross, A. D. Msg. Concrete Res. 1962, 14, (40), 5 Suan, R. H. PhD thesis, King’s College, Univ. London, 1964 Stanculescu, G. and Ionescu, M. Review Roumaine des Sciences Techniques 1968,9, (7), 161 Krishnamoorthy, S. PhD thesis, Imperial College, Univ. London, 1969 Clarke, J. L. Oceanol. International ‘78 O.ffshore Strucrures Conference, Brighton, March 1978 England, G. L. and Jordaan, I. J. Msg. Concrete Rex 1975,27, (92), 131 Moharram, A. PhD thesis, King’s College, Univ. London, 1979 England, G. L. Nucl. Eng. Design 1966, 3,54 and 1966, 3, 246 Andrews, K. R. F. et al. Conf.: Environmental Forces Eng. Struct., London, July 1979 Allen, S. J. PhD thesis, King’s College, University of London, 1971 England, G. L. and Allen, S. J. Proc. SMiRTI Conf: Berlin 1971, Paper H 3/8 Jordaan, I. J. In IABSE Seminar: Concrete Structures Subjected to Triaxial Stresses. Bergamo, Italy, May 1974 Jordaan, I. J. et al. Proc. ASCE, 1977, 103 (ST3), 475 CEB-FIP Code. Model code for concrete structures, 1978 Phok, M. MSc thesis, King’s College, Univ. London, 1968 England, G. L. et al. Research Seminar of the Cement and Concrete Association, September 1977, Report No. 48 Lebek, D. E. (Private communication), 1979 England, G. L. et al. BOSS ‘79 Con5 London, August 1979
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