Creep and Shrinkage

When calculating the effects of creep and shrinkage on the behavior of concrete structures, three problem areas are of particular importance: the knowledge of all possible effects and thus the practical problems requiring a creep and shrinkage analysis [1].the development of realistic expedients for estimating the coefficient of creep and shrinkage;information on reliable and simple analytical methods to calculate the effects.

In order to facilitate an understanding of the effect that creep and shrinkage have on a given structure, it is useful to break down the stresses and action effects1 occurring in a reinforced concrete structure into the following groups: Load-induced stresses and action effects result from externally applied loads. Their most significant feature is that they are necessary to satisfy conditions of equilibrium. In statically indeterminate structural systems they must, in addition, satisfy conditions of compatibility.The generally accepted designation ”load“ strictly speaking, encompasses only stresses that are caused by loads (such as dead weight or snow) but not those which are caused by forces (such as deceleration or wind pressure). But since the consequences are the same, force induced stresses are included among load ind uced stresses, as well.

Shrinkage is defined as the reduction in volume of an unloaded concrete at constant temperature. Its primary cause is the loss of water during a drying process. The magnitude of this deformation is described by the shrinkage strain ɛ_s The inverse process is called swelling, but it is of little significance in actual practice.

The creep characteristics of construction materials are defined by the creep coefficient ϕ = ɛ_c/ɛ_e (cf., Section 4).

If the creep deformation is expressed in terms of the ratio ϕ = ɛ_c/ɛ_e, i.e., the creep strains under constant stress ɛ_c and the elastic strain ɛ_e,while shrinkage is expressed in terms of the shrinkage strain ɛ_s, the foregoing explanations about the effect of creep and shrinkage under working loads can be summarized as follows. Estimates of these effects are given for greater clarity.

The realization that concrete not only shrinks but also creeps when subjected to sustained stresses is a relatively recent insight. Early in this century, Woolson [1] and others (cf., [2]) had recognized the creep phenomenon in principle and had begun the first investigations. However, it was not until 1930that a systematic study of this property was undertaken through the work of Davis [3] and of Glanville [4]. Thereafter, interest grew very rapidly. Very soon it became barely possible to retain an overview of the innumerable publications on the subject. By 1946, the American Concrete Institute Committee 209 was entrusted with organizing studies on “volume changes and plastic flow in concrete” and to insure a critical review of the findings. In 1958,Wagner [5] published his analysis of most of the studies published up to that time. This provided the first overview. In the more than 20 years since then, countless further test reports have been published. They can be found, for example, in the ACI bibliographies ([6] and [7]). Beyond that, there developed a knowledge of the underlying physical principles which can offer the key to the multiplicity of observed phenomena. It therefore became necessary to assemble and review critically the insights and findings accumulated to date. A notable contribution along these lines was Neville’s book [2]. In the following exposition, there is also a rough overview of the new insights and their effect on the suggestions regarding the evaluation of the magnitude of creep and shrinkage. This includes particular attention to the needs of the engineer faced with practical design problems.

The rate of hydration, and consequently the time-related process of strength development of concrete, depends on the type of cement used. In Fig. 2.1,1 the coefficient $$ {{\beta }_{t}} = {{f'}_{{ct}}}/{{f'}_{{c28}}} $$ for concrete made of normal cement (Type I), of rapid Fig. 2.1,1The relations given in Fig. 2.1 are based on experiments on concrete made with German cements. Figure 2.1. Coefficient $$ {{\beta }_{t}} = {{f'}_{{ct}}}/{{f'}_{{c28}}} $$to describe the effect of concrete age on concrete compressive strength.

As a rule, it is not necessary to know the modulus of elasticity of concrete E_c with precision in order to determine the stresses and action effects caused by loads. It is enough to use the approximations according to Section 3.1.4. However, it is another matter when imposed or internal stresses occur, for instance, in the case of prestressing. This applies particularly when the concrete is strained at a very early age, since the data given in Section 3.1.4 are valid for 28-day-old concrete. In addition, the type of aggregates and the rate of loading often have an effect that can no longer be disregarded, even when considering only the domain of normal concrete. More precise information on E, is also of importance for an estimate of deflections.

Prediction methods are intended to provide design engineers with the means to estimate creep and shrinkage strain for a given concrete rapidly, with sufficient precision and using known parameters.

Like concrete, steel may creep. Since, however, creep in steel is ascribable to other causes than in concrete, e.g., to stress-induced dislocation movements, it becomes noticeable only under very high stresses. Consequently, creep can be completely disregarded in the reinforcing steel used in reinforced concrete construction. In prestressing steel, the manufacturing process has a great influence. In practice, however, we refer to relaxation rather than creep because this behavior is studied experimentally as stress reduction under constant strain. However, in prestressed concrete the stress loss of the prestressing steel can only partially be described by relaxation because of the added effect of concrete creep.

The validity of the data provided for the prediction of creep and shrinkage was investigated by means of measurements made on 15 bridges. The dimensions of these structures and other details can be found in [52]. For brevity, only the average Table 6.1.Comparison of creep and shrinkage measurements on 15 bridges. results are given here. To this effect, those dimensions and environmental conditions that differed only slightly were averaged, and the final strain values extrapolated in accordance with Ross [35] were adjusted to a mean stress of 4.5 N/mm2. (640 psi). The resulting data are given in Table 6.1. They were compared to the values computed using the various methods. Reasonable agreement has been obtained with all methods

Sections 1, 3.3, and 3.4 developed rheological concepts based on the analysis of test results; these concepts point at the possibility that the principles used to predict creep strains can be greatly perfected in the years to come. Since continued development of knowledge in most technological fields is in constant flux, any newly proposed approach—including the one introduced in Section 4.3—can only be an imperfect and temporary solution to existing problems.

In the followingthe Rüsch-Jungwirk method to estimate concrete strains as given in the German prestressed concrete code DIN 4227 is presented in more detail because the examples to demonstrate the effect of creep on structural behavior given in the subsequent chapters are based on this method. However, it should be pointed out that other methods such as the one proposed by ACI committee 209 are equally suited for the numerical methods presented in this book.

In this example, the concrete arch represents the structural member subject to shrinkage and creep. The steel tie which takes up the horizontal thrust, is the elastic coupling element. The force that is induced in it can be increased through prestressing. The prestraining required creates a constraint. The system is indeterminate to the first degree.

A system is internally indeterminate to the first degree only as long as all the reinforcement may be represented by one tendon. For the analysis of prestressing, tension members may be combined into one member only if they are in approximately the same location within the cross section and if they have the same prestress. On the other hand, when studying the effect of a sustained load or a constraint, including shrinkage, neighboring reinforcing bars may be combined into one tendon, regardless of the degree of prestressing.

In the foregoing sections, it was assumed that the reinforcement is not rigid. However, in the analysis of stress redistributions in a composite beam, a considerable role is played not only by the area of the structural steel member, but also by its moment of inertia.

In many cases, particularly when the live-load component is large, top as well as bottom reinforcement must be used to take up varying moments. Then, Eqs. (2.10) and (4.3) are no longer valid. However, they can be used as approximations for the prestress loss in the main tendon when the force in the adjoining tendon is small.

Section 3.2 dealt with the case of an unbonded prestressed beam which is supported externally statically determinate, but which is internally statically indeterminate. Now we will examine the case of a post-tensioned beam under externally statically indeterminate support conditions.

In this section, we will show how one can avoid having to solve the coupled differential equations which normally result for systems with multiple degrees of indeterminacy. The approaches particularly suited for this purpose have been described in Appendix II, Section 7.1. The two examples presented below of externally statically indeterminate cases are intended to illustrate in detail how to proceed.

Most constraints of this type are unintentional, such as the result of non-uniform foundation settlements. They are very markedly reduced by creep. This is desirable in the case of settlement, but it is undesirable in those cases where the constraint was created on purpose in order to improve the distribution of action effects. For this reason, an artificially created constraint is used only in exceptional cases when the supports are near rigid.

Up to now the basis for the solution of these problems has been almost exclusively the engineer’s experience and instinct. This approach is entirely acceptable for normal structures that are divided by joints into sufficiently short segments. In the domain of engineering design, however, there are frequent cases where the validity of such rules of thumb must be checked by a special analysis. The following is intended to show how one proceeds under such circumstances.

For such cases, the Dischinger/Kupfer method is particularly suited. However, this method presupposes a similar development over time for the various creep values. Since there are only small deviations, this simplification has no marked consequences. Only a coupling of very thin structural components with very thick components will lead to more prominent deviations from similarity. We will demonstrate in Section 12 how to proceed in case the deviations are prominent.

A similar problem appeared in Section 11.12. However, for the sake of simplicity, it was assumed that the time-dependent deformations in both spans were not the same, but that their development over time was similar. In the case at hand, this assumption is avoided while everything else remains the same.

An engineer will always try to keep the redistribution of action effects as low as possible if the effects on the structural behavior are adverse. Sections 9.4, 11.1.2, and 12.1, for example, dealt with this redistribution caused by creep in structures consisting of prefabricated elements or of concrete cast in segments. In a two-span continuous girder, the extreme case can occur that the moment under sustained load in the span is $$ {{M}_{d}} = d{{l}^{2}}/8 $$ at time t = 0, and that the center support moment at time t = ∞ has almost the same value (see, also, Fig. 9.22). In the following, we will describe possible ways to avoid such an over-dimensioning.

Above all, deformation calculations serve two purposes. They permit an assessment of the dangers which deformation may cause, and they provide a base for calculating the required camber of the scaffolding. This is shown in Fig. 14.1, where a is counted from the stress-free condition, while the scaffolding has to be raised by u with respect to the required baseline.

In Part B, Section 3.1.8, relations to estimate the modulus of elasticity of concrete are summarized. In the following, the effect of type of aggregate and age at time of load application upon the modulus of elasticity, as well as on the modulus of deformation, will be illustrated.

This method has been described in detail in Part B, Section 4.3. In the following, the choice of parameters taken into account in this prediction method as well as the particular formulation will be explained. Furthermore, the method will be compared with other prediction methods.

Various methods to predict creep coefficients of concrete have been compared with each other and with experimental data. Some of the results of these studies are given in [23J Part (B). They are summarized in the following. The following methods have been analyzed: (1)The method proposed by Bažant-Panula (BP), Part B [34J];(2)The method proposed by CEB-1970 (CEB-70), Part B [44];(3)The method given in the CEB/FIP model code 1978 (CEB-78);(4)The Rüsch – Jungwirth method (RJ), Part B [46];(5)The method proposed by the British Concrete Society (BCS), Part B [47];(6)The ACI 209 method (ACI), Part B [50].

A strictly mathematical formulation of the consequences (see Part A) of true timedependent strains (see Part B) of concrete is difficult. The mathematical effort can grow to such an extent that it becomes unworkable. Just consider the inhomogeneous behavior of reinforced concrete cross sections in a multiple statically indeterminate structure which is erected in segments. Therefore, the analytical methods developed for use in engineering practice always contain more or less rough approximation, either of the mathematical treatment or of the material properties. One of the least accurate approximations, though frequently used in the past is the consideration of creep by reducing the modulus of elasticity with the formula $$ {{E}_{{c\,id}}} = {{E}_{c}}/(1 + \varphi ) $$.

The differential equation solved by Dischinger in 1937 [l] describes the redistribution of action effects and/or stresses developing in a structure as the result of creep and shrinkage, if purely elastic structural members interact with others subject to creep. The coupling of these members can be local (e.g., a concrete arch with a steel tie or unbonded prestressing) or continuous (e.g., reinforced concrete, prestressing with bond, composite girders). Since the use of the equation is not limited to concrete construction, we use the subscripts e for the elastic element (e.g., reinforcement or prestressing steel); and c for the member which is subjected to creep.

If very complex conditions have to be considered for special cases, such as a time-dependent change in the sustained load, the interaction of concretes of different ages at loading, or a systems change during the creep and shrinkage process, the step-by-step method frequently offers the only, if rather laborious, way to solve the problem.

Trost [6], [10] suggested to express the time-dependent strain ɛ_t, which develops under a variable stress, with the aid of a simple algebraic equation: (4.1)$$ {{\varepsilon }_{t}} = \frac{1}{{{{E}_{c}}}}[{{f}_{0}}(1 + {{\varphi }_{t}}) + ({{f}_{t}} - {{f}_{0}})(1 + \rho {{\varphi }_{t}})] + {{\varepsilon }_{{st}}}. $$

Sometimes, it is still common practice to consider the effects of creep by using a reduced modulus $$ {{E}_{{{\text{eff}}}}} = {{E}_{c}}/(1 + \varphi ) $$ to take into account creep effects. Since in this way the permanent deformations are rated equal to the elastic ones, the results were unsatisfactory initially. Bažant [12] suggested an improvement, analogous to Trost, by adapting, through correction factors, the value of the effective modulus E“ to the respective requirements of the relevant differential and/or integral equations.

For multiple statically indeterminate composite problems, the Dischinger approach leads to coupled linear differential equations which can be solved, but only with an extensive effort as Knittel demonstrated with the example of two-strand prestressing [19]. This problem can be avoided by using either an iteration method (see Part C, Section 6.2) or the method of the mean creep-inducing stress shown in Section 3.2.

In the previous sections, we primarily examined the basic approaches and explained their use by means of common problems. In the following, more complicated cases are described.

The creep-induced reduction (relaxation) of stresses caused by imposed constant deformation is a particularly sensitive measure for limiting the range ofapplicability of approximate methods. In Fig. 8.1, the results of six previously described methods are compared. We can see that the method for mean creep inducing stresses is reliable over a wide range when using Eq. (3.15). It is reliable only over a narrow range when using Eq. (3.16).

It is apparent that for simply statically indeterminate systems and small stress changes, all discussed methods can be used. The simple, clear formulas of Dischinger and the method for mean creep-inducing stress recommend themselves.