EI is the flexural rigidity of a beam on elastic foundation that is analogous to the cylindrical shell (equation 2.5):
It is interesting to note that, in spite of the eccentricity of the tendon, the major height of the wall is subjected to N, with M close to zero, Owing to hydrostatic pressure, a wall fixed at the base is subjected to a relatively high bending moment at the bottom edge. Eccentricity of vertical prestressing cannot be used to alleviate this bending moment.
6.7 Time-dependent internal forces
The internal forces caused by creep and shrinkage of concrete and relaxation of prestressed reinforcement are considered here for an axisymmetrical shell of revolution, assuming that the structure is made of elastic isotropic material. All loads and prestressing are assumed to be axisymmetrical. Analysis of the effect of creep and
shrinkage is similar to the procedure used to determine the effect of temperature in section 5.7. First assume that creep and shrinkage can occur freely in an elemental shell shown in Figure 5.2(c) and define free generalized strain vector (equation 5.13):
(6.38)
where w and u are, respectively, translations in direction normal to shell surface and along a meridian line (Figure 5.2a); α is the angle defined in Figure 5.2(a); and the subscript ‘free’ refers to the values of w and u when they can occur without restraint. The free strain can be expressed as the sum of creep and shrinkage effects:
{ε}free={ε}creep+{ε}shrinkage, (6.39)
where
(6.40)
with being the vector of generalized instantaneous strain due to loads introduced at t0. and sustained to a later instant t; and is the creep coefficient.
Assuming that the value of the free shrinkage εcs(t,t0) in the period t0 to t is linearly variable through the shell thickness between (εcs)i and (εcs)o at inner and outer faces of the shell, respectively, the generalized strain vector due to free shrinkage is:
(6.41)
A numerical analysis example of the effect of differential shrinkage between the inner face of a water tank wall, in contact with water, and the outer face, exposed to air, is presented in Example 7.1.
The vector of artificial generalized stress that, when gradually introduced in the period t0 to t, would prevent the occurrence of the generalized strain {ε}free is:
(6.42) The subscript r refers to the restraint state; the generalized stress vector is defined by
(6.43)
and is the age-adjusted elasticity matrix related to generalized stress and generalized strain (equation 5.14):
(6.44)
where is the age-adjusted elasticity modulus (equation 6.7):
(6.45)
Equation (6.44) differs from equation (5.14) in that E is replaced by to take into account the effect of creep due to gradually applied stress. The nodal forces that are required at the local co-ordinates of the element in Figure 5.2(a), to maintain equilibrium in the restrained state, are given by equation (5.24). These forces are transformed to statical equivalent element nodal forces in global directions (equation 5.25). The element nodal forces are assembled to give a vector of restraining forces for the structure, which are subsequently applied in reversed direction to eliminate the artificial restraint and produce nodal displacements due to creep and
individual elements due to creep and shrinkage is given by:
(6.46)
where [B] is a matrix of functions expressing the generalized strain at any point within the element when the displacement is unity at coordinate i (Figure 5.2a). The matrix [B] is given by equation (5.16).
Relaxation of prestressed reinforcement can be accounted for in the time-dependent analysis by the application of nodal forces equivalent to the relaxed prestress forces on a structure whose elasticity modulus is The relaxed pre-stress force in a tendon is equal to where Aps is the cross-sectional area of the tendon and is the reduced relaxation value (equation 6.9).
A computer program performing elastic analysis, that can account for the prescribed displacement at any node and the effects of temperature variation, can also be used to determine the time-dependent effects as the sum of results from three computer runs:
(a) In the first computer run, apply the loads introduced at t0 on a structure whose modulus of elasticity is Ec(t0).
(b) The second analysis is for a structure whose modulus of elasticity is subjected to prescribed nodal displacements where {D(t0)} is the vector of instantaneous nodal displacements determined in (a).
The results of the second analysis give {σr} and a vector of restraining nodal forces in global directions due to creep. In the input data for this computer run, the prescribed nodal displacement should be entered with as many significant figures as available (preferably six or more); otherwise, the vector of restraining nodal forces can be significantly erroneous. The vector of restraining forces obtained in the second analysis will be the same as the forces on the nodal lines in the first analysis (including the reactions) multiplied by: Also {σr} obtained in the second analysis will be the same as the stress resultants {σ(t0)} obtained in the first analysis factored by the same multiplier. Thus, by using the results of the first analysis in this way, the second computer run can be omitted.
(c) In the third computer run apply the restraining forces determined in (b) in reversed direction on a structure whose elasticity modulus is In the same run apply nodal forces that are statical equivalents to the time-dependent change in prestress forces. Also enter the free shrinkage values εcs(t,t0) as temperature rise
T=εcs(t,t0)/µ, where µ is the coefficient of thermal expansion, and εcs(t,t0) is the value of free shrinkage, commonly a negative quantity.
The sum of the results of computer runs (b) and (c) gives the time-dependent effects of creep, shrinkage and relaxation. Adding the results of run (a) gives the instantaneous effects combined with the time-dependent effects.
Some comments are needed: If the material of the structure is homogeneous, with constant elasticity modulus Ec(t0) and constant creep coefficient creep increases the deformations by a multiplier but the stresses in the structure remain unchanged. However, creep changes stresses when the structure is composed of parts having different E and/or . This is the case of a circular prestressed concrete wall on elastomeric pads; the pads have different material properties from those of the concrete. A structure cast and prestressed in stages is another example where internal forces develop due to creep; the ages of concrete in the parts, the elasticity moduli and
the creep coefficients are different.
Example 6.3 Time-dependent internal forces in a cylindrical wall on elostomeric pads
A circular-cylindrical wall, supported on elastomeric pads, is subjected at time t0 to post-tensioned
circumferential prestress. It is required to find the radial translation and the shearing force at the bottom edge and the bending moment M in the wall at time t0. and at a later instant t, accounting for the effects of creep and shrinkage of concrete and relaxation of prestressed steel. Assume that the forces on the concrete at time t0 due to circumferential prestressing are equivalent to linearly varying inward pressure whose intensity p is: p1=−7.5 and p=−80.0kN/m2 (−160, −1670 lb/ft2) at the top and bottom edges, respectively; the minus sign indicates inward pressure. The wall dimensions are given in Figure 6.11(a). The following data are given: Ec(t0)=30GPa (4350ksi);
χ(t,t0)=0.8; εcs(t,t0)=−300×10−6; Poisson’s ratio v=1/6.
Assume that the loss of prestress due to creep and shrinkage of concrete and relaxation of prestressed steel during the period t0 to t is 15 per cent of the value at t0. The stiffness of the elastomeric pad is K=2.0MPa (0.29ksi); this is the radial reaction of the pad at the bottom edge of the wall per unit length of the perimeter per unit radial horizontal translation. Ignore creep of the pad.
The problem is solved below by hand calculations using the force method6 of structural analysis and tables of Chapter 11. The same results, presented in Figures 6.11(c) and (d), are determined by computer (finite-element) analyses, as discussed above.
INSTANTANEOUS DEFORMATIONS AND STRESSES
If the bottom edge of the wall is separated from the pad (Figure 6.11a), the wall will be subjected to hoop force pr, without moment, and the radial translation at bottom edge, immediately after prestressing, will be:
Relative radial translation of the wall edge and top of pad due to F1=1 applied on the released structure in Figure 6.11(b) is:
with the coefficient 0.3285. This is read from Table 11.16 for a wall having [l2/(2rh)]=72/{2×40×0.3)=2.04. Thus
The reaction at the bottom edge of the wall at time t0 is:
The radial translation at the bottom edge at time t0=22.25×103/(2× 106)=11.1mm (0.437 inch) (inward). Figure 6.11(c) shows variations of the hoop force and the bending moment in the vertical direction M at time t0
These are determined using coefficients read from Tables 11.8 and 11.9 and superposition of the effect of the force F1=1 at the bottom edge and the effect of prestressing on the wall with both its edges free.
TIME-DEPENDENT CHANGES IN DEFORMATIONS AND STRESSES BETWEEN t0 AND t
Again, if immediately after prestressing the wall is separated from the pad, creep and shrinkage of concrete and the loss of prestresses will, gradually in the period t0 to t, move away from the pad in the radial direction a distance:
The three terms on the right-hand side of this equation represent creep, shrinkage and prestress loss, in this order.
Again the coefficient is 0.3285 (read from Table 11.16). The age-adjusted elasticity modulus of concrete is used in the last term because the prestress loss develops gradually in the period t0 to t.
The age-adjusted modulus is calculated by equation (6.7):
Figure 6.11 Analysis of time-dependent changes in prestressed concrete tank wall (Example 6.3). (a) Tank dimension and distribution of circumferential prestressing. (b) Released structure and co-ordinate system. (c) Hoop force and bending moment at t0. (d) Changes in hoop force and bending moment between time t0 and t.
The term in the square brackets is the radial translation at bottom edge at t0 (=11.1mm). A unit radial force F1=1 gradually introduced in the period t0 to t at the edge of the wall and the top of the pad produces at time t a relative radial displacement:
with the coefficient 0.3285. Again is used in this equation because the force F1=1 is gradually introduced. Thus
The change in reaction at the bottom edge of wall between t0 and t:
Change in radial translation at the bottom edge in the period t0 to t is
∆F1(t, t0)/K=34.60×103/(2×106)=17.3mm (0.681 inch) inward
The changes in the period t0 to t in and M are plotted in Figure 6.11(d). The values in the figure can be verified using Tables 11.8 and 11.9 and by superposition of the effects of an outward radial force of 34.60kN/m) on the bottom edge and an outward pressure whose intensity equals 12 per cent that of the prestress.
Example 6.4 Time-dependent internal forces in a cylindrical tank wall monolithic with base
The tank wall in Figure 6.12(a) is circumferentially prestressed at time t0, while its bottom edge is free to rotate and slide on a footing, At a later time t1, a ring-shaped part of the base is cast to connect the wall to an existing central part of the base, thus making the wall monolithic with the base. It is required to determine the time-dependent changes and M(t2, t1) in the hoop force and the vertical bending moment in the wall in the period t1 to t2, where t2 is an instant much later than t1. The circumferential prestressing at time t0 is equivalent to a radial inward pressure, whose intensity is varying linearly as shown in Figure 6.12(b); the symbol p in this figure is the pressure intensity at the top edge. The tank dimensions are given in Figure 6.12(a) in terms of wall
thickness h The following data are given: Poisson’s ratio=1/6;
days (to be defined below).
Assume that the loss of prestress between time t0. and time t2 is 15 per cent of its value at t0; also assume that half the loss occurs in the period t0 to t1. Conduct the analysis on the idealized structure in Figure 6.12(c). The data are chosen to represent a practical case in which h=0.3m (1ft); p=7.5kN/m2 (160 lb/ft2); t0, t1,t2={28, 100, ∞} days;
the creep coefficients are approximately in accordance with CEB–FIP Model Code MC-907, for concrete strength 30MPa (4400 psi) and 50 per cent relative humidity.
Figure 6.12 Analysis of time-dependent internal forces in a prestressed tank cast in stages (Example 6.4). (a) Tank dimensions. (b) Radial pressure equivalent of circumferential prestressing applied at time t0. (c) Structure idealization. (d) and (e) Changes in hoop force and vertical bending moment M occurring between time t1 of casting the ring-shaped floor and t2=∞.
Use of the idealized structure in Figure 6.12(c) implies ignoring the subgrade reaction on BC and neglecting the in-plane strain in the part of the base between B and the centre. These assumptions simplify the presentation, but may not represent the conditions in practice. While the bottom edge is free to slide, the linearly varying pressure due to prestressing produces no moment, but results in linearly varying hoop force:
where x is the distance from B to any point. Creep and shrinkage of concrete and relaxation of prestressed steel in the period t0 to t reduce the hoop force to 92.5 per cent of its value at t0 thus,
After casting and hardening of part BC of the base, the time-dependent translation and rotation at the bottom edge of the wall are restrained, thus causing changes in and M. The age-adjusted elasticity moduli required for the analysis are (equation 6.45):
The modulus of elasticity of part BC is here considered to affect the analysis when the age of concrete of this part is days; the time-dependent deformation of part AB in the first three days after casting BC is here treated as if it were developed gradually between time (t1+3) and time t2=∞. Two computer runs of finite-element analyses are performed to give the changes in hoop force and meridional bending moment in the structure in the period t1
to t2. The elasticity moduli, the loads and the support conditions in the two runs (a) and (b) are:
(a) The cylindrical wall AB, having elasticity modulus Ec(t0), is subjected to inward radial pressure caused by prestressing at t0 (Figure 6.12b). Only the vertical translation at the bottom edge is prevented.
(b) The continuous structure ABC is analysed with elasticity moduli for Part AB and for part BC.
The vertical translation is prevented at B, while both the vertical and horizontal translations are prevented at C.
The loading is composed of:
• the nodal forces that can prevent displacements in run (a) multiplied by the factor
• 7.5 per cent of the prestress loading in Figure 6.12(b) in reversed direction, representing prestress loss between t1 and t2;
• temperature rise uniform through the thickness, the magnitudes of the rise being εcs(t2, t1)/µ=(−300×10–6)/μ for
AB and for BC.
The time-dependent changes in hoop force and M(t2,t1) in bending moment in the vertical direction are plotted in Figures 6.12(d) and (e). These are equal to the sum of the result of run (b) and result of run (a) multiplied by the same factor (−0.4059).
Notes
1 Graphs for χ(t, t0) are given in Ghali, A. and Favre, R. (1994). Concrete Structures: Stresses and Deformations, 2nd edn, E & FN Spon, London, 444 pp. The graphs are based on the assumption that the time variation of the absolute value of the stress increment has the same shape as the relaxation function for concrete.
2 Comité Euro-International du Béton (CEB)—Fédération Internationale de la Précontrainte (FIP) (1990). Model Code for Concrete Structures (MC-90), CEB, Thomas Telford, London, 1993.
American Concrete Institute (ACI) Committee 209 (1992). Prediction of Creep Shrinkage and Temperature Effects in Concrete Structures, 209R-92, ACI, P.O. Box 9094, Farmington Hills, Michigan, USA, 47 pp.
Eurocode 2 (1991). Design of Concrete Structures, Part 1: General Rules and Rules for Buildings, European Prestandard, ENV 1992–1:1991E. European Committee for Standardization, Rue de Stassard 36, B-1050 Brussels, Belgium.
3 See references mentioned in note 2.
4 An equation and graph for χr are given in Ghali and Favre, ibid.
5 See Ghali and Favre, ibid.
6 See Ghali, A. and Neville, A.M. (1997). Structural Analysis: A Unified Classical and Matrix Approach, 4th edn, E & FN Spon, London, 831 pp.
7 See note 2 of this chapter.
Chapter 7