Effects of cracking of concrete
9.9 Influence of cracking on internal forces
Mean crack width (equation 9.29):
EFFECT OF REDUCTION OF THE NON-PRESTRESSED STEEL ON CRACK WIDTH
If in this example the non-prestressed steel area at each face of a wall is reduced from 1000mm2 (1.6 inch2); to 500mm2, what will be the effect on crack width? The reduction in steel area has a significant effect on the crack spacing Sr and strain εs2 in the bottom reinforcement in the fully cracked states; both parameters increase. Thus wm, by equation (9.30), will increase even further, Assuming that sr increases to 250mm and repeating the analysis3 gives εs2=589×10−6 and wm=0.11mm (4.3×10−3inch). When the analysis is repeated once more with the same non-prestressed steel area, the same crack spacing, the same N=240kN (54kip), but with M changing sign to become M=−50kNm (−440kipinch), the results are:
εs2=1370×10−6, wm=0.25mm (10×10−3 inch).
9.8 Time-dependent strain and stress in cracked sections
The analysis presented above applies to reinforced concrete sections with or without prestressing. Four steps of analysis are presented in section 6.5 to give the time-dependent changes in stresses and strains due to creep, shrinkage and relaxation. The flow chart in Figure 6.5 shows how the four steps can be applied for time-dependent analysis of a cracked section. In the chart it is assumed that forces N and M are applied on a section for which the initial stress is known. When these forces produce cracking, the chart indicates additional calculations to be done before proceeding with the same four steps as for a non-cracked section. The only difference is that, with cracking, the cross-section properties exclude concrete in tension.
9.9 Influence of cracking on internal forces
A cracked zone of a reinforced concrete tank wall is less stiff than the part away from the cracks. This causes redistribution of the internal forces. Values of the internal forces in a cracked zone are generally smaller than the values obtained by linear analysis in which cracking is ignored. This is shown by the example of Figure 9.4.
The cylindrical reinforced concrete water tank wall shown is prestressed only in the circumferential direction.
The top edge of the wall is free, while the bottom edge is totally fixed (Figure 9.4a).
The bending moment M and the hoop force diagrams shown in Figure 9.4(b) are due to hydrostatic pressure with the tank full to the top. Results of three analyses are discussed below. The data used in the analyses are:
specific weight of water=9.81kN/m3 (62.4lb/ft3); Ec=32GPa (4600ksi); fct=2.5 MPa (360psi); Poisson’s ratio=1/6; vertical reinforcement ratios, As/d=0.006 and 0.002 at the inner and outer faces, respectively, where As is the steel area per unit length of the parameter and d=0.25m (10 inch); Es=200GPa (29 000ksi). The wall dimensions are shown in Figure 9.4(a).
The curves labelled A in Figure 9.4(b) result from linear analysis, ignoring cracking. At the inner face at the bottom edge, the moment M=62kN (14kip). It can be verified that this moment produces tension at the inner face of the wall greater than the tensile strength of concrete, indicating that cracking occurs.
A non-linear analysis4, which takes into account the effect of the reduction in stiffness due to cracking, gives the results presented by the curves labelled C. Cracking reduces the rigidity over a short distance (0.3m, 1ft)
adjacent to the base, causing the bending moment at the edge to drop to M=42kN (9.4kip); this is accompanied by a small increase in the hoop force in the lower part of the wall.
The curve labelled B is the result of linear analysis that is the same as for curves A, but with the bottom edge hinged instead of fixed. As expected, M at the bottom edge is reduced to zero; at the same time the hoop force increases in the lower part of the wall, compared to curve C. The above comparison indicates that cracking at the bottom causes M and diagrams to lie between the two extremes of bottom-edge rotation prevented or allowed to rotate freely.
The bending moment value at the base in this example depends upon the steel reinforcement ratio at the inner face. Non-linear analysis for the same problem with reinforcement ratio 0.006, 0.010 and 0.014 gives values of M=42, 48 and 52kN (9.4, 10.8 and 11.7kip), respectively.
Figure 9.4 Effects of cracking on internal forces in a
reinforced concrete cylindrical tank wall free at the top edge and encastré at the bottom. The wall is prestressed in the vertical direction only.
Notes
1 Graphs and tables giving the depth of compression zone in a reinforced concrete rectangular or T-shaped section are provided in Ghali, A. and Favre, R. (1994). Concrete Structures: Stresses and Deformations, 2nd edn, E & FN Spon, London.
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 I: General Rules and Rules for Buildings, European Prestandard, ENV 1992–
1:1991E. European Committee for Standardization, Rue de Stassard 36, B-1050 Brussels, Belgium.
American Concrete Institute (1999). Building Code Requirements for Structural Concrete and Commentary, ACI 318–99/ACI31R-99, ACI, P.O. Box 9094, Farmington Hills, Michigan, USA.
3 Analysis is done by computer program CRACK: Ghali, A. and Elbadry, M. (undated). Manual and Computer Program CRACK, Research Report No. CE 85–1, Department of Civil Engineering, University of Calgary, Calgary, Canada T2N 1N4. This program calculates time-dependent stresses and deformations of reinforced concrete members, with or without prestressing.
4 See Elbadry, M. and Ghali, A. (undated). User’s Manual and Computer Program CPF: Cracked Plane Frames in Prestressed Concrete, Research Report No. CE85-2, Department of Civil Engineering, University of Calgary, Calgary, Canada T2N 1N4.
Chapter 10