# CRACK FORMATION AND CRACK PROPAGATION INTO THE COMPRESSION ZONE ON REINFORCED CONCRETE BEAM STRUCTURES

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2 S. Kakay et al. Int. J. Comp. Meth. and Exp. Meas. Vol. 5 No. (017) 117 However an important parameter which controls the strength/resistance of the concrete against crack formation is the water/cement ratio. Technical literature studies and calculations are performed according to Eurocode Design of concrete structures [ 3]. 1 THEORETICAL CALCULATION OF CRACK FORMATION IN A STATICALLY LOADED REINFORCED CONCRETE BEAM From a theoretical point of view it is well known that crack formation is caused by externally applied loads. However there are also several other factors that influence the formation of cracks in concrete. Cracks will be formed when s 1 = f ctk0.05 where: s 1 = principal stress f ctk0.05 = concrete characteristic tensile strength. The state of stress on element 1 of Figs. 1 and is given as: = x τ τ = x 0 y 0 0 x 0 (Assuming that the strain is sufficiently low for the concrete to take tension under the neutral axis). We considered the principal stress given in eqn (1): The Principal stress is calculated from the normal and shear stresses in the x-y plane: 1 = x + y x y ± + τ (1) x x 1 = ± 4 Equation () gives the crack-propagation angle. The crack angle for element 1 will be: τ tan( θ ) = Considering element the state of stress is given as: x y θ = 0 θ = 180 => θ = 90 Whereas = 1 x = 0. () = x τ τ = 0 y τ τ 0 Figure 1: Simply supported reinforced concrete point loaded beam cross section.

3 118 S. Kakay et al. Int. J. Comp. Meth. and Exp. Meas. Vol. 5 No. (017) Figure : Stress distribution for elements 1 and 3. The principal stresses will be: 1 =± τ If we imagine that ( x y) is infinitely small then tan( θ ) = and the crack angle will be θ = 90 => θ = 45 Considering element 3: the state of stresses and the principal stresses are given as: = x τ τ = x y τ 0 τ x x = > 1 = ± + τ 4 Since element 3 is located in the compression zone the normal stress along the x-direction is s x and by sign convention the stress will have a negative value. We assume that t has a large enough value compared to s x. This implies according to eqn () that: x y τ tan( θ ) = = > 1 θ < tan ( ) => θ < 90 => θ < 45 ELEMENT MODELLING AND CRACK PROPAGATION OVER NEUTRAL AXIS The method for estimating crack propagation is based on simple beam theory. Since the concrete is not homogeneous the strength varies from place to place within the beam element. A crack will be formed at the location where crack initiation requires the least amount of energy. This method does not estimate how the crack will start such as FEM software s are capable of. However we could use the formula to estimate the crack propagation angle when we know where the crack is formed. Figure 3 shows the comparison between the calculated and the experimental observation of the crack propagation. Modelling of Element 1: We look at eqn (3) which is derived from the internal moment of resistance of the beam.

4 S. Kakay et al. Int. J. Comp. Meth. and Exp. Meas. Vol. 5 No. (017) 119 Figure 3: Numerical calculation of crack propagation into the compression zone compared with the beam tested in the laboratory. Figure 4: Internal forces in the beam. M z ( y) x = I (3) x = 1 = f ctk 005 I Then M = * f ( y ) ctk 005 The height of the cross section of the beam is defined as h. The distance from the Neutral Axis (N.A) down to the bottom side of the beam will be ( h α. d) (see Fig. 4). α Is = As( 1 α). 1. d z = d α (4) M R 1 b d α... α = 3 * fctk ( ( h α. d)) 005 (5)

5 10 S. Kakay et al. Int. J. Comp. Meth. and Exp. Meas. Vol. 5 No. (017) 1.. b. d 3 1 α. 1. α M = 3 * 0 567* f Rd ( ( h α. d)) ctk 005 Equation (5) describes the external moment that α can be applied to the beam before it will form a crack on the underside of the beam f = ct. f = f td γ ctk 005 ctk 005 α ct = 085 c and γ c = 15 are factors given in the National Application document to EC [3]. Equation (6) is the design moment capacity. Modelling of Element : Vecchio & Collins theory of plates describes shell elements subjected to membrane shear forces. It is well known that concrete subjected to shear force cannot resist compression shear failure. This is when the compression diagonal in the truss-member-model fails. We observe from element that s 1 will represent the tension force and s will represent compression. The combined effect of these forms a crack see Fig.. We have now the basis for an inclined crack if we consider Element. The shear stress can be calculated from eqn (7) as: (6) V τ = = = f bz. 1 ctk 005 (7) V R = b* d( 1 1 * )* f 3 α ctk 005 (8) The formula describes how large is the shear force that the beam can take before an α inclined crack will occur at the support point of the beam f = ct. f = f td γ ctk 005 ctk 005 c (α ct = 085 and γ c = 15 are factors given in the N.A EC-NA [3]). Equation (9) is used to find the applied shear force for a concrete structure: V = b* d( 1 1 * α )* 0 567* f (9) Rd. 3 ctk 005 Modelling of element 3: Element 3 is a little more complex because it varies with x and y. M We insert eqns (10) and (11) P = x η = E E ( x). ρ = A s bd. s cm V y. Q y τ = I s. b α = ( η. ρ) +. η. ρ η. ρ α Is = As( 1 α). 1. d 3 V y P = (10) Q = d d y y y b y + α. ( α. ). First moment of inertia (11)

6 S. Kakay et al. Int. J. Comp. Meth. and Exp. Meas. Vol. 5 No. (017) 11 τ P α. d y. ( α. d y). by+ = I. b Inserting eqns (4) and (1) into eqn (1) and using the crack formation criteria 1 = f ctk005 one can obtain the equation below. From the equation the applied load P is solved using Mathcad the result being given as in eqn (13): s (1) f ctk = P x ( y ) Is + P x ( y ) Is 4 P α d y [ (α d y] b y+ + solve I s b 8 I f P 8 I f α d α d y x y y I f x y ( y α d ) α d α d y x y y I f x y ( y d ) α s ctk s ctk s ctk s ctk 8 I f P ( ) = α d α d y x y y I f x y ( y α d ) s ctk s ctk The crack propagation angle is obtained by inserting eqns () and (1) and the result is given in eqn (14). P α d y ( α ) + d y b y 1 θ = 1 I tan s b P x y I s Equation (14) is used to determine the crack angle under various loadings P in the x-y coordinate system. s 1 and s are dependent on the distance x from the support and the distance y relative to the neutral axis. A crack will theoretically form along a critical path where the first principal stress exceeds the characteristic tensile strength of the concrete i.e. when 1 = f ctk 005. (13) (14)

7 1 S. Kakay et al. Int. J. Comp. Meth. and Exp. Meas. Vol. 5 No. (017) The angle of the crack can be calculated considering the values of t and s 1 along this critical path. x h d L ( α. ) y [ 0 α. d] y = 0 => τ = max x = 0 y = α. d => τ = 0 x = max 3 CRACK FORMATION IN THE COMPRESSIVE ZONE FOR ELEMENT 3 When the concrete beam is at the stage when the concrete cracks in the compressive zone the reinforcement is yielding. This situation leads to the needs for a non-linear stress analysis. Such analysis is beyond the scope of this paper. Our study is based on a linear stress analysis which results in some errors for yielding reinforcement. Using the equations derived above we can estimate the crack path as shown in Table 1 below. E s = MPa f = 500 MPa E yk cm = MPa c = 5 mm f = 31 MPa ctk A h s = 3.p j => A s = 943mm η = 5 18mm 4 The following (Figs 5 and 6) illustrates how to calculate the crack propagation angle and load for given x-y values. The complete calculated results are shown in Table 1. x = h α. d => x = 18 mm x = x+ 10mm.cos( q) => x = 4. 9mm y = 10 0 y = y+ 10mm.sin( q) => y = mm 1 1 The final summary is that the method presented is innovative and it cannot be compared with results of high-tech software. It provides information and understanding why a crack is formed at high loads and how the crack pattern develops. The intention of this paper is to show how a simple method could predict crack propagation angle based on simple beamelement theory. Figure 5: Effect of loading on crack propagatin angle.

8 S. Kakay et al. Int. J. Comp. Meth. and Exp. Meas. Vol. 5 No. (017) 13 Figure 6: Crack propagatoin in x-y axses. Table 1: Calculation of x y q and P according to Formulas given in eqns (13) and (14). Crack nr. x[mm] y[mm] q[deg.] P[kN] CONCLUSIONS The cracking of a structure reduces the performance of the structure. In the worst-case scenario cracking may lead to failure. Furthermore cracks to the depth of the reinforcement lead more easily to corrosion of the reinforcing bars. The mechanism of crack formation is challenging and difficult to fully understand. However this paper investigates the problem using mathematical and experimental methods. The method is based on a study of the crack formation when s 1 exceeds the characteristics tensile strength of the concrete structure. The commonly accepted understanding about concrete structures is that the crack formation angle is 45 in the tensile zone of a supported beam. In this paper the load dependent crack formation is experimentally and theoretically presented. Both show that as the load increases the crack angle decreases. Finally a proposal on how crack is propagating into the compressive zone has been presented. Notation As Min minimum cross sectional area of reinforcement mm E cm Secant modulus of elasticity of concrete I s Second moment of area with respect to reinforcement L Length mm Q y First moment of inertia S Internal forces and moments T Torsional moment Design value of the applied shear force V Ed

9 14 S. Kakay et al. Int. J. Comp. Meth. and Exp. Meas. Vol. 5 No. (017) c d f ck f tk f yk xy z a h r f Concrete cover mm Effective depth of a cross-section mm Characteristic compressive cylinder strength of concrete at 8 days Characteristic tensile strength of reinforcement Characteristic yield strength of reinforcement Coordinate system x and y variables in x and y directions Lever arm of internal forces mm Ratio Ratio Reinforcement ratio for longitudinal reinforcement Diameter of a reinforcing bar mm REFERENCES [1] Beer F.P. Russel J. Jr E. & De Wolf J.T. Mechanics of Materials New York 008. [] Sørensen S.I. Concrete construction calculating and design according Eurocode Trondheim 010. [3] Eurocode : Table 8. of Eurocode : Design of concrete structures; part 1-1: General rules and rules for buildings and Norwegian Application document. Standards Norway Oslo 008.

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