Crack Para Formula 1 2014 12 !!LINK!!

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Now consider the case of an ultrasonic pulse f(t) with a duration of 2T as shown in Figure 3. A second identical pulse is added and that the total delay between the two pulses is 2t0 as shown in Figure 4. They are shown separated in time for clarity. It is worth to note that these two pulses may overlap. In this study, the two pulses represent the reflection waves from the tip and mouth of the surface crack as shown in Figure 2.

We obtain the analytic solution of a singular integrodifferential equation of the thermoelastic problem for a three-dimensional body weakened by an elliptic crack. Heat flows of opposite directions act on the crack surfaces. The formula for the stress intensity factor K I on the crack edges is deduced.

Suppose that the beam has cracks at positions xj with the depths of aj, , where (Figure 2). The cracks are modeled as rotational springs with the stiffness kzj calculated by converting formulas [15, 16]. In the combination with equation (7), we have the following compatible relations at the crack positions xj:

The expansive soil is a material of homogeneity that is formed by the mixture of clay minerals such as the montmorillonite, the illite, and other organic materials. Neither the original state nor the laboratory sample preparation could guarantee their homogeneity, and there must be some natural defects in the internal structure of the soil body. With decrease of free water in the soil, the bound water film around the soil particles becomes thinner, the pores become bigger, and further more the matrix suction increases, when it reaches a certain limit value, namely, the maximum tension strain that the soil can withstand, cracks will appear in where the natural defects exist.

In the formula, is an elastic constant associated with . When the unsaturated soil is under the static condition of , the strain occurs in a fixed direction; that means that , , before the cracks appears on the horizontal surface. When they are put into (3), it is seen that

The purpose of this study is to investigate the flexural crack development of high-strength reinforced concrete (HSRC) beams and suggest the design equations of the flexural crack control for HSRC beams. This study conducts two full-size simply-supported beam specimens and seven full-size cantilever beam specimens, and collects the experimental data of twenty full-size simply-supported beams from the past researches. In addition to high-strength reinforced steel bars of specified yielding stresses of 685 and 785 MPa, these specimens are all designed with the high-strength concrete of a specified compressive stress of 70 or 100 MPa. The experimental data is used to verify the application of the flexural crack control equations recommended in ACI 318-14, as reported by AIJ 2010, as reported by JSCE 2007 and as reported by CEB-fib Model Code 2010 on HSRC beam members; then, this study concludes the design equations for the flexural crack control based on ACI 318-14. Additionally, according to the experimental data, to ensure the reparability of an HSRC beam member in a medium-magnitude earthquake, the allowable tensile stress of the main bars can be set at the specified yielding stress of 685 MPa.

Chiu et al. (2014) and (2016) proposed formulas for determining the allowable stresses of stirrup that ensure the serviceability and reparability of HSRC beam members. However, with respect to controlling flexural cracks of HSRC beam specimens, the development of such cracks must be investigated by performing full-scale experiments. Therefore, in this work, two four-point loading simply-supported beam tests and seven cantilever beam tests are performed, and 20 four-point loading simply-supported beam tests that were performed previously are considered. All specimens herein include a high-strength main reinforcement (with a specified yielding strength of 685 MPa) with high-strength stirrups (with a specified yielding strength of 785 MPa), and the specified compressive strengths of the concrete that is used herein are 70 and 100 MPa. The purpose of this work is to investigate flexural crack control with a view to ensure the serviceability and reparability of HSRC beam members.

Since ACI 318 assumes that \\( d_{1}^{*} \\) exceeds \\( d_{2}^{*} \\), Eqs. (6) and (8) can be substituted into Eq. (5); then, Eq. (9) can be obtained to estimate the maximum flexural crack width. On the basis of Eqs. (9), (10) provides the original formula of the tensile reinforcement spacing, which can be used to control the flexural crack width. If the allowable maximum flexural crack width is set to 0.40 and 0.50 mm under the various tensile stresses of the reinforcement (0.4f y and 0.6f y ), respectively, the corresponding curves for the limiting value of the reinforcement spacing considering can be obtained as shown in Fig. 4. For convenience, the distance from the center of the reinforcement to the concrete surface is replaced by the thickness of the concrete cover of the reinforcement C c ; then, the related design formulas of ACI 318-02 (2002) are obtained as Eqs. (11) and (12). In ACI 318-05, Eqs. (11) and (12) are replaced by Eqs. (13) and (14). If the reinforcement spacing is set equal to the limiting values of Eqs. (11) and (12) in ACI 318-05, then the maximum flexural crack width can be controlled within the range from 0.40 to 0.50 mm. ACI 318-14 also uses the design equations that were recommended by ACI 318-05 for flexural crack width control.

Since the strength of bonding between the concrete and reinforcement influences the flexural crack width, JSCE (2007) uses parameters k 1 and k 2 to account for the effects of the surface geometry of the reinforcement and the strength of concrete on the bonding strength. The parameter k 1 is set at 1.0 for deformed bars and 1.3 for non-deformed bars or prestressed bars. Equation (23) for parameter k 2 indicates that stronger concrete is associated with a smaller maximum flexural crack width; however, k 2 cannot be less than 0.9. Equation (24) for parameter k 3 indicates that more layers of the tensile reinforcement are associated with a smaller maximum flexural crack width.

This section describes the setup for testing HSRC beam specimens. Nine full-size beam specimens are tested to investigate the relationship between flexural crack development and deformation of the member. Some of the experiments that were conducted by Chiu et al. (2014, 2016) are also investigated in this work. All tests were performed at the National Center for Research on Earthquake Engineering, Taiwan (NCREE).

In the symmetric monotonic loading test, the mechanical behavior of the equivalent shear region is similar to that of a beam member with a single curvature. It can also be assumed to be a half of the middle region of a beam member in the antisymmetric loading test based on the moment and shear distribution diagrams. Zakaria et al. (2009) used the symmetric monotonic loading method to investigate the shear crack behavior of RC beams with shear reinforcement. Therefore, Chiu et al. (2014, 2016) adopted the symmetric monotonic loading test to investigate shear crack behavior. For the 20 specimens listed in Table 1, the spacing of flexural cracks in the equivalent moment regions is investigated in this work. Additionally, some flexural-shear cracks in the equivalent shear regions are used to investigate the relationship between crack width and the stress of the reinforcement.

According to the reference in Sect. 2, the thickness of the concrete cover significantly influences the flexural crack width of concrete. Since the specimens that are listed in Table 1 have almost the same thickness as the concrete cover, 40 mm, seven cantilever beam specimens with the thicknesses of concrete cover of 20, 30, 40, and 50 mm, as listed in Table 2, are designed. Figure 6 shows the system for applying a load to the cantilever beam specimens. The loading system of Chiu et al. (2014) is also applied to two simply-supported beam specimens with two thicknesses of concrete cover, 20 and 30 mm. The two simply-supported beam specimens are of the same design and material as those used by Chiu et al. (2014).

Shear crack cracking occurs where the shear stress is at its maximum. Furthermore, the width at the intersection between the shear crack and the stirrup, which includes the shear crack width and the width parallel to the stirrup, is measured.

Figure 15 compares the measured average flexural crack spacing with the values calculated using various specifications (Table 3). According to Fig. 15, the average crack spacings that are calculated using various specifications are more conservative than the experimental values. Furthermore, according to Frosch (1999), ACI 318 requires that larger of the values of \\( d_{1}^{*} \\) and \\( d_{2}^{*} \\) shall be used to calculate the crack spacing (Eq. (6)). Table 4 lists the required parameters in Eq. (6) for each selected specimen. However, since \\( d_{2}^{*} \\) is identified as a main control item in Eq. (9), which is used in ACI 318 from 2002, the values for some specimens fall in the non-conservative region. Therefore, in this work, the average flexural crack spacing is calculated for all specimens on the basis of the larger of \\( d_{1}^{*} \\) and \\( d_{2}^{*} \\), as shown in Fig. 16; then, all of the results are in the conservative region. Unlike those in AIJ (2010) and fib Model Code (2010), the equations that are recommended by ACI 318 and JSCE (2007) for the average flexural crack spacing are convenient for use by engineers or designers to control the width of cracks in HSRC beam members. Additionally, on the basis of the limited experimental data in this work, the average crack spacing can be predicted conservatively only using the concrete cover thickness and tensile reinforcement spacing, as shown in Fig. 16. 1e1e36bf2d