Example: Assuming the concrete is uncracked, compute the bending stresses in the extreme fibres of the beam for a bending moment of 25 ft- k. the normal weight concrete has an fc of 4000 psi and a modulus of rupture fr = 7.5 ( 1.0 ) = 474 psi. determine cracking moment of the section.
If the t beam is uncracked, calculate the stress in concrete at top and bottom of extreme fibres under a positive bending moment of 80 ft-k. if fc = 3000 psi and normal weight of concrete is used, what is the maximum uniformly distributed load the beam can carry if it is used as a simple beam with 24 ft span without exceeding the modulus of rupture of concrete?
It is permissible to design minimum steel areas for T-beams subjected to negative bending as ifthey were rectangular beams with b = bw in spite of the large area of concrete in the tension flange that could, in fact, sustain a much larger "uncracked" moment. The reason for this appears to havesomething to do with the redundancy of continuous (indeterminate) T-beam floor systems: redistributionof moments from supports to midspan is possible if failure at the negative-moment supportsrenders them incapable of sustaining bending stress, essentially turning the system into a series ofsimply-supported spans with positive moment only. This logic does not apply in the following twosituations. First, for statically determinate T-beams (such as precast cantilevered tees) where redistributionof moments is not possible, the minimum negative steel is calculated based on the flangewidth or twice the web (stem) width, whichever is smaller (see Appendix Table A-5.9d). Second, forany other negative moment where a T-beam cantilever occurs (i.e., where moment redistributioncannot occur), the minimum steel should be increased as it is for determinate T-beams.
The MSJC does not prescribe a method of determining the cracked moment of inertia, Icr. As such, any rational method of determining cracked section properties is permitted. TEK 14-1B, Section Properties of Concrete Masonry Walls (ref. 14), provides typical section properties for various uncracked wall sections. For use in Equations 1 and 2, the cracking moment can be taken as:
Abstract:The safety and reliability of bridges gradually decrease over time under the influence of disadvantageous environmental factors, primarily due to reinforcement corrosion caused by chloride ingress. The traditional lateral load distribution (LLD) theory does not consider the influence of corrosion, which degrades the accuracy of bridge performance and reliability calculation. A time-dependent reliability assessment method for simply supported T-beam bridges is proposed in this paper, which considers the influence of reinforcement corrosion on LLD. Firstly, the steel corrosion process and degree are predicted based on the chloride ingress model, into which the water/cement ratio and concrete strength are innovatively introduced in order to improve the prediction accuracy. Secondly, the effective stiffness calculation method for corroded reinforcement bridges is established with the moment of inertia and section crack condition employed. Thirdly, the modified eccentric compression method is improved by the effective stiffness and iterative algorithm, which is suitable for the LLD calculation of corroded reinforcement bridges. The time-dependent vehicle load effect can be computed combined with the probability distribution of live load. Finally, the time-dependent reliability of the flexural bearing capacity is obtained by the Monte Carlo method and Bayesian theory without prior information. A simply supported bridge with five T-beams is taken as an example for analysis. It is indicated that the results calculated by the traditional reliability method are conservative, which cannot make a true and accurate evaluation. The method proposed in this paper can effectively reduce the assessment error caused by model uncertainty while considering the interaction between reinforcement corrosion and vehicle live load effect.Keywords: reliability; simply supported T-beam bridge; chloride ingress; reinforcement corrosion; load lateral distribution; effective stiffness
Specifically, it can be found in Fig. 12a that the compressive and tensile strain at the post-cracking phase decreases significantly with the increase of reinforcement ratio, which is similar to regular reinforced concrete beams. Moreover, while Fig. 12b shows that fiber slenderness has no significant effect on the strain values at the same load, Fig. 12c indicates that the usage of hooked-end fiber can effectively reduce the compressive and tensile strain of the specimens at the post-cracking phase. This is due to the fact that the effective moment of inertia of the specimen at the crack development stage could be significantly enhanced by the usage of hooked-end fibers.
In addition, Fig. 13 also displays the selected reinforcement strain (500 με, 1000 με, 1500 με and 2000 με) of all specimens. Specifically, it can be found from Fig. 13 that compared with specimen B-S65-16, the corresponding load of B-S65-20 with the same reinforcement strain increased by at least 38%. In another word, the stress of reinforcement, especially at the post-cracking phase, can effectively be reduced by increasing the reinforcement ratio, which is beneficial to control the crack width of the UHPC (NF P18-710 2016b). In addition, Fig. 13 indicates that the usage of hooked-end fiber leads to higher load values at the same strains, and thus reduces the reinforcement strain of the specimens after cracking at the same load. This is because the effective moment of inertia of the specimen at the crack development stage could be effectively improved by the utilization of hooked-end fibers. However, Fig. 13 illustrates that the reinforcement strain in the specimens with various fiber lengths is nearly similar, which demonstrates the fact that the fiber length has little influence on the reinforcement strain of the specimens. This might be attributed to the fact that little difference in UHPC tensile behavior, resulting from the change in fiber slenderness, has no significant effect on the reinforced UHPC behaviors.
Section Designer is an integrated utility, built into SAP2000, CSiBridge, and ETABS, that enables the modeling and analysis of custom cross sections. These section definitions can then be assigned to frame objects. Section Designer is useful for the evaluation of member properties and nonlinear response, including nonlinear hinge and PMM-hinge behavior. Nonstandard or composite sections of arbitrary geometry may be created in Section Designer and then incorporated into a structural model. Sections may include one or more materials and a user-defined rebar layout. Modification factors may be assigned to simulate cracked-section behavior. All sections are assumed to be noncompact. Additional information is available through Context Help.
Ig here is the moment of inertia of the gross concrete section about its neutral axis, yt is the distance from the neutral axis to extreme tension fiber prior to cracking, and Mcr is the cracking moment.
After the section cracks, tension is resisted only by the steel, and the neutral axis shifts to a new position. Within the service load range, the member continues to behave linearly under short-term loading, but the moment of inertia is markedly lower than it was for the section before it cracked. To calculate deflection under short-term loading, ACI 318 employs an effective moment of inertia, Ie, that weights the gross and cracked moments of inertia.
Cracking Moment for Reinforced Concrete Beams calculator uses Cracking moment = (Modulus of Rupture of Concrete*Moment of inertia of the gross concrete section)/(Distance) to calculate the Cracking moment, The Cracking Moment for Reinforced Concrete Beams formula is defined as the moment, which when exceeded causes the cracking of concrete. Cracking moment is denoted by Mcr symbol. How to calculate Cracking Moment for Reinforced Concrete Beams using this online calculator? To use this online calculator for Cracking Moment for Reinforced Concrete Beams, enter Modulus of Rupture of Concrete (fr), Moment of inertia of the gross concrete section (Ig) & Distance (yt) and hit the calculate button. Here is how the Cracking Moment for Reinforced Concrete Beams calculation can be explained with given input values -> 29.70297 = (30000*20)/(20.2).
The crack presence causes nonlinear stress distributions along the sections of a beam, which change the neutral axis of the sections and further affect the beam stiffness. Thus, this paper presents a method for the stiffness estimation of cracked beams based on the stress distributions. First, regions whose stresses are affected by the crack are analyzed, and according to the distance to the crack, different nonlinear stress distributions are modeled for the effect regions. The inertia moments of section are evaluated by substituting these stress distributions into the internal force equilibrium of section. Then the finite-element technique is adopted to estimate the stiffness of the cracked beam. The estimated stiffness is used to predict the displacements of simply supported beams with a crack, and the results show that both static and vibrational displacements are accurately predicted, which indicates that the estimated stiffness is precise enough. Besides, as the section shape of beam is not limited in the process of modeling the stress distributions, the method could be applicable not only to the stiffness estimation of cracked beams with a rectangular section, but also to that of the beams with a T-shaped section if the crack depth ratio is not larger than 0.7.
Various continuous flexibility models were also proposed for modeling cracks to develop structural vibration equations. Christides and Barr  firstly assumed that the stress around a crack decayed exponentially with the distance from the crack and exhibited a stiffness reduction in the region near the crack tip with an exponential variation. However, Sinha et al.  proposed a stiffness reduction with a local effect governed by a triangular variation. Bilello  analyzed the effect of a crack in terms of the ineffective area delimited by a linear reduction of its height, starting from the cracked section. Chondros et al.  modeled a crack as continuous flexibility by using the displacement field in the vicinity of the crack tip, found with fracture mechanics methods. Yang et al.  computed the equivalent bending stiffness by using the strain energy variation around the crack and considered the cracked beam as a continuous system with varying moments of inertia. Abdel Wahab et al.  described the stiffness reduction by using a cosine function with three parameters that were determined by the vibration characteristics extracted from the experimental data. 2b1af7f3a8