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2444-8656
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The Complexity of Virtual Reality Technology in the Simulation and Modeling of Civil Mathematical Models

Published Online: 15 Jul 2022
Volume & Issue: AHEAD OF PRINT
Page range: -
Received: 08 Feb 2022
Accepted: 10 Apr 2022
Journal Details
License
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Introduction

The bonding interaction between steel and concrete in the civil engineering building structure is the essential mechanical mechanism for resisting external loads together. The components and the structure as a whole give full play to their strength. The research on the bonding and anchoring performance of ordinary steel bar and concrete under static load has been in-depth [1]. After many buildings reach the aging period, the steel bars in the structure or components will rust due to specific environmental impacts. The component or structure is also subjected to dynamic loads and even impact loads, such as a car crashing into a roadside concrete highway barrier. Existing experimental studies have shown a qualitative conclusion that the bond performance of concrete and steel bars will be improved under fast loading conditions. Still, there is no quantitative analysis of the effect of loading rate on bond strength. We will first use mathematical modeling, a powerful tool, to establish a formula describing the effect of loading rate on bond bearing capacity. Combining experiments, the relevant parameters in the formula are determined, and an inference is made about the correlation between the strength of the concrete itself and the influence of the loading rate. The methods and techniques used in this article are essentially the continuous Markov process part of the mathematical random process theory. The basic idea is to treat the physical quantity studied as a random variable or a random process. Combining the physical and mechanical laws of the research object to obtain the required probability characteristic quantities [2]. Due to the randomness of concrete materials in industrial buildings, random mathematical modeling is suitable.

Model building

Assuming that the size of a group of components, the concrete label, the type of steel bar, the degree of corrosion of the steel bar, the thickness of the protective layer, the form of stirrups, the loading rate, and other artificially controllable factors are all the same. Then the bearing of this group of components force's bonding area will certainly not be a constant [3]. The strength of this group can be used as a representative value of the bearing capacity or strength. The rate of change of the bond stress is the rate of change of the average bond stress of the entire bond and anchor section. The viscous stress is the average bond stress of the bond anchoring section. We assume that a random variable is the average bond stress (bond load) that causes the bond area to fail under other conditions. Combining the assumptions of the Markov process to get the density distribution function of this random variable [4]. The density distribution function obtains the expected failure stress.

The basic assumptions of the mathematical model

1) Assume that there is an equivalent bonding area around the steel bars. There is continuous and uniform bond stress along the length of the steel bar on the inner surface of this area. The magnitude of which is τ and the rate of change is τ. Before reaching the bond failure stress τu, the entire bonding area will not be damaged [5]. When the bond failure stress (ultimate load) is reached, the bond area is destroyed, and unloading begins. Where F is the external load, L is the length of the bond zone, and d is the net diameter of the steel bar. τ=Fπdl,τ=Fπdl \tau = {F \over {\pi dl}},\,\tau = {F \over {\pi dl}}

2) Under the same conditions in all aspects that human resources can control, we make the load (bonding stress) for the bond failure of the bond and anchor section to be a random variable rather than a definite value [6]. Theoretically, its distribution interval is the open interval (0, ∞), and the bond strength should be a statistically significant expected value. From the beginning of loading to the beginning of the unloading of the components, when the bonding failure occurs in the real bonding and anchoring section, the whole process is random. Therefore, we use the failure probability Pc(t) of the bonded area or the non-destructive probability Pnc(t) to describe the bonding performance of the anchored section. Obviously Pc(t) + Pnc(t) = 1. Where Pc(t) is the probability of bond failure at loading time t, and Pnc(t) is the probability of no bond failure at loading time t. It is assumed that the bonding performance of the bonding area is a continuous and irreversible Markov process as the loading progresses. Irreversibility refers to the irreversibility of state transition. As the loading progresses, the state transition of the bonding area always develops from non-destructive to destructive. In this process, there is no transition from the destroyed state to the nondestructive state. Because from the material point of view, the destruction of the bonding area is a process of continuous accumulation of local concrete damage, and the destruction of concrete is brittle and irreversible.

3) The space of the state probability of the bonding area at the beginning of loading is shown in equation (2). Among them, 1 is the probability of undamaged, and 0 is the probability of damage. Assume that the probability of transition from the entire state to the damaged state in a short period dt at time t is Ptr(t). The Markov state transition probability matrix in the process is equation (3) p(0)=[10] p\left( 0 \right) = \left[ {1\,\,0} \right] P=[1ptr(t)Ptr(t)01] P = \left[ {\matrix{ {1 - {p_{tr}}\left( t \right)} & {{P_{tr}}\left( t \right)} \cr 0 & 1 \cr } } \right]

Assuming that the period [0, t] is evenly divided into n divisions, according to the nature of the Markov process, the absolute probability distribution at time t is approximately p(t)[10][1ptr(tn)ptr(tn)01][1ptr(itn)ptr(itn)01][1ptr(t)ptr(t)01] p\left( t \right) \approx \left[ {1\,\,0} \right]\left[ {\matrix{ {1 - {p_{tr}}\left( {{t \over n}} \right)} & {{p_{tr}}\left( {{t \over n}} \right)} \cr 0 & 1 \cr } } \right]\ldots\left[ {\matrix{ {1 - {p_{tr}}\left( {{{it} \over n}} \right)} & {{p_{tr}}\left( {{{it} \over n}} \right)} \cr 0 & 1 \cr } } \right]\ldots\left[ {\matrix{ {1 - {p_{tr}}\left( t \right)} & {{p_{tr}}\left( t \right)} \cr 0 & 1 \cr } } \right]

When the interval of each period tends to zero (i.e. n → ∞), the equation (4) changes from a discrete Markov chain to a continuous Markov process.

4) When formula (4) changes from a discrete Markov chain to a continuous Markov process, it cannot be directly used to perform effective calculations. Now write the probability of the transition from the undamaged state to the damaged state in a small period dt at time t as Ptr(t) as equation (5) ptr(t)=Tr(t)dt {p_{tr}}\left( t \right) = Tr\left( t \right)dt Where Tr(t) is the state transition probability per unit time at time t, so there is p(t+dt)p(t)=[pnc(t),pc(t)][1ptr(t)ptr(t)01][pnc(t),pc(t)][1001] p\left( {t + dt} \right) - p\left( t \right) = \left[ {{p_{nc}}\left( t \right),\,{p_c}\left( t \right)} \right]\left[ {\matrix{ {1 - {p_{tr}}\left( t \right)} & {{p_{tr}}\left( t \right)} \cr 0 & 1 \cr } } \right] - \left[ {{p_{nc}}\left( t \right),\,{p_c}\left( t \right)} \right]\left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right] We take the probability of destruction as the object of investigation, and we can get the differential equation with the probability quantity as the unknown number. dpc(t)=pc(t+dt)pc(t)=pnc(t)Tr(t)dt=(1pc(t))Tr(t)dt d{p_c}\left( t \right) = {p_c}\left( {t + dt} \right) - {p_c}\left( t \right) = {p_{nc}}\left( t \right)Tr\left( t \right)dt = \left( {1 - {p_c}\left( t \right)} \right)Tr\left( t \right)dt

5) According to the brittle damage model, the stress expression of the damage rate is as follows D=(σK)a D = {\left( {{\sigma \over K}} \right)^a}

The above formula is obtained by taking the damage threshold c = 0. Where K, a is a constant related to the material. When a is forced to 1 instead of a variable parameter, it becomes a linear damage model expressed in stress instead of a non-linear damage model in the general sense [7]. Among them, K is also called damage modulus, and its unit is the same as the unit of stress and the unit of elastic modulus. We assume that concrete material, steel reinforcement, pouring and curing, and loading rate are all determined and constant. Tr(t) The fundamental reason for the change in the loading time is only related to the increasing load or stress causing material damage and accelerating the deterioration of material properties. We believe that the conditional probability of transition from the state of non-destructive bonding to the state of bonding failure at time t is proportional to Tr(t) and t. Because the greater the damage rate, the greater the probability of damage transition, so Tr(t) can be expressed as equation (9) Tr(t)=cτ(t)¯a Tr\left( t \right) = {\overline {c\tau \left( t \right)} ^a}

6) The law of the influence of steel bar corrosion on the bond performance and the law of the influence of the loading rate on the bond performance is independent of each other. The effect of loading rate on bonding performance under different corrosion rates of steel bars obeys the same form of law. We can express the effect of strain rate or loading rate on bond strength in the same form. However, the parameters may vary with the corrosion rate of steel bars.

Load rate influence function

Integrate equation (7). There is an initial condition that the probability of no bond failure in the bonding area when the loading time is zero is 1, and there is τ = τt at the same time, so the failure probability at time t is pc(t)=1exp(cτata+1a+1) {p_c}\left( t \right) = 1 - \exp \left( { - c{{{\tau ^a}{t^{a + 1}}} \over {a + 1}}} \right)

Change the form of expression pc(τ)=1exp(cτa+1(a+1)τ) {p_c}\left( \tau \right) = 1 - \exp \left( { - {{c{\tau ^{a + 1}}} \over {\left( {a + 1} \right)\tau }}} \right)

Equation (11) shows that the failure probability can be expressed as a function of the bond stress. Therefore, the density function of the failure probability expressed by the cohesive force is f(τ)=dpc(τ)dτ=cexp(cτa+1(a+1)τ)τa+1τ f\left( \tau \right) = {{d{p_c}\left( \tau \right)} \over {d\tau }} = c\,\exp \left( { - {{c{\tau ^{a + 1}}} \over {\left( {a + 1} \right)\tau }}} \right){\tau ^{a + 1}}\tau

The fracture bond stress distribution interval is (0 + ∞), so the expected bond failure stress is E(τa)=0τf(τ)dτ E\left( {{\tau _a}} \right) = \int_0^\infty {\tau f\left( \tau \right)d\tau }

The detailed calculation process of the points is as follows E(τa)=0exp(τa+1c(a+1)τ)τa+1τ1dcτ E\left( {{\tau _a}} \right) = \int_0^\infty {\exp \left( { - {{{\tau ^{a + 1}}c} \over {\left( {a + 1} \right)\tau }}} \right){\tau ^{a + 1}}{\tau ^{ - 1}}dc\tau } E(τa)=0exp(τa+1c(a+1)τ)τdτa+1c(a+1)τ E\left( {{\tau _a}} \right) = \int_0^\infty {\exp \left( { - {{{\tau ^{a + 1}}c} \over {\left( {a + 1} \right)\tau }}} \right)\tau d{{{\tau ^{a + 1}}c} \over {\left( {a + 1} \right)\tau }}}

In equation (15), suppose τa+1c(a+1)τ=T {{{\tau ^{a + 1}}c} \over {\left( {a + 1} \right)\tau }} = T

But τ=(1cT(a+1)τ)1a+1 \tau = {\left( {{1 \over c}T\left( {a + 1} \right)\tau } \right)^{{1 \over {a + 1}}}}

Substituting equation (15) can completely separate the stress rate influence from the integral number E(τ)(1c(a+1)τ)1a+10eTT1a+1dT E\left( \tau \right){\left( {{1 \over c}\left( {a + 1} \right)\tau } \right)^{{1 \over {a + 1}}}}\int_0^\infty {{e^{ - T}}{T^{{1 \over {a + 1}}}}dT}

Assuming that the loading rate under static load is τ0c { \cdot \over {\tau {0_c}}} , then according to (18), E0c(τ))(1c(a+1)τ0c)1a+10eTT1a+1dT {E_{0c}}\left. {\left( \tau \right)} \right){\left( {{1 \over c}\left( {a + 1} \right){ \cdot \over {{\tau _{0c}}}}} \right)^{{1 \over {a + 1}}}}\int_0^\infty {{e^{ - T}}{T^{{1 \over {a + 1}}}}dT}

(18) Compared with (19), the bond bearing capacity is affected by the stress rate, and the expression is (20) τc¯τ0c=(τc¯τoc) {{\overline {{\tau _c}} } \over {{\tau _{0c}}}} = \left( {{{\overline {{\tau _c}} } \over {{\tau _{oc}}}}} \right)

Parameter determination

The author of this paper conducted a rapid loading test with multiple rust rates and multiple loading rates in the disaster prevention and load reduction laboratory [8]. The concrete strength of the components is 25Mpa, and the rebar is threaded steel with a diameter of 16mm. The basic form of the component is shown in Figure 1. Assume that the average bond stress model in 1 is shown in Figure 2.

Figure 1

The basic form of the component

Figure 2

Average bond stress model

In the test, the ratio of the measured dynamic and static load peak values of the components with rust rates of 2% and 4%, respectively, and the test fitting curves obtained according to equation (19) are shown in Figs. 3 and 4. It can be seen that the influence of the loading rate on the bond strength can be approximated by the power function of the loading rate. The test data is in good agreement with the fitted curve.

Figure 3

The relationship between dynamic and static strength ratio and loading rate ratio (corrosion rate 2%)

Figure 4

The relationship between dynamic and static strength ratio and loading rate ratio (corrosion rate 4%)

Comparing the above two figures, it can be seen that the influence index b value of the ratio of dynamic and static strength under different rust rates is not much different. The b value with a rust rate of 2% is 0.0136. The b value with a rust rate of 4% is 0.0143. The difference between the two is about 5%. Therefore, it can be concluded that the influence index b of the rust rate on the loading rate is very small in the smaller rust rate range [9]. Then the value of b can be written as a constant. For the member in this test, b is 0.014. Since the b value in formula (20) has nothing to do with the degree of corrosion, the following method can be used to introduce the influence of concrete material strength on the dynamic and static strength ratio with the loading rate.

Inference

Since the damage modulus K and index in (8) are both material-related parameters, K=fca1 K = f_c^{a1} and β=fca2 \beta = f_c^{a2} can be introduced. Both a1 and a2 are unknown constants. The calculation formula (21) describing the influence of loading rate on the strength of concrete materials of the same strength is calculated through the same steps as follows. The second term contains a main research variable-stress rate. The coefficient includes the strength of the concrete material itself. lnτc¯=φ+11+fca2lnτ0c \ln \overline {{\tau _c}} = \varphi + {1 \over {1 + f_c^{a2}}}\ln \,{\tau _{0c}}

Among them φ is the amount that contains the strength of concrete fc. It is a constant for the same kind of concrete. Therefore, the strength ratio under the dynamic and static load of the same strength concrete has different loading rates. The influence of the K value is eliminated by calculation, so the unknown parameter a1 does not require specific consideration. However, the influence of the stress rate ratio on the strength includes the influence of concrete strength 11+fca2 {1 \over {1 + f_c^{a2}}} . In the rust range of this test, the b value in formula (20) can be regarded as irrelevant to the degree of rust, and b can be rewritten as b(fc). Then formula (20) can be re-expressed as τc¯τ0c=(τc¯τoc)11+fca2 {{\overline {{\tau _c}} } \over {{\tau _{0c}}}} = {\left( {{{\overline {{\tau _c}} } \over {{\tau _{oc}}}}} \right)^{{1 \over {1 + f_c^{a2}}}}} Where fc is the standard compressive strength. According to the results of this experiment and analysis, a2 ≈ 1.3 can be obtained. Because of this a2 > 0, it reflects the increase in concrete strength [10]. Under the same dynamic loading rate, the increase in strength decreases instead. This is consistent with Sparkes described in the experimental study that “the decrease in the strength of the concrete aggregate will increase the sensitivity to the loading rate.” This is because the material properties of high-strength concrete are more brittle than low-grade concrete. As the loading speed increases, the increase in fracture energy it can absorb is less than that of ordinary concrete. This also shows that the above numerical simulation method has grasped the stress rate, the material properties of the concrete itself, and the essence of dynamic load to a considerable extent. The method is reasonable.

Conclusion

The concrete itself is random. The failure of the bonded area is a complex physical process. The effect of loading rate on the bearing capacity of the bonded area is a more complex issue. This paper uses the method of random mathematics to establish the overall failure model based on the bearing strength. After mathematical analysis and calculation, it is expressed as a power function of the loading rate ratio. It represents the effect of loading rate on bearing capacity. The curve fitted according to formula (20) better reflects the changing trend of the bearing capacity of the bonding area as the loading rate changes in the test. And the relative influence of concrete strength and loading rate on the bond bearing capacity obtained by this method is completely consistent with the current test results and qualitative conclusions.

Figure 1

The basic form of the component
The basic form of the component

Figure 2

Average bond stress model
Average bond stress model

Figure 3

The relationship between dynamic and static strength ratio and loading rate ratio (corrosion rate 2%)
The relationship between dynamic and static strength ratio and loading rate ratio (corrosion rate 2%)

Figure 4

The relationship between dynamic and static strength ratio and loading rate ratio (corrosion rate 4%)
The relationship between dynamic and static strength ratio and loading rate ratio (corrosion rate 4%)

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