1 Introduction

Various cracks can easily appear in reinforced concrete beam,and in fact,the reinforced concrete beam with harmless cracks is in service state frequently. However,initiations and propagations of certain type of crack in the reinforced concrete beam may make the beam’s flexibility increase dramatically and render the reinforced concrete beam out of use due to its large deflection. Meanwhile,there also exist harmless cracks in the reinforced concrete beam, which have little influence on the bearing capacity of reinforced concrete beam. Therefore,it is important for the performance analysis and the rehabilitation design of the reinforced concrete beam with cracks. Nowadays,damage detection^{[1, 2, 3]} of the cracks in the beam and the analysis of the static/dynamic performances^{[4, 5, 6, 7, 8]} of the cracked beam have attracted much attention of the researchers and engineers in civil engineering and mechanical engineering fields,and a lot of research achievements have been made in the last decades.

The discrete spring model of the crack,which neglects the initiation and propagation of the crack,is usually used in the analysis of the global mechanical performance of the cracked beam,in which the crack in the beam is equivalent to a linear or nonlinear internal rotational spring. At present,the linear open crack model,in which it is assumed that the crack is always open,and the rigidity of the rotational spring is independent of the couple,has been applied extensively in studying the static deformation^{[6, 9, 10, 11]},stability analysis^{[12, 13, 14]},and dynamic characteristics^{[3, 14, 15, 16]} of the cracked beam. However,the open crack model can only be used in the U-type or V-type crack with the relative large crack gap,in which it is assumed that the crack cannot be closed during the deformation. Nevertheless,for the U-type or V-type crack with relative small crack gap,or for the capillary crack,the crack can easily be closed during the beam deformation. In such a case,the open crack model is no longer applicable, and the breathing crack model or the switching crack model should be used. For the breathing crack model^{[7, 8, 17, 18]},it is assumed that the transition between the open and closed states of the crack is a smooth process. For the switching crack model^{[6, 19, 20]},it is assumed that the crack has only two states (the open state and the closed state),and the crack state transition is abrupt. Although the breathing crack model provides the abundant information on the crack, it is difficult to determine the constitutive relation between its rigidity and the crack state. Furthermore,the final state of the crack and the ultimate capacity of the beam are related only to the open or closed state of the crack. Therefore,the switching crack model,which not only considers the open and closed states of the crack but also has the advantage of being simpler mathematical treatment and easier physical parametric measurement,has been adopted extensively in the performance analysis of cracked beam.The classical analytical approach for analyzing the bending deformation of the cracked beam with the discrete spring model is based on the partition of the cracked beam into several sub-beams between two consecutive cracks. Then,the general bending of the sub-beam is analyzed,and the bending of the global cracked beam can be obtained by imposing the boundary conditions and the pertinent continuity conditions of the rotational spring between two adjacent sub-beams. With this approach,Challamel and Xiang^[6] investigated the stability of the cracked column with one or two cracks,and the influence of the crack closure on the critical load of the cracked column was examined. Bakhtiari-Nejad et al.^[21] studied the dynamical characteristics of the beam with two cracks,and Khaji et al.^[10] dealt with the dynamical characteristics of the Timoshenko beam with one crack. However,the numbers of the continuity conditions at the crack locations increase rapidly with the crack number. The solution approach becomes tedious,and the computation effort increases. It is difficult to get a simple explicit solution. A completely different strategy is to make use of the so-called Dirac delta function to handle the discontinuities along the beam and to obtain the simpler explicit solution. Buda and Caddemi^[9] and Caddemi and Cali´o^{[12, 15, 16, 22]} introduced the minus Dirac delta function to describe the rigidity damage at the crack of the beam,but this description resulted in a negative rigidity of the beam which is impossible in reality. Therefore,this strategy is imperfect in physics. To deal with this imperfection,Palmeri and Cicirello^[5] and Cicirello and Palmeri^[19] used the positive Dirac delta function to describe the bending flexibility,which avoids the physical imperfection completely.At present,for the switching crack model,the open or closed state of the crack is determined by the positive or negative value of the bending curvature^{[20, 23]} or of the axial strain^{[19, 24]} at the crack location of the beam,and the effect of the crack gap has not been taken into consideration. In this study,the effect of the crack gap on the bending deformation of the cracked beam is investigated with the switching crack model. At first,considering the effect of the crack gap, the equivalent flexural rigidity is derived physically for the beam with the unilateral switching crack by the Dirac delta function. Then,a general closed-form solution for the bending of the Timoshenko beam with an arbitrary number of unilateral switching cracks was derived, and a general computational strategy for determining the undermined constants was presented. Based on the closed-form solution,three examples of the cracked beam bending are explored, and the influence of the beam’s slenderness ratio and the crack’s depth on the crack state andbending performance of the Timoshenko beam is discussed. It is shown from the numerical results that there exists a cusp on the deflection curve and a jump on the rotation angle curve at a crack location due to the existence of crack,and the deflection-load curve of the cracked beam is bilinear. Furthermore,for the open crack,the flexibility of the cracked beam decreases with the increase of the beam’s slenderness ratio and the decrease of the crack depth.

2 Equivalent flexural rigidity of beam with switching crack model

First,the constitutive relation of the unidirectional rotational spring is discussed. As shown in Fig. 1,consider a linkage mechanism composed of two rigid straight rods connected by a unidirectional rotational spring. It is assumed that,when the linkage is subjected to a counterclockwise couple M,the linkage mechanism can undergo an arbitrary rotation angle θ = M/K in the counter-clockwise direction with the rigidity K. However,for the clockwise couple M, its rigidity is still K when the couple M<M₀,but when the couple M ≥ M₀,the rotation angle θ₀ = M₀/K of the rotational spring is independent of the couple M and keeps constant. Therefore,the constitutive relation of the linkage mechanism (the unidirectional rotational spring) can be expressed as

where H(x) is the Heaviside function^{[5, 19]}.

Suppose that there exists an open crack on the bottom surface of a beam at the location x = x₀ (see Fig. 2(a)),and it is assumed that the bending rigidity of the perfect beam is (EI)₀. Now,the effect of the crack with gap is equivalent to a nonlinear unidirectional rotational spring^{[18, 24]} satisfying the constitutive relation (1). It is assumed that the crack is closed after the rotation angle θ reaches θ₀ = M₀/K,where M₀ is the critical bending moment for crack closure. When M >M₀,the crack disappears,and the cracked beam becomes a perfect beam. In accordance with the sign standard of the beam bending model,when the cracked beam is subjected to a couple M at its end,the rotation angle φ of the beam’s cross section can be expressed as

Therefore, where δ(x) is the Dirac delta function^{[5, 20]},defined as δ(x) = H'(x).

Denote the equivalent flexural rigidity of the beam with a unilateral switching crack on the bottom surface of the beam by (EI)_e. Then,from the formula

where

Similarly,for the beam with a unilateral switching crack on the top surface of the beam (see Fig. 2(b)),the equivalent flexural rigidity (EI)_e can also be expressed as (4),but the parameter α takes the form of

3 Bending deformation of Timoshenko beam with unilateral switching crack

Suppose that there exist N unilateral switching cracks on a Timoshenko beam with the flexural rigidity (EI)₀,the shear rigidity (GA)₀,and the length L subjected to a transversal load q(x). Denote the equivalent rigidity of the crack i at the location x = x_i by K_i (i = 1,2,· · · ,N). Then,the equivalent flexural rigidity (EI)_e can be expressed as

If the bending moment of the beam at the crack i is M_i,and the corresponding critical bending moment for the crack closure is M_ij,then it has

Neglecting the effect of the crack on the shear deformation and denoting the deflection and the rotation angle of the cross section of the cracked Timoshenko beam by w(x) and φ(x), respectively,the governing equations are as follows^[25]:

The bending moment M(x) and the shear force F_s(x) on the beam’s cross section are

where κ is the shear correction factor of the Timoshenko beam^[25].

Introduce the following dimensionless variables and parameters:

Then,the dimensionless equivalent flexural rigidity $\overline {{{\left( {EI} \right)}_e}} $ can be expressed as

Denote m_i = M_iL/(EI)₀ and m_i0 = M_i0L/(EI)₀ = k_iθ_i0. The expression (8) is still valid in the dimensionless sense if M_i and M_i0 are replaced by m_i and m_i0,respectively.

The dimensionless governing equations of the cracked Timoshenko beam are

From the governing equations (13),it is easy to obtain

Therefore,it has where C_i (i = 1,2) are undetermined constants,and the functions Q^[i](ξ) (i = 1,2,3,4) are defined as

Denote the Laplace transform of the functions W(ξ) and φ(ξ) by${\bar W}$(s) and ${\bar \phi }$(s),respectively,

With the properties of the function δ(ξ),the Laplace transform of (15) becomes

where C₃ = φ(0),and ${\bar Q}$^[2](s) is the Laplace transform of the function Q^[2](ξ).

Furthermore,from the second equation of (13),we can obtain

where C₄ = W(0),and ${\bar Q}$^[1](s) is the Laplace transform of the function Q^[1](ξ).

With the inverse Laplace transform,it is easy to obtain

With (10),the dimensionless bending moment m(ξ) and the dimensionless shear force f_s(ξ) can be expressed as

Usually,there exist four boundary conditions for a single span beam,which are used to determine the four constants C_i (i = 1,2,3,4). With these boundary conditions,the equation for determining the constants C_i (i = 1,2,3,4) can be expressed as

where A is a 4×4 matrix,C=(C₁,C₂,C₃,C₄)^T is a vector,and b is a known vector determined by the load Q(ξ).

It should be pointed out that the elements in the matrix A are dependent on the constants C_i (i = 1,2,3,4),because the coefficient α_i is dependent on the constants C_i (i = 1,2,3,4),as expressed in (8) and (20)-(22). Therefore,except for several simple cases,it is necessary to use the iteration method to solve (23). Here,given the crack locations ξi,their equivalent rigidity ki,the critical bending moment m_i0 (i = 1,2,· · · ,N),and the external load Q(ξ),the following iteration strategy is put forward:

(i) Given the allowance error ε > 0,suppose that all the cracks are open at the initial states, i.e.,α_i = α_i⁰ = 1 (i = 1,2,· · · ,N),and denote Ω₀ = (α₁⁰ ,α₂⁰,· · · ,α_N⁰).

(ii) Given Ω₁^k = (α^k,α₂^k ,· · · ,α_N^k) (k = 0,1,2,· · · ),solve (23),and obtain the constants C_i^k (i = 1,2,3,4). Then,use (22) and (8) to compute Ω^k+1.

(iii) If ║Ω^k+1 - Ω^k║ < ε,where ║·║ denotes the Euclidean norm of a vector,then the approximate solution of (23) is obtained,and the bending deformation of the cracked Timoshenko beam is obtained,otherwise go to step (ii) for the next iteration.

It is worth mentioning here that the above iteration calculation is unnecessary for the simple case as shown in the example of Subsection 4.1,in which the coefficient α₁ can be determined easily,and the convergence of the above iteration calculations can be achieved within several iterations in the examples of Subsections 4.2 and 4.3 when the allowance error ε = 10^-6.

4 Examples

In this section,the formulae presented in the preceding section will be applied to three cracked Timoshenko beams to illustrate their bending performances in detail. Suppose that the beam has a rectangular cross section. Then,we have^[26]

where ν = 0.3 is Poisson’s ratio of the beam material,and L and h are the length of the beam and the height of beam’s cross section,respectively.

For a crack i with a depth d_i,the rigidity K_i and its dimensionless equivalent rigidity k_i can be expressed as^{[5, 15, 19]}

4.1 Bending of simply-supported Timoshenko beam with crack at its top surface

As shown in Fig. 3,consider a simply-supported Timoshenko beam with a crack subjected to a uniform load q₀. The crack with the depth d₁ is located on the top surface of the beam at the location x = x₁. Denote the dimensionless load and the dimensionless rigidity by Q₀ = q₀L³/(EI)₀ and k₁,respectively.

With the boundary conditions W(0) = W(1) = m(0) = m(1) = 0,we can obtain

Then,the dimensionless deflection,the rotation angle,and the dimensionless bending moment can be expressed as,respectively,

Since the bending moment m₁ at the crack location ξ = ξ₁ is independent of the constants C_i (i = 1,2,3,4),the iteration method presented above is not needed,and the coefficient α₁ can be determined as follows. Suppose that the crack is closed when the rotation angle of the crack reaches θ₁₀. Then,the dimensionless critical couple m₁₀ for the crack closure is m₁₀ = k₁θ₁₀.When m₁ = m₁₀,the crack is closed,and the critical load Q_cr = 2m₁₀/(ξ₁(1 - ξ₁)) for the crack closure can be obtained from (29). Thus,it has α₁ = 1 when Q < Q_cr and α1 = m₁₀/m₁ when Q ≥ Q_cr.

Suppose that the crack is closed when the rotation angle θ₁ = θ₁₀ = 1°,and take the location and depth of the crack to be ξ₁ = 0.5 and d₁/h = 0.5,respectively. The distributions of the deflection W(ξ) and the rotation angle φ(ξ) along the axial line ξ are given in Fig. 4 for different loads Q₀ and beam’s slenderness ratios L/h. It can be seen that the deflection W(ξ) and the rotation angle φ(ξ) increase with the load Q0,and there is a cusp on the deflection curve and a jump on the rotation angle curve at the crack location ξ₁ = 0.5. Furthermore,when the load Q₀ < Q_cr,the jump value of the rotation angle φ at the crack location ξ₁ = 0.5 increases as the load Q₀ increases,while when the load Q₀ ≥ Q_cr,i.e.,the crack is closed,this jump value remains constant and equals the rotation angle θ₁₀. It should be pointed out that the numerical results also show that these properties are also valid for other values of ξ₁ and d₁/h.

The responses of the deflection W₀ = W(0.5) at the mid-span of the simply-supported Timoshenko beam versus the load Q₀ when ξ₁ = 0.5 for different slenderness ratios L/h and crack depths d₁/h are presented in Fig. 5,in which the curve with d₁/h = 0 corresponds to the deflection of the perfect beam without crack. It is revealed that for the simply-supported Timoshenko beam with a crack at its top surface,i.e.,d₁/h ≠ 0,the relation between the deflection W₀ and the load Q₀ is bilinear. When Q₀ < Q_cr,the crack is open,and the flexibility between the deflection W₀ and the load Q₀ is the resultant flexibility of the beam’s flexibility and equivalent rotational spring,which increases with the crack depth d₁/h. When Q₀ ≥ Q_cr, the crack is closed,and the effect of the crack disappears. The flexibility between the deflection W0 and the load Q₀ is one of the perfect beam only. Obviously,the flexibility of the beam with the crack closure is less than that of the cracked beam when the crack is open.

The responses of the deflection W0 versus the load Q₀ when ξ₁ = 0.5 and d₁/h = 0.5 for different slenderness ratios L/h are presented in Fig. 6 for the Timoshenko beam and the Euler-Bernoulli beam,respectively. Here,the results for the Euler-Bernoulli beam are obtained by letting $\overline {\left( {GA} \right)} $₀→ +∞ in (27). It is shown that the resultant flexibility of the simplysupported beam decreases with the slenderness ratio L/h when the crack is open (Q₀ < Q_cr). Furthermore,when Q₀ ≥ Q_cr,different from the response of the Euler-Bernoulli beam,the flexibility of the simply-supported Timoshenko beam does not remain constant and decreases with the slenderness ratio L/h due to the effect of shear deformation.

4.2 Bending of cantilever Timoshenko beam with two cracks

As shown in Fig. 7,consider a cantilever Timoshenko beam with a bottom surface crack I at x = x₁ with the depth d₀ and a top surface crack II at x = x₂ with the depth d₀. The cantilever beam is subjected to a force P0 and a couple T₀ at its free end. Denote the dimensionless force and couple by P = P₀L²/(EI)₀ and T = T₀L/(EI)₀,respectively.

Therefore,it has Q^[i](ξ) = 0(i = 1,2,3,4). With the boundary conditions W(0) = φ(0) = 0, f_s(1) = P,and m(1) = T,it gets C₁ = -P,C2 = P - T,and C₃ = C₄ = 0. The bending moments m₁ = m(ξ1) at the location ξ = ξ₁ and m₂ = m(ξ2) at the location ξ = ξ₂ are

Obviously,the crack I at ξ = ξ₁ is closed when m₁ = -m₁₀,and the crack II at ξ = ξ₂ is closed when m₂ = m₂₀. The two conditions give the relations

which divide the force-couple plane OPT into four portions Ω₁,Ω₂,Ω₃,and Ω₄,as shown in Fig. 8. It is easy to prove that when the point (P,T) ∈ Ω₁,the cracks I and II are all open; when (P,T) ∈ Ω₂,the crack I is closed,and the crack II is open; when (P,T) ∈ Ω₃,the cracks I and II are all closed; and when (P,T) ∈ Ω₄,the crack I is open,and the crack II is closed.

Take ξ₁ = 0.25,ξ₂ = 0.75,θ₁₀ = θ₂₀ = 0.5°,d = d₁ = d₂,and d/h = 0.5. Then,it has m₁₀ = m₂₀ = m₀. Letting T = P/2 and the allowance error ε = 10^-6,the iteration numbers for convergence of the preceding iteration processes are less than 6 for different loads (P,T) and beam’s slenderness ratios L/h,and the corresponding distributions of deflection W(ξ) and rotation angle φ(ξ) along the axial line ξ are given in Fig. 9 for different loads (P,T) and beam’s slenderness ratios L/h. It can be seen that the deflection W(ξ) and the rotation angle φ(ξ) increase with the load P,and there are two cusps on the deflection curve and two jumps on the rotation angle curve at the crack locations ξ₁ = 0.25 and ξ₂ = 0.75. Furthermore,the cusp phenomenon for the short beam is more evident than that for the slender beam.

4.3 Bending of continuous beam with two cracks

Consider a simply-supported continuous beam with two subspan lengths L1 and L2,subjected to a concentrated force P₀ at the locations x = x₀ (see Fig. 10). There exist two bottom surface cracks with the depth d₁ and d₂ at the locations x = x₁ and x = x₂,respectively. In this case,the load acting on the beam is given by the concentrated force P₀ and the reaction R₁ at the intermediate support,i.e.,

Let L = L₁ + L₂,and define the dimensionless parameters by

Then,we have

With the boundary conditions W(0) = W(1) = m(0) = m(1) = 0 and a constraint condition W(l₁) = 0 at the intermediate support,it gets

where

The bending moment at the crack ξ = ξ_i (i = 1,2) is

Suppose that ξ₀ < ξ₁ < l₀ < ξ2. It is easy to prove m₁ > 0. Thus,the crack at ξ = ξ₁ is always open,and α₁ = 1. Furthermore,we can also obtain

Then,the critical force P_cr for the crack closure at ξ = ξ₂ is

That is,when P < P_cr,the crack at ξ = ξ₂ is open,and α₂ = 1,while when P ≥ P_cr,the crack at ξ = ξ₂ is closed,and

Take ξ₀ = 0.125,ξ₁ = 0.25,ξ₂ = 0.75,l₀ = 0.5,θ₁₀ = θ₂₀ = 1°,d = d₀ = d₀,and d/h = 0.5. Then,it has m₁₀ = m₂₀ = m₀. The distributions of the deflection W(ξ) along the axial line ξ of the Timoshenko and Euler-Bernoulli cracked beams are given in Figs. 11(a) and 11(b), respectively,for different concentrated loads P. It can be seen that,for both the Timoshenko and Euler-Bernoulli beams,there are two cusps on the deflection curve at the crack locations ξ₁ = 0.25 and ξ₂ = 0.75. However,there also exists a cusp on the deflection curve of the Timoshenko beam at the location ξ₁ = 0.125. This is because that the transversal shearing force f_s of the beam is discontinuous at the location ξ₁ = 0.125,and the shearing deformation caused by the shearing force f_s induces the additional deflection in the Timoshenko beam model,while for the Euler-Bernoulli beam,there exists no shearing deformation,and therefore, its deflection curve is smooth at the discontinuous shearing force f_s.

5 Conclusions

This paper offers a new macro-scale switching crack model which takes account of the effect of the crack gap on the bending analysis of the cracked beam. The crack with a gap is represented by means of an equivalent nonlinear internal rotational spring,i.e.,the crack effect is equivalent to a linear rotational spring when the crack is open,while the crack effect disappears when the crack is closed. The equivalent flexural rigidity for bending of the cracked beam is presented by the generalized Dirac delta function. Then,a closed-form general solution for the static bending of the Timoshenko beam with an arbitrary number of cracks is obtained by the Laplace transform,and a computational strategy for determining the unknown constants is devised. The numerical results are presented for three examples,namely,a simply-supported Timoshenko beam with the top surface crack,a cantilever Timoshenko beam with the top and bottom cracks,and a simply-supported two subspan continuous beam with two bottom cracks. The influence of the beam’s slenderness ratio,the crack’s depth,and the load on the crack states and bending performances of the beam is analyzed,and the following conclusions are obtained.

(i) Based on the Dirac delta function,the closed-form solution for bending of the cracked Timoshenko beam is universal,which can be used for analysis of single beam and multisupported beam.

(ii) The convergence of the iteration strategy for determining unknown constants can be achieved in several iterations in the presented numerical examples for the given allowance error.

(iii) There is a cusp on the deflection curve and a jump on the rotation angle curve at every crack location. Furthermore,the jump value of the rotation angle increases with the load when the crack is open,and this jump value remains constant when the crack is closed.(iv) Generally,the relation between the deflection and the load is bilinear. When the crack is open,the flexibility between the deflection and the load increases with the crack depth, while the flexibility increases as the slenderness ratio decreases. When the crack is closed,the flexibility between the deflection and the load is one of the perfect beam only.