An algorithm for the automatic design of concrete shell reinforcement

© 2014 IBRACON An algorithm developed for the design of reinforcement in concrete shells is presented in this text. The formulation and theory behind the development is shown, as well as results showing its robustness and capability of application on fairly large-scale structures. The design method is based on the three-layer model for reinforced concrete shell elements. A material model is also proposed in order to improve the numerical stability of the algorithm. Comparisons of single element design show that the modifications made to the material model don’t effect significantly the final results while making for better numerical stability.


Introduction
Shell type structural elements are used to model a great number of reinforced concrete structures, these elements can be found in structures such as nuclear power plants, offshore structures, and tunnel linings.This document focuses on finding the necessary reinforcement for a shell or plate element subjected to membrane forces N x , N y , N xy and flexural forces M x , M y and M xy .In the last decades many researchers (Baumann [1], Brodum-Nielsen [2] and Gupta [3]) have proposed solutions for this type of problem.The basic idea behind these solutions is that the forces and moments are resisted by the resultant tensile forces of the reinforcement and the resultant compressive forces of the concrete blocks.More complex models, which focus on the behavioral analysis of reinforced shell elements, have also been developed (Scordelis [4], Hu [5], Cervera [6], Polak [7], Wang [8], Liu [9], Schulz [10] and Hara [11]).These formulations are fundamental for the development of new design techniques, on the other hand the complexity of the material models and analysis procedures involved in these, make them very difficult to apply in practical design situations.For these situations a simpler procedure is more favorable.Gupta [3] presented a general solution procedure and an automatic solution algorithm based on it and on the CEB [12] formulation was presented by Lourenço [13], these authors proposed a general method of solution, including cases where there is no need for reinforcement.Tomás [14] used optimization techniques to design elements using this formulation.More recently Fall [15] suggested the same algorithm to reinforce tailor-made concrete structures.The algorithm presented here is proposed as an alternative to the one presented by Lourenço [13], it diverges on the algorithm structure and some modifications were made to the material model adopted.

Formulation
This section will discuss the basic theoretical concepts necessary in order to comprehend the proposed algorithm.

Three-layer model
Shells are two-dimensional elements that are subjected to combined membrane and plate load components.Generally the state of internal stresses in the shell can be described in function of eight resultant force components shown in Figure 1.An idealized shell composing of three layers is proposed by CEB [12], in this idealization the outer layers resist the bending moments and membrane forces acting on the shell and the inner layer resists the transverse shear.The membrane forces for the outer layers of the model can be found by applying equilibrium between the shell's membrane forces and bending moments and the membrane forces for the outer layers, as shown in Figure 2. Equations (1) -( 6) are obtained through this equilibrium, in these equations, h (z+) , h (z- ) , h sx(z+) , h sy(z+) , h sx(z-) and h sy(z-) are the indicated dimensions in Figure 3.
(1) As shown in Figure 2, the three-layer model can be thought of as being comprised of two membrane elements that resist the acting forces of the shell.A problem with this idealization is that, for this to be true, the reinforcement must always be in the mid-plane of the membranes, which is often not the case.This issue is overcome by correcting the reinforcement values (4) (5)

Reinforcement for membrane elements
As shown in the previous item, determining the necessary reinforcement for membrane elements is an essential part of the solution for the three-layer model.Reinforcement in membrane elements has been studied by many authors including Baumann [1], Brodum-Nielsen [2] and Gupta [16].The design method proposed by the structural code CEB [12] incorporates the main ideas proposed by these authors.The formulation shown below is derived for a generic membrane element shown in Figure 5, of thickness a.These equations can be applied to both the top (z+) and bottom (z-) membranes.Gupta [16] shows that by using the principal of minimum resistance it's possible obtain the following governing equations for the and membrane forces in order to account for the real reinforcement depth.The real forces acting on the reinforcement can be found by imposing equilibrium between the calculated values at the mid-plane of the membrane and the values at the real position of the reinforcement, as shown in Figure 4, this provides the following equations for the real reinforcement values. ) In the case where there is no membrane reinforcement in one of the layers, it's possible to find a correction value for the membrane forces as shown in Figure 4, based on this we find the following equations.
(9) (10) These equations are an important part of the proposed algorithm because the correction factor ΔN changes the membrane forces in a given layer, making it necessary to find new reinforcement values for this layer.These new values need to be corrected again to account for the real reinforcement position, creating in some cases an iterative process, this will be discussed in a latter part of this text.( The optimal design is obtained for q = 45°, so long as N sx > 0 and N sy > 0, meaning that the reinforcement is needed in both x and y directions.This case will be referred to as Case I, applying the q value above to eq. ( 13)-( 15) yields the following equations. ( Where N sx and N sy are the force by unit length in the reinforcement in the x and y directions, N c is the force per unit length, acting on the concrete parallel to the crack.If eq.( 16) gives a negative value for N sx , setting N sx = 0 to this equation gives the following equations which constitute Case II. ( In a similar form if eq. ( 14) yields a negative value for N sy , its' possible to obtain the following equations for Case III. ( (26) If eq. ( 20) or (23) result in a negative value no reinforcement is needed.This case will be referred to as Case IV.The concrete force N c in this case is the minimum principal force N c2 , also the maximum principal stress N c1 has to be less than or equal to zero.The equations for the principal forces acting on the membrane are stated below. ( The necessary reinforcement area per unit length can be found by dividing the reinforcement forces by the reinforcement tension. An algorithm for the automatic design of concrete shell reinforcement (29) When working with the three-layer model, the solution to the necessary reinforcement problem is finding the thickness of the outer layers.This thickness is found by dividing the compression force acting parallel to the crack direction of the concrete membrane N c by a limit stress, the evaluation of this limit stress will be described in the next item.Eq. ( 31) may be used to find the thickness of the layer. (31)

Material properties
Due to the high complexity of concrete a full description of the material behavior would imply in the use of a great amount of variables.For design purposes CEB [12] suggests values for average concrete strength based on the state of cracking of the structural element.In uncracked zones, the average concrete strength is given by f cd1 in eq.( 32). (32) For concrete subjected to biaxial compression, f cd1 may be increased by the factor K given below, where α = σ 1 /σ 2 and σ 1 and σ 2 are the principal stresses at failure.For cracked zones the average strength is given by f cd2 in eq. ( 34). ( The simplicity of the above model makes it ideal for practical use in structural design.A more complex model that represents the behavior of concrete in a better manner was presented by Vec-chio [17].This model was based on experimental results from reinforced concrete panels.In this model the maximum compressive strength decreases as the maximum tensile strain ε 1 increases, this compression softening equation is shown in eq. ( 35). ( The material model proposed in this paper adapts the above expression to interpolate between the values given by the CEB.In this model as e 1 increases, the compressive strength is reduced from the value given by eq. ( 32) down to a minimum value given by expression (34).When the element is subjected to biaxial compression the increase in the concrete strength given by eq. ( 33) is considered.Equations ( 36)-( 38) give a mathematical representation of the proposed model.In order to use this model, it is necessary to be able to estimate ε 1 at failure for a membrane element.The study of the state of strain in membrane elements had great contributions by Gupta [16], the author presents equations (39) and (40) where the strains in the x and y reinforcements are related to the principal strains, ε 1 and ε 2 , and the crack angle, q.For more information on the development of these equations the reader may refer to Chen [18].
(39) (40) Using these equations it's possible to find ε 1 by estimating values for ε 2 and ε sx or ε sy .Expressions for ε 1 will be developed for the four cases of membrane reinforcement shown in item 2.2.For Case I where the reinforcement is needed in both the directions, setting ε 2 = ε cp (where ε cp is the concrete peak compression strain), ε sx = ε yi (where ε yi is the steel yield strain), q = 45°, and substituting these values in eq.(39) yields eq. ( 41).The same result is obtained using a similar approach with eq. ( 40).
(42) (43) Finally for Case IV ε 1 is set to zero, this is done in order to obtain concrete strength value equal to f cd1 .

Algorithms
The main objective of this procedure is finding the thickness of the outer layers of the three-layer model.In Appendix A three algorithms are presented, Algorithm 1 is the main algorithm and it calls the other two algorithms.Algorithm 1 establishes an initial value for a (z+) and a (z-) , calculates membrane forces acting on the outer layers and uses Algorithm 2 to evaluate the reinforcement forces for the outer membranes.These initial estimates for the reinforcement forces are inputted into Algorithm 3, this procedure reevaluates the membrane forces and reinforcement forces to take into account the difference between the position of the mid-plane of the outer layers and the actual reinforcement position.Using the values of N c(z+) and N c(z-) , new thickness values a (z+) and a (z-) are obtained.This procedure is repeated until the thickness values converge.To better illustrate the algorithms a numerical example is given in Appendix B. A complete iteration for element 3 from Table 2 is shown in the appendix.

Results
A computer routine that implements the algorithm shown in the previous item was developed using the Java programming language.Elements A and B, described below, were processed by Lourenço [13].Comparisons between the results presented by these authors with the results obtained by the algorithm proposed here are shown in Table 1.

Element A Element B
With the computer program developed it is also possible to find reinforcement values for processed finite element models.The program was used to design the reinforcement for a model of a subway station.The contour maps in Figure 6 show the reinforcement results for this model, in order to present numerical results for this model, some elements were chosen (see Figure 7) and the results for these elements are shown in Table 2.

Conclusions
Shell-type elements can be used to model a large number of structures, for a designer, being able to determine the necessary reinforcement and check the concrete tension for these elements is fundamental.The three-layer model is a simplified model that has been adopted by structural codes such as the CEB Model Code 1990 [12].The fundamental concept of this model is that internal forces of two outer membranes resist the shell's active forces.The presented procedure calculates trough an iterative method the thickness of these outer membranes, and therefore the necessary reinforcement.The material model presented incorporates aspects of the CEB model and the compression softening model by Vecchio [17].This was done in order to improve convergence of the algorithm, since the discontinuity introduced by the CEB model when the material goes from an uncracked state to a cracked state caused numerical difficulties.Using the compression softening equation it was possible to introduce some continuity to the material model which resulted in a much more stable behavior.Results on Table 1 show a comparison between this algorithm and the one presented by Lourenço [13].Both results are in equilibrium with the applied forces and the reinforcement values are consistent.For practical use in engineering the two methods yield basically the same results.Overall this algorithm has proven to be reliable and to give consistent results.The material model developed reduced numerical  (cm /m)     Since Since it is necessary to find the correct reinforcement forces for using eq.( 7)-(8).a correct reinforcement force for the (z-) membrane will be calculated as well as a correction to the membrane force for the (z+) membrane.Using eq. ( 11)-( 12

Figure 2 -
Figure 2 -Equilibrium between membrane forces in the outer layers and the shell active forces

Figure 3 -
Figure 3 -Dimensions used in the three-layer model formulation

Figure 4 -
Figure 4 -Equilibrium conditions used to find the correct reinfocement force

Figure 7 -
Figure 7 -Selected elements shown in Table 2 of the (z+) and (z-) layers (Fig.3); f ck = characteristic strength for concrete; f cd1 = average concrete strength for uncracked concrete; f cd2 = average concrete strength for cracked concrete; c f = average concrete strength given by material model; h = shell thickness; ) ( + z h = distance from the shell mid-plane to the (z+) layer mid-plane; ) ( − z h = distance from the shell mid-plane to the (z-) layer mid-plane; h sx(z+) = distance from the shell mid-plane to the x direction (z+) reinforcement;h sy(z+) = distance from the shell mid-plane to the y direction (z+) reinforcement; h sx(z-) = distance from the shell mid-plane to the x direction (z-) reinforcement;h sy(z-) = distance from the shell mid-plane to the y direction (z-) reinforcement;An algorithm for the automatic design of concrete shell reinforcement8.Appendix A -Flowcharts for the algorithmsAlgorithm 1 -Shell reinforcement design algorithm An algorithm for the automatic design of concrete shell reinforcement Algorithm 3 -Algorithm for finding real reinforcement force 9. Appendix B -Numerical example of an iterative step 3rd iteration for Element 3 from Table 2: The previous iteration yielded the following thickness values: 3rd iteration for Element 3 from Table 2: The previous iteration yielded the following thickness values: From eq. (27) Reinforcement is needed in at least one direction.X-Direction: Y-Direction: ): The new forces in the (z+) membranes are: Recalculating the reinforcement for the (z+) membrane with these new values yield: and and Since reinforcement is needed in both directions.and Since reinforcement is needed in the y direction only.Reinforcement is needed in at least one direction.Calculating Reinforcement for the (z+) membrane:Calculating Reinforcement for the (z-) membrane: Calculate Reinforcement considering real depth:

Table 2 -
Numerical results from the finite element model