Thin-Walled Concrete Bridge Piers

Thin-Walled Concrete Bridge Piers
(Local and Overall Stability)

("Roads" Magazine, Bulgaria, issue No.3, 1990)

Marcho Minev & Mario Behar

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Jump to: Page 1 ... Page 2 ... Page 3 ... Page 4 ... Page 5 ... Translation in English

Jump to: Page 1 ... Page 2 ... Page 3 ... Page 4 ... Page 5 ... Translation in English

Thin-wall reinforced concrete bridge piers –
local and overall stability

(Translation from Bulgarian)

Ass. Prof. Martcho Minev, Eng. Mario Behar
         Formulation of the Problem

   In contemporary bridges and viaducts for highways and railways the cost-wise participation of the substructure in the overall cost of the structure is very frequently significant. Known are foreign and local solutions for beam type structures where the substructure participates with up to 80% of the overall cost.
         In conventional solutions, the piers of these structures take the form of significantly elongated prisms or truncated pyramids, with considerably different dimensions of their cross sections in both principal directions.
   In order to reduce the material expenses for this type of bridge piers, while preserving sufficient overall dimensions of their cross sections, the trend is to proceed toward significant reduction in thickness of the walls that make them up.
   Carrying huge longitudinal and transversal forces and moments in their two principal di-rections, these thin-walled structures – bridge piers, more frequently than before require investigation of the local stability of their walls, besides the overall stability studies. Therefore, the topic of the present review are the most general principles that encompass the problem of local stability of thin-walled reinforced concrete piers.

         Review of structures erected up-to-date locally and abroad

         The tall piers are made hollow and thin-walled with the aim to increase their stiffness characteristics [while pushing the structural material away from the centroid]. Depending on the form and cross section, from geometric and structural point of view, the piers can present the following main characteristic features:
         - variable or constant thickness of the walls within the cross section;
         - variable or constant thickness of the walls along the height of the pier, where the variation can be linear, more rarely in steps, or following any other [mathematic] law;
         - constant or variable external dimensions along the height of the pier (i.e. vertical or sloped walls);
         - prestressed piers, or such with regular reinforcement;
         - piers with or without diaphragms, whose introduction (of the diaphragms) is rarely governed by stability requirements. In most of the cases, these are incorporated in the structure following not well substantiated technological requirements.
         Most frequently the ratio “thickness of the wall v. external dimension” (t/b) is within the limits of 1/10 to 1/20; the “slenderness” λ is within the limits of 50 to 120 depending on the support conditions; the concrete primarily used in Bulgaria is of grade B25, abroad – grade B25 to B45 and higher.

         Review of some national norms and rules

         The review of the basic international and national norms and rules with respect to stability problems and the stress-strain diagram of concrete leads to the following conclusions: the differences in approach and criteria are too much; some norms refine the issues related to the physical characteristics of the concrete, relying probably on the notion that the characteristics of the structural material should be known in depth, in order to obtain greater freedom with respect to structural requirements; others take the opposite approach – impose specific, rigorous structural limits, being in the same time less meticulous in clarifying the characteristics of the material; third ones stress their efforts on both factors.
         Basic criterion for the concrete strength in some norms is the resistance of a prismatic concrete sample, while in other norms criterion is the resistance of a cylindrical sample, depending on the accepted method of quality of concrete control.

         Types of local stability failure

         Two types of local stability failure – characteristic to the thin-walled elements, can be distinguished (fig.1): tilting and buckling. What unites them is the way the external load is ap-plied, namely – along the mid-plane of the element. What distinguishes them from each other is the fact that the first phenomenon takes place in bending elements, while the second takes place in elements under compression. The one under consideration in this article is the buckling.
         Further on when we mention “stability failure” we will mean buckling. [In thin walls] its primary characteristic is that it is a wave-like form of stability failure, where the half-wave length is approximately equal to the width of the element, i.e. a/b ≈ 1. Zones of inward and out-ward buckling follow each other separated by zero-curvature sections. Another typical character-istic is that in the case of two adjacent walls, always one is buckled inward and the other – out-ward, which is the result of the tendency of the angle between them to keep its form, being the stiffest part of the cross section.

         Depending on the magnitude and type of deformations assumed admissible from practical point of view [for any given structure], according to [1] there are four consecutive criterions used to determine the critical loads:
         - buckling in the elastic range [of deformations]. In this case it is assumed that the deformations perpendicular to the mid-plane are negligibly small and do not affect the stressed state. The undistorted shape is being used [for design purposes];
         - buckling in the presence of load perpendicular to the mid-plane, or in the presence of initial curvature. In this case the deformations are of such a magnitude, that already cannot be neglected when the stressed state of the plate is being determined. The distorted shape, or the second order theory, is being used. There is already a geometric non-linearity;
         - buckling in the post-critical range. In this case the membrane effect comes in help to the bearing capacity of the element. Thanks to the membrane effect, the thin plate becomes able to sustain greater loads. This phenomenon is widely used in metal structures. The idea herein is to investigate precisely this effect in the reinforced concrete as structural material, by introducing the concept of effective width (bef), described further on;
         - buckling in the plastic range. In this case the difficulties from mathematical point of view become enormous as a result of the introduction of the viscosity conditions in the differential equations. Basic criterion in this approach is the rate of change of the deformations. From the point of view of structural engineering practice this case is still not of interest;

         Types of pier behavior depending on their overall slenderness

         - In the cases of great overall slenderness λ, an overall buckling is observed – with the form of the cross section unchanged;
         - In the middle range of overall slenderness λ, especially in the case of open profiles, the loss of stability consists of torsion without change in the form of the cross section;
         - In the cases of small overall slenderness λ, the loss of local stability (local buckling) takes precedence.

         Solving for stresses and deformations in thin-walled elements

         As a result of the composite character of the reinforced concrete material, these elements are considered as orthotropic. Obviously, this has to be reflected in the mathematical apparatus in use. Based on the laws of physics, the laws of geometry, and the different stiffness characteristics of the material in its different directions, valid is the following relation between deformations and external load:

Eq. (1) … (see the Bulgarian original)
         The general equations for thin plates (thin-walled elements) with significant transverse deformations are as follows:

Eq. (2.1) … (see the Bulgarian original) … (equation of statics)

Eq. (2.2) … (see the Bulgarian original) … (equation of geometry)

where: Dx, H, Dy and D are stiffnesses in the corresponding equations;
           E – the modulus of elasticity of the material;
           w – the transverse deformation (deflection) of the thin plate;
           ψ – the stress function.

         Particular solutions of the problem

         The easiest for solution, and closest to the considered herein structures, is the case with load in one single direction (fig. 3). It is well established that to this form of deformation the best to correspond is the following expression:

Eq. (3) … (see the Bulgarian original)

         It is evident from the equation, that for this deformation a waving is assumed both in the direction of loading, as well as in direction transverse to this of the load. Practically, the smallest critical value of the external load is being reached when the number of transverse waves n = 1. The number of longitudinal waves m depends on the ratio of the sides (edges) of the plate a/b.

         Energy method [2]

         It is based on the requirement to balance the work of the external and internal forces, the last of which equals the potential energy of the deformation.
         In pure bending, i.e. in the case of absence of forces normal to the mid-plane, the deformation energy (Vo) can be expressed as follows:

Eq. (4) … (see the Bulgarian original)
         If there are, though, forces normal to the mid-plane, their contribution has to be added to the deformation energy of the pure bending (equations 4.1 & 4.2). By doing so we get the deformation energy V1 in the unbent element, plus such V2 due to action of the forces in the bent thin plate. The last one can be subdivided to energy that goes for deformation along the mid-plane V2a and energy for deformation in direction perpendicular to said mid-plane V2b:

Eqs. (4.1) & (4.2) … (see the Bulgarian original)

where the first addition represents V2a, and the second – V2b.
         The total deformation energy V is a sum of the three energies Vo, V1 and V2 (equation 5). On the other hand, though, it is equal to the work of external forces A, i.e. V = A. The last, in its turn, can be divided in work of the forces in the mid-plane Ah (equation 6.1), and work that goes to bend the thin-plate Av (equation 6.2). The latter of the last two is the one to be used to obtain the smallest value of pcr.

Eq. (5) … (see the Bulgarian original)

Eqs. (6.1) & (6.2) … (see the Bulgarian original)

         Physical non-linearity

         The physical non-linearity is an inherent property of the concrete as a structural material. Two approaches are involved in the solution of the [stability] problem, while taking into account said property:
         First approach – using the Method of “Anisotropic thin-plate” (non-linearly-elastic or slightly plastic). Introduced is a coefficient of reduction of the stiffness λ, while assuming that the plastic properties of the material reveal themselves in the direction of the main stress, i.e. in the case of σx > σprop. In the y-direction the material remains with elastic properties. According to [3], the relation between stress and deformation can be expressed as follows:

Eq. (7) … (see the Bulgarian original)

where λ < λ' < 1.
         Second approach – using the Theories of plasticity. In this case the above mentioned coefficients λ are being chosen not by intuition, and two theories are being used: 1) the theory of small elasto-plastic deformations, or as it is better known – the Theory of deformations; 2) the theory of visco-elasticity. In the case of the first theory, the total deformations are employed, while for the second, basic criterion is the rate of change of deformations.
         Assumptions for the non-linear theory of thin plates:
         assumption # 1 – the relative deformations, in direction perpendicular to the mid-plane, are equal to zero, i.e. Ez = 0;
         assumption # 2 – the volume deformation e = Ee + Ey + Ez is linearly elastic, i.e.

Eq. (8) … (see the Bulgarian original)

where θ = σx + σy + σz.

         Basic differential equation

         The appearance of plastic deformations is an indication that the elastic ones are already exhausted, i.e. the material has become elastically non-shrinkable.
         In order to obtain the basic differential equation, two approaches are employed:
         Approach # 1 (of Engesser-Karman) – it consists in dividing the thin plate in its buckled state in zones of relaxation and surcharge. It is assumed that, in the first zone the material con-tinues to work with its initial elastic modulus Eo, while in the second zone it works with the tan-gent modulus EH (fig. 4). The basic differential equation for plates made of non-linearly-elastic material (physical non-linearity) in this case is (9) [given below], obtained by Ilyushin (1948) [4], where D is the cylindrical stiffness in case of μ = 0.5 (non-shrinkable material).

Eq. (9) … (see the Bulgarian original)
         Approach # 2 (theory of Shanby) – this approach does not take into account the relaxation effect, i.e. both zones of the cross section are being deformed according to the same modulus – the secant modulus Ee. Particular solutions for thin plates made of non-linearly elastic material using this approach are offered in [3] & [4].

         The Method of Effective Widths as a way to take into account the post-critical strength reserve [5]

         One of the goals of the current investigation is to make clear whether this theory, applied to metal structures, is also applicable to reinforced concrete structures.
         The philosophy behind after-critical work is as follows:
         1. The load limit, under which the thin-plate is being destroyed is times-fold higher than the critical one.
         2. The rate of increasing of the deformations is being reduced when the load increases beyond the critical limit.
         3. The supported edges of the thin-plate, parallel to the direction of the applied load, provide additional stiffness, because they do not have the possibility to deform to the same extent the thin-plate itself does.
         4. There is a stress redistribution.
         5. Balancing membrane forces come into effect.
         6. Until reaching σcr, the thin-plate works with its complete cross section. After reaching and surpassing this limit, it is assumed that the thin-plate starts working post-critically with its effective width bef, with a uniform distribution of the edge stress σe along it (fig. 5).
         The term “relative slenderness” β is being introduced, and “critical relative slenderness” βm respectively. The limit state of the thin-plate is reached, when in the strips bef /2 the stresses reach the yield point.
         The coefficient of resistance against buckling K depends directly on the degree of elastic end-fixing ξ and on the dimensions of the cross section. In relation to the term bef, the terms Fef, Ief, ief, def and overall effective slenderness of the element, are being introduced.

         Practical Experiments on models of Dural & Brass with square cross section [6]

         Figures 6.1, 6.2, 6.3a, 6.3b and 6.3c represent respectively: the theoretic formulation of the problem; the “relative stress – relative strain” diagram; as well as the diagrams of relative stress in the walls of the model. On fig. 6.2 can be seen that until the two-fold surpass of the critical stresses p* there is almost complete correspondence between theoretical and experimental results. The discrepancies occur after this level until reaching the destruction at 3.3-times surpassing of p*. An indication for destruction has been the crumple of the edge-corners in one of the cross-sections.
         From the shown on figs. 6.3 it is evident the almost 5-times surpassing of σx over its criti-cal value at the edges.

         Conclusion

         Based on the above presented achievements of the theory of structures, the engineering methods for investigation of the local stability of thin-walled reinforced concrete bridge piers are being developed.

         REFERENCES (missed in the publication due to a printing error):
         1. Dimitrov, N., Strength studies, Betonkalender, part 1, 1987, Wilhelm Ernst & Son Editorial, Berlin, p. 435.
         2. Timoshenko, S.P., S. Voynovski-Krieger, Thin plates and shells, Editorial “Science”, Moscow, 1966, pp. 380, 426.
         3. Volmir., A., Stability of the deformable systems, Editorial “Science”, Moscow, 1967, pp. 394, 398, 409, 417, 426.
         4. Bijlaard, P.P., Basic considerations related to the Buckling of thin plates and shells in the plastic range, Studies of the Engineering Statics Institute, Zurich, 1948, p. 23.
         5. Prodanov, K., Effective width of centrically compressed thin-plates, Civil Engineering magazine (Bulgaria), 1984, issue No. 4, p. 11.
         6. Reinitzhuber, F., Design of thin-walled profiles compressed over the buckling limit, Aeronautical Research.

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