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

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

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

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

Force (stress) calculations for structures subject to temperature influence (Translation from Bulgarian) Ass. Prof. eng. Ivan Iakimov, Ph.D., Sc.A. (Scientific Assistant) eng. Mario Behar          Introduction          The influence of the surrounding air temperature [considered] in the design of bridges and other structures, in the general case is divided in two separate effects: 1) the change of the average temperature of the cross-section; and 2) the disparity in temperature between upper and lower, and/or external and internal surfaces.          The first of these two effects is very clear physically, and is not a design problem. The second one, in its simplest form (as it is reflected in the Bulgarian Norms [1]), is physically represented as a linear change of the temperature from one surface to the other (fig. 1). It generates deflection of the elastic line in statically determinate structures [2]. In addition to it, internal forces and support reactions are generated in the statically indeterminate structures. The same deflections would be generated by substitute bending moments, applied to the ends of the [struc-tural] element, and computed using the equation: Eq. (A) … (see the Bulgarian original) … where          α is the thermal expansion [contraction] factor;          Δt – the temperature difference between the two external boundary surfaces;          h – the height of the cross section of the element;          E – the modulus of elasticity;          I – the moment of inertia.          The research that has taken place in the last years shows that the temperature does not change linearly along the height of the section, rather, it changes following a curve, whose approximation is a broken line. As a specific example, a measurement presented by Fritz Kehbeck in [3] is given herein (fig. 3).          Of course, this state of the matter requires certain complication in the mathematical apparatus, which with the existence of modern computing devices (programmable calculators and computers) does not make the work of the engineer more difficult.          1. Design model          The term temperature disparity is not limited only to the difference between the edge temperatures of the element (upper and lower end, respectively external and internal edge of the cross section), but also encompasses the disparity between edge temperatures and the temperatures in the internal portions of the cross section.          The explanation in [4] reads as follows:          - positive temperature disparities appear, when the conditions are such, that the sun radiation and other effects contribute to the accumulation of heat through the upper face of the su-perstructure;          - negative temperature disparities appear, when the conditions are such, that heat is being lost from the upper face of the bridge plate due to re-radiation (radiation of heat in the opposite direction – from the structure to the atmosphere) and other effects.          In the same standard [4] the schemes of distribution of the temperature along the height of the element are four (fig. 3). These schemes have been incorporated – practically without any change, in the Euronorms for design of reinforced concrete structures.          The basics of the design model are as follows:          The height of the element, submitted to the load effect “temperature disparity” (according to [4]), is divided into sections (layers) with linear change in temperature (fig. 3) within. This way, each of these layers, if neglecting their interconnection, would deflect itself according to the model incorporated in the Bulgarian Norms, i.e. linear change in temperature along the whole height of the cross section. The complication arises from the fact, that the interconnection between the layers cannot be neglected. The [deformation] scheme shown on fig. 4 is assumed under the following conditions:          a) valid is the hypothesis of Bernoulli for flatness of the cross section [in bending], from where it follows that: Eqs. (1), (a), (b), (c), (d) … (see the Bulgarian original)          b) the free thermal deformation of the material is linearly proportional to the change in temperature, from where it follows: Eqs. (2), (3), (4), (5) … (see the Bulgarian original)          c) valid are the equilibrium conditions ΣN = 0 and ΣM = 0, from which it follows that: Eqs. (6) and (7) … (see the Bulgarian original)          d) the elastic behavior of the material presupposes an interrelation between the substitute internal force (bending moment M) and the strain ε through the equations for curvature of the elastic line – eqs. (B) and (C), from [5] and [6] respectively (fig. 1 of this article): Eqs. (B) and (C) … (see the Bulgarian original) that united lead to the following equation: Eq. (8) … (see the Bulgarian original)          e) the free linear strain (εFRi) of the idealized layers consists of reduced strain (εi) and restrained (unrealized) strain (εLIMi) – (fig. 4b).          From the last two above mentioned, the first is related to the geometry of the element, i.e. to the position of the layer with respect to the centerline. The second strain is related to the internal forces. It determines the “stressed state” of the respective layer, and is the one that increases or decreases the free strain, in order to bring it [in magnitude] to the linearly-increasing along the height strains (εi) that have actually taken place.          2. Basic equations          The equations (1) to (8), quoted in section 1, subsections a), b), c), d) and e), are a system of eight equations with eight unknowns, used in the matrix calculations aimed to obtain the substitute internal force M. The meaning of the notations is as follows:          ΔTi – the temperature disparity between the (i-th) layer and the layer with the lowest tem-perature;          TºiMEAN – the average temperature in the layer;          ai – ordinate of the centerline of the layer with respect to the centerline of the cross section;          aUP and aDOWN – ordinates respectively of the upper and lower edges of the whole section;          εUP, εDOWN and εMID – strains at levels up-most, down-most, and at the centerline, resp.;          εi – the strain that has actually taken place;          εLIMi – the restricted (not realized) strain;          εFRi – the free strain;          αt – factor of thermal expansion (contraction);          Fi – area of the cross-section of the respective layer;          E – modulus of elasticity;          h – height of the total cross section;          I – moment of inertia of the total cross section;          ρ – radius of curvature.          3. Substitute internal forces          The way the influence of the thermal differences is presented in [4] is not suitable for direct use in the design calculations. This is why the current algorithm is being offered. The result from it is a moment as an internal force with constant magnitude along the length of the element under design.          This approach is applicable for statically determinate structures, as well as for statically indeterminate ones – as continuous beams, beam-grates, etc. From fig. 5 it is evident, that the influence of the thermal differences (substitute moment) is under consideration here, which is superimposed with the other effects – in the specific case, the reaction in the mid-support.          4. Example          The approach presented in this article was applied in the lay-out design project for two of the bridges in Tikrit, Iraq – continuous reinforced concrete orthotropic plates, solved as beam-grates using the Method of Finite Elements (MFE) and erected by the engineering company “Transkomplekt” in 1983 (fig. 6).          A programmable calculator Texas Instruments TI 59 has been used as a computing device for the matrix solution for the unknowns, including the substitute moment M. The matrix in its general form (parametric form) and in numerical form for the end longitudinal strip (“end longitudinal beam”) [of the beam-grate solution] are presented in the tabulogram … Tabulogram ... (see the Bulgarian original).          REFERENCES:          1. Loads and load-effects, Design norms, Bulletin for civil engineering and architecture, 1980, p. 63.          2. Kvartirnikov, A., Structural statics, “Technika” Editorial, 1978, p. 12.          3. Fritz Kehbeck, Influence of sun radiation on bridge structures, “Werner” Editorial, Düsseldorf, 1975.          4. British standard BS-5400, part 2, 1978, “Specification for loads”, p. 21.          5. Kisliakov, S., Strength of Materials, “Technika” Editorial, 1975, p. 208.          6. Manoilov, L., Reinforced concrete, “Technika” Editorial, 1983, p. 302. Back to Top home ... other publications