A structural member with a thin-walled open cross-section is often used as a core in multi-story reinforced concrete (RC) buildings or as a roof in folded plate RC structures. In the general case, this cross-section is asymmetric, having strongly spatial behavior that significantly affects the torsional-translational behavior of the structure. Examples of the use of open cross sections are folded plates loaded with gravity loads that are used as roofs, especially in structures with large spans, as well as vertical cores that are usually used in buildings to carry lateral loads (such as seismic or wind loads). The elastic center of the asymmetric cross-section is generally located away from its geometric center and this distance is called “cross sectional eccentricity”. Due to the existence of the partial legs of the thin wall cross-section, the thin-wall open cross-section presents important translational stiffness and significant resistance in bend-from-torsion (phenomenon of torsion-warping).

A folded plate consists of -plane surface disks that are not the same, which are connected, as shown in (Fig. 1). This is a prismatic surface structure in which each-one disk operates in a mixed way since it is loaded into its plane and, also perpendicular on its surface. In addition, it is common to set an end-diaphragm at each end of a folded plate. The distance between the two end-diaphragms is the longitudinal dimension L of the folded plate, while the width is the b-dimension, and the height is the H-dimension (Fig. 1). Folded plates are used for the cover of large areas. The exact analysis procedure of the folded plates according to the theory of elasticity is particularly hard, harder than the analysis procedures of shell structures, while for many types of folded plates (especially when they have non-parallel edges), the unique solution comes from experimental results [1, 2]. The numerical solutions using the finite element method are easy, but the issue of checking the reliability of the numerical results remains a major problem, especially at such special structures. On the other hand, special finite elements have been proposed for a better numerical analysis, where a higher order mixed-based Bernoulli element (HMB) used with very good results in symmetric thin-walled open cross-sections [3]. However, in non-symmetric thin-walled open cross-sections, the question point remains. For this reason, the comeback to approximate methodologies of the past is a convenient way. The first approximate methodologies were presented for the first time in 1930 by Ehlers [4] and Craemer [5]. Afterward there were many others as Gruber [6], Gruning [7], Vlasov [8], Hartenbach [9], Winter and Pei [10], Girkman [11], Gaafar [12], Aichinger [13], Valentin [14], and Yitzhaki [15]. All the researchers have used various assumptions.

The present article proposes a new hybrid procedure, combining a recently presented new technique [16] that is based on the Vlasov torsion theory [17, 18] of thin-walled prismatic beams, with the Bernoulli bending theory. The Vlasov torsion theory has been widely used in the past for the study of open thin-walled cross-section, since it is the unique procedure that examines the torsion-warping phenomena of such structures [19-21]. Furthermore, in another recent article, the Vlasov theory was used in order to define the equivalent torsional-warping stiffness of thin-walled open cross-section [22].

In more detail, the following subjects are determined: (a) the location of the elastic center (or stiffness center) as well the center of gravity, (b) the cross-sectional eccentricity of members of the folded plate, (c) the orientation of the principal elastic axes of the cross-section, (d) the position of the principal start point of the cross-section, (e) the exact diagram of the sectorial coordinates of the cross-section, (f) the warping moment of inertia of the cross-section and, last but not least, (g) the normal stresses on the cross-section of the folded plate due to bi-moment (this is the first source of the normal stresses). Afterwards, we use the Bernoulli bending theory to determine the normal stresses at the critical cross-sections

along the longitudinal direction of a folded plate (this is the second source of the normal stresses). It is worth that all the above-mentioned properties explicate qualitatively and quantitatively the spatial behavior of the folded plate due to gravity loads and, for this reason, can be used to verify the numerical results produced from various types of the finite element method. The present procedure is very simple in practice, giving exact results (which are based on closed mathematical relationships), which come from the double-bending around the two perpendicular axes and the warping phenomenon of the thin-walled open cross-section of the folded plate. On the contrary, is well-known that in such structures, the finite element method gives approximate results, because the torsion-warping phenomenon is ignored by this method. Hence, the main advantage of the present procedure is that it is directly based on the superposition of Bernoulli bending theory and Vlasov torsion theory (i.e. solving closed mathematical relations derived from differential equation solutions), while on the one hand, the finite element method is approximate and on the other hand the abovementioned classical approximate methodologies [4-15] use many additional assumptions. The present procedure is based on the findings of a recently developed technique [16] for the calculation of the elastic characteristics of the thin-walled open cross-section structures (elastic center of the open cross-section, principal axes of the open cross-section, principal start point of the open cross-section, sectorial coordinates of the cross-section and wrapping moment of inertia of the open cross-section).

An easy procedure based on a recently developed technique [16], which examines the torsion-warping phenomenon of cores or of other thin-walled open cross-section structures, is presented in this article. The main steps of the proposed procedure are given below:

a) A temporary Cartesian three-orthogonal reference system OXYZ is used for the determination of the gravity center, G, as well as for the orientation of the principal axes ξ and η of the thin-walled open cross-section.

b) Calculation of the principal moments of inertia I_ξ and I_η of the thin-walled open cross-section about the principal axes ξ and η passing through the gravity center G of the cross-section.

c) Calculation of diagrams of coordinate-functions ξ(s) and η(S) of the thin-walled open section with regard to the gravity reference system Gξηz.

d) Determination of the location of the elastic center K (which is the stiffness center) of the thin-walled open section using a repetitive mathematical procedure.

e) Determination of the location of the principal start point M_O(x_O, y_O) of the thin-walled open section as well as of the sectorial coordinates ω(s) with respect to the pole K (that is, the Elastic Center of the cross-section) and based on the principal start point M ₀ of the thin-walled open cross-section.

f) Determination of the numerical value of the warping moment of inertia (or warping constant according to other researchers, I_ω, of the thin-walled open section, according to Vlasov torsion theory.

g) The Bernoulli bending theory is applied, and for this reason, all loadings have to be moved to the Principal Reference System K(I,II,III) of the cross-section. Hence, there are two planes (I,III) and (II,III) of pure bending and a warping effect around the III-axis.

h) Calculation of the normal stresses on the critical cross-sections of the folded plate using the Vlasov torsion theory in combination with the Bernoulli bending theory.

It is noted that the theory of Bending from the Torsion (torsion-warping effect) of a structural member with a thin-walled open section, which was developed by V. Vlasov (1959), uses the following assumptions:

i. The cross-sections of the structural element remain undeformed into their planes (disk behavior). This is the well-known Bernoulli assumption (or Bernoulli-Navier assumption) from the Technical Theory of Beam Bending.

ii. The shear deformations of the considered structural member are assumed to be zero (Bernoulli assumption).

iii. The perpendicular lines that belong at the disk of the thin-walled open section remain also perpendicular at the deformed state. This is the Kirchhoff's assumption in the study of thin plates.

According to [16], if the displacements of the point P are known, then the displacements u_s, u_n, u_z of the random point M on the mean line of a thin-walled open cross-section, in the local coordinate system Μsnz (Fig. 2):

where:

are the degrees of freedom in the plane X^OY ^O of the solid disk of the open section at the start point P.

u_s, u_n, u_z are the displacements of point M along the local s, n and z-axis.

^α is the angle of the local s axis with the X ⁰ axis of the global coordinate system PX ⁰Y ⁰Z ⁰, which is a function of the dimension s.

S_m, N_m are the coordinates of point M in the global coordinate system PSNZ, which are functions of the dimension s.

are the angles of the open cross-section about the horizontal axes x and y, respectively. These angles are functions of the dimension z.

gives the change of the θ_z per unit length along Z-axis. This change is called twist and it constitutes the seventh degree of freedom, instead of the six known degrees of freedom of a common joint of a spatial structure.

is the function of the Sectorial Area or Sectorial Coordinate, of the point M («function of warping» or «measurement of warping»), Fig. (3). The parameter dω(s) is equal to the double area that is formed from the moving radius PM, where point M is shifted along the infinitesimal element on the mean line of the open section-leg. This area is considered positive when PM is rotated about Z-axis according to the rule of clockwised-screw (Cartesian System). The graphical display of the sectorial coordinate defines the distribution of the normal stresses on the open cross-section in the case of external bi-moment loading of the section, which is restrained against free torsion.

Eq. (3) yields that if a core with a thin-walled open cross-section is purely loaded with a torque about Z-axis, causing a unit twist of the section, i.e.

then axial displacements are produced, which are numerically equal to the distribution of the sectorial coordinate ω(s):

By using the constitutive law of the material from the theory of elasticity, it is proved that the normal stresses acting on the cross-section are determined by:

When the gravity principal directions ξ and η of the cross-section are rotated with respect to the gravity reference system Gxyz, Eq. (5) becomes:

where I_ξ and I_η are the principal moments of inertia of the thin-walled open section about the principal axes ξ and η passing through the center of gravity G of the cross-section (local reference system Gξηz). The normal stresses resulting from the axial loading (1^st term of Eq. 6) are superimposed and, also the normal stresses developed in the thin-walled open section due to bi-moment are superimposed (4^th term of Eq. 6). It is noted that the positive sign in Eqs. (5-6) represents tensile stresses, while the negative sign represents compressive stresses on the cross-section. The sign of bending moments is taken positively when the moment vectors have the same sign with the positive semi-axis of the Cartesian reference system Gξηz.

In Eqs. (5-6), the meaning of the symbols is as follows:

ξ(s) and η(s) are functions of the cross-section coordinates in the local reference system Gξηz.

N(z) is the axial force (along the Z-axis) of the open section with area A and Modulus of Elasticity E:

M_x(z) and M_y(z) are the bending moments on the cross-section about the x and y-axis at the z level, according to Bernoulli's technical bending theory for prismatic beams. The quantity I_y represents the moment of inertia of the cross-section about the y-axis passing through the center of gravity G of the thin-walled open section, so that both the product second and first moments of area are zero:

M_ξ(z) and M_η(z) are the bending moments acting on the thin-walled open section about the principal axes ξ and η, respectively, passing through the center of gravity G of the cross-section.

B_K(z) is the bi-moment (or warping moment) at the z-level acting on the open cross-section.

I_ω represents the warping or sectorial moment of inertia of the open cross-section with unit of length to the sixth power (m⁶). To calculate the torsional moment of inertia I_ω, the following two conditions must always be met:

(i) I_ω is calculated with respect to the Main Pole P of the cross-section, which coincides with the elastic center K (or stiffness center) of the thin-walled open cross-section and,

(ii) The principal start point M_O(x_O, y_O) of the cross-section must always be used.

As long as these two cross-section conditions are satisfied, the torsional moment of inertia I_ω becomes numerically minimal because both the product sectorial moments I_x,ω, I_y,ω (or I_ξ,ω, I_η,ω depending on the reference system we are working in) as well as the sectorial first order moment of inertia S_ω of the open cross-section become equal to zero:

where:

To determine the location of the elastic center K (or center of stiffness) of the thin-walled open section, the following repeated procedure is proposed:

The elastic center K of the cross-section presents an eccentricity with respect to the pole P (which coincides with the center of gravity G of the same cross-section at the first approximation). This eccentricity is determined by the coordinates δ_ξ and δ_η with respect to G along the principal directions ξ and η, which are calculated from Εqs. (12 and 13), which are repeated below:

where e(s) is the constant thickness of the section, ω(S)⁽¹⁾ is the diagram of the sectorial coordinates (sectorial area) of the cross-section with respect to the temporary start point M (Fig. 9), while ξ(s) and η(s) are known from the diagrams of Fig. (8). As regards the temporary start point M of the cross-section, it is recommended that some corner of the thin-walled cross-section can be chosen that is approximately in the middle of the mean line length of the cross-section, while the location of this start point M must be kept fixed for all the iterations to be performed. The location of the principal start point M_O of the thin-walled open section will be calculated at the end, after finalizing the location of the elastic center K. In Tables 3 and 4, the calculations for finding the product warping moments I_η,ω and I_ξ,ω are shown. The symbols a, b, c, d are used to apply Eq. (14). It is also emphasized here that the values of the two product warping moments (I_ξ,ω, I_η,ω) are becoming zero when these are calculated with respect to the real elastic center K of the thin-walled open cross-section.

Therefore, using Eqs. (12 and 13) in a first approximation, the location of the elastic center K⁽¹⁾ with respect to the pole P (which here coincides with the center of gravity G) is (Fig. 9):

In the second approximation, the elastic center K⁽¹⁾ of the first approximation, which was calculated in the 1^st approach, is used as the pole P, while the same temporary start point M of the cross-section (at point B) is used. In this second approximation, the sectorial diagram coordinates ω(S)⁽²⁾ of the cross-section with respect to the temporary start point M, is given in Fig. (10). The calculations for finding the two product warping moments I_η,ω, I_ξ,ω are shown in Tables 5 and 6.

Therefore, using Eqs. (12 and 13) in a second approximation, the location of the elastic center K⁽²⁾ with respect to the pole P (which here coincides with K⁽¹⁾) is (Fig. 10):

So, the corrected location of K from the center of gravity G is (Fig. 10):

The accuracy achieved is very satisfactory and therefore no further approximation is required. However, if a third approximation is applied, then the final location of K is given as δ_ξ^(1+2+3) = 0.98m and δ_η^(1+2+3) = 0.242m, and this position of the Elastic Center K is used in the following.

The final location of the elastic center K has been determined. Using pole K and with the same temporary start point M of the cross-section already obtained at point B, we correct the diagram of the sectorial coordinates ω(S)^(M) of the cross-section (Fig. 11).

With the final location of the elastic center K known, and given the temporary start point M of the cross-section, we form the ideal triangle KMM_o, with the point M_O as unknown that must be determined. This is achieved by calculating the unknown distance ρ. The area A_r of the triangle KMM_o is:

where υ is the height of the triangle which is known geometrically (Fig. 12).

Therefore, the distance ρ is given by:

The magnitude 2A_r (i.e. twice the area of the triangle) represents the sectorial coordinate ω(s) of the triangle KMM_o with the elastic center K as a pole and will be denoted by Ω(ρ) which is calculated from the following equation:

where A = 1.3165 m² is the total area of the thin-walled cross-section (Τable 1), ω(S)^(M) is the sectorial coordinates based on the temporary start point M and υ is the height of the triangle KMM_o from K (Fig. 12), where υ is measured geometrically. The calculations for finding Ω(ρ) are shown in Table 7, in which the quantity is calculated.

If the quantity Ω(ρ) is negative, then the principal start point M_O will be located with a negative rotation of the radius KM of the triangle KMM_o (which is the case here).

Therefore, from Eq. (29):

The height υ of the triangle KMM_o is measured geometrically as 2.685 m, and consequently the distance ρ is calculated from Eq. (28):

Being known the principal start point M_O, the final corrected diagram of sectorial coordinates ω(s) is calculated, using the elastic center K as the pole (Fig. 13).

Then the value of the warping moment of inertia I_ω of the thin-walled open cross-section is calculated, which will be used to calculate the positive normal stresses due to the external load of bi-moment B_K for a specific level of the cross-section along the longitudinal dimension of the thin-walled structure.

The warping moment of inertia I_ω (Eq. 11) of the thin-walled open section is calculated in Table 8 as:

Fig. (14) shows the elastic principal system (K,I,II,III) of the thin-walled open cross-section with respect to the original reference system OXYZ, where the III-axis is the vertical axis passing through the elastic center K and together with the two horizontal axes I and II constitute the principal elastic reference system of the thin-walled open cross-section, with origin the elastic center K.

We consider the reinforced concrete (R/C) folded plate of Fig. (1), which has longitudinal dimension L = 20.00 m and is loaded with two loadings; (a) the self-weight (relative value 25 kN/m³) and (b) the projected-in-plan snow-weight q_s = 4.00 kN/m². Thus, the consultant forces of the two above-mentioned loadings are given (Fig. 15):

Next, the forces P₁ and P₂ are moving to Elastic Center K, hence, a torsional moment m_t around the horizontal longitudinal axis III is the result:

m_t = P₁ x 0.75 + P₂ x 0.685 = 32.92 x 0.75 + 16.80 x 0.685 = 36.20 kNm/m

’In addition, the consultant force F of P₁ and P₂ at the Elastic Center is:

F = P₁ + P₂ = 32.92 + 16.80 = 49.72 kN/m along the longitudinal direction L.

Furthermore, the force F is analyzed along the two Principal I and II-axes of the thin-walled open cross-section ABCDE (Fig. 15). Along the longitudinal direction, the folded plate is a prismatic beam (having thin-walled open cross-section) with hinges at both ends. Hence, there are two flexural moment diagrams at planes (II,III) and (I,III), as well as a bi-moment diagram due to torsional moment m_t around the horizontal longitudinal axis III, as are shown in Fig. (16), where:

Therefore, the normal stresses σ_z(z,s) of the thin-walled open cross-section at the middle section of the longitudinal length of the folded plate, due to the simultaneous action of the forces P₁ and P₂ and their bi-moments, can be directly calculated from Eq. (6):

In this equation, the first term is zero since there is not axial loading on the folded plate along the horizontal III-axis (i.e. N(z) = 0). All the necessary data to use the equation at the six points from A to E of the cross-section have already been calculated (ξ(S) and η(S) from Fig. (8), ω(S) from Fig. (13), I_ξ = 0.0786m⁴, I_η = 3.4447m⁴ and I_ω = 0.32248m⁶), while the normal σ_z(0.5L, S) diagram at the middle section of the folded plate with open thin-walled structure is as shown at Fig. (17). This is considered as the exact solution because all these come from the solving of closed mathematical relations derived from differential equations of the abovementioned “Bernoulli and Vlasov” theories.

Point A of the cross-section at the middle section of the folded plate along the longitudinal dimension (z = 0.5L):

Point B of the cross-section at the middle section of the folded plate along the longitudinal dimension (z=0.5L):

Point C of the cross-section at the middle section of the folded plate along the longitudinal dimension (z=0.5L):

Point D of the cross-section at the middle section of the folded plate along the longitudinal dimension (z=0.5L):

Point E of the cross-section at the middle section of the folded plate along the longitudinal dimension (z=0.5L):

For comparison reasons, Fig. (18) shows the normal stress diagram σ_z(0.5L,s) derived from the F.E.M. analysis software SAP2000 [23] -using Modulus of Elasticity E = 30 GPa- with reference to the exact solution that comes from by the proposed present methodology. In the same Figure, we can see big differences where at point E is against safety (-39.42%).

It is worth noting that the analysis of the folded plate along the transverse dimension can be achieved as an inclined plate, with width w = 1.00 m, while we consider simple supports at the edges (at join of the folds) of the folded plate (Fig. 19).

In this paper, the exact determination of the principal elastic reference system of an RC folded plate having an asymmetric, thin-walled open section, and the calculation of the warping constant of this section, as well as the normal stresses (due to self-weight plus snow-weight) along the longitudinal dimension was presented. All calculations are carried out according to a recently modified technique [16] that examines the warping phenomenon of cores and is based on the Vlasov torsion theory [17, 18].

In the present article, the calculation of the warping properties of a thin-walled open cross-section is achieved by applying the following steps: (a) the location of the center of gravity G and the orientation of the principal axes ξ and η of the thin-walled open section are determined with respect to a temporary cartesian reference system OXYZ, (b) the principal moments of inertia I_ξ and I_η of the thin-walled open section about the principal axes ξ passing through the gravity center G of the cross-section are calculated, (c) the diagrams of the coordinate-functions ξ(S) and η(S) of the thin-walled open section relative to the gravity reference system Gξηz are drawn, (d) the location of the elastic center K (stiffness center) of the thin-walled open section is calculated using a repetitive mathematical procedure, (e) the location of the principal start point M_O(x ₀, y ₀) of the thin-walled open section is determined. It is worth noting that this start point is needed for the right calculation of the sectorial coordinates and for the calculation of the minimum numerical value of the warping moment of inertia I_ω of the thin-walled open section, which is the exact value of I_ω according to Vlasov torsion theory, (f) the numerical value of the warping moment of inertia I_ω of the thin-walled open section is calculated, (g) the warping stiffness of the core is calculated and, last but not least, (h) the normal stresses at the middle section of a folded plate along the longitudinal dimension are calculated, considering the bi-axes flexure with the bi-moment phenomenon. All the above-mentioned properties give the exact solution of the folded plate in longitudinal dimension according to Bernoulli bending theory for prismatic beams (bending moments and axial force) and, additionally, according to Vlasov torsion theory (i.e. about the warping moment or bi-moment concept) for thin-walled open sections. This exact solution can be used for checking and assessment of the reliability of the results of various F.E.M. analysis software, since it is directly based on the superposition of Bernoulli bending theory and Vlasov torsion theory while the finite element method is approximate.