Second-order effects in horizontally loaded reinforced concrete columns

A major dynamically growing trend in construction is high bay racked warehouses. Their principal structural components are frames consisting of high reinforced concrete columns on which steel girders are laid. For economic reasons, the main load-bearing frames are spaced as much as 12.0 m apart. Consequently, the wind load transmitted via the enclosure panels to the columns becomes a significant factor. The columns are rigidly fixed in the foundations and their connection with the steel structure is practically of the hinged type, but allowing horizontal displacement. The use of static models with elastic constraints, limiting the displacements of the column's upper joint, introduces the arbitrariness of assuming the stiffness of the constraints, the way of considering the effects of wind action on the building's opposite surface, and so on. With the current state of knowledge, it seems safest to treat such columns as corbel columns loaded with a relatively weak vertical force – a reaction from the roof girder – and a horizontal load (wind pressure).

Previously, standard PN-B-03264 (2002) was used in Poland. The standard recommended that the slenderness (l₀/h) of the columns should not exceed 30. For example, for height h=60 cm, the buckling length l₀ should not exceed 0.6 m•30=18 m, which, in the case of corbel columns, comes down to the condition that the column length l_col≤9.0 m. In high bay warehouses, the heights of the columns are much greater and their slenderness considerably exceeds the former recommendations. There is no such slenderness limitation in the current European standard EN 1992-1-1 (2004), and very slender members can be designed, but one should properly take into account the resulting second-order effects (see, e.g., Klempka and Knauff, 2005; Knauff, 2022).

Second-order effects in static load calculations are taken into account by rejecting the structural system solidification assumption. In other words, the internal forces are calculated taking into account the values of the displacements they induce. As regards an elastic system with invariable stiffness, this comes down to a relatively simple iterative process. In the first step, internal forces and the displacement they generate are determined. In general, both moments and axial and shearing forces can be considered (Aristizabal-Ochoa, 2007). In the next step, the additional moments generated by the displacements are taken into account. The computations can be terminated when the moment increments are infinitesimally small and it is certain that their sequence is convergent.

This procedure becomes more complex when subsequent changes in moment values entail changes in the stiffness of the cross sections. Additional difficulties arise when it is necessary to take into account changes in stiffness over time – rheological phenomena. All these problems must be addressed in the case of reinforced concrete structures.

Standard EN 1992-1-1 (2004) is the main source of information for designers as regards determination of second-order effects. It presents two approximate calculation methods: a nominal stiffness method and a nominal curvature method. How to use the methods and their basic relationships are described in quite detail. Also, an outline of the use of the general method and recommendations concerning its use are given. The description contained in the standard is rather general, requiring much inventiveness from the designer in its use.

The source of the two approximate methods is an analysis of the ultimate strain of a beam with constant stiffness EI, axially compressed with force N, pinned at both its ends – the Euler model (Fig. 1) (Timoshenko and Gere, 1961).

Figure 1:

For the above system, one can write a similar relation as when determining deflections in beam under bending with moment M (1). (1) d2ydx2=−MEI=−N·yEI. {{{d^2}y} \over {d{x^2}}} = - {M \over {EI}} = - {{N \cdot y} \over {EI}}. This is a homogenous differential equation of the second order, which can be expressed as follows: (2) d2ydx2+−N·yEI=0. {{{d^2}y} \over {d{x^2}}} + - {{N \cdot y} \over {EI}} = 0. The general solution of this equation has the form: (3) y=A·cosk·x+B·sink·x=0,where k2=NEI. \matrix{ {y = A \cdot {\rm{\;cos\;}}\left( {k \cdot x} \right) + B \cdot {\rm{\;sin\;}}\left( {k \cdot x} \right) = 0,} \cr {{\rm{where}}\;{k^2} = {N \over {EI}}.} \cr } In the considered case, the boundary conditions are y(x=0) = y(x=l) = 0, resulting in an equation with unknowns A and B (note: constants A, B, C, … have the character of “local variables” and when recalled in the subsequent equations, assume other values): (4) (5) A·1+B·0=0A·cosk·l+B·sink·l=0. \left\{ {\matrix{ {A \cdot 1 + B \cdot 0 = 0} \cr {A \cdot {\rm{\;cos\;}}\left( {k \cdot l} \right) + B \cdot {\rm{\;sin\;}}\left( {k \cdot l} \right) = 0.} \cr } } \right. Equations (4) and (5) form the so-called indeterminate system. Either A = B = 0, which means that the beam will not deform, or the system's main determinant is equal to 0 and there will be an infinite number of equations. But in a nontrivial case, the following must hold true: (6) 1·sink·l−0·cosk·l=0⇒sink·l=0. {\rm{1}} \cdot {\rm{sin\;}}\left( {k \cdot l} \right) - 0 \cdot {\rm{cos\;}}\left( {k \cdot l} \right) = 0 \Rightarrow {\rm{\;sin\;}}\left( {k \cdot l} \right) = 0. Function sinus equals zero when α=π. Thus, one gets k•l=π. Taking into account (3), one can calculate critical force N_B: (7) NB=π2EIl2. {N_{\rm B}} = {{{\pi ^2}EI} \over {{l^2}}}. The solutions of equation (2) have the form: (8) y=B·sink·x=B·sinπ·xl. y = B \cdot {\rm{sin\;}}\left( {k \cdot x} \right) = B \cdot {\rm{sin\;}}\left( {{{\pi \cdot x} \over l}} \right). The modified form of equation (7) constitutes a basis for the nominal stiffness method and that of equation (8) a basis for the nominal curvature method.

It should be noted that in the Euler problem, there are no horizontal loads acting on the beam and affecting the displacements of its axis and, therefore, the curvature. Generalized solutions for a beam with constant stiffness were presented by Yoo and Lee (2011).

This method has no connection whatsoever with the assumptions made in the simplified methods. Its idea was most fully described in CEB/FIP Manual (1978). In Poland, the method was presented in outline by Knauff et al. (2006) and in a much more extended form by Pędziwiatr (2019) and Pędziwiatr and Musiał (2023). It consists in solving the equation: (18) d2ydx2=−MxEIx. {{{d^2}y} \over {{dx^2}}} = - {{M\left( x \right)} \over {EI\left( x \right)}}. This entails the following:

• 1)

  the determination of the relationship between bending moment and stiffness for a given value of the axial force;

• 2)

  the solution of the differential equation. In the adopted approach, it is changed into a difference equation, and solutions (beam axis displacements y(x_i)) in selected column cross sections are obtained. On this basis, one can determine the current moment paths and solve the difference equation again. This procedure ends when the differences between the results (moment values) in successive steps are already infinitesimally small or if the sequence of the results becomes divergent – the column fails under the set configuration of load, geometry, and reinforcement.

For determining the moment–stiffness relationship or the moment–curvature relationship in a second-order analysis, the standard recommends the following relation for stress in concrete: (19) σc=σcεc=fcmκη−η21+κ−2η, {\sigma _{\rm c}} = {\sigma _{\rm c}}\left( {{\varepsilon _{\rm c}}} \right) = {f_{\rm cm}}{{\kappa \eta - {\eta ^2}} \over {1 + \left( {\kappa - 2} \right)\eta }}, where f_cm is the mean compressive strength of the concrete. The coefficients occurring in equation (19) are defined by the following relations: (20) k=1.05Ecmεc1fcm, k = 1.05{E_{{\rm{cm}}}}{{{\varepsilon _{{\rm{c}}1}}} \over {{f_{{\rm{cm}}}}}}, (21) η=εcεc1, \eta = {{{\varepsilon _{\rm{c}}}} \over {{\varepsilon _{{\rm{c}}1}}}}, where E_cm is the mean E-modulus of the concrete and ɛ_c1 is the strain corresponding to the largest deformation. It is convenient to write equation (19) in the form with the highlighted variable ɛ_c representing strains in the concrete: (22) σc=fcmAεc+Bεc21+Cεc {\sigma _{\rm{c}}} = {f_{{\rm{cm}}}}{{A{\varepsilon _{\rm{c}}} + B\varepsilon _{\rm{c}}^2} \over {1 + C{\varepsilon _{\rm{c}}}}} Table 1 gives values of the parameters which make it possible to use relation (22) for a large group of concrete grades. However, the form of this equation is problematic, leading to quite complicated relations for the value of force in the concrete. It is much convenient to replace this formula with its polynomial approximation. This is illustrated in Fig. 2 for C30/37 concrete.

Figure 2:

One can use relation σ_c=σ_c(ɛ_c) to determine the force in the compressed concrete for all the stress intensity levels. Then the relations between the moment value and the beam axis curvature can be determined.

It is convenient to use nondimensional quantities, whereby much higher universality of the results can be achieved. The following denotations and relations were assumed for the parameters (the axial force, the bending moment, and the relative curvature, respectively): (23) n=NEdfcdbd, n = {{{N_{{\rm{Ed}}}}} \over {{f_{{\rm{cd}}}}bd}}, (24) m=MEdfcdbd2, m = {{{M_{{\rm{Ed}}}}} \over {{f_{{\rm{cd}}}}b{d^2}}}, (25) κd=1rd. \kappa d = {1 \over r}d. To plot the κd=f(m) curve, one must determine this relation for a given value of n at changing values of strain ɛ_c in the concrete in the cross section's outer fibers. The initial value is ɛ_c=ɛ_c,min, which perfectly matches the axially compressed cross section. The subsequent values increase gradually up to ɛ_cu1. When calculating the successive values of κd and m, one uses the equation for the equilibrium of forces and moments and the law of flat sections. Steel yield is taken into account at appropriate stress intensity levels when strain ɛ_s in the steel reaches plastic strain ɛ_pl=f_yd/E_s. Exemplary results of such calculations are presented in Table 2.

The independent variable in the calculations is ɛ_c. The strain (ɛ_d) of the least compressed fibers (if the cross section is compressed) or the relative height (ξ) of the compression zone is determined on the basis of the equilibrium-of-forces condition, whose particular form depends on the stage of stressing. Thanks to the law of flat sections, one can calculate the strains in the reinforcement – ɛ_s1, ɛ_s2 – and the corresponding forces – n_s1, n_s2. The force in the compressed concrete is calculated by integrating σ_c=σ_c(ɛ_c) over ζ within the limits ζ=0 to ζ=ξ, having previously written relation ɛ_c=ɛ_c(ζ) stemming from the law of flat sections. In the next step, using the equilibrium conditions, one can calculate the value of relative moment m corresponding to determined strain ɛ_c. Relative curvature κd is calculated from appropriate relation (26) when the whole cross section is compressed or (27) when a tension zone appears in the cross section. (26) κd=εc−εd1+ad, \kappa d = {{{\varepsilon _{\rm{c}}} - {\varepsilon _{{d}}}} \over {1 + {a \over d}}}, (27) κd=εcξ. \kappa d = {{{\varepsilon _{\rm{c}}}} \over \xi }. On the basis of the calculation results, one can draw graphs as the ones shown in Figs 3 and 4. The relation of the type shown in Fig. 4 is used for further calculations aimed at determining the displacements of the column axis. Equation (18) can be written in the form: (28) κx·d=mxfcdbd3EIx=mxfcdbd3EIx=Fm. \kappa \left( x \right) \cdot d = {{m\left( x \right){f_{{\rm{cd}}}}b{d^3}} \over {EI\left( x \right)}} = m\left( x \right)\left[ {{{{f_{{\rm{cd}}}}b{d^3}} \over {EI\left( x \right)}}} \right] = F\left( m \right). Function F(m) in (28) is represented by the relations shown in Fig. 4. By replacing the differential equation with a difference equation, one gets: (29) κid=wi−1−2wi+wi+1li2d=Fm. {\kappa _i}d = {{{w_{{\rm{i}} - 1}} - 2{w_{\rm{i}}} + {w_{{\rm{i}} + 1}}} \over {l_{\rm{i}}^2}}d = F\left( m \right). One should follow the procedure:

• 1)

  divide the column using points (the more points there are, the higher the solution precision),

• 2)

  determine the value of relative moment m in each of the points,

• 3)

  calculate the relative curvature on the basis of appropriate relations (see Fig. 4),

• 4)

  determine displacements w_i – according to (29) with the boundary conditions taken into account,

• 5)

  in successive points, calculate moment increments N_Edw_i and the corresponding nondimensional quantities, and

• 6)

  calculate new moment values and repeat the procedure.

Figure 3:

Figure 4:

Figure 5a shows a corbel beam. In point 0, the displacement and the angle of rotation are equal to zero. This leads to the following two relations: w₀=0, w₁=w₋₁. The forms of equation (29) in the particular points are as follows:

• – 0:κ0d=2w1li2d⇒w1=κ0dli22d=κ0li22, {\bf{0}}:{\kappa_0}d = {{2{w_1}} \over {l_{\rm{i}}^2}}d \Rightarrow {w_1} = {\kappa _0}d{{l_{\rm{i}}^2} \over {2d}} = {{{\kappa _0}l_{\rm{i}}^2} \over 2},

• – 1:κ1d=w0−2w1+w2li2d=2w1+w2li2d⇒w2=2w1+κ1li2, {\bf{1}}:{\kappa_1}d = {{{w_0} - 2{w_1} + {w_2}} \over {l_{\rm{i}}^2}}d = {{2{w_1} + {w_2}} \over {l_{\rm{i}}^2}}d \Rightarrow {w_2} = 2{w_1} + {\kappa_1}l_{\rm{i}}^2,

• – 2:κ2d=w1−2w2+w3li2d⇒w3=2w2−w1+κ2li2, {\bf{2}}:{\kappa_2}d = {{{w_1} - 2{w_2} + {w_3}} \over {l_{\rm{i}}^2}}d \Rightarrow {w_3} = 2{w_2} - {w_1} + {\kappa_2}l_{\rm{i}}^2,

• – n-1:κn−1d=wn−2−2wn−1+wnli2d⇒wn=2wn−1−wn−2+κn−1li2. \left( {{\bf{n {\text -} 1}}} \right):{\kappa_{n - 1}}d = {{{w_{n - 2}} - 2{w_{n - 1}} + {w_n}} \over {l_{\rm{i}}^2}}d \Rightarrow {w_{\rm{n}}} = 2{w_{{\rm{n}} - 1}} - {w_{{\rm{n}} - 2}} + {\kappa_{{\rm{n}} - 1}}l_{\rm{i}}^2.

Table 3 shows an exemplary iteration sequence illustrating the presented procedure. The column is rigidly fixed in the bottom node. It is 6.0 m high and its effective cross-sectional height amounts to d=h-a=0.5 m. The relative axial force value amounts to n=0.5, and the relative value of the first-order moment varies linearly from 0.25 in the fixing to zero at the free end. Seven consecutive cross sections spaced l_i=1 m apart are distinguished in the column. The number of the iteration step is given in the first column of the table. For the initial state, the values of relative curvatures κd, horizontal displacements w_i, and moment increments Δm_i=n(w₆-w_i) induced by the displacements were determined one by one in all the selected cross sections. Six iterations were performed. The last column shows the relative moment increments in the successive iteration steps. In the considered case, the sequence is decreasing very quickly. Taking into account second-order effects, the relative moment is m=0.2766. The increment relative to initial value m₀=0.25 amounts to about 11%.

Figure 5:

It is important to pay careful attention during the successive iterations. Too early termination of the iteration process may turn out to be a wrong decision. Table 4 shows fragments of the iteration for a 12.0-m-high column with maximum first-order moment m₀=0.18. Up to iteration 14, the moment increments slightly decreased. But after iteration 14, the tendency changed. The eccentricity increments began to gradually increase. This meant that the column failed. This was due to the fact that a load level was reached at which the increment in column curvature significantly accelerated.

To examine the differences between the results yielded by the approximate methods and the ones obtained using the general method, calculations were carried out assuming the following: concrete C30/37 (f_ck=30 MPa), steel with f_yk=500 MPa, and reinforcement ratio ρ₁=ρ₂=1%. Four corbel columns with heights l=6.0,8.0,10.0,12.0 m were considered. To increase the generality of the results, the nondimensional parameters relative axial force n, relative first-order moment m₀, relative curvature κd, and moment increment md induced by second-order effects were used. Results for three stress intensity levels induced by axial force n=0.1,0.5,0.8 are given in the examples. At these levels of relative force n, the maximum values of moment m_Rd can be determined from the equilibrium equations whose specific form depends on the assumed concrete model:

• a)

  The nonlinear model from the general method

  • –

    For n=0.1, the steel in the compression zone is not utilized and the equilibrium-of-forces condition has the form of a quadratic equation with regard to ξ: (30) n=ξfcd13Aεc2+12Bεc+C+εcEsfcdξ−adξρ2−fydfcdρ1. n = {\xi \over {{f_{{\rm{cd}}}}}}\left( {{1 \over 3}A\varepsilon _{\rm{c}}^2 + {1 \over 2}B{\varepsilon _{\rm{c}}} + C} \right) + {\varepsilon _{\rm{c}}}{{{E_{\rm s}}} \over {{f_{{\rm{cd}}}}}}{{\xi - {a \over d}} \over \xi }{\rho _2} - {{{f_{{\rm{yd}}}}} \over {{f_{{\rm{cd}}}}}}{\rho _1}. The solution is ξ=0.149. After determining the appropriate strains and forces, one gets the maximum moment m_Rd=0.233.

  • –

    For n=0.5, both the steels can be utilized, but m_Rd=0.381 is reached just before the yielding of the steel in the compression zone at ɛ_c=3.10‰.

  • –

    For n=0.8, both the steels can be utilized, but m_Rd=0.419 is reached just before the yielding of the steel in the compression zone at ɛ_c=3.10‰.

• b)

  The simplified model

  • –

    For n=0.1, the steel in the compression zone is not utilized, the equilibrium-of-forces condition has the form (31), and the relative resistance to bending is calculated from equation (32). (31) n=ξeff−ρ1fydfcd⇒ξeff=0.1+0.0143521.4=0.3033, n = {\xi _{{\rm{eff}}}} - {\rho _1}{{{f_{{\rm{yd}}}}} \over {{f_{{\rm{cd}}}}}} \Rightarrow {\xi _{{\rm{eff}}}} = 0.1 + 0.01{{435} \over {21.4}} = 0.3033, (32) mRd=ξeff1−0.5ξeff−0.45n=0.1223. {m_{{\rm{Rd}}}} = {\xi _{{\rm{eff}}}}\left( {1 - 0.5{\xi _{{\rm{eff}}}}} \right) - 0.45n = 0.1223. If the bilinear model were used, m_Rd =0.232 would be obtained.

  • –

    For n=0.5, both the steels are utilized, the equilibrium-of-forces condition has the form (33), and the relative resistance to bending m_Rd calculated from (32) amounts to 0.3329. (33) n=ξeff⇒ξeff=0.5. n = {\xi _{{\rm{eff}}}} \Rightarrow {\xi _{{\rm{eff}}}} = 0.5.

  • –

    For n=0.8, only the steel in the compression zone is utilized, relative effective height ξ_eff of the compression zone amounts to 0.5967, and relative bending strength m_Rd=0.2637.

For some time now, a dynamic development of the branch of construction connected with the design and building of high bay racked warehouses has been observed. The basic load-bearing member of such structures is a frame consisting of high, slender, reinforced concrete columns and a lightweight steel girder forming the girt of the frame. Because of the height of the columns and the increasing spacing of the frames, the load acting on the column as a result of wind pressure on the curtain walls of a high bay warehouse becomes a critical factor.

For the safe and at the same economical design of such columns (and their foundations), the second-order effects occurring in this case must be precisely determined. In practice, simplified methods implemented in computer programs are used for this purpose. Most often, the nominal stiffness method is employed. But as demonstrated in this paper, this is an incorrect approach when horizontal forces occur. A somewhat better solution is to use the nominal curvature method, where one can distinguish a part of the first-order moment which is not subject to unwarranted increase.

It is characteristic of the two methods that the calculation results very significantly depend on (l₀/d)². The use of a fictional value of design length l₀ in a situation when horizontal loads occur is a source of a significant overestimate of the second-order effects. The error increases with column height and the stress intensity level induced by the vertical force. Calculations done in this paper show that the benefits stemming from the use of the general method amount to as much as 200%. Only for l≤8.0 m and n=0.1, the differences between the results yielded by the approximate methods and the general method, respectively, are slight.

The general method is much more labor intensive, requiring a relative moment-relative curvature relation for the given axial force value, reinforcement ratio, and concrete grade to be derived. Moreover, the iterative process, consisting in tracking the sequence of strain increments, is time-consuming. However, in the case of very slender columns, the benefits stemming from the use of general methods are very substantial. As part of further work, one can try to create detailed algorithms to be implemented in relevant computer programs.