Moment Magnifier Method Research Articles

Behavior of Eccentrically Loaded Slender High-Strength Concrete Columns.(Dept.C)

The buckling behavior of slender High-Strength Concrete (HSC) columns under axial load with different end eccentricities has been studied experimentally. The results of these tests have been used to examine the applicability of the design requirements of Normal Strength Concrete (NSC) columns of the Eurocode 2 (EC-2), the Egyptian code (ECC-9S) and ACI 318-99 building code, when applied to slender HSC columns with a characteristic compressive strength of 85 MPa. The program included tests of22 columns with slenderness ratios λ1 ranged between 17.3 and 69.3. The main parameters examined in this study were the values of end eccentricity of the applied axial load (e/t=O.3, 0045 and 0.90), the transverse reinforcement ratio (p v) and the longitudinal reinforcement ratio (Pt ). The results of this experimental work showed that, slender columns constructed with HSC increased the ultimate load of the tested columns in comparison with the same columns constructed with NSC. The Model Column method. used by EC·2 and also the Moment Magnifier method used by ACl 318-99 building code for the design of HSC slender columns showed to be conservative for r design of HSC slender columns. The slenderness ratio limit between short and long columns required by the ECC-95 need to be modified when applied for HSC columns. Based on the results or these tests. the second order effect should be taken into account for HSC columns when λ1> 40 (λb> 12 ).

Non-linear Analysis of Slender High Strength Concrete Column

This article shows the influence of axial force eccentricity on high strength concrete columns design. The behavior of columns made of normal, middle and high strength concrete with slenderness values between 20 and 60 under an eccentric axial force has been studied. Structural analysis has been developed by means of software which considers both geometrical and mechanical non-linearity. The sequence of points defined by increasing values of axial force and bending moment produced by eccentricity has been represented on the cross-section interaction diagram until failure for each tested column. Then, diagrams depicting the relationship between failure axial force and column's slenderness have been drawn. The loss of bearing capacity of the member for normal and middle strength columns when compared with the bearing capacity of their cross-section is more noticeable as axial force eccentricity assumes higher values. However, this situation reverses for high strength columns with high slenderness values. On the basis of results obtained, the accuracy level for the moment magnifier method was checked. Despite the good concordance in most of the cases, it was verified that the moment magnifier method leads to excessively tight results for high strength concrete columns with high slenderness values. In these specific cases, a coefficient which amends the column rigidity is proposed so as to obtain safer values.

Stiffness Analysis of Recycled Self‐Compacting Concrete‐Filled Circular Steel Tubular Columns under Eccentric Compression

In this study, twenty‐one recycled self‐compacting concrete‐filled circular steel tubular (RSCFCST) columns are designed and tested under eccentric compression. The studied parameters including the replacement ratio of the recycled coarse aggregate (RCA), strength grade of concrete, eccentricity, and length‐diameter ratio (L/D) of specimens are considered. The load‐stiffness curves of the specimens are obtained by observing the whole process of loading, and the effects of various parameters on the stiffness of the specimens are analyzed. Test results demonstrate that the RSCFCST short columns (L/D ≤ 4) under eccentric compression exhibit drum‐like bending failures, while the RSCFCST long columns (L/D > 4) under eccentric compression experience the global flexural buckling failure modes. With the replacement ratio of RCA, the length‐diameter ratio or eccentricity increases, and the bearing capacity of specimens under eccentric compression decreases. However, the increase in the strength grade of concrete increases the bearing capacity. The stiffness of the RSCFCST columns under eccentric compression gradually increases as the strength grade of concrete increases, while the eccentricity had an adverse effect on stiffness of specimen. With the increase of load, the increase of the length‐diameter ratio would accelerate the stiffness degradation of specimen. The effect of the replacement ratio of RCA on stiffness of specimen in the elastic stage is not obvious. A validated FE model is employed to conduct parametric studies to widen the available test results. Additionally, an analytical model for predicting the effective stiffness of the RSCFCST columns under eccentric compression is proposed based on the moment magnifier method, and verification of this method is performed using the test data and FE analysis.

Effective Length of Reinforced Concrete Columns in Braced Frames

The American Concrete Institute (ACI) 318-11 permits the use of the moment magnifier method for computing the design ultimate strength of slender reinforced concrete columns that are part of braced frames. This computed strength is influenced by the column effective length factor K, the equivalent uniform bending moment diagram factor C m and the effective flexural stiffness EI among other factors. For this study, 2,960 simple braced frames subjected to short-term loads were simulated to investigate the effect of using different methods of calculating the effective length factor K when computing the strength of columns in these frames. The theoretically computed column ultimate strengths were compared to the ultimate strengths of the same columns computed from the ACI moment magnifier method using different combinations of equations for K and EI. This study shows that for computing the column ultimate strength, the current practice of using the Jackson–Moreland Alignment Chart is the most accurate method for determining the effective length factor. The study also shows that for computing the column ultimate strength, the accuracy of the moment magnifier method can be further improved by replacing the current ACI equation for EI with a nonlinear equation for EI that includes variables affecting the column stiffness and proposed in an earlier investigation.

Nonlinear Second-Order Effect of Non-Sway Columns

The moment magnifier method was introduced to analyze the second-order effect of slender columns, which can eliminate extensive calculations based on the solution to a differential equation. The formula adapted in this method incorporates the effective flexural stiffness (EI) and equivalent uniform moment diagram factor (Cm), which are influenced by eccentricity and slenderness of columns. For a non-sway reinforced concrete column, by subdividing the column into segments with different stiffness ratios and establishing a differential equation for each segment, a set of nonlinear equations illustrated by the deformation of segments was derived based on the deformation compatibility. With the force equilibrium conditions, new nonlinear equations for EI and Cm were then developed for the design of the slender reinforced concrete column. Comparison between the experimental data from different sources showed that the proposed equations can provide a better response prediction than the equations adopted in the ACI (American Concrete Institutue) code for columns.

Effective flexural stiffness of slender reinforced concrete columns under axial forces and biaxial bending

Most of the design codes (ACI-318-2008 and Euro Code–2-2004) propose the moment magnifier method in order to take into account the second order effect to design slender reinforced concrete columns. The accuracy of this method depends on the effective flexural stiffness of the column. This paper proposes a new equation to obtain the effective stiffness EI of slender reinforced concrete columns. The expression is valid for any shape of cross-section, subjected to combined axial loads and biaxial bending, both for short-time and sustained loads, normal and high strength concretes, but it is only suitable for columns with equal effective buckling lengths in the two principal bending planes. The new equation extends the proposed EI equation in the “Biaxial bending moment magnifier method” by Bonet et al. (2004) [6], which is valid only for rectangular sections. The method was compared with 613 experimental tests from the literature and a good degree of accuracy was obtained. It was also compared with the design codes ACI-318 (08) and EC-2 (2004) improving the precision. The method is capable to verify and design with sufficient accuracy slender reinforced concrete columns in practical engineering design applications.

Experimental study of reinforced masonry walls subjected to combined axial load and out-of-plane bendingThis article is one of a selection of papers published in this Special Issue on Masonry.

Twelve reinforced masonry wall specimens with nominal dimensions of 2400 mm × 800 mm × 150 mm were tested under eccentric compressive loading with varying eccentricity to thickness ratios, e/t, and end eccentricity ratios, e1/e2. Pinned-pinned support conditions resulted in a slenderness ratio of 17.1 for all specimens. Test results showed that the variation of ultimate load, Pu, and effective modulus of rigidity values, EIeff, at failure depended on the type of failure mode, which was influenced by e/t and e1/e2ratios and their interaction. Comparing ultimate loads obtained by test against those calculated using the EIeffvalues from the Canadian standard CSA S304.1-04 and against the ones calculated using the EIeffvalues proposed herein indicates that, while the moment magnifier method used in the current Canadian design standard to account for secondary effects is effective, the standard underestimates EIeffvalues, especially in regions where compression-controlled failure tends to predominate and, thus, leads to a conservative design. However, the use of proposed EIeffvalues in combination with the moment magnifier method provides estimations of ultimate loads in reasonably good agreement with test results.

Nonlinear EI Equation for Slender Composite Columns Bending about the Minor Axis

The moment magnifier method was introduced into design practice to eliminate the need for extensive calculations, based on the solution to a differential equation, for the second-order moments in slender columns. This method is strongly influenced by the effective flexural stiffness (EI) , which varies due to the nonlinearity of the concrete stress–strain curve and the cracking along the column length among other factors. The EI equations given in the ACI building code are approximate when compared to the EI values computed from the axial load–bending moment–curvature relationships. This study was undertaken to determine the influence of a full range of variables on EI used for the design of slender, tied, composite columns in which the bending moment was applied about the minor axis of the steel shape encased in concrete, and also to examine the existing ACI EI equations. Approximately 12,000 isolated square composite columns, each with a different combination of specified properties of variables, were s...

Biaxial bending moment magnifier method

This paper proposes a simplified method to design slender rectangular reinforced concrete columns with doubly symmetric reinforcement subjected to combined axial loads and biaxial bending, which is valid for short-time and sustained loads and for both normal and high strength concretes. The method is based on the so-called moment magnifier method (ACI-318-02) and EuroCode-2 (draft for stage 49) and it is only suitable for columns with equal effective buckling lengths in the two principal bending planes. A new equation of the column stiffness EI is proposed with which to obtain the moment magnification factor for single and double curvature. The method was compared with 371 experimental tests from the literature and a good degree of accuracy was obtained. Precision for sustained loads was improved in comparison with the method proposed by the American ACI-318 (02) and the European EC-2 (September 2003). The method allows slender reinforced concrete columns to be both checked and designed with enough accuracy for practice engineering.

Flexural Stiffness of Reinforced Concrete Columns and Beams: Analytical Approach

The present ACI code 318-02 provisions on effective stiffnesses of beams and columns are reviewed. Factors influencing moments of inertia of beams and columns are discussed. The primary variables considered are: the reinforcement ratio, the axial load ratio, the eccentricity ratio, and the compressive strength of concrete. On the basis of a parametric study, simple formulas are proposed to determine the effective stiffnesses of reinforced concrete columns and beams. The proposed stiffness expressions are applicable for all levels of applied loading, including both service and ultimate loads. The analytical results show that the flexural stiffness assumption in the current ACI code procedure for design of slender columns using the moment magnifier method (Eq. 10-11 and 10-12) is extremely conservative. Recommendations are made concerning stiffness assumptions in the analysis of reinforced concrete frames under lateral loads.

Reinforced Concrete Columns under Unequal Load Eccentricities

This paper gives experimental results and findings from analysis of high-strength concrete (HSC) columns subjected to either equal or unequal load eccentricities. 18 HSC columns were tested, and a computer-aided method to analyze reinforced concrete columns is described. Using the results of a parametric study based on the analytical method, the simplified formula for equivalent eccentricity recommended in the ACI 318 and AS 3600 Codes is evaluated. It was found that the equation recommended in these design codes for estimating the equivalent moment correction factor for use in the moment magnifier method may result in an unsafe prediction of the failure load of reinforced concrete columns. A modified simple equation that predicts the failure load conservatively is proposed.

Moment Magnifier Method for Long-Term Behavior of Flat Plates Subjected to In-Plane Compression

Floor slabs of deep basement structures are subjected to large compressive forces due to soil and hydraulic lateral pressures. In this study, nonlinear finite-element analyses were carried out to improve the existing moment magnifier method so that it is applicable to the long-term behavior of the flat plates. For this purpose, a computer program that can address creep and shrinkage as well as geometrical and material nonlinearity was developed. Parametric studies were performed to investigate the characteristics of the long-term behavior and the load-carrying capacities of flat plates. In the proposed moment magnifier method, the creep factor was introduced to accommodate the effect of creep on the load-carrying capacity. A design example was presented to demonstrate the proposed design method.

The behavior of reinforced concrete columns subjected to axial force and biaxial bending

When stress is beyond elastic limit or cracking occurs in a reinforced concrete member subjected to axial force and biaxial bending, curvature about each principal axis of gross section may be influenced by axial force and bending moments about both major and minor principal axes. It is mainly due to the translation and rotation of principal axes of the cross section after cracking. In this study, a numerical method for predicting the behavior of reinforced concrete columns subjected to axial force and biaxial bending is proposed considering curvature localization. To verify the proposed numerical method, a series of tests was also carried out for 16 tied reinforced concrete columns with 100×100 mm square and 200×100 mm rectangular sections under various loading conditions. The boundary conditions at both ends of the column were hinged and eccentricities (40 mm) were equal and of the same directions. The angles between the direction of eccentricity and the major principal axis of gross section were 0°, 30°, 45° for the square section and 0°, 30°, 45°, 60°, 90° for the rectangular section. A comparison between the numerical predictions and the test results shows good agreements in ultimate loads, axial force-lateral deflection relations, and lateral deflection trajectories. It is also found, in this limited investigation, that ACI's moment magnifier method is conservative in both uniaxial and biaxial eccentric loading conditions.

RC Flat Plate under Combined In-Plane and Out-of-Plane Loads

A numerical study using nonlinear finite-element analysis is performed to investigate the behavior of reinforced-concrete flat plates subject to combined in-plane compressive and out-of-plane floor loads. For the nonlinear finite-element analysis, a computer program addressing material and geometric nonlinearities is developed. The flat plates to be studied are designed according to the direct design method. Through studies on the effects of different load combinations and loading sequence, the load condition that governs the strength of the flat plates is determined. And, for the flat plates under the governing load condition, parametric studies are performed to investigate the variations of the buckling coefficient and the effective flexural rigidity. As a result, this paper provides rational design rules for the moment magnifier method that is applicable to the flat plates.

Design of slender shear connected masonry cavity walls

In recent work the writers carried out extensive testing and numerical analysis investigation of slender cavity masonry walls constructed with vertically aligned shear connectors and subjected to eccentric vertical compressive loads. This type of masonry tie has been shown to provide increased stiffness and strength characteristics to the wall system. In this paper a multilinear regression analysis is carried out based on the data generated by that investigation. The effective stiffness of the shear connected cavity walls is evaluated. A design approach for these walls when subjected to eccentric compressive loads is proposed. This design approach makes use of the effective stiffness in the context of the moment magnifier method. It is then verified against the code requirements and available test data by the writers and others.

Buckling behaviour of slender high-strength concrete columns

A numerical method for predicting the behaviour of a reinforced concrete column is proposed, using the layered finite element method. To investigate the effects of concrete strength and longitudinal steel ratio on the ultimate load and axial force-bending moment relation of a column, a series of tests was carried out for 30 tied reinforced concrete columns with a 80 mm square cross-section and three slenderness ratios of 10, 60 and 100. Three different concrete strengths of 25.5, 63.5 and 86.2 MPa, and two different longitudinal steel ratios of 1.98 and 3.95% were used. The boundary conditions at the ends were both hinged and end eccentricities (24 mm) were equal and of the same sign. Experimental results revealed that the ultimate load for a short high-strength concrete column was significantly increased but not for a slender column. The possibility of stability failure for a slender column was increased with the increase of concrete strength. The increment of ultimate load due to increase in longitudinal reinforcement for the short column was less than for the slender one, and the heavier reinforcement for the slender column led to a more stable column. The predicted behaviours of the concrete columns by the numerical method proposed herein show good agreement with the test results, they also show that the ACI's moment magnifier method may be unconservative for very slender high-strength concrete column.

Approximate Moment‐Curvature Relationships for Slender Columns

This paper presents the derivation of an approximate second‐order method for the analysis and design of slender cantilever compression members. The method is based on the generation of equivalent linear representations for the nonlinear moment‐curvature (M‐Φ) relationships for the column cross section. The linear M‐Φ equations are derived in such a way that they closely predict the nominal strength of the slender column for a prescribed axial load. The derivation is based on the guidelines of the German Code and Heft 220—which, unlike the moment magnifier method, takes into account the influence of the axial load on the stiffness of the column. The results of the proposed method are in excellent agreement with experimental data and predictions from sophisticated computer programs.

Slender Reinforced Concrete Bridge Towers under Cyclic Lateral Load

The nonlinear behavior of a cable‐stayed‐bridge tower with slender columns is generally affected not only by material nonlinearity, but by geometric nonlinearity as well. This geometric nonlinear effect is accentuated by a sustained gravity load, an increase of lateral load and a stiffness reduction of the structure. A series of static analyses up to failure are conducted with various parameters that govern the nonlinear behavior of an A‐shaped bridge tower, under sustained gravity load combined with seismic lateral load perpendicular to the longitudinal axis of the bridge. The present study particularly focuses on the influence of creep deformation due to gravity load and of cyclic lateral load on the ultimate strength and deformation characteristics of the tower and the applicability of the current “moment magnifier method” in the American Concrete Institute (ACI) code to the analysis and design of this type of structure. Analytical results indicate that no significant effect of creep and cyclic load is...