Slope-deflection Equations Research Articles

Design method for determining appropriate stiffness for end point braces

The high dependence of end point braces on rotational restraint conditions differentiates them from intermediate point braces. Although the point brace stiffness equation of the American Institute of Steel Construction (AISC) design specifications applies to both braces, it restricts the stiffness to only the buckling load based on the unbraced length and an effective length factor of 1, which is too conservative for end point braces. This study investigated the relationship between end point brace stiffness and column buckling loads through buckling analyses using slope–deflection equations with stability functions, considering diverse supports, end rotation constraints, and column counts. Irrespective of the column count, the columns with end point braces exhibited a non-sway elastic buckling load of up to four times the Euler buckling load depending on the rotational restraint at the column ends. The proposed procedure enhances the AISC regulations by introducing a new equation that allows up to 90% utilization of the buckling load based on the unbraced length and an effective length factor of 1 or less, considering column end rotational constraints. The procedure was applied to six frames undergoing elastic or inelastic buckling and validated by assessing the buckling loads and related point brace stiffness. The proposed procedure significantly enlarged the load-bearing capacity of the column compared with the AISC design procedure, although both approaches obtained comparable point brace cross-sectional areas. It extends the buckling load limit for columns or frames with an end point brace, providing predictive insights into brace stiffness to improve structural design.

Lateral displacement mode of column-supported modular steel structures with semi-rigid connections

The lateral displacement of a column-supported modular steel structure with semi-rigid connections (CSMSS-SRC) owing to lateral loading was investigated to determine the corresponding lateral displacement mode and establish a foundation for a direct displacement-based seismic design method. Theoretical equations for the lateral displacement caused by the bending deformation of beams and columns, axial deformation of columns, and shear deformation of the vertical inter-module connections were theoretically derived using slope–deflection equations, unit-load method, and shear deformation definition, respectively. Two simplified single frame finite element models were then established to verify the theoretical equations and study the effects of the rotational and horizontal shear stiffnesses of the vertical inter-module connections on the lateral displacement of the CSMSS-SRC. The results showed that the proposed lateral displacement equations are highly accurate. Furthermore, although the rotational stiffness of the vertical inter-module connection did not exert an appreciable effect on the lateral displacement of the CSMSS-SRC, its horizontal shear stiffness did. These results are expected to provide a reference for the engineering design of CSMSS-SRC.

Effective buckling length of frames with tapered columns and partially tapered beams

This paper presents stability analysis of steel frames composed of linearly tapered columns and partially tapered restraining beams. In the first part of the stability analysis, closed-form equations for the bending stiffness of the symmetrical and unsymmetrical partially tapered restraining beam are derived and presented in form of design charts. Subsequently, slope deflection equations of linearly tapered column subjected to axial compressive load are then assembled along with the bending stiffness equations of partially tapered beams to determine the elastic critical load and the effective buckling length factor for non-sway and sway steel frames. Numerical examples and efficient design charts are presented for several braced and unbraced steel frames. Moreover, the effect of gravity loads acting on leaning columns connected to sway frames is investigated. Based on this investigation it is concluded that correction of effective buckling length factor based on story buckling concept can be extended to cover steel frames with tapered column as well.

Impact of shear forces on variable cross section shaft stiffness

Influence of shear forces is presented in this paper for static analysis of a shaft with variable cross-section. A stiffness matrix method is used to evaluate displacement and reaction forces on the shaft in regard to the slope deflection equations. The Timoshenko beam theory is addressed according to the shear coefficient and the shear correction factor. Bending moments are calculated.

Semi-analytical solution for gable roofs under uplift wind loads

A common type of failure observed during hurricane events is the detachment of the roofs of light-frame wood houses because of the inability of the roof-to-wall connection (RTWC) to transfer uplift wind loads acting on the roofs. This paper introduces a computationally efficient procedure for conducting nonlinear analysis of the roofs to determine the connections forces using a semi-analytical solution. The new procedure simulates an entire roof as a continuous beam resting on elastic supports through the use of statically indeterminate slope deflection equations that include shear deformations. It accounts for the two-dimensional spatial variation of the uplift wind pressure. The new model was validated against a three-dimensional finite element model and also through a comparison with the results of experimental testing of a gable roof house previously conducted at the University of Western Ontario, Canada. The model was used to determine the load sharing among the trusses under a realistic load. It was also used to assess the adequacy of using the tributary area method under a code load. Based on the validation results, the semi-analytical solution model demonstrated an ability to predict the response of a roof under uplift wind loads that vary both in time and space. The main advantage of the new model is the efficient computational time compared to three-dimensional finite element modeling. As such this model can be used in the future for conducting probabilistic analysis of the roof taking into account the variation of the connection properties.

Behavior of Sway Two-Bay, Two-Story Composite Steel Frames in Fire

The experimental results of three full-scale two-story, two-bay, composite steel frames under furnace loading are described briefly. During tests, the composite beams and steel columns were exposed to fire, and only the beam-to-column connections of the tested frames were protected. The furnace temperature, composite steel beam temperature, steel column temperature, horizontal displacements of the columns, and vertical deflections of the composite beams as well as the complete deformation process of the test frames observed during the heating phase and the cooling phase are given. By introducing the slope-deflection equations and joint equilibrium conditions at each joint, the critical temperatures of the buckling of two-story, two-bay, composite steel frames are determined for different furnace scenarios; therefore, this method can consider the effect of the axial force at different levels. The calculated critical temperatures agree well with the experimental results.

Tangent Stiffness Matrix of Spatial Beam Element Considering Bend and Torsion Coupling Effect

Based on the spatial beam-column differential equations, the slope deflection equations considering second-order and bend-torsion coupling effect are established. The additional moment caused by torsion and bend deflection are taken into account. Then the finite element pattern of spatial beam-column considering the couple effect of torsion and bend is given. The tangent stiffness matrix and relevant program for nonlinear analysis are further obtained. By the nonlinear calculation and stability analysis of single component which has high accuracy or precise solution, the precision of the FEM model given in this paper is verified by comparing the results with that given in references.

Out-of-plane buckling of Y-braced frames with rigid joints

Because of the complicated buckling characteristics of Y-shaped bracings, calculation of their strength is beyond routine engineering procedures. A method based on slope–deflection equations incorporating stability functions has been used for computation of buckling eigenvalues and effective length factors. Lateral strength is computed based on the least buckling strength of bracing members. This study is limited to bracings with similar sections and fixed end connections resisting out-of-plane rotation. Lateral strengths predicted by the analytical method for two cases are compared with experimental results on full-scale specimens. It is shown that the proposed analytical method predicts the real strength with a good precision for practical purposes.

Elastic buckling load of multi-story frames consisting of Timoshenko members

The objective of this paper is to propose a method for the evaluation of the elastic critical buckling load of columns in frames consisting of members susceptible to non-negligible shear deformations, such as built-up members in steel frames, based on Engesser's approach. To that effect, a stability matrix is proposed and three general stability equations are derived for the cases of unbraced, partially braced and braced frames. Indicative graphic interpretation of the solutions for the stability equations of the braced and unbraced cases is shown. Slope-deflection equations for shear-weak members with semi-rigid connections are also derived and used for the presentation of a complete set of rotational stiffness coefficients, which are then used for the replacement of members converging at the bottom and top ends of the column in question by equivalent springs. All possible rotational and translational boundary conditions at the far end of these members, as well as the eventual presence of axial force, are considered. Five examples are presented, dealing with braced, unbraced and partially braced frames, with rigid and semi-rigid beam to column connections, loaded with concentrated or uniformly distributed loads, in a symmetrical or non-symmetrical pattern. In all cases the proposed approach is in excellent agreement with finite element results.

Computer Program for Exact Analysis of Flat Slabs and Beam-Supported Slabs

Two-way slab structures, owing to their economical and architectural advantages, are used extensively in civil engineering. In addition, these systems not only resist gravity loads and transfer them to the columns, but also sometimes in combination with other structural members make a lateral resisting system. Elastic analysis of equivalent frame method is a precise and comprehensive method which is widely used for analysis of two-way slab structures. Because of non-prismatic sections of slab-beams and columns, carry over and stiffness factors and fixed end moments of members with constant moment of inertia could not be used. Some tables have presented these values for different types of the slab-beams, columns and torsional members. Due to low precision of these tables and complicated and time-consuming procedure of equivalent frame method, precise analysis of two-way slab systems using equivalent frame method including stiffness degradation in all members, using slope-deflection equations, in the form of a prepared software is pursued in this research. Two analyses of example structures were performed and the efficiency and accuracy of the program were verified by comparing the hand-calculated results, results of a finite element approach and the developed computer program's results, which shows some differences in the support moments.

Second-order slope–deflection equations for imperfect beam–column structures with semi-rigid connections

A new set of second-order slope–deflection equations for Euler–Bernoulli beam–columns of symmetric cross-section including the effects of initial imperfections (i.e., initial curvature, out-of-plumbness and axial load eccentricities) and semi-rigid connections is developed in a classical manner. The proposed method has the following advantages: (1) it can be utilized in the stability and second-order analysis of framed structures made of Euler–Bernoulli imperfect beam–columns with rigid, semi-rigid, and simple connections subject to axial and transverse loads; (2) the effects of semi-rigid connections and member imperfections are condensed into the slope–deflection equations for tension and compression axial loads; (3) it is more accurate than any other method available in the technical literature and capable of capturing not only the first-order and second-order elastic responses of frames made of imperfect beam–columns but also the phenomenon of reversals of deflections along the members as the axial loads are increased; and (4) it is powerful, practical, versatile and easy to apply. Analytical studies indicate that the initial imperfections (a) act as if they were additional transverse loads proportional to the bending stiffness and magnitudes of the imperfections of the corresponding beam–column; and (b) are detrimental to structures, increasing the lateral deflections, moments, and shears, and also reducing the critical axial loads of beam–columns and framed structures. This is particularly critical in structures made of beam–columns with initial crookedness and low stiffness connections subjected to high compressive axial loads. In addition, the effects of initial curvature are amplified by the compressive axial loads applied at the ends of the member. Four comprehensive examples are included that show the effectiveness of the proposed method.

Timoshenko beam–column buckling. Does Dario stand the test?

In the paper [Aristizabal-Ochoa JD. Slope–deflection equations for stability and second-order analysis of Timoshenko beam–column structures with semi-rigid connections. Eng Struct 2008;30(9)] the Haringx stability theory is chosen. In the present paper it is shown that this theory yields a wrong limit value for very weak-in-shear buckled beam–columns. The theory is derailing seriously in a very unsafe manner. The failure is shown in three ways, first by considering the case of an ideal weak-in-shear beam–column, next by reasoning arguments and finally by referring to older studies. It has not been sufficiently recognized that Haringx studied the lateral buckling of helical springs, and did not consider the critical buckling load for extremely shear-weak structural members. His theory appears to be reliable for helical springs, but is inadequate for shear-weak beam–columns. Results in several beam–column papers by Dario were obtained on the basis of the Haringx theory and, therefore, must be used with utmost care, if not extremely distrusted.

Slope-deflection equations for stability and second-order analysis of Timoshenko beam–column structures with semi-rigid connections

Slope-deflection equations for stability and second-order analysis of Timoshenko beam–column structures with semi-rigid connections

Slope-deflection equations for stability and second-order analysis of Timoshenho beam–column structures with semi-rigid connections

A new set of slope-deflection equations for Timoshenko beam–columns of symmetrical cross section with semi-rigid connections that include the combined effects of shear and bending deformations, and second-order axial load effects are developed in a classical manner. The proposed method that also includes the effects of the shear component of the applied axial force as the member deflects laterally (Haringx Model) has the following advantages: (1) it can be utilized in the stability and second-order analyses of framed structures made up of Timoshenko beam–columns with rigid, semi-rigid, and simple end connections; (2) the effects of semi-rigid connections are condensed into the slope-deflection equations for tension or compression axial loads without introducing additional degrees of freedom and equations; (3) it is more accurate than any other method available and capable of capturing the phenomena of buckling under axial tension forces; and (4) it is powerful, practical, versatile and easy to teach. Analytical studies indicate that shear deformations increase the lateral deflections and reduce the critical axial loads of framed structures made of members with low shear stiffness. The effects of shear deformations must be considered in the analysis of beam–columns with relatively low effective shear areas (like laced columns, columns with batten plates or with perforated cover plates, and columns with open webs) or with low shear stiffness (like elastomeric bearing and short columns made of laminated composites with low shear modulus G when compared to their elastic modulus E ) making the shear stiffness GA s of the same order of magnitude as EI / L 2 . The shear effects are also of great importance in the static, stability and dynamic behavior of laminated elastomeric bearings used for seismic isolation of buildings. Four comprehensive examples are included that show the effectiveness of the proposed method and equations.

Analytical and experimental study on the behavior of elastically supported reinforced concrete decks

Current design specifications prescribe that the upper and lower reinforcement mat is required in the same amount to resist negative and positive moment in bridge decks. This design concept is primarily based on the unrealistic assumption that the girder plays a role of rigid support against deck deflection. In reality, however, girders are flexible and the deflection of girders affect the behavior of deck slabs. In the present study, an analytical method was developed to take the effect of the girder flexibility on the deck behavior into account. The method was formulated based on the slope-deflection equations of plates and harmonic analysis. Unlike the conventional finite element analysis, the input and output schemes are simple and convenient. The validity of the presented study was verified by a series of comparative studies with finite element analyses and experimental tests. It was shown from the analyses that the negative transverse moments of decks were significantly reduced in many cases when the girder flexibility were appropriately taken into consideration whereas the positive moments tend to increase. This poses a strong need to improve the conventional design concept of decks on rigid girders to those on flexible girders.

Inverse Design of Frames for Specified Second-Order Bending Strains

A new inverse design method for frames has been presented by means of an inverse formulation including frame stability effects. Elastic building frames in non-earthquake-prone areas, designed primarily for light horizontal wind loads, are considered. Governing equations based on the well-known second-order slope-deflection equations are derived for fundamental problems for a set of specified member-end fiber bending strains, as a target performance. An explicit expression for the solution stiffnesses is obtained for a hybrid-inverse problem. Further consideration is given for a class where the member stiffness set is determined only by a strain mode such as for a frame consisting of columns under central compression. It is demonstrated that the strain mode of an entire multistory frame can be synthesized by overlapping of the strain modes of decomposed unit frames. The design formulas for the simple models in this paper not only have disclosed the essential characteristics of design solutions but also play the role of initial backbone solutions in the neighborhood of which perturbed solutions can be derived for problems under more practical conditions.

Tangent Stiffness Equations for Laterally Distributed Loaded Members

Tangent stiffness equations for a beam-column, which is subjected to either uniformly or sinusoidally distributed lateral loads, are presented. The equations have been derived by differentiating the slope-deflection equations under axial forces for a member. Thus, the tangent stiffness equations take into consideration axial forces, bowing effect, and laterally distributed loads. As a numerical example, elastic buckling behavior of parallel chord latticed beams with laterally distributed loads is investigated to compare the results obtained from the present method with those from the conventional matrix method in which the distributed loads are considered as a series of concentrated loads at additional intermediate nodes of a member. Furthermore, buckling tests were carried out to confirm the equations derived as well as to clarify the buckling behavior of space frame structures. In conclusion, it can be said that the new equations can provide a good efficient way of estimating the equilibrium paths and buckling loads. They can also lead to a significant savings in core storage and computing time required for the analysis of space frame structures.

Stiffness values and static analysis of flat plate structures

Flat plate constructions are structural systems which are directly placed on columns without any beams. Various solution methods have been introduced for the solution of flat plate structures under horizontal and vertical loads. In most of these solution methods, models comprising of one column and one plate have been studied. In other solutions, however, co-behavior of two reciprocal columns has been investigated. In this study, interrelations of all the columns on one storey have been examined. At the end of the study structure consisting of nine columns and four plates has been chosen as a model. Then unit moment has been successively applied to each of these columns and unit moments carried over the other columns have been found. By working out solutions far plates and columns varying in ratio, carry-over factors have been found and these factors given in tables. In addition, fixed-end moment factors on the columns arising due to vertical load were also calculated. Then citing slope-deflection equations to which these results could be applied, some examples of moment and horizontal equilibrium equations have been given.

Connection element method for the analysis of semi-rigid frames

A general procedure for incorporating the effects of joint flexibility into standard methods for the analysis of frames is presented. The method can allow for joint flexibility associated with all six degrees of freedom normally considered when analysing three-dimensional frames, including coupling between deformations. It can also allow for the finite size and exact location of connections. Simple examples using the moment distribution technique, slope deflection equations and matrix stiffness method illustrate the application of the proposal. Additional examples to illustrate the effects of connection length on the frame moment distribution, lateral drift and column load capacities are also present.

Some methods for structural analysis

Solutions based on fundamental aspects of beam theory are used for establishing some new methods in the structural analysis of continuous beams and frames. New slope-deflection equations are given which represent an improvement over the conventional slope-deflection method. A method using end moments and deflections as primary quantities is presented as an alternative procedure to the slope-deflection method. This alternative method is referred to as the moment - or curvature-deflection method. The method and some results based on the alternative procedure are compared with the slope-deflection method through illustrative examples. A third procedure of using Fourier series in conjunction with the Stokes' transformation is discussed. The Stokes' transformation procedure suggests that the choice of end curvatures and deflections as primary quantities used in the moment-deflection method is more natural than choosing the end slopes and deflections.