ELASTIC BUCKLING ANALYSIS OF BEAM-COLUMN

Abstract

Beam-Columns are the essential elements of a frame structure that simultaneously carry axial and transverse loads. When the axial loads are compressive and large, then loss of stiffness and instability due to buckling may occur. Modern computer programs are often incapable of capturing the effect of buckling instability and loss of stiffness due to compressive axial loads and increase in stiffness due to tensile axial loads. The reason for this debility in modern computer programs arises, because the structure state matrices like the mass matrix and the stiffness matrix are based upon material properties (E, G, 𝜌 , 𝜇 ) and geometric properties (𝐿, 𝐴, 𝐼􀯫􀯫 , 𝐼􀯬􀯬, J). Neither the mass matrix, nor the stiffness matrix depends upon loads. The axial stiffness of a hinged-hinged truss element, for example, is based upon 𝐸𝐴/𝐿; whereas, Euler (1757) showed that hinged-hinged truss element undergoes buckling instability when compressive load P reaches 𝜋􀬶𝐸𝐼⁄𝐿􀬶. Here the buckling load is not dependent on x-sectional area 𝐴 of the member but on 2nd moment of area 𝐼 of the member. Therefore, it is obvious as to why modern computer programs cannot capture the effects of axial forces on stiffness. Buckling of beam-columns is very different from simple compression members. Although some non-linear analysis for calculation of buckling load in beam-columns had been done in the past. But if we search for linear analysis methods, there is some research gap in this field. While analysing, the stiffness matrix often does not include the effect of axial force. It only depends on material and geometric properties. It is presumed that stiffness is not changing throughout loading due to loads, and at the instant of buckling, when the axial load reaches Euler’s Load, it directly becomes zero. Although Buckling is an instantaneous phenomenon, the formulation is quite a difficult task. But we had tried to formulate it using rotational stiffness of Beam-columns, which also includes the effect of axial load. In this thesis, work emphasis has been given to incorporating the effect of axial load and slenderness of each member in stiffness calculation. I tried to develop a MATLAB program that calculates the buckling load for each member in the 2D frame structure. To validate, my results were compared with the work of Timoshenko and Clough and Penzien. Further study is being done to see how much there is a change in the results of analysis of a 2D frame structure after incorporation of the effects of axial load in stiffness. Both the program runs satisfactorily, and there is a significant change in the results of the analysis due to our new modification.