Tham khảo tài liệu 'plastics product design part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 220 Plastics Engineered Product Design In the following formula p axial load I length of column I least moment of inertia k least radius of gyration E modulus of elasticity y lateral deflection at any point along a larger column that is caused by load p. If a column has round ends so that the bending is not restrained the equation of its elastic curve is H g - -p 4-12 dx2 When the origin of the coordinate axes is at the top of the column the positive direction of X being taken downward and the positive direction of y in the direction of the deflection. Integrating the above expression twice and determining the constants of integration give 4-13 which is Euler s formula for long columns. The factor Í2 is a constant depending on the condition of the ends. For round ends Q 1 for fixed ends Q 4 for one end round and the other fixed Í2 . Pis the load at which if a slight deflection is produced the column will not return to its original position. If p is decreased the column will approach its original position but if p is increased the deflection will increase until the column fails by bending. For columns with value of l k less than about 150 Euler s formula gives results distinctly higher than those observed in tests. Euler s formula is used for long members and as a basis for the analysis of the stresses in some types of structural parts. It always gives an ultimate and never an allowable load. Torsion A bar is under torsional stress when it is held fast at one end and a force acts at the other end to twist the bar. In a round bar Fig. with a constant force acting the straight-line ab becomes the helix ad and a radial line in the cross-section ob moves to the position ad. The angle bad remains constant while the angle bod increases with the length of the bar. Each cross section of the bar tends to shear off the one adjacent to it and in any cross section the shearing stress at any point is normal to a radial line drawn through the point. Within the .
