Buckling

aircraft. Thin skin panels buckle at very low loads. In the case shown here, the weight of the forward fuselage structure ahead of the nose undercarriage is sufficient to cause the panels to buckle. Buckled panels are still effective in carrying shear by diagonal tension. railway coach made of stainless steel, unevenness caused by buckling becomes apparent because of the reflection on the polished surface. In structural engineering, buckling is the sudden change in shape (deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a gradually increasing load, when the load reaches a critical level, a member may suddenly change shape and the structure and component is said to have buckled. Euler's critical load and Johnson's parabolic formula are used to determine the buckling stress of a column. Buckling may occur even though the stresses that develop in the structure are well below those needed to cause failure in the material of which the structure is composed. Further loading may cause significant and somewhat unpredictable deformations, possibly leading to complete loss of the member's load-carrying capacity.…

aircraft. Thin skin panels buckle at very low loads. In the case shown here, the weight of the forward fuselage structure ahead of the nose undercarriage is sufficient to cause the panels to buckle. Buckled panels are still effective in carrying shear by diagonal tension. ]] railway coach made of stainless steel, unevenness caused by buckling becomes apparent because of the reflection on the polished surface.]] In structural engineering, buckling is the sudden change in shape (deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a gradually increasing load, when the load reaches a critical level, a member may suddenly change shape and the structure and component is said to have buckled. Euler's critical load and Johnson's parabolic formula are used to determine the buckling stress of a column. Buckling may occur even though the stresses that develop in the structure are well below those needed to cause failure in the material of which the structure is composed. Further loading may cause significant and somewhat unpredictable deformations, possibly leading to complete loss of the member's load-carrying capacity. However, if the deformations that occur after buckling do not cause the complete collapse of that member, the member will continue to support the load that caused it to buckle. If the buckled member is part of a larger assemblage of components such as a building, any load applied to the buckled part of the structure beyond that which caused the member to buckle will be redistributed within the structure. Some aircraft are designed for thin skin panels to continue carrying load even in the buckled state. F_c = \frac{\pi^2 EI}{(KL)^2} where F_c , maximum or critical force (vertical load on column), E , modulus of elasticity, I , smallest area moment of inertia (second moment of area) of the cross section of the column, L , unsupported length of column, K , column effective length factor, whose value depends on the conditions of end support of the column, as follows. For both ends pinned (hinged, free to rotate), K = 1.0 . For both ends fixed, K = 0.50 . For one end fixed and the other end pinned, K \approx 0.699 . For one end fixed and the other end free to move laterally, K = 2.0 . K L is the effective length of the column. Examination of this formula reveals the following facts with regard to the load-bearing ability of slender columns. The elasticity of the material of the column and not the compressive strength of the material of the column determines the column's buckling load. The buckling load is directly proportional to the second moment of area of the cross section. The boundary conditions have a considerable effect on the critical load of slender columns. The boundary conditions determine the mode of bending of the column and the distance between inflection points on the displacement curve of the deflected column. The inflection points in the deflection shape of the column are the points at which the curvature of the column changes sign and are also the points at which the column's internal bending moments of the column are zero. The closer the inflection points are, the greater the resulting axial load capacity (bucking load) of the column. A conclusion from the above is that the buckling load of a column may be increased by changing its material to one with a higher modulus of elasticity (E), or changing the design of the column's cross section so as to increase its moment of inertia. The latter can be done without increasing the weight of the column by distributing the material as far from the principal axis of the column's cross section as possible. For most purposes, the most effective use of the material of a column is that of a tubular section. Another insight that may be gleaned from this equation is the effect of length on critical load. Doubling the unsupported length of the column quarters the allowable load. The restraint offered by the end connections of a column also affects its critical load. If the connections are perfectly rigid (not allowing rotation of its ends), the critical load will be four times that for a similar column where the ends are pinned (allowing rotation of its ends). Since the radius of gyration is defined as the square root of the ratio of the column's moment of inertia about an axis to its cross sectional area, the above Euler formula may be reformatted by substituting the radius of gyration A r^2 for I : \sigma = \frac{F}{A} = \frac{\pi^2 E}{(l/r)^2} where \sigma = F/A is the stress that causes buckling in the column, and l/r is the slenderness ratio. Since structural columns are commonly of intermediate length, the Euler formula has little practical application for ordinary design. Issues that cause deviation from the pure Euler column behaviour include imperfections in geometry of the column in combination with plasticity/non-linear stress strain behaviour of the column's material. Consequently, a number of empirical column formulae have been developed that agree with test data, all of which embody the slenderness ratio. Due to the uncertainty in the behavior of columns, for design, appropriate safety factors are introduced into these formulae. One such formula is the Perry Robertson formula which estimates the critical buckling load based on an assumed small initial curvature, hence an eccentricity of the axial load. The Rankine Gordon formula, named for William John Macquorn Rankine and Perry Hugesworth Gordon (1899 – 1966), is also based on experimental results and suggests that a column will buckle at a load F max given by: \frac{1}{F_\max} = \frac{1}{F_e} + \frac{1}{F_c} where F_e is the Euler maximum load and F_c is the maximum compressive load. This formula typically produces a conservative estimate of F_\max . h_\text{crit} = \left(\frac{9B^2}{4}\,\frac{EI }{\rho gA}\right)^\frac{1}{3} where g is the acceleration due to gravity, I is the second moment of area of the beam cross section, and B is the first zero of the Bessel function of the first kind of order −1/3, which is equal to 1.86635086... q_{cr}=\frac{3EI}{R^2} where E is the Young's modulus of the material and I is the second moment of area of the ring cross-section. For so-called dead loads, which remain constant in both magnitude and direction during deformation, the corresponding critical buckling pressure is q_{cr}=\frac{4EI}{R^2} \frac{\partial^4 w}{\partial x^4} + 2\frac{\partial^4 w}{\partial x^2 \partial y^2} + \frac{\partial^4 w}{\partial y^4} = \frac{12\left(1 - \nu^2\right)}{E t^3}\left(-N_x \frac{\partial^2 w}{\partial x^2}\right) where w , out-of-plane deflection N_{x} , uniformly distributed compressive load \nu , Poisson's ratio E , modulus of elasticity t , thickness The solution to the deflection can be expanded into two harmonic functions shown: w = \sum_{m=1}^\infty \sum_{n=1}^\infty w_{mn}\sin\left(\frac{m\pi x}{a}\right)\sin\left(\frac{n\pi y}{b}\right) where m , number of half sine curvatures that occur lengthwise n , number of half sine curvatures that occur widthwise a , length of specimen b , width of specimen The previous equation can be substituted into the earlier differential equation where n equals 1. N_x can be separated providing the equation for the critical compressive loading of a plate: N_{x, cr} = k_{cr} \frac{\pi^2 Et^3}{12\left(1 - \nu^2\right)b^2} where the buckling coefficient k_{cr} , is given by: k_{cr} = \left(\frac{mb}{a} + \frac{a}{mb}\right)^2 The buckling coefficient is influenced by the aspect of the specimen, a / {b} , and the number of lengthwise curvatures. For an increasing number of such curvatures, the aspect ratio produces a varying buckling coefficient; but each relation provides a minimum value for each m . This minimum value can then be used as a constant, independent from both the aspect ratio and m . Given stress is found by the load per unit area, the following expression is found for the critical stress: \sigma_{cr} = k_{cr}\frac{\pi^2 E}{12\left(1 - \nu^2\right)\left(\frac{b}{t}\right)^2} From the derived equations, it can be seen the close similarities between the critical stress for a column and for a plate. As the width b shrinks, the plate acts more like a column as it increases the resistance to buckling along the plate's width. The increase of a allows for an increase of the number of sine waves produced by buckling along the length, but also increases the resistance from the buckling along the width. This creates the preference of the plate to buckle in such a way to equal the number of curvatures both along the width and length. Due to boundary conditions, when a plate is loaded with a critical stress and buckles, the edges perpendicular to the load cannot deform out-of-plane and will therefore continue to carry the stresses. This creates a non-uniform compressive loading along the ends, where the stresses are imposed on half of the effective width on either side of the specimen, given by the following: \frac{b_\text{eff}}{b} \approx \sqrt{\frac{\sigma_{cr}}{\sigma_{y}} \left(1 - 1.022 \sqrt{\frac{\sigma_{cr}}{\sigma_y}}\right)} where b_\text{eff} , effective width \sigma_y , yielding stress As the loaded stress increases, the effective width continues to shrink; if the stresses on the ends ever reach the yield stress, the plate will fail. This is what allows the buckled structure to continue supporting loadings. When the axial load over the critical load is plotted against the displacement, the fundamental path is shown. It demonstrates the plate's similarity to a column under buckling; however, past the buckling load, the fundamental path bifurcates into a secondary path that curves upward, providing the ability to be subjected to higher loads past the critical load. C_b = \frac{12.5M_\max}{2.5M_\max + 3M_A + 4M_B + 3M_C} where M_\max , absolute value of maximum moment in the unbraced segment, M_A , absolute value of maximum moment at quarter point of the unbraced segment, M_B , absolute value of maximum moment at centerline of the unbraced segment, M_C , absolute value of maximum moment at three-quarter point of the unbraced segment, The result is the same for all unit systems. \sigma L = \rho c^2 h where \sigma is the impact stress, L is the length of the rod, c is the elastic wave speed, and h is the smaller lateral dimension of a rectangular rod. Because the buckle wavelength depends only on \sigma and h , this same formula holds for thin cylindrical shells of thickness h . U_\text{applied} = U_\text{strain} In this method, there are two equations used (for small deformations) to approximate the "strain" energy (the potential energy stored as elastic deformation of the structure) and "applied" energy (the work done on the system by external forces). \begin{align} U_\text{strain} &= \frac{E}{2} \int I(x)(w_{xx}(x))^2 \, \mathrm{d}x \\ U_\text{applied} &= \frac{P_\text{crit}}{2} \int (w_{x}(x))^2 \, \mathrm{d}x \end{align} where w(x) is the displacement function and the subscripts x and xx refer to the first and second derivatives of the displacement. V = U - P\Delta where U is the strain energy stored in the structure, P is the applied conservative load and \Delta is the distance moved by P in its direction. Using the axioms of elastic instability theory, namely that equilibrium is any point where V is stationary with respect to the coordinate measuring the degree(s) of freedom and that these points are only stable if V is a local minimum and unstable if otherwise (e.g. maximum or a point of inflection). These four forms of elastic buckling are the saddle-node bifurcation or limit point; the supercritical or stable-symmetric bifurcation; the subcritical or unstable-symmetric bifurcation; and the transcritical or asymmetric bifurcation. All but the first of these examples is a form of pitchfork bifurcation. Simple models for each of these types of buckling behaviour are shown in the figures below, along with the associated bifurcation diagrams.

See also

References

Further reading

External links

The complete theory and example experimental results for long columns are available as a 39-page PDF document at http://lindberglce.com/tech/buklbook.htm

Text derived from https://en.wikipedia.org/w/index.php?oldid=1362996017 under CC BY-SA 4.0; modifications have been made.