Chapter 7. ELASTIC INSTABILITY Dr Rendy Thamrin; Zalipah Jamellodin

Transcription

1 Chapter 7 ESTIC INSTIITY Dr Rendy Thamrin; Zalipah Jamellodin 7. INTRODUCTION TO ESTIC INSTIITY OF COUN ND FRE In structural analysis problem, the aim is to determine a configuration of loaded system, which satisfies the condition of: Equilibrium Compatibility, and Force-displacement relations of material For a structure to be satisfactory whether the equilibrium configuration so determined is stable. The stability loss under compressive load is usually termed structural or geometrical instability commonly nown as bucling. In the case of a column loaded by a compressive load, the ability of the column to resist axial load terminates as soon as bucling occurs and, consequently, the critical load of the column is its failure load. It is to be emphasized that the load at which instability occurs depends upon the stiffness of structure (or parts of the structure), rather than on the strength of material. 7.. Concept of Stability Initial state osition after displacing force is applied osition after displacing force is removed

2 (a) Stable equilibrium (b) Neutral equilibrium (c) Unstable equilibrium 7. TYES OF INSTIITY Instability is a condition wherein a compression member loses the ability to resist increasing loads and exhibits instead a decrease in load carrying capacity. Generally, the types of instability of a structure (or a member of structure) can be classified as follows : Bucling with respect to principal minor axis, Bucling with respect to principal major axis, ure torsional instability. Classification of instability can be also provided as follows : Flexural bucling, Torsional bucling, Torsional flexural bucling, and Snap-through bucling Figure 7.: ode of Failures

3 7. ETHODS OF NYSIS Stability analysis consists in determining the mode of loss of structural stability and corresponding load called critical load. Four definitely different classical methods available for the solution of bucling problems are: Non-trivial equilibrium state approach, or approach, Energy approach, and Kinetic or dynamical approach 7. CRITIC OD NYSIS USING STIITY FUNCTIONS The slope deflection equation of a member related moments and slopes can be written as follows: EI θ + θ B Δ here; moment at EI flexural stiffness member length; θ slope at θ B slope at B Δ member deformation The equation of a member subjected to axial load can be written as follows: EI sθ + scθ B s Δ ( + c) here; s c member stiffness carry-over factor

4 The parameters s and c are called stability functions. The values of parameter s and c are functions of the ratio of axial load,, and Euler s load, e. This ratio can be expressed and notate as: e π EI The selected values of these functions (s and c) are tabulated and will be shown in the next following pages. 7.. Stability Functions Derivation (Non-Sway) Consider a member with length subjected to flexural moment and compression load. In linear elastic analysis, if the axial load is neglected (only flexure): EI θ i.e. θ EI θ i.e. θ where : EI + V For a member with axial load, the stiffness coefficient of moment and should be changed and can write: sθ and csθ where: s stiffness factor and c carry-over factor hence: ( + c) sθ + csθ s V or V θ The parameters s and c are nown as stability functions.

5 From the figure above, for each location x from : + y V x. () x From pure bending: d y x EI dx d y substitute to Eq. (): EI + y V x dx d y V x + y dx EI EI EI d y replace: μ V x + μ y. () EI dx EI EI The homogeneous differential equation (DE) has the form: d y + μ y 0 or y" + μ x 0 dx The solution of homogeneous DE, y, consist of the part: y y c + y (i) The complementary solution is: y c sin μx + Bcosμx. () (ii) The particular solution may be assumed to be the form: y Cx + D V x μ ( Cx + D) ) EI EI V x V V Tae the coefficient of x: μ Cx C EI μ EI Coefficient of constant: μ D EI μ EI V x Substitute to particular solution: y. () Thus, the complete solution is: V x sin μx + Bcosμx + D 5 y The constants and B may be determined by applying the end boundary conditions: (i) x 0 ; y 0 B V (ii) x ; y 0 ; sin μ + cosμ V

6 sin μ + cosμ + tan μ sin μ + 0 ( + ) y sin μx + cosμx + x tan μ sin μ. () dx (iii) x ; 0 dy substitute sθ and cs (iv) θ sθ sin μ μ sin μ μcosμ + c 0 sin μ sin μ sθ 0 sin μ μ c. (7) μcosμ sin μ replace μ π and assume π α we find: x 0 ; dx dy sin α α c. (8) α cos α sin α θ θ cot cosec μ μ μ + + substitute sθ and csθ sθ μ μ μ μ μ θ sin cos sin + c sin μ sin μ replace μ μ ( μcot μ) s. (9) μ μ tan replace μ α π we obtain: ( α cot α) α s. (0) tan α α The values of parameter s and c are functions of the strength ratio,, which can be expressed as: π EI e

7 7.. Stability Functions for the Compression and Tension ember (Non-Sway) For the compression member, the stability functions parameter s and c as: sin α α c. () α cos α sin α ( α cot α) α s... () tan α α here, α π The selected values of the stability functions for axial compression are listed in Table and for intermediate values interpolation may be adopted. For the tension member, the stability functions parameter s and c can be expressed as: α ( α C S) ( C + α S) α ( S α) ( C + α S) s... () sc... () here S sinh α, C cosh α, and, α π The selected values of the stability functions for axial tension are listed in Table and for intermediate values interpolation may also be adopted. 7.. Stability Functions Curve The curve below shows the plot of Equation ~ against the strength ratio,, note that : if s and c (+) stable, and if s and c (-) instable. Equation Equation /s Equation Equation s c

8 Table : Tabulated Selected Values of Stability Functions (Compression) s c s( c ) (sc) 8

9 Table : Tabulated Selected Values of Stability Functions (Compression) s c s( c ) (sc) Table: Tabulated Selected Values of Stability Functions (Tension) s c s( c ) (sc) 9

10 EXE Considering a frame structure shown in figure below, the internal forces in members and can be calculated as follows: v hv + h ; v h v + h Summing moment at point B : s + s θ B ( ) B Note that : + (stable) B B (unstable) B 0 (critical) Instability condition requires: s + s ( ) 0 B Given:.0 m,. m, h. m, h I 00.0 cm, I cm, v.5 m.5 m. Calculate the internal forces, assuming that the frame is pin-jointed :.5. x.5.7 ;.5.5 x x.5.. x.. Calculate the Euler s load and the strength ratio: π EI π x 07 x 00 x 0 x 0 π EI π x 07 x 000 x 0. x 0 x E x E E E I 00 x 0 x 0 I 000 x 0. x mm 77 mm 75 N 57 N

11 . Instability criteria of the structure: B ( s + s ) s + s 0 st Trial nd Trial rd Trial th Trial 5 th Trial th Trial s s s s + s Calculate the critical load, cr : cr. x N 75 cr.7 x N 507 Different due to rounded of calculation EXE rigid jointed steel frame C carry a vertical load at B as shown in figure beside. Formulate the instability equation and find the critical load (cr) for the frame. Given : h.0 m, h.0 m, v.0 m I 00.0 cm, I cm. Calculate the internal forces, assuming that the frame is pin-jointed: v h v.0.0 x h x.0 h v.0.0 x h x.0 v.0.0

12 . Calculate the Euler s load and the strength ratio: π EI π x 00 x 00 x 0 E π EI π ( ) ( ) x 0 x 00 x 000 x 0 E x N.8 N x x.0.8 E E I 00 x 0 x 0 I 000 x 0 x 0. mm 77.8 mm Instability criteria of the structure: B ( s + s ) 0.s + s 0 st Trial nd Trial rd Trial th Trial 5 th Trial th Trial s s s s + s Instability occur when:.87 and.. Calculate the critical load, cr: cr.87 x N 09. cr. x N 8.0 Different due to rounded of calculation

13 EXE rigid jointed steel frame C carry a vertical load at B as shown in below. Formulate the instability equation and find the critical load (cr) for the frame. Given: h h m, v m, I 00 cm, I 000 cm. Calculate the internal forces, assuming that the frame is pin-jointed: v h v.0.0 x h x.0 h v.0.0 x h x.0 v. Calculate the Euler s load and the strength ratio: π EI π π EI π x 00 x 00 x 0 5 x 0 x 00 x 000 x 0 5 x 0 E E N 789. N x x E E I 00 x 0 5 x 0 I 000 x 0 5 x mm mm

14 . Instability criteria of the structure: B ( s + s ) 0.s + s 0 st Trial nd Trial rd Trial th Trial 5 th Trial th Trial s s s s + s Instability occur when:.87 and.. Calculate the critical load, cr: cr.87 x N 5.0 cr. x N. Different due to rounded of calculation