5 Stabilit Of Structures: Continuous Models SEN 311 ecture 5 Slide 1
Objective SEN 311 - Structures This ecture covers continuous models for structural stabilit. Focus is on aiall loaded columns with various end support conditions. Columns are treated with inearized reuckling () analsis assumptions to set up the equilibrium equation in a perturbed shape. Formulas for beam deflection (art 3 of course) reused. Equilibrium equation is a linear second order ODE - ma be homogeneous or not. Seeking nontrivial ODE solutions leads to a trascendental eigenproblem, from which critical loads and buckling mode shapes can be obtained. For prismatic columns with standard C, the stabilit analsis can be entirel done b hand. SEN 311 ecture 5 Slide
Three Eample roblems SEN 311 - Structures Three column buckling problems are worked out in this ecture: 1. The Euler column: a prismatic, elastic, aiall loaded column pinned (same as SS) at both ends. The pinned-fied column: a prismatic, elastic, aiall loaded column pinned at one end and fied at the other 3. The pinned-elasticall-restrained column: a prismatic, elastic, aiall loaded column SS at both ends but connected to a torsional spring at the other These 3 C cases were tested Frida, and are used in Eercise 10.6 of HW#10 SEN 311 ecture 5 Slide 3
The Three C Cases Treated in This ecture elastic constant EI constant EI constant EI elastic elastic k inned-pinned (a.k.a Euler column) inned-fied inned-restrained SEN 311 ecture 5 Slide 4
The inned-inned (Euler) Column elastic constant EI (b) X v() X' assumed buckled shape (c) X v() X' M() = v() (+ as drawn) SEN 311 ecture 5 Slide 5
inned-inned Column: ssumed uckling Shape Figure (b) of Slide 4, reproduced on the right for convenience, is a FD done in an admissible perturbed configuration for the Euler column. It shows reactions at end, which reduce to just a vertical force balancing. There are no end moment reactions because the column is hinged at both and. Furthermore, taking moments with respect to and shows that both horizontal reactions (along ) vanish. X v() X' assumed buckled shape SEN 311 ecture 5 Slide 6
inned-inned Column: FD at Variable Net, do the FD of a segment X, with X located at distance from, as shown in (c) of last slide, reproduced on the right for convenience. The displaced X is labeled X', and the distance from X to X' is the lateral column deflection v(). t this cross section we will have a bending moment M (), which is positive as drawn z z (reason: a positive M () compresses the "beam top surface", which b convention lies on the + side.) Taking moments with respect to X' for convenience, equilibrium requires X v() X' M() = v (+ as drawn) z M () + v() = 0 z ut according to beam deflection theor (developed in art 3 of the course) M () = E I v''() = E I v''() zz SEN 311 ecture 5 Slide 7
inned-inned Column: ODE Derivation Here I = I zz denotes the minimum moment of inertia of the column cross-section. This defines the z direction. The ais is taken normal to and z forming a right-handed RCC frame. The column will buckle in the direction; that is, the deflected shape will be in the - plane. Replacing M () = E I v''() gives z E I v''() + v() = 0 This is the ODE that governs stabilit analsis. It is a linear, homogeneous, second-order ODE in the deflection v(). For convenience in the reduction to standard (canonical) form, introduce λ def = + EI Replacing b E I λ and dividing through b E I ields the canonical ODE form v''() + λ v() = 0 SEN 311 ecture 5 Slide 8
inned-inned Column: General ODE Solution The general solution of the foregoing ODE is v() = cos λ + sin λ Coefficients and are determined from the kinematic C of simple support at both ends: v(0)=0 and v() =0. The first one requires = 0, whence the general solution reduces to v() = sin λ This is called a characteristic solution. Two cases can be distinguished: = 0 0 v() = 0: the column remains straight for an load v() 0: the column buckles with shape sin λ These are called the trivial and nontrivial solutions, respectivel, in the literature. Since the case = 0 or v() = 0 is that of the reference (undeformed) state at 0 load, it represents the primar equilibrium path. SEN 311 ecture 5 Slide 9
inned-inned Column: Critical oads SEN 311 - Structures The critical loads are the values of at which nontrivial solutions are possible. These are determined b appling the second end condition v() = 0. Since is nonzero (else we would get a trivial solution) we must have sin λ = 0 This characteristic equation is actuall a trascendental eigenproblem in λ. Its solutions are λ = n π, in which n = 1,,... is a positive integer. Square both sides for convenience: λ = n π, replace λ b /(E I), solve for, and tag those loads as critical: cr,n = n π EI, n = 1,,... s can be observed, there is an infinite number of critical loads. These are the nontrivial solutions of the characteristic equation. SEN 311 ecture 5 Slide 10
inned-inned Column: Critical oad The lowest critical load, denoted simpl b, corresponds to n = 1: cr = = cr cr,1 π EI This is called the Euler critical load. The buckling mode shapes associated with the set of critical loads are given b v c cr,n r,n() = sin n π If n = 1 the mode shape is a half sine wave, similar to the one pictured as assumed mode shape. If n =, we get a complete sine wave, if n = 3 a one-and-a-half sine wave, etc. Those buckling shapes are drawn for n = 1,, 3 in the net slide. SEN 311 ecture 5 Slide 11
inned-inned Column: uckling Mode Shapes for n = 1,,3 SEN 311 - Structures = = cr cr1 π E I cr= / 4 π E I cr3= /3 9 π E I /3 / /3 Mode shape 1 (n=1) Mode shape (n=) Mode shape 3 (n=3) SEN 311 ecture 5 Slide 1
inned-inned Column: Determinant Form of Stabilit Equation The boundar condition equations = 0 and sin λ = 0 can be recast in matri form as 1 0 0 = 0 sin λ 0 For a nontrivial solution in which and are not simultaneoul zero the matri determinant, which is simpl sin λ, must be zero. Thus we recover the characteristic equation sin λ = 0. For this particular problem this detour is a waste of time. ut for more complicated cases it can shortcut error prone hand derivations especiall with the help of a computer algebra sstem (CS), which can directl compute and simplif the determinant. SEN 311 ecture 5 Slide 13
inned-fied Column: ssumed uckling Shape (a) (b) (c) elastic X constant EI R X v() M X' assumed buckled shape R = R R X v() X' R M() (+ as drawn) In the configuration shown in (a) the column is pinned at and fied at. The FD of a kinematicall admissible buckling shape is illustrated in (b). The most notable difference with respect to the pinned-pinned column case is the presence of additional reaction forces: the lateral reactions R and R, and the fied end moment M. These are positive as pictured above. SEN 311 ecture 5 Slide 14
inned-fied Column: ODE and its Solution Equilibrium of forces in (b) of the previous slide is satisfied b the vertical reaction R = at. Equilibrium of forces gives R = R. Equilibrium of moments taken with respect to either or ields M = R, or R = M /. For now M will be kept as an unknown until the ODE is solved. Net consider the FD of the portion X shown in (c) of the previous slide Moment equilibrium with respect to X' gives the non-homogeneous ODE E I v''() + v() = R = M / The HS is the same as that of the pinned-pinned column, but the RHS is no longer zero. Dividing through b E I and setting λ = /(EI) gives the canonical form v''() + λ v() = M /(E I ) = λ M /( ) The general solution of this ODE is the sum of the homogeneous solution v H() = cos λ + sin λ and the particular solution v () = λ M /(λ ) = M /( ): v() = cos λ + sin λ + M /( ) SEN 311 ecture 5 Slide 15
inned-fied Column: Critical oad The three kinematic C are v = v(0) = 0, v = v() = 0, and v' = v'() = 0. These provide 3 equations. One is trivial: = 0, but the other two are not M M cos λ + sin λ + = 0, λ sin λ + λ cos λ = 0 Soving these equations simultaneousl gives the characteristic equation tan λ = λ The smallest root of this trascendental equation, to 4 places, is λ = 4.493. The corresponding critical load is cr = 0.19 EI Since 0.19 ~.05 π, this critical load is approimatel twice that of the pinned-pinned Euler column. Thus fiing one end has substantiall increased the critical load. SEN 311 ecture 5 Slide 16
inned-fied Column: Determinant Form of Stabilit Equation SEN 311 - Structures The three boundar condition equations of the previous slide can be recast in matri form as 1 0 0 0 cos λ sin λ 1/ = 0 λ cos λ λ sin λ 1/() M 0 For a nontrivial solution in which, and M are not all simultaneousl zero the matri determinant must vanish. Epanding gives ( sin λ λ cos λ ) / ( ) = 0 Since and are nonzero, the factor 1 / ( ) ma be removed. That leaves sin λ = λ cos λ. Dividing through b cos λ we recover the characteristic equation tan λ = λ which was found in the previous slide b an elimination method SEN 311 ecture 5 Slide 17
The inned-elasticall Restrained Column (a) (b) (c) (d) R constant EI v() elastic,v R M()= v+r X R beam-column k M = k θ C elastic beam θ R This is relevant to one of the C tested in ab 4: the restraining elastic beam can be replaced b an equivalent torsional spring For analsis details, see Sec 4 of ecture 5 SEN 311 ecture 5 Slide 18