Plate Buckling ref. Hughes Chapter 12

Size: px

Start display at page:

Download "Plate Buckling ref. Hughes Chapter 12"

Homer George
5 years ago
Views:

1 Plate Buckling ref. Hughes Chapter 1 Buckling of a plate simply supported on loaded edges treated as a wide column results in similar Euler stress, with EI replaced byd*(b): dividing by area (b*t): π Db π Et 3 eqn 9.1.5, 1/b D D := P e := σ e := a a 1( 1 ν t ) implied Buckling of a simply supported plate. i.e. simply supported on all four sides deflected shape is represented by sine waves in x and y: wx, y) := C m and n are the number of half waves in ( mn m n π a D m + n argument in text for minimum stress a b can specify n = 1, but m not clear σ acr := tm with n = 1, can express in form where k is buckling coefficient in equation eqn σ acr := k π D N.B. shift to plate width b in denominator b t m := 1.. n := 1.. a_over_b := 0.5, k(a_over_b, m) := m + a_over_b eqn sin mπx sin nπy, a b deflection eqn k(a_over_b, 1) k min (a_over_b) := min k(a_over_b, ) k(a_over_b, 3) k(a_over_b, 4) 6 a_over_b m k min (a_over_b) 5 therefore for long plates, simply supported on loaded ends k = 4 4 σ acr := 4 π D b t a_over_b very wide i.e. a/b -> 0, approaches Euler as a<<b σ acr (a_over_b) := k min (a_over_b) π D k -> b a + = b + a -> b b t a b ab a 1 notes_9_plate_buckling.mcd

2 a square column meets these boundary conditions hence will buckle with each edge forming half sine wave deflection in half sine waves approaching square for large a/b; long plate loaded on end simply supported on all sides Plate loaded on all four sides; σ ax in a direction, σ ay in b direction Again taking two half wave sine series using energy methods, results in combination expression for both stresses, again taking minimum of straight lines from: critical m σ ax + n σ ay 1 = m + n σ acr := 4 π D stress with α σ axcr1 σ axcr1 b t 4 α only σ ax σ axcr1 := σ acr σ σ ymn, ( ay ( ax, α), α) := α := a := xmn, σ axcr1 σ axcr1 b 1 y 1 ( xm,, n, α ) := m + n m x 4n α nα x := 1.5, fig 9.5 in T&G y 1 ( xm,, n, α ), y1 ( xm,, n, α ), (y 1 ( xm,, n, α )), y 1 ( xm,, n, α ), y1 ( xm,, n, α ), y1 ( xm,, n, α ) y 1 (x1,, 1, 1) 4 y 1 (x,, 1, 1) ( y 1 (x1,,, 1) ) y 1 (x,,, 1) y 1 (x3,, 1, 1) 0 y 1 (x1,, 3, 1) x notes_9_plate_buckling.mcd

3 Buckling stresses of biaxially loaded simply supported plates only taking m=n=4 terms. Effect of higher terms would be seen in lower right corner asymptote. x := 1, y 1 ( x1,, 1, α ) y1 ( x1,,, α ) y1 ( x1,, 3, α ) y1 ( x1,, 4, α ) y ( 1 ( x,, 1, α ) y1 ( x,,, α ) y1 ( x,, 3, α ) y1 ( x,, 4, α ) yx, α) := min y 1 ( x3,, 1, α ) y1 ( x3,,, α ) y1 ( x3,, 3, α ) y1 ( x3,, 4, α ) y1 ( x4,, 1, α ) y1 ( x4,,, α ) y1 ( x4,, 3, α ) y1 ( x4,, 4, α ) 4 3 y(x, 0.5) y(x, 0.75) yx (, 1) yx (, ) yx (, 3) x 3 notes_9_plate_buckling.mcd

4 Biaxial loading all edges clamped: σ ax is relative to π *D/(b *t) α + σ ax ( α, σay_over_σax ) := ( α eqn in T&G form α σay_over_σax) tbl. 1-3 σ ax (1., 0) = σ ax (1, ) = , 0 σ ax (11), = σ ax (1, 0.5) = , 8.80 per T&G good for shape close to square, data does not compare well with Hughes tbl. 1.3 or T&G tbl 9-15 plate buckling due to pure shear: simply supported on four sides: T&G article 9.7, page 379ff τ cr := π π D 1 σ ref := π D h = thickness = t 3 b h λ b h from five equations := 1, = a/b π ( ) := π D 3 ( ) b h λ τ cr 1 ( ) := λ ( ) k in τ cr := k π D π 1 becomes k ( ) := close for = <, no good elsewhere ( ) b 3 λ h it's probably ok by T&G, for few equations 4 compare with simple curve fit k 1 ( ) := exact solution for very long plates = a/b -. is k = At = 1 k = 9.34 exact. k1 ( ) 1 = less than 1% difference more complexity doesn't buy anything 9.34 k (b_over_a) := b_over_a := 0., π b_over_a 1 3 λ 1 k 1a (b_over_a) := b_over_a b_over_a k (b_over_a) 8 k 1a (b_over_a) 6 k ( ) k 1 ( ) b_over_a, b_over_a 4 notes_9_plate_buckling.mcd

5 table 9-10 in T&G page 38; values of k in τ cr := k π D from the results of the solution by determinants = 0. b h tbl_9_10 := i := k3 i := tbl_9_10 1i, b_over_a 3i := tbl_9_10 0i 1, 10 k (b_over_a) k 1a (b_over_a) k clamped on four sides critical shear stress b_over_a, b_over_a, b_over_a 3 T & G only provides infinitely long value of τ cr 8.98 π D := and then plots combination b h 5 notes_9_plate_buckling.mcd

6 Interaction formula for combination of shear stress and axial load from ref 10 (in AA library) when a/b > 1, problem is one of axial stress, and parabola is good model. When a/b is less than one, it is the equivalent of a transverse stress with shear. Parabola is not a good fit. Hence Hughes model based on a/b. R c (R s, α) := if 1 R s α > 1, 1 R s, if R s > 3 ( 1 α ), 8 ( α ) 1.6, 1 eqn R s := 0, fr ( s, α) := 1 R s α := 0.5 looks a transverse stress with a/b = 4. since α <1, parabola doesn't fit well with Rc(Rs,α). Curve lays on top when α>1, (by definition) R s R s R c ( R s, α), f( R s, α ) 6 notes_9_plate_buckling.mcd

7 Ultimate strength of plates: ( ) := := ξ Es_over_E ( ) := ξ ( ) ( ) 10.4 ξ b t σ Y σr_over_σy := 0.0 for curve in text: set this parameter to 0.1 σau_overσy ( ) ( ) ξ ( ) 10.4 := 0.5 4σr_over_σy + ξ Faulkner 1 E with reduction σr_over_σy = 0 σau_over_σy F ( ) := if < 1, 1, Ets_over_E ( ) := if < 1, 0, if >.5, 1, ( 1) to correct for residual stress σau_overσy E ( ) := 3.6 for reference := 0.01, σau_overσy ( ) σau_over_σy F ( ) 0.6 σau_overσy E ( ) similar to fig 1.5 in text 7 notes_9_plate_buckling.mcd

8 bringing in some other standards: AISI and UK (new) from Professor Wirrzbicki's Manual for Crash Wothiness Engineering Feb 1989 CTS MIT and in handout validation of plate buckling note: be/b is another way of representing peak load b e σ Y = bσ cr σ cr_over_σy ( ) := 1.9 σ Y_over_σcr ( ) := 1.9 C ( ) := if > 1.5,.5 1.5, 1 ABS Alaa σau_overσy E ( ) := if > 1.9, 3.6, 1 for reference 1 := ( ) := if > 1, 1 14( ( ) 0.35) 4, 1 be_over_b uk + σ Y_over_σcr be_over_b aisi ( ) := if > 1, σ cr_over_σy ( ) ( σ cr_over_σy ( )), 1 := 0.5, σau_overσy ( ) σau_over_σy F ( ) σau_overσy E ( ) C ( ) be_over_b uk ( ) 0.4 be_over_b aisi ( ) notes_9_plate_buckling.mcd

1.1 Graphing Quadratic Functions (p. 2) Definitions Standard form of quad. function Steps for graphing Minimums and maximums

1.1 Graphing Quadratic Functions (p. 2) Definitions Standard form of quad. function Steps for graphing Minimums and maximums Quadratic Function A function of the form y=ax 2 +bx+c where a 0 making a u-shaped