MIT - 16.20 Fall, 2002 Unit 15 Shearing and Torsion (and Bending) of Shell Beams Readings : Rivello Ch. 9, section 8.7 (again), section 7.6 T & G 126, 127 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Paul A. Lagace ? 2001 MIT - 16.20 Fall, 2002 Thus far, we have concentrated on the bending of shell beams. However, in the general case a beam is subjected to: ? axial load, F ? bending moments, M ? shear forces, S ? torque (torsional moments), T Figure 15.1 Examples of general aerospace shell beam structures Aircraft Wing Space Habitat Shell connecting nodes Paul A. Lagace ? 2001 Unit 15 - 2 MIT - 16.20 Fall, 2002 Idealize the cross-section of the shell beam into two parts: ? Parts that carry extensional stress, σ xx (and thus the bending and axial loads) ? Parts that carry shear stress σ xs (and thus the shear loads and torques) Two examples again … ? high aspect ratio wing with semi- monocoque construction Notes : ? monocoque construction ?– all in one piece without internal framing ?– from French “ coque ” meaning “eggshell ” ?– “mono ” = one piece ? semi- monocoque – stressed skin construction with internal framework – still have “ eggshell ” to carry shear stresses, σ xs – internal framework to carry axial stress, σ xx Paul A. Lagace ? 2001 Unit 15 - 3 MIT - 16.20 Fall, 2002 Figure 15.2 Representation of wing semi-monocoque construction web + rib skin stiffeners flanges web flanges = spar Idealize this section as: Figure 15.3 Idealization of wing semi- monocoque construction Paul A. Lagace ? 2001 Unit 15 - 4 MIT - 16.20 Fall, 2002 → Skins and webs are assumed to carry only shear stress σ xs → Flanges and stringers are assumed to carry only axial stress σ xx ? Space habitat Figure 15.4 Representation of space habitat semi- monocoque construction wall wall stiffeners spar flanges Idealize as for wing: Paul A. Lagace ? 2001 Unit 15 - 5 MIT - 16.20 Fall, 2002 Figure 15.5 Idealization of space habitat semi-monocoque construction → Outer skin and walls are assumed to carry only shear stress σ xs → Flanges and stiffeners are assumed to carry only axial stress σ xx Analyze these cross-sections as a beam under combined bending, shear, and torsion. Utilize St. Venant assumptions: 1. There are enough closely spaced rigid ribs to preserve the shape of the cross-section (or enough stiffness in the internal bracing to do such) 2. The cross-sections are free to warp out-of-plane Start to develop the basic equations by looking at the most basic case: Paul A. Lagace ? 2001 Unit 15 - 6 MIT - 16.20 Fall, 2002 Single Cell “ Box Beam ” Figure 15.6 Representation of geometry of single cell box beam modulus-weighted centroid of flange and stiffener area used as origin Breakdown the problem … (a) Axial Bending Stresses : Each flange/stiffener has some area associated with it and it carries axial stress only (assume σ xx is constant within each flange/stiffener area) Paul A. Lagace ? 2001 Unit 15 - 7 MIT - 16.20 Fall, 2002 The axial stress is due only to bending (and axial force if that exists --leave at zero for now ) and is therefore independent of the twisting since the wing is free to warp (except near root -- St. Venant assumptions) * Find M, S, T from statics at any cross-section x of the beam Consider the cross-section: Figure 15.7 Representation of cross-section of box beam Area associated with flange/stiffener i = A i Find the modulus-weighted centroid ( Note : flange/stiffeners may be made from different materials) Paul A. Lagace ? 2001 Unit 15 - 8 MIT - 16.20 Fall, 2002 ? Choose some axis system y, z (convenience says one might use a “ corner” of the beam) ? Find the modulus-weighted centroid location: * ∑ A y i * i y = * ∑ A i * ∑ A z i * i z = * ∑ A i n ( ∑ = sum over number of flanges/stiffeners) i = 1 number = n ( Note : If flanges/stiffeners are made of the same material, remove the asterisks) ? Find the moments of inertia with reference to the coordinate system with origin at the modulus-weighted centroid * * * 2 I y = ∑ A i z i * * * 2 I z = ∑ A i y i * * * * I yz = ∑ A i y i z i Paul A. Lagace ? 2001 Unit 15 - 9 MIT - 16.20 Fall, 2002 ? Find the stresses in each flange by using the equation previously developed : E ? F TOT ? σ xx = ? * ? Ef y ? E f z ? E 1 α ? T ? 12 13 E 1 ? A ? 0 for no axial force (Will do an example of this in recitation) (b) Shear stresses : assume the skins and webs are thin such that the shear stress is constant through their thickness. Use the concept of “ shear flow ” previously developed: q = σ xs t [Force/length] shear thickness flow shear stress (called this the shear resultant in the case of torsion) Look at the example cross-section and label the “joints ” and “ skins ” Paul A. Lagace ? 2001 Unit 15 - 1 0 MIT - 16.20 Fall, 2002 Figure 15.8 Representation of joints, skins, and shear flows in cross-section of box beam Figure 15.9 Look at the equilibrium of joint 1 Representation of skins and stringer and associated loads and shear flows at joint 1 stringer skin skin : Paul A. Lagace ? 2001 Unit 15 - 1 1 MIT - 16.20 Fall, 2002 Notes : ? The stringer only carries axial load ? The skin carries only shear flow ?? The shear flow at the “end ” of the skin (where it is “cut” ) must be the same as at the edge (the cross-section cut). This is due to equilibrium ( σ xy = σ yx ) ? Apply equilibrium: ∑ F x = 0 dP ? ? P + P + dx + q 1 dx ? q 6 dx = 0 dx dP ? q 1 ? q 6 = ? dx More generally note that: Paul A. Lagace ? 2001 Unit 15 - 1 2 MIT - 16.20 Fall, 2002 1. Angle that skin comes into joint doesn ’ t matter since q along edge is always in x-direction Figure 15.10 Representation of skins at joint coming in at any angle 2. Stringer alone gives dP as its contribution dx Figure 15.11 Representation of stringer isolated at joint Paul A. Lagace ? 2001 Unit 15 - 1 3 MIT - 16.20 Fall, 2002 3. If shear flows “ into ” joint, its contributions is in the negative x- direction; if shear flows “ out of ” joint, its contribution is in the positive x-direction Figure 15.12 Representation of shear flowing (left) into and (right) out of joint Adding all this up: dP ? q in + q out = 0 dx dP ? q out ? q in = ? dx Use this in general Paul A. Lagace ? 2001 Unit 15 - 1 4 MIT - 16.20 Fall, 2002 For a more complicated joint, use superposition Figure 15.13 Representation of joint with multiple skins --> Need an expression for P -- start with: P = A σ xx differentiating: dP A d σ xx + σ dA ? = xx dx dx dx = 0 since are considering stringers with a uniform cross-section Most general case : q out ? q in = ? A d σ xx Joint Equilibrium dx Paul A. Lagace ? 2001 Unit 15 - 1 5 MIT - 16.20 Fall, 2002 Consider a simpler case: ? I yz = 0 (symmetric section) ? M z = 0 know that: Mz y σ xx = ? I y dP = A d ? Mz ? y ? dx dx ? ? ? I y ? ? dP ? Az dM y = dx I dx y Recall that: dM y = S z (shear resultant) dx Az = Q y (moment of area about y) So for this case, the joint equilibrium equation becomes: Paul A. Lagace ? 2001 Unit 15 - 1 6 MIT - 16.20 Fall, 2002 QS q out ? q in = y z ? Symmetric section I y ? M z = 0 Now have an equation for the equilibrium of shear stresses at the joints. Shear stresses arise due to two reasons: ? Shear resultant ? Twisting In general have at any cross-section: It is convenient to break up the problem into two separate problems: Paul A. Lagace ? 2001 Unit 15 - 1 7 MIT - 16.20 Fall, 2002 (1) “ Pure Shear ” (2) “ Pure Twist ” shear resultant acts at shear center so there is no twisting ? --> Solve each problem separately, then add the results (use superposition) Condition : The two force systems ( S z , T and S z ’ , T ’ ) must be equipollent Figure 15.14 Demonstration of equipollence of force systems Paul A. Lagace ? 2001 Unit 15 - 1 8 MIT - 16.20 Fall, 2002 Define : d = distance from where shear resultant acts to shear center ′ ∑ F z = same ? S = S z z ∑ T = same ? T ′ ? S d = T z careful : sign could be + or - depending upon direction S z is moved! Figure 15.15 Representation of positive and negative contribution of S z to torque Hint : Add up torques about line of S z action in each case. They must be the same! ( ? d has magnitude and sign) Paul A. Lagace ? 2001 Unit 15 - 1 9 MIT - 16.20 Fall, 2002 Solution procedure Given: ? section properties ? loading [T(x), S z (x)] Find: ? shear stresses (flows) (n joints) ? shear center ? (n + 1) variables 1. Consider “ Pure Shear ” case a) Apply joint equilibrium equation at each joint Note : n joints will yield n-1 independent equations. (one is dependent since the section is closed) b) Use Torque Boundary Condition ∑ T internal = T applied This is torque equivalence , not equilibrium Do this about the line of action of S z Paul A. Lagace ? 2001 Unit 15 - 2 0 MIT - 16.20 Fall, 2002 Then: T applied = S z d ∑ T internal = ∑ q i ( moment arm ) i ( skin length ) i c) Specified no twist (Pure Shear Case), so apply the No Twist Condition Recall from Torsion Theory: ∫ τ ds = 2 A G d α dx d α Here: = 0 dx and: τ = q t ? q ∫ ds = 0 t Paul A. Lagace ? 2001 Unit 15 - 2 1 MIT - 16.20 Fall, 2002 This gives: a) n - 1 equations b) 1 equation c) 1 equation n + 1 equations for n + 1 variables Solving these gives: ? q’ s due to “ Pure Shear ” case ? d --> when complete, check via: ∑ Internal Shear loads = Applied Shear (Horizontal & vertical) 2. Consider “ Pure Torsion ” case a) Apply joint equilibrium equation at each joint Note : again, n joints give n-1 equations Since no shear: q out ? q in = 0 Paul A. Lagace ? 2001 Unit 15 - 2 2 MIT - 16.20 Fall, 2002 b) Use Torque Boundary Condition ∑ T internal = T applied T applied = T + o r ? S z d found in part 1 This gives: a) n - 1 equations b) 1 equation ? Need: n q ’ s ? Solve these for q ’s due to “Pure Torsion ” case 3. Sum results for “ Pure Shear ” and “ Pure Torsion ” cases (could use q i s , q i T : q i due to pure shear = q i s q i due to pure torsion = q i T ) Paul A. Lagace ? 2001 Unit 15 - 2 3 MIT - 16.20 Fall, 2002 Now have: shear flows (get shear stresses via: τ xs = q ) t location of shear center (will be important to determine deflection) (will go over sample problem of handout #6 in recitation) for: Unsymmetric Shell Beams ? Cannot make simplifying assumptions (use equations coupling bending in y and z) ? See handout #4B Now that the stresses (due to bending, shear and torsion) are determined, proceed to find the … Deflections of Shell Beams Have divided the problem and used superposition for the stresses. Again use the: St. Venant Assumption : Bending and Torsion are independent Paul A. Lagace ? 2001 Unit 15 - 2 4 MIT - 16.20 Fall, 2002 Thus, can add the effects of bending, shearing, and torsion deflections to get the total deflections. General Approach: Figure 15.16 Representation of loading resultants and deflections for general cross-section ? Place S y and S z at the shear center ? T ’ acts about the shear center ? v, w = deflection of shear center ? α = twist about shear center Paul A. Lagace ? 2001 Unit 15 - 2 5 MIT - 16.20 Fall, 2002 Figure 15.17 Representation of loading and resultants in x-z plane ? From the loadings p y , p z obtain the resultants S y , S z , M y , M z , T as functions of x ? Use the deflection equations previously developed: for bending : (subscript “ B ” = bending) 2 IM y dv B = IM z ? yz y dx 2 EI I ( yz ? I y 2 z ) ( yz ? I y 2 z ) EI I 2 IM z dw B = IM y ? yz z dx 2 EI I ( yz ? I y 2 z ) ( yz ? I y 2 z ) EI I Paul A. Lagace ? 2001 Unit 15 - 2 6 MIT - 16.20 Fall, 2002 for shearing : (subscript “ s ” = shearing) dv S S s z = ? y ? dx GA GA yy yz dw s = ? S z ? S y dx GA GA zz yz for torsion/twisting d α T = dz GJ then get: v = v B + v s w = w B + w s α = rotation about center In order to do this, need to know the … Paul A. Lagace ? 2001 Unit 15 - 2 7 MIT - 16.20 Fall, 2002 SECTION PROPERTIES a) Bending Stiffness, EI (as before) *2 I z = ∑ A y *2 I y = ∑ A z * I yz = ∑ A y z b) Shear Stiffness, GA (have not done this before) Consider the deflections ? v and ? w for a segment ? x with only shear forces S y and S z acting at the shear center Figure 15.18 Representation of shell beam segment with only shear forces acting at the shear center Paul A. Lagace ? 2001 Unit 15 - 2 8 MIT - 16.20 Fall, 2002 Can express the shear flows in each member as contributions due to S y and S z : q (s) = q y (s) S y + q z (s) S z where: q s y () = shear flow due to S y of unit magnitude q s z () = shear flow due to S z of unit magnitude --> To determine ? v and ? w, it is best to use an Energy Method It can be shown that: 1 A = yy 2 q y ∫ () ds t 1 A = zz 2 q z ∫ () ds t 1 A = yz qq z y ∫ ds t Paul A. Lagace ? 2001 Unit 15 - 2 9 qs MIT - 16.20 Fall, 2002 c) Torsional Stiffness, GJ Previously saw that: d α 1 dx = 2 A G ∫ q t ds (for closed section) Let: (as just did) qq () T = s where: () = shear flow due to T of unit magnitude Then, using this in the above: ds d T q α = ∫ dx GA t 2 Recalling: d α T = dx GJ 2 A ? J = q ds ∫ (for closed section) t Paul A. Lagace ? 2001 Unit 15 - 3 0 MIT - 16.20 Fall, 2002 --> Summarizing, to find the deflections: 1. Obtain the section properties (EI, GA, GJ) and the location of the shear center 2. Decompose load into moments, shears at shear center, and torque about shear center 3. Find (independently) bending, shearing, and twisting deflections about line of shear centers (elastic axis) 4. Sum deflections to obtain total deflection Figure 15.19 Representation of shell beam showing elastic axis Place x-axis along locus of shear centers --called Elastic Axis Paul A. Lagace ? 2001 Unit 15 - 3 1 MIT - 16.20 Fall, 2002 Then, vertical deflection is: x x x w = w B () + w S () ? y s i n α () found from integrating appropriate differential equation Notes : ? Shear center important for aeroelastic analysis ? Shearing deflection ( w s ) appreciable for shell beams (often 10-20% of total) ? Generally – Stiff areas provide bending (EI) – Skin thickness provides torsion (GJ) Have done this analysis for closed single-cell thin-skin shell beams. But can also apply this for other cases (3). First consider … Paul A. Lagace ? 2001 Unit 15 - 3 2 MIT - 16.20 Fall, 2002 Multi-Cell Shell Beams Figure 15.20 Representation of general multi-cell shell cross-section (looks like single-cell beam with additional webs) --> Bending stresses: calculated same way as before --> Shear stresses: divide into two problems as before (1) Pure Shear and (2) Pure Twist Proceed as before: 1. Consider “ Pure Shear ” case a) Apply joint equilibrium equation at each joint Paul A. Lagace ? 2001 Unit 15 - 3 3 MIT - 16.20 Fall, 2002 Note : Still get (n - 1) equations but there are now (n + m -1) variables n = number of joints m = number of cells (extra q ’ s are in extra webs) b) Apply Torque Boundary Condition c) Apply No Twist Condition for each cell ( Note : no twist condition applies for any closed loop: q ∫ ds = 0 ) t This gives: a) (n - 1) equations b) (1) equation c) (m) equations (n + m) equations for (n + m) variables: (n + m -1) q’ s (1) d (n + m) variables Paul A. Lagace ? 2001 Unit 15 - 3 4 MIT - 16.20 Fall, 2002 2. Consider “ Pure Torsion ” case a) Apply joint equilibrium equation at each joint get: (n -1) equations b) Apply Torque Boundary Condition New c) Every cell must twist the same amount: This is the Equal Twist Condition : ? d α ? ? ? = C for each cell j, this is a constant ? dx ? j (True since assume shape of the cross- section is preserved) This gives: a) (n - 1) equations b) (1) equation c) (m - 1) equations (n + m - 1) equations for (n + m - 1) q’ s 3. Sum results for “Pure Shear ” and “ Pure Torsion ” Paul A. Lagace ? 2001 Unit 15 - 3 5 MIT - 16.20 Fall, 2002 Have considered closed shell beams, but could also have: Open Section Shell Beams Consider (for example): Figure 15.21 Representation of Cross-Section of Open Section Shell Beam --> Bending stresses: calculated same way as before --> Shear stresses: divide into two problems as before … (1) Pure Shear and (2) Pure Twist Paul A. Lagace ? 2001 Unit 15 - 3 6 MIT - 16.20 Fall, 2002 1. Consider “ Pure Shear ” case a) Apply joint equilibrium equation at each joint thus only (n - 1) q’ s Difference: Since the section is open, there are only (n - 1) skins and ? So, get (n - 1) equations in (n - 1) unknowns and the ? shear flows are now determined b) Apply Torque Boundary Condition --> This yields the location of the shear center, d ( Note : Step (c) not necessary here) 2. Consider “ Pure Torsion ” case --> Cannot use shear flow technique. Why? ? in a closed (thin) section, assume the shear stress is constant through-the-thickness ? in an open section, the shear stress is linear through-the- thickness Paul A. Lagace ? 2001 Unit 15 - 3 7 MIT - 16.20 Fall, 2002 --> Use Membrane Analogy 2 T τ = x J J = 1 ∑ b h 3 3 for each skin Note : closed sections have greatest stiffness 3. Sum results for “Pure Shear ” and “ Pure Torsion ” Note : Shear stress is now composed of constant and linear variation (with respect to thickness) All other methods apply (deflections, etc.) Paul A. Lagace ? 2001 Unit 15 - 3 8 MIT - 16.20 Fall, 2002 Analysis of Thick Skin Shells All that ’s been done so far apply to “thin skin ” shells. How does one handle shells with “thick skins”? --> Approximate as a “thin skin ” shell. Figure 15.22 Representation of Cross-Section of Thick Skin Shell Thick skin section: skin carries normal stresses as well as shear stresses. Idealize the section by breaking up the thick skin into a finite number of “bending areas ” : See Rivello , pp. 146-147 Kahn, P., “Stresses in Aircraft and Shell Structures”, McGraw-Hill, NY, 1956 Paul A. Lagace ? 2001 Unit 15 - 3 9 MIT - 16.20 Fall, 2002 Figure 15.23 Representation of idealization of thick skin shell into bending areas and skin Bending area shear stress Idealization Thick skin Skin here carries only Wing cross-section becomes: Figure 15.24 Representation of idealization of overall wing thick skin shell cross-section Now the problem reduces to previous one (stiffened shell with ineffective skin) Paul A. Lagace ? 2001 Unit 15 - 4 0 MIT - 16.20 Fall, 2002 Question : How to idealize section? (different levels of approximation/complication) Use concept of “Effective Width ” Figure 15.25 Generic idealization of shell with thick skin If skin thickness is large, create a fictitious stringer Lump skin area in with flange and stiffness area and analyze as stiffened shell with ineffective skin Effective area: A e = A s + bt width of thickness of skin skin Paul A. Lagace ? 2001 Unit 15 - 4 1 MIT - 16.20 Fall, 2002 Notes : ? For compressive loads, not all skin area is “effective ” due to buckling. There is an “ effective widt h ” that resists buckling: A = A s + b e t “effective width ” (various expressions for b e -- depends on buckling, etc.) ? For some thick sections, there are analytical expressions to find section properties (e.g., circular) ? Skin is still “ thin ” in regards to shear, so that analysis (and the skin thickness used for it) stays the same. General Comments on St. Venant Assumptions in Shell Beam Analysis ? Apply to high aspect ratio structures where – section retains same shape – sections free to warp (i.e., no axial stresses due to twisting) Paul A. Lagace ? 2001 Unit 15 - 4 2 MIT - 16.20 Fall, 2002 ? Cross-section shape is maintained by ribs, particularly where concentrated load is applied Figure 15.26 Representation of shell beam and possible rib configurations Ribs might look like Sheets w/ hole for a pressure vessel: rings Paul A. Lagace ? 2001 Unit 15 - 4 3 MIT - 16.20 Fall, 2002 Ribs are acted on by local concentrated loads and reactions from skin Figure 15.27 Representation of load(s) acting on rib skin rib Rib will deform somewhat under loads ? Near root (or attachment), structure is not free to warp Figure 15.28 Representation of shell cross-section near attachment Axial loads result in flanges due to T Paul A. Lagace ? 2001 Unit 15 - 4 4 MIT - 16.20 Fall, 2002 ? For low aspect ratio structures, root sections, sections with cutouts, etc., refinements must be made to St. Venant Theory of Shell Beams --> use energy methods, finite element Final Note : Approximate Shell Beam Analysis is important in early stages of the iterative process of design ( “ relatively ” quick and easy) Paul A. Lagace ? 2001 Unit 15 - 4 5