美国麻省理工大学：《结构力学》（英文版） Unit 15 Shearing and Torsion (and Bending) of Shell Beams

MT-1620 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 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

MT-1620 al.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 Air T T connecting nodes Paul A Lagace @2001 Unit 15-2

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

MT-1620 al.2002 Idealize the cross-section of the shell beam into two parts Parts that carry extensional stress, O(and thus the bending and axial loads Parts that carry shear stress Oxs(and thus the shear loads and torques TWo examples again igh aspect ratio wing with semi-monocoque construction Notes monocoque construction all in one piece without internal framing from French "coque" meaning" eggshell “mono”= one plece seml-monocoque stressed skin construction with internal framework still have "eggshell to carry shear stresses, o internal framework to carry axial stress,O Paul A Lagace @2001 Unit 15-3

MIT - 16.20 Fall, 2002 Idealize the cross-section of the shell beam into two parts: • Parts that carry extensional stress, σxx (and thus the bending and axial loads) • Parts that carry shear stress σxs (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, σxs – internal framework to carry axial stress, σxx Paul A. Lagace © 2001 Unit 15 - 3

MT-1620 al.2002 Figure 15.2 Representation of wing semi-monocoque construction web flanges spal skin web flanges Stiffeners realize this section as: Figure 15. Idealization of wing semi-monocoque construction Paul A Lagace @2001 Unit 15-4

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

MT-1620 Fall 2002 Skins and webs are assumed to carry only shear stress o Flanges and stringers are assumed to carry only axial stress Space habitat Figure 15.4 Representation of space habitat semi-monocoque construction flanges wall spar wall stiffeners Idealize as for wing Paul A Lagace @2001 Unit 15-5

MIT - 16.20 Fall, 2002 → Skins and webs are assumed to carry only shear stress σxs → Flanges and stringers are assumed to carry only axial stress σxx • 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

MT-1620 Fall 2002 Figure 15.5 Idealization of space habitat semi- monocoque construction Outer skin and walls are assumed to carry only shear stress oxs Flanges and stiffeners are assumed to carry only axial stress o 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 Figure 15.5 Idealization of space habitat semi-monocoque construction → Outer skin and walls are assumed to carry only shear stress σxs → Flanges and stiffeners are assumed to carry only axial stress σxx 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

MT-1620 al.2002 Single ce‖ Box Beam Figure 15.6 Representation of geometry of single cell box beam modulus-weighted centroid of 4 flange and stiffener area used as origin Breakdown the problem Axial Bending stresses Each flange /stiffener has some area associated with it and it carries axial stress only(assume oxx is constant within each flange/stiffener area) Paul A Lagace @2001 Unit 15-7

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 σxx is constant within each flange/stiffener area) Paul A. Lagace © 2001 Unit 15 - 7

MT-1620 al.2002 The axial stress is due only to bending(and axial force if that exists ave 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 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 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 = Ai Find the modulus-weighted centroid (Note: flange/stiffeners may be made from different materials) Paul A. Lagace © 2001 Unit 15 - 8

MT-1620 al.2002 Choose some axis system y, z(convenience says one might usea“ corner of the bean Find the modulus-weighted centroid location A y A (2=sum over number of flanges/stiffeners number (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 =∑4 Paul A Lagace @2001 Unit 15-9

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 = * ∑Ai * ∑A z * i i z = * ∑Ai 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 Iy = ∑Ai zi * * *2 Iz = ∑Ai yi * * * * Iyz = ∑Ai yi zi Paul A. Lagace © 2001 Unit 15 - 9

MT-1620 al.2002 Find the stresses in each flange by using the equation previously developed E F 10T E,fy-E,f2-E1a△T 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=0xst [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""skins Paul A Lagace @2001 Unit 15-10

MIT - 16.20 Fall, 2002 • Find the stresses in each flange by using the equation previously developed: E FTOT  σ xx =  * − E f 12 y − E1 3f z − E1 α ∆T  E1  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 - 10