1 UNIT- I Thin plate theory, Structural Instability:
2 Analysis of thin rectangular plates subject to bending, twisting, distributed transverse load, combined bending and in-plane loading
5 Thin plates having small initial curvature
6 Energy methods of analysis. Strain energy produced by bending and twisting
7 Buckling of thin plates- elastic, inelastic Inelastic buckling of plates
9 experimental determination of critical load for a flat plate
10 local instability, instability of stiffened panels,
11 failure stresses in plates and stiffened panels.
13 Tension field beams- complete diagonal tension,
14 incomplete diagonal tension,
15 post buckling behaviour.
16 Unit-II Bending and Shear and Torsion Of Thin Walled Beams:
17 Symmetrical bending arises in beams which have either singly or doubly symmetrical cross-sections
18 UNSYMMETRICAL BENDING Let the axis origin coincide with the centroid G of the cross section, and that the neutral axis is a distance p from G. The direct stress s z on element d A at point (x,y) and distance x from the neutral axis is: Look at the beam in a plane parallel to the neutral axis with two segments ij and kl which are of equal length when the beam is undeflected: Derivation of bending stress using general method
19 Once the beam has been deflected this section will look like this where: R = the radius of curvature dq = angle between planes ik and jl The strain in plane kl can be defined as: As the beam supports pure bending, the resultant load on the end section must be zero. Hence
20 If the inclination of the Neutral Axis (N.A.) is at an angle from the x-axis then : x = x sin( α ) + y cos(α )
21 Neutral Axis: When a homogeneous beam is subjected to elastic bending, the neutral axis (NA) will pass through the centroid of its cross section, but the orientation of the NA depends on the orientation of the moment vector and the cross sectional shape of the beam. When the loading is unsymmetrical (at an angle) as seen in the figure below, the NA will also be at some angle - NOT necessarily the same angle as the bending moment. ( Realizing that at any point on the neutral axis, the bending strain and stress are zero, we can use the general bending stress equation to find its orientation. Setting the stress to zero and solving for the slope y/x gives
22 SHEAR FLOW AND SHEAR CENTRE Restrictions: 1. Shear stress at every point in the beam must be less than the elastic limit of the material in shear. 2. Normal stress at every point in the beam must be less than the elastic limit of the material in tension and in compression. 3. Beam's cross section must contain at least one axis of symmetry. 4. The applied transverse (or lateral) force(s) at every point on the beam must pass through the elastic axis of the beam. Recall that elastic axis is a line connecting cross-sectional shear centers of the beam. Since shear center always falls on the cross-sectional axis of symmetry, to assure the previous statement is satisfied, at every point the transverse force is applied along the cross-sectional axis of symmetry. 5. The length of the beam must be much longer than its cross sectional dimensions. 6. The beam's cross section must be uniform along its length.
23 Shear Center If the line of action of the force passes through the Shear Center of the beam section, then the beam will only bend without any twist. Otherwise, twist will accompany bending. The shear center is in fact the centroid of the internal shear force system. Depending on the beam's cross-sectional shape along its length, the location of shear center may vary from section to section. A line connecting all the shear centers is called the elastic axis of the beam. When a beam is under the action of a more general lateral load system, then to prevent the beam from twisting, the load must be centered along the elastic axis of the beam.
24 Shear Center The two following points facilitate the determination of the shear center location. 1. The shear center always falls on a cross-sectional axis of symmetry. 2. If the cross section contains two axes of symmetry, then the shear center is located at their intersection. Notice that this is the only case where shear center and centroid coincide.
25 SHEAR STRESS DISTRIBUTION RECTANGLE T-SECTION
26 SHEAR FLOW DISTRIBUTION
27 EXAMPLES For the beam and loading shown, determine: (a) the location and magnitude of the maximum transverse shear force 'Vmax', (b) the shear flow 'q' distribution due the 'Vmax', (c) the 'x' coordinate of the shear center measured from the centroid, (d) the maximun shear stress and its location on the cross section. Stresses induced by the load do not exceed the elastic limits of the material. NOTE:In this problem the applied transverse shear force passes through the centroid of the cross section, and not its shear center.
28 Shear Flow Analysis for Unsymmetric Beams SHEAR FOR EQUATION FOR UNSUMMETRIC SECTION IS
29 SHEAR FLOW DISTRIBUTION For the beam and loading shown, determine: (a) the location and magnitude of the maximum transverse shear force, (b) the shear flow 'q' distribution due to 'Vmax', (c) the 'x' coordinate of the shear center measured from the centroid of the cross section. Stresses induced by the load do not exceed the elastic limits of the material. The transverse shear force is applied through the shear center at every section of the beam. Also, the length of each member is measured to the middle of the adjacent member.
30 Beams with Constant Shear Flow Webs Assumptions: 1. Calculations of centroid, symmetry, moments of area and moments of inertia are based totally on the areas and distribution of beam stiffeners. 2. A web does not change the shear flow between two adjacent stiffeners and as such would be in the state of constant shear flow. 3. The stiffeners carry the entire bending-induced normal stresses, while the web(s) carry the entire shear flow and corresponding shear stresses.
31 Analysis Let's begin with a simplest thin-walled stiffened beam. This means a beam with two stiffeners and a web. Such a beam can only support a transverse force that is parallel to a straight line drawn through the centroids of two stiffeners. Examples of such a beam are shown below. In these three beams, the value of shear flow would be equal although the webs have different shapes. The reason the shear flows are equal is that the distance between two adjacent stiffeners is shown to be 'd' in all cases, and the applied force is shown to be equal to 'R' in all cases. The shear flow along the web can be determined by the following relationship
32 Important Features of Two-Stiffener, Single-Web Beams: 1. Shear flow between two adjacent stiffeners is constant. 2. The magnitude of the resultant shear force is only a function of the straight line between the two adjacent stiffeners, and is absolutely independent of the web shape. 3. The direction of the resultant shear force is parallel to the straight line connecting the adjacent stiffeners. 4. The location of the resultant shear force is a function of the enclosed area (between the web, the stringers at each end and the arbitrary point 'O'), and the straight distance between the adjacent stiffeners. This is the only quantity that depends on the shape of the web connecting the stiffeners. 5. The line of action of the resultant force passes through the shear center of the section.
33 EXAMPLE For the multi-web, multi-stringer open-section beam shown, determine (a) the shear flow distribution, (b) the location of the shear center
34 Torsion of Thin - Wall Closed Sections Derivation Consider a thin-walled member with a closed cross section subjected to pure torsion.
35 Examining the equilibrium of a small cutout of the skin reveals that
37 Angle of Twist By applying strain energy equation due to shear and Castigliano's Theorem the angle of twist for a thin-walled closed section can be shown to be Since T = 2qA, we have If the wall thickness is constant along each segment of the cross section, the integral can be replaced by a simple summation
38 Torsion - Shear Flow Relations in Multiple-Cell Thin- Wall Closed Sections The torsional moment in terms of the internal shear flow is simply
39 Derivation For equilibrium to be maintained at a exterior-interior wall (or web) junction point (point m in the figure) the shear flows entering should be equal to those leaving the junction Summing the moments about an arbitrary point O, and assuming clockwise direction to be positive, we obtain The moment equation above can be simplified to
40 Shear Stress Distribution and Angle of Twist for Two-Cell Thin-Walled Closed Sections The equation relating the shear flow along the exterior wall of each cell to the resultant torque at the section is given as This is a statically indeterminate problem. In order to find the shear flows q1 and q2, the compatibility relation between the angle of twist in cells 1 and 2 must be used. The compatibility requirement can be stated as where
41 The shear stress at a point of interest is found according to the equation To find the angle of twist, we could use either of the two twist formulas given above. It is also possible to express the angle of twist equation similar to that for a circular section
42 Shear Stress Distribution and Angle of Twist for Multiple-Cell Thin- Wall Closed Sections In the figure above the area outside of the cross section will be designated as cell (0). Thus to designate the exterior walls of cell (1), we use the notation 1-0. Similarly for cell (2) we use 2-0 and for cell (3) we use 3-0. The interior walls will be designated by the names of adjacent cells. the torque of this multi-cell member can be related to the shear flows in exterior walls as follows
43 For elastic continuity, the angles of twist in all cells must be equal The direction of twist chosen to be positive is clockwise.
44 TRANSVERSE SHEAR LOADING OF BEAMS WITH CLOSED CROSS SECTIONS
45 EXAMPLE For the thin-walled single-cell rectangular beam and loading shown, determine (a) the shear center location (ex and ey), (b) the resisting shear flow distribution at the root section due to the applied load of 1000 lb, (c) the location and magnitude of the maximum shear stress
46 Unit-III Structural Idealisation of Thin Walled Beams
47 Structural Idealization Consider the two-spar wing section shown. The stringers and spar carry most of the direct stresses while the skin carries the shear stresses. Since variation in stress over the stringer or spar flange due to bending of the wing would be small, it can be assumed to be constant.
48 Structural Idealization Stingers and spar flanges can then be replaced by concentrations of areas known as booms. It can be assumed that all direct stresses are carried by booms and the skin carries only shear. Direct stress carrying capacity of the skin may be accounted for by increasing the boom cross-section area.
49 Structural Idealization If skin does carry direct stress, we idealize it as a section that carries only shear stress, and add effective area to the booms. Right Side Similarly: Effective boom area due to skin carrying direct stress these add to the boom areas of flanges, strings, spars, etc..
50 Shear of open-section beams In the expression for shear flow in the cross-section: t D is the direct stress carrying thickness of the skin t D = t, if the skin is fully effective in carrying direct stress t D = 0, if the skin is assumed to carry only shear stress If we idealize skin as shown previously, then shear flow in the skin due to bending of the skin = 0. The above expression does not account for booms. How can we deal with booms that cause discontinuity in the skin and therefore interrupt the shear flow?
51 Shear of open-section beams Sx and Sy produce direct stresses due to bending in the booms (and skin) and shear stresses in the skin
52 Shear of open-section beams r th boom has a cross-sectional area B r Shear flows in skin adjacent to it are q 1 and q 2. Equilibrium in z direction of previous figure This gives:
53 Recall: Shear of open-section beams
54 Shear of open-section beams This gives the change in shear flow induced by a boom. Each time a boom is encountered, the shear flow is incremented by this amount. If at some distance s at the profile, if n booms have been passed:
55 Open C/S Sample Problem Calculate the shear flow distribution in channel due to a 4.8 kn vertical shear load acting through the shear center. Booms carry all the direct stresses (B r = 300 mm 2 ) q s S I y xx n r 1 B r y r
56 Open C/S Sample Problem Calculate I xx : (Only consider direct stress carrying areas) I.e. Booms I xx Ad mm 2 q s n r 1 B r y r 10 4 n r 1 B r y r At the outside of boom 1, q s = 0. As boom 1 is crossed, the shear flow changes to: 4 q N/mm q 1
57 Open C/S Sample Problem There will be no further changes in shear flow until the next boom (2) is crossed. 4 q N/mm q q N/mm q 3 At the outside of boom 4, the shear flow is zero (q s = 0) as expected N/mm
58 Open C/S Sample Problem S 1 S 2 12 N/mm 6 N/mm q q q N/mm 12 N/mm 6 N/mm How come all the signs are negative? 6 N/mm S 3
59 Closed C/S Sample Problem For the single cell beam, the booms carry the direct stresses and the walls carry only the shear stress. A vertical shear load of 10 kn acts through the vertical plane between booms 3 and 6. Calculate the shear flow distribution
60 Closed C/S Sample Problem Centroid on Horizontal Axis of Symmetry. I xy = 0 Also S x = 0, t D = 0 q I xx s S I y xx n r 1 B r y r q s,0 I xx can be calculated from the direct stress carrying area of the booms 2 B 30 B 100 B 100 B Substituting B 1, B 4 gives I xx = x 10 6 mm 4 q s n r 1 B r y r q s, n r 1 B r y r q s,0
61 Closed C/S Sample Problem Introduce a cut in the wall 23 and calculate the basic shear flow around the walls q q q b,23 b,34 b,45 0 Since the t D = B 3 y N/mm B 4 y N/mm q q q q b,56 b,67 b,21 b,18 q q q b q b,34 b, N/mm (by symmetry) (by symmetry) , 87 b,21 B y N/mm B 1 y N/mm 18.1 N/mm (by symmetry)
62 Closed C/S Sample Problem Taking moments about the intersection of the line of action of shear load and horizontal axis: 0 pqbds 2Aqs,0 Solve for q s,0 is broken up into segments where each q b is constant Draw out the shear flow distribution to determine the sign of the moment generated by the shear flow on each segment
63 [ q b,81 2q q p s b,43 Closed C/S Sample Problem 2q b, q b,54 2q ] b, ,200q s,0 0 Substituting for the basic shear flow gives: q s, N/mm Enclosed Area, A Add q s,0 to the basic shear flow to get the total shear flow in every wall. In any wall the final shear flow is given by q s =q b + q s,0 so that q q q q q N/mm 5.4N/mm q 34.3 N/mm q 37.9 N/mm 17 N/mm q 87
64 UNIT- IV Structural and Loading Discontinuities in Thin Walled Beams
65 Closed section beams- shear stress distribution of a closed section beam built in at one end under bending, shear and torsion loads. If the moment centre is chosen to coincide with the lines of action of Sx and Sy then
66 Open section beams
68 Shear lag- effect of shearing strains in beamsredistribution of bending stresses due to restraining of warping, limitation of elementary bending theory, effect of accounting for shear lag on the estimated strength.
69 UNIT- V Stress Analysis of Aircraft Components- Wing, Fuselage:.
70 Wing spars and box beams- tapered wing spar, open and closed section beams, beams having variable stringer areas. Wings- Three-boom shell in bending, torsion, shear, tapered wings, deflections, cut-outs in wings. Bending, shear, torsion, cut-outs in fuselages. Fuselage frames and wing ribs- principles of stiffener/ web construction, fuselage frames, wing ribs
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