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Fuselage Frame Analysis

MECH 6471 Aircraft Structures

Fuselage Frame Analysis

by

Dr. Mohammed Abdo

Copyright of Course Materials: © Instructor generated course materials (e.g., handouts, notes, summaries, exam questions, etc.) are protected by law and may not be copied or distributed in any form or in any medium without explicit permission of the instructor. Note that infringements of copyright can be subject to follow up by the University under the Code of Student Conduct and Disciplinary Procedures.

Concordia University

MECH 6471 Aircraft Structures

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Fuselage Frame Analysis

FUSELAGE FRAME ANALYSIS References:

1) Analysis and Design of Flight Vehicles Structures, E.F.Bruhn, Sec.A.20 & C.9. 2) Engineering Sciences Data Unit, Structures Series 2

1. GENERAL INTRODUCTION In general the purpose of a transport aircraft is to carry a commercial or a military payload. In commercial service airliners are designed to carry many passengers and their baggage. They are designed to fly at high altitudes where temperatures may be far below freezing and where the air density is such as not to sustain human life. These facts mean that the vessel which carries the passengers, commonly termed the cabin, must be heated, ventilated and pressurized to provide the necessary safety and comfort. The airplane body must shield the passengers from excessive noise and vibration. The aircraft stress engineer is responsible for the strength, rigidity, fatigue life and light weight design of the fuselage structure. 2. FUSELAGE LOADS The wing being the lifting body is subjected to large distributed surface air forces, whereas the fuselage is subjected to relatively small surface air forces. The fuselage is subjected to large concentrated forces such as the wing reactions, landing gear reactions, empennage reactions, etc. In addition, the fuselage houses many items of various sizes and weights which therefore subject the fuselage to large inertia forces. Because of high altitude flight, the fuselage must also withstand internal pressures, and to handle these internal pressures efficiently requires a circular cross-section or a combination of circular elements. The basic fuselage structure is essentially a single cell thin walled tube with many transverse frames (or rings) and longitudinal stringers designed to provide a combined structure which can absorb and transmit the many concentrated and distributed applied forces safely and efficiently. The fuselage is essentially a beam structure subjected to bending, torsion and axial forces. Frames primarily serve to maintain the shape of the fuselage and to reduce the column length of the stringers to prevent general instability of the structure. Bulkheads are provided at points of introduction of concentrated forces. Unlike frames, the bulkhead structure is quite substantial and serves to distribute the applied load into the fuselage skins. Figure 1 shows examples of frames serving different functions. i) Intermediate frames are the most numerous and primarily serve to maintain the fuselage cross sectional shape, and to serve as attachment points for floor beams and posts. ii) Main frames or bulkheads are designed to carry concentrated loads such as those from the nose landing gear bay, engine mounts, pressure bulkheads and door surround structures. iii) Tail bulkheads and main spar frames are designed to carry highly concentrated loads from the wing and empennage.

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Fuselage Frame Analysis

Fig.1 Types of frame functions 3. ULTIMATE STRENGTH OF STIFFENED CYLINDRICAL STRUCTURE There are three types of instability failure of semi-monocoque structures: a) Skin Instability: Thin curved sheet skins buckle under relatively low compressive stress and shear stress and if the design requirements specify no buckling of the skin under load, it would have to be relatively thick or the section must have closely spaced stringers. This results in an inefficient design with poor strength to weight ratio. Note that the internal pressure tends to alleviate the condition. Longitudinal stringers provide efficient resistance to compressive stresses and buckled skins can transfer shear loads by diagonal tension field action. The design criteria will specify that buckling is not allowed below a certain percent of limit load. b) Panel Instability (Fig.2a): The internal rings or frames in a semi-monocoque structure, such as fuselage, divide the longitudinal stringers and their attached skin into length called panels. If these frames are sufficiently rigid, a semi-monocoque structure if subjected to bending will fail on the compression side as shown. The stringers act as columns with an effective length equal to that of the frame spacing which is the panel length. Initial failure thus occurs in a single panel and is referred to as panel instability failure. c) General Instability (Fig.2b): (over 2 or more frames) Failure by general instability extends over a distance of two or more frames as shown and is not confined between two adjacent frames as is the case for panel instability. General instability occurs when the frames are not rigid enough to support the stringers. The analogy would be to have stringer on very flexible supports.

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Fuselage Frame Analysis The design objective is to ensure that the systems will fail as a result of panel instability rather than general instability.

Fig.2 Panel (a) and general (b) instability of a fuselage shell 4.

FRAME CONFIGURATIONS Frames can be categorized in three groups: a) b) c)

Light Frames (Formed Frames): sheet metal construction, mainly to support shell; Machined Frames: to distribute concentrated loads; Bulkheads: to react pressure loads or high concentrated loads.

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Fuselage Frame Analysis The formed-frame can at best be 38% effective in helping to reduce the maximum hoop stress in the skin. Considering their inefficient cross-sectional shape for resisting radial loads applied to their outer flange, and due to the small amount of material they contain in relation to the skin, their contribution to the reduction of the skin hoop-stress is, in practice, inappreciable. The actual fuselage skin stress of modern transports are approximately 85% of the calculated hoop stress: The circumferential, or hoop, stress is determined by

. (Fig.3)

The longitudinal stress is given by Where P= cabin pressure (typically 8 psi), R=fuselage radius and t= skin thickness

Fig.3 Shell stresses due to internal pressure Clips are used to connect the stringer to the frame as shown in Fig.4 The purpose of stringer clips is to: i) transfer skin panel normal pressure loads to frame; ii) help break up excessive column length of stringers; iii) provide some degree of compressive strength of frame inner chord (cap); iv) act as frame web panel stiffener.

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Fuselage Frame Analysis

Fig.4 Frame-to-stringer attachment methods Due to the pressure load, the skin wants to assume a radially deflected shape as shown in Fig. 5a. The shear tie holds the skin in the desired shape locally between stringers to the desired contour. The frame will therefore experience a bending load, i.e. the skin and shear ties are trying to bend the frame from the desired contour into the more natural radially deflected shape for carrying pressure. The essence of the frame requirements shows up in the areas where pressure loads combine with concentrated loads (e.g. floor beams loads) to produce large bending moments. The radial deflections due to the pressure load are then restrained by the concentrated loads such as shown in Fig. 5a.

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Fuselage Frame Analysis

Fig.5 Frame bending under pressurization (a) and frame/skin section (b) The effect of the frame in helping to carry pressurization loads can be illustrated by the following analysis (refer to Fig.5b). Consider a cabin pressure P = 8.4 psi. The frame spacing, b=15 inches. The effective skin width is taken to be from mid-bay to mid-bay ( ). The frame cross sectional area Af = 0.050(3+2x0.7) = 0.22 in2. If the frame area is spread over the15 inch frame bay, its thickness would be tf =

= 0.0147 in. If only the skin is considered to be

carrying pressure loads only, the skin hoop stress is given by

=

= 8900 lb/in2.

However, if the frame is also considered to be effective in carrying pressure loads, the equivalent skin thickness increases to t = 0.050 + 0.0147 = 0.0647 in. and the hoop stress drops by 23% to 6880 lb/in2. Load sharing between the frame and skin can thus be considered to contribute towards a lightweight design. 5.

PRESSURE BULKHEADS

The cylindrical shell of pressurized-cabin is closed at the rear by a dome in preference to a flat bulkhead (Fig.6). For small aircraft, the space saving obtained from flat bulkheads are nonnegligible. From a structural point of view, a hemispherical shell provides an ideal rear dome because of the membrane stresses for a given amount of material are the least. The problem of choosing the most efficient method of joining the hemispherical dome to the forward cabin shell and to the rear fuselage is challenging work.

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Fuselage Frame Analysis

Fig.6 Types of fuselage bulkheads.

6.

WING AND FUSELAGE INTERSECTIONS

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Fuselage Frame Analysis If the airplane is of the low wing or high wing type, the entire wing structure can be continuous within the fuselage. Typical wing to fuselage interfaces are shown for the Lockheed L-1011 Tristar and Boeing 747 in Fig.7. A similar configuration is used for the Bombardier Canadair Challenger and Regional Jet (CRJ) aircraft.

Fig.7 Wing and fuselage intersections

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Fuselage Frame Analysis Figure 8 shows the finite element modeling of the CRJ engine mount structure and intersection with the frame. The engine mount beam is a much more efficient way of carrying engine thrust, inertia and gyroscopic loads onto the bulkhead frame than if the engine nacelles were individually mounted on the frame. Such a configuration could impose very high concentrated loads on the frame resulting in a heavier design.

Fig.8 CRJ engine mount bulkhead frame FEM. 7.

STRESS ANALYSIS METHODS – SKIN SHEAR FLOW

7.1

Introduction The stiffened-shell type of fuselage structure is quite similar to the wing construction with distributed bending material. One of the essential differences is that in the wing shear webs are provided to carry the greater part of the vertical shear, whereas in the fuselage the shell of the structure is relied upon to resist the shear loads. The fuselage frames and bulkheads are equivalent to ribs in the wing, for they transmit the shear loads to the covering and maintain the shape of the structure. The bending loads are resisted by the sheet covering and longitudinal stiffening elements. The stiffening elements are usually stringers such as angles, zees, hat sections either bent up or extruded.

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Fuselage Frame Analysis

In calculating the bending stresses it is usually assumed that the elementary beam theory is sufficiently accurate resulting in a bending distribution given by the equation

.

In order to agree with the bending theory the fuselage shear distribution over the frames should be in accordance to the shear equation It is common practice to use the simplified beam theory in calculating the stresses in the skin and stringers of a fuselage structure. If the fuselage is pressurized, the stresses in the skin due to this internal pressure must be added to the stresses which resist the flight loads (ie σsk = σb + σL.) Figure 9 illustrates a distributed stringer type of fuselage section. Up to the point of buckling of the curved sheet between the skin stringers, all the material in the beam section can be considered fully effective and the bending stresses can be computed by the general flexure formula

, where Iy is the centroid moment of inertia of the section.

Fig.9 Effective fuselage section in upward bending

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Fuselage Frame Analysis

On the tension side, the entire sheet is effective in resisting the bending moment. When a bending compression stress is reached which causes the curved sheet between stringers to buckle, a re-arrangement takes place in the stress distribution on the section as a whole. The following assumptions are made: i)

A small width of sheet W1 on each side of the attachment line of the skin to stringer is considered as carrying the same compressive stress as the stringer. (W1 = 30tskin)

ii)

In practice, since the thin curved skin between stringers normally buckles under a compressive stress far below the buckling strength of the stringers, the curved sheet between stringer outside of the effective skin area given in Item (i) above is not included in the calculation of the moment of inertia. (conservative assumption).

One can see that when an asymmetrical fuselage cross section (as shown in Fig.9) is subjected to pressurization loads, the entire section would experience tensile stresses. This implies that the skin can be treated as fully effective in the calculation of the moment of inertia. Thus, for various flight cases the top arch could be in tension and the bottom arch in compression, such as in an up-gust case. The reverse could also be true as in a down-gust or dynamic landing case, or for something totally different, such as in a lateral gust case. This means that one would have to initially calculate the stresses around the cross-section for a given critical load case by estimating which areas are in tension or compression and then iterating for the true condition until the most optimum cross-section can be obtained. This could be an enormous task considering the number of load cases that must be considered in the analysis of a fuselage structure. To remedy the problem, an effective stringer area is calculated which usually considers an effective skin width of approximately 30t to be used on the compression and tension sides.

7.2

Shear Flow Analysis for Fuselage Structures

The shear flow analysis can be made once the effective cross-sections of the fuselage are obtained. The procedures are given in the example below: Example Problem 1: Find the skin shear flows acting on a frame due to fuselage bending loads for a symmetrical tapered section. Figure 10 represents a tapered circular fuselage structure that might be representative of the rear portion of a small airplane. For simplicity, the stringers are the only effective material. Solution is by the Change in Stringer Loads Between Adjacent Stations. ΔP Method

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Fuselage Frame Analysis The problem will be to determine the stringer stresses and the skin shear flow stress system at Station (0) under a given load system at Station (150). Moment of inertia of section at stations (0) and (30) is determined by Iy=ΣAizi2: At Station (0): Iy = 2(0.1 x 152) + 4[0.1(13.862 + 10.612 + 5.742)] = 180 in4 At Station (30), (see Fig.11) : Iy = 2(0.1 x 142) + 4[0.1(12.932 + 9.92 + 5.362)] = 157 in4

Fig.11 Fuselage section at Sta.30

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Fuselage Frame Analysis

Fig.10 Analysis of a tapered fuselage under bending to obtain flexural skin shear flows MECH 6471 Aircraft Structures

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Fuselage Frame Analysis

Fig.12 Skin shear flows acting on a frame due to fuselage bending and torsion loads The shear flow between stringers arising from flexural stress is shown in Column 13 in Figure 10. But since the 2000 lb vertical load is offset by 5 inches, a torsion load results which adds another shear flow of

=-7.06 lb/in., where A = area of fuselage cross-

section enclosed area. The total shear flow acting on the frame at the skin line is given in Fig.12. Since there is an offset between the shear flow acting along the skin line and the frame neutral axis, a bending moment will be induced in the frame as shown in Fig.13. The frame stress analysis can now be conducted.

Fig.13 Skin shear flow induced bending in the frame

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Fuselage Frame Analysis 7.3

Supporting Boundary Forces on Fuselage Frames

Example Problem 2: Find the skin shear flows acting on a frame due to concentrated loads acting on the fuselage. The cross-section of the fuselage is shown in Fig.14. Two concentrated loads of 2000 lbs each are applied to the fuselage frame at the points indicated. The problem is to find the reacting shear flow forces in the fuselage skin which will balance the two externally applied loads. The fuselage stringer arrangements are assumed symmetrical. Solution: It is assumed that the fuselage skin resists the forces according to the beam theory. The general flexural shear stress equation for bending about the y-axis is given by flexural shear flow is given by q = τt =

and the

, where V is the vertical load.

Assuming that the effective stringer areas are the same around the frame and taken as 0.15 in , the moment of inertia is calculated as: 2

Iy = 4x0.15(17.62 + 16.22 + 13.52 + 13.52 + 102 + 52)= 637 in4. (Note that there are two back-toback angles at Str.3, 3’, 9 and 9’)

Fig.14 Skin shear flows acting on a frame due to symmetrical concentrated loads

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Fuselage Frame Analysis Due to the symmetry of the effective section and external loading, the shear flow in the fuselage skin on the z-axis or between stringers 1 and 1’ and between 11 and 11’ will be zero. The general shear flow solution for this problem is q (1-2) = -6.275 ( 0.15 )( 17.6 ) = -16.57 lb/in q (2-3) = -16.57 - 6.275 ( 0.15 )( 16.2 ) = -31.82 lb/in q (3-4) = -31.82 - 6.275 ( 2 x 0.15 )( 13.5 ) = -57.22 lb/in q (4-5) = -57.22 - 6.275 ( 0.15 )( 10 ) = -66.62 lb/in q (5-6) = -66.62 - 6.275 ( 0.15 )( 5 ) = -71.32 lb/in As a check on the above work, the summation of the z components of the shear flow on each skin panel between the stringers should equal to the external load of 4000 lbs. Asymmetric Load Case Assume that the external loading is asymmetrical as shown in Fig.15. A torque would be generated about the c.g of the fuselage section, T = 1500 ( 11.5 ) – 2500 ( 11.5 ) = -11500 in-lb. The shear load Vz=4000 lbs produces the same shear flow pattern. To balance the moment, a constant shear flow q1 around the frame is necessary, where where A = area of fuselage cross-section enclosed area. Adding this constant force system to that calculated in the symmetrical example gives the results in Fig.15.

Fig.15 Skin shear flows acting on a frame due to asymmetric concentrated loads MECH 6471 Aircraft Structures

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Fuselage Frame Analysis

8.

STRESS ANALYSIS METHODS – DIRECT LOADS

The previous section considered loads induced into the frame due to skin shear flows. In this section, concentrated loads will be analyzed. Concentrated loads arise from adjoining structure such as floor beams and posts, empennage, engine mounts and wing spars. Even though finite element modeling has become the common tool to determine internal frame loads, classical methods such as those presented here can be used not only in preliminary sizing (before an FEM is generated) but in detailed analysis if an FEM is crude or not available. In the decades preceding the practical application of FEM (mid-1960s), classical stress methods were used in structural design - from the first all-metal aircraft of the 1930s, to the first generation jetliners (including the Concorde) and high performance aircraft such as the Mach 3.5 SR-71 (1962) and Mach 6.7 X-15 (1959). Some of the more commonly used analytical tools today are those found in Engineering Sciences Data Unit data sheets. The ESDU (based in the UK) has compiled data from all sectors of the industry, universities, and government research centers (such as NASA) and have prepared series dealing with stress, fatigue, thermodynamics and other fields. Some of the ESDU data sheets dealing with frame analysis are contained in the Appendix. A simple analysis will be used here to illustrate the use of some ESDU data sheets. Consider the frame in Fig.16 having radius 60 inches, subjected to concentrated loads arising from its attachment to a floor beam and posts. The frame is considered to be a rigid circular ring having no adjacent structure. The applied external forces are resolved into normal, moment and tangential components to fit the convention of the data curves on the pages that follow (Figures 16a to 16c). The frame has been given a radial coordinate system. Each external force that is applied to the frame is reacted by a skin with uniform shear flow, q (lb/in) The internal axial force (N), shear force (S) and bending moment (M) at any point (θº) along the frame can now be determined by reading the coefficients on the vertical scale of Figures 16a to 16c and solving for N, S and M. A table (Figures 16d to 16f) is then created to tabulate the internal loads for each of load P1, Q1, T1, P2, Q2, etc. A final summary table (Figure 16g) is then used to determine the total loads and plotted (Figure 16h) to give the max and min loads that will then be used to conduct a stress analysis and sizing of the frame.

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Fuselage Frame Analysis

Fig.16 Direct frame loads

R = 60 in RP1 = 80040 in-lb RP2 = 79080 in-lb RP4 = -80040 in-lb

RQ1 = 219900 in-lb RP3 = -79080 in-lb RQ4 = 219900 in-lb

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Fuselage Frame Analysis

Fig.16a Internal loads for a rigid frame due to a tangential load P

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Fuselage Frame Analysis

Fig.16b Internal loads for a rigid frame due to a radial load Q MECH 6471 Aircraft Structures

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Fuselage Frame Analysis

Fig.16c Internal loads for a rigid frame due to a bending moment T

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Fuselage Frame Analysis

Fig.16d Frame shear loads

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Fuselage Frame Analysis

Fig.16e Frame axial loads (tension+) MECH 6471 Aircraft Structures

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Fuselage Frame Analysis

Fig.16f Frame bending moments MECH 6471 Aircraft Structures

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Fuselage Frame Analysis

Fig.16g Summary of frame internal loads

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Fuselage Frame Analysis

Fig.16h Plot of frame loads

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Fuselage Frame Analysis 9

FLEXIBLE FRAMES

In the previous section, a method of determining the load distribution in circular frames was given in which it was assumed that the shear stresses in the shell were distributed in accordance with the engineers theory of bending. However, this assumption is very approximate in the region of the frame where concentrated loads are applied. The amount of divergence from the simple engineer’s theory increases with frame flexibility and spacing. The example problem that follows will demonstrate the differences between rigid frames and flexible frames, the latter being analyzed using ESDU data sheets and by a NASTRAN finite element model.

Fig. 17

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Fuselage Frame Analysis STEP 1: Determine the Bending Moment Distribution Assuming Rigid Frame (Fig.17). P = 10,000 lbs. From ESDU 03.06.01 “Moments in Circular Frame due to Concentrated Loads and Couples”. Note (1): ESDU sign convention for +M is tension on inside of frame cap Note (2): converted to FEM sign convention for +M is compression on inside cap angle θº 0 20 44 66 90 120 150 180 210 240 270 293 317 340 360

M/(Na*R) -0.24 -0.08 0.10 0.09 0.09 0.02 -0.05 -0.08 -0.05 0.02 0.09 0.10 0.03 -0.08 -0.24

M (Note 1) -144000 -48000 18000 60000 54000 12000 -30000 -48000 -30000 12000 54000 60000 18000 -48000 -144000

M (Note 2) 144000 48000 -18000 -60000 -54000 -12000 30000 48000 30000 -12000 -54000 -60000 -18000 48000 144000

STEP 2: Bending Moment Distribution due to radial load P assuming a flexible frame. Reference (1): ESDU 03.06.17 “Flexible Circular Frames Supporting a Shell. The Effect of Adjacent Frames and the Longitudinal Flexibility of the Shell”. Reference (2): ESDU 03.06.08 “Flexible Circular Frames supported be a Shell. Moments in a Frame due to Concentrated Radial Loads”. Using Ref (1) with R = 60 in. E = 10.5x106 lb/in2 G = 3.99x106 lb/in2 A (str) =0.29 in2 MECH 6471 Aircraft Structures

I = 0.388 in4 t = 0.050 in (shell thickness) L = 20 in (frame pitch)

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Fuselage Frame Analysis

Assume

the

stringers

are

equally

spaced

around

the

circumference:

=29107 If all the stringer areas (total number N=14) are smeared into the skin, the new shell thickness is t΄=

=

and

= 0.061 in = 916 887. From Fig1 in Ref.1,

M+ produces tension on inside frame cap angle θº 0 20 44 67 90 120 150 180

M/(Na*R) -0.07 0.015 0.010 -0.005 0 0.002 0 0

M+ 42000 -9000 -6000 3000 0 -12000 0 0

STEP 3: Finite Element Model To demonstrate how a flexible frame analysis does not conform to the engineers bending theory, the frame analysis in Step 1 will be analyzed using NASTRAN FEM. The model consists of 6 frame bays, 20 inches apart, fore and aft stringers and t=0.050 in. fuselage skin. The frames are three inches deep with 0.070 inch flanges and 0.050 thick typical. I (frame) = 0.38 in4 including effective skin A (frame) = 0.29 in2 A (stringer) = 0.29 in2 The frame was modeled using CBAR elements. The stringers and floor posts were modeled using CROD elements. The skin shell was modeled using CSHEAR elements (Figs. 18 and 19) The load applied was 10, 000 lbs in the –ve y-axis on Node 101. MECH 6471 Aircraft Structures

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Fuselage Frame Analysis

Fig.18 CBAR element coordinate system and element forces

Fig.19 Frame identification, nodes (left) and elements (right)

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Fuselage Frame Analysis The graphs in Fig.20 show the bending moment distribution around the frame for the following methods of analysis: a) Rigid Frame b) Flexible Frame using Finite Element c) Flexible Frame analysis using ESDU The Engineers Bending Theory assumes all the load is dissipated within one frame bay. The finite element analysis shows that the moments are much lower than those predicted by the rigid analysis. In addition, as shown in the graph, the curve labeled frm1-fem represents the bending moment distribution for the frame at station 100 which is directly loaded. The curve identified as frm2-fem is the frame next to the loaded frame at station 100 which shows that it also picks up some of the load; with the Engineers theory this will not occur. The analysis using ESDU curves for flexible frame (step 2) are consistent with the FEM analysis. The shear distribution in the first bay is vastly different from the Engineers Bending Theory. It is not till Bay 5 is reached that the shear distribution approaches that given by the Engineers Bending Theory. The change in shear distribution between Bay 1 and Bay 2 also indicated that the second frame is also loaded. Hence, the members nearer to the applied loads are being subjected to greater loads. Bar ID

FEM Angle Rigid Bar ID FEM Flexible sta100 θº Frame sta200 frame (note 1) ESDU 03.06.01 (note 2) ESDU 03.06.08 101 47980 0 -144000 201 14805 42000 102 -13994 19.47 -48000 202 -341 -9000 103 -16612 43.96 18000 203 -11784 -6000 104 10749 66.44 60000 204 7033 3000 105 122 90 54000 205 -319 0 106 -2133 120 12000 206 -1334 -12000 107 285 150 -30000 207 229 0 108 1472 180 -48000 208 1061 0 109 285 210 -30000 209 229 110 -2133 240 12000 210 -1334 111 122 270 54000 211 -319 112 10749 293.55 60000 212 7033 113 -16612 317.04 18000 213 -17784 114 -13994 340.52 -48000 214 -341 47980 360 -144000 14805 Note 1: P=10000 lb applied at θº = 0 on node 101 sta.100 Note 2: Bending moment distribution one frame bay from directly loaded frame

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Fuselage Frame Analysis

Fig. 20 MECH 6471 Aircraft Structures

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Fuselage Frame Analysis

Appendix ESDU 03.06.01 Moments in Circular Frame due to Concentrated Loads and Couples ESDU 03.06.02 Direct Forces in Circular Frame due to Concentrated Loads and Couples ESDU 03.06.03 Shear Forces in Circular Frame due to Concentrated Loads and Couples ESDU 03.06.08 Flexible Circular Frames Supported by a Shell. Concentrated Radial Loads

ESDU 03.06.17 Flexible Circular Frames Supporting a Shell. and the Longitudinal Flexibility of the Shell

Moments in a Frame due to

The Effect of Adjacent Frames

ESDU 02.04.04 Initial Buckling of Flat Plates Under Bending and Compression or Tension.

ESDU 02.04.05 Initial Buckling of Flat Plates Under Compression, Bending and Shear

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