Lecture #12 Stress state of sweptback wing

STRUCTURAL LAYOUT OF SWEPTBACK WINGS 2 Boeing 757

STRUCTURAL LAYOUT OF SWEPTBACK WINGS 3

STRUCTURAL LAYOUT OF SWEPTBACK WINGS 4

STRUCTURAL LAYOUT OF SWEPTBACK WINGS 5

STRUCTURAL IDEALIZATION 6

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1 – front fuse- lage beam; 2 – rear fuse- lage beam; 3 – fuselage rib; 4 – front spar continuation; 5 – root rib; 6 – front spar; 7 – ribs; 8 – rear spar; 9 – wingbox; 10 – end rib. STRUCTURAL LAYOUT OF SWEPTBACK WING 8

9 STRUCTURAL IDEALIZATION

DESIGN MODEL OF SWEPTBACK WING 10

ASSUMPTIONS AND SIMPLIFICATIONS 11 a) deformations are linear; b) displacements are small; c) wingbox has absolutely rigid cross section; d) the axial loads are carried only by spar caps; e) spar webs and skins carry only shear loads; f) the elements of the root triangle ABC and the fuselage structure (RR, FR, FSC, FFB, RFB) are planar beams, they are finitely rigid in their planes and absolutely flexible outside them; g) upper and lower skins of the root triangle do not carry any loads; h) the fuselage structure composed of beams FR, FFB, RFB is a spatial statically determinate system.

STRUCTURAL IDEALIZATION 12 Spar caps Normal forces only Quite robust idealization Skins (spar webs, upper and lower panels) Shear flows only Too robust idealization Root triangle beams Bending moments and shear forces Appropriate idealization

AIM OF THE PROJECT 13 The aim is to find the distribution of bending moments in root triangle beams. Other data (normal forces, shear flows) could not be used since it is obtained using very robust idealization. Actually, the wingbox is studied just to take its rigidity into account.

ANALYSIS OF THE MODEL 14 Kinematical analysis:

ANALYSIS OF THE MODEL 15

Matrix for statical analysis: ANALYSIS OF THE MODEL 16

Conclusion: The system is twice statically indeterminate. The force method will be used as one being optimal for systems with small degree of statical indeterminacy. ANALYSIS OF THE MODEL 17

FLOWCHART OF SOLUTION USING FORCE METHOD 18 Classification of the problem Basic system Loaded and unit states Canonical equations Total stress state Forces in removed constraints are determined Displacements corresponding to removed constraints are determined for each state In loaded state, external load is applied. In unit states, unit force is applied instead of constraint. Redundant constraints are removed

BASIC SYSTEM 19

EQUIVALENT SYSTEM 20

BASIC SYSTEM IN LOADED STATE 21

FORCES IN LOADED STATE 22

STRESS STATE OF WINGBOX – NORMAL FORCES 23 The stress state of wingbox is a problem inside a problem, twice statically indeterminate. In contrast to general problem, it is solved using Papkovich’ theorem.

STRESS STATE OF WINGBOX – SHEAR FLOWS 24

25 STRESS STATE OF WINGBOX – SUPERPOSITION

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LOADS ACTING ON ROOT TRIANGLE BEAMS 27

STRESS STATE OF ROOT TRIANGLE BEAMS 28

BASIC SYSTEM IN 1 ST UNIT STATE 29

FORCES IN 1 ST UNIT STATE 30

FORCES IN 1 ST UNIT STATE 31

LOADING OF ROOT TRIANGLE IN 1 ST UNIT STATE 32

MOMENTS IN ROOT TRIANGLE IN 1 ST UNIT STATE 33

TABLE FOR MOMENTS IN DIFFERENT STATES 34

SYSTEM OF CANONICAL EQUATIONS 35 We have twice statically indeterminate problem:

Each of coefficients has three terms; last term is from bending moments: TABLE FOR MOMENTS IN DIFFERENT STATES 36

EXAMPLE FOR A TOTAL STRESS STATE