Lecture #12Stress state of sweptback wing

STRUCTURAL LAYOUT OF SWEPTBACK WINGS

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Boeing 757

STRUCTURAL LAYOUT OF SWEPTBACK WINGS

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STRUCTURAL LAYOUTOF SWEPTBACK WINGS

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STRUCTURAL LAYOUTOF SWEPTBACK WINGS

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STRUCTURAL IDEALIZATION

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STRUCTURAL IDEALIZATION

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

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STRUCTURAL IDEALIZATION

DESIGN MODEL OF SWEPTBACK WING

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ASSUMPTIONS AND SIMPLIFICATIONS

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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

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Spar capsNormal forces

onlyQuite 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

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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

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Kinematicalanalysis:

ANALYSIS OF THE MODEL

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Matrix for statical analysis:

ANALYSIS OF THE MODEL

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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

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FLOWCHART OF SOLUTION USING FORCE METHOD

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Classificationof the problem

Basic system

Loaded andunit states

Redundant constraints are removed

In loaded state, external load is applied. In unit states, unit force is applied instead of constraint.

Canonical equations

Total stress state

Forces in removed constraints are determined

Displacements corresponding to removed constraints are

determined for each state

BASIC SYSTEM

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EQUIVALENT SYSTEM

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BASIC SYSTEM IN LOADED STATE

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FORCES IN LOADED STATE

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STRESS STATE OF WINGBOX – NORMAL FORCES

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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

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STRESS STATE OF WINGBOX – SUPERPOSITION

STRESS STATE OF WINGBOX – SUPERPOSITION

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LOADS ACTING ON ROOT TRIANGLE BEAMS

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STRESS STATE OF ROOT TRIANGLE BEAMS

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BASIC SYSTEM IN 1ST UNIT STATE

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FORCES IN 1ST UNIT STATE

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FORCES IN 1ST UNIT STATE

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LOADING OF ROOT TRIANGLE IN 1ST UNIT STATE

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MOMENTSIN ROOTTRIANGLEIN 1ST UNITSTATE

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TABLE FOR MOMENTS IN DIFFERENT STATES

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SYSTEM OF CANONICAL EQUATIONS

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We have twice statically indeterminate problem:

Each of coefficients has three terms; last term is from bending moments:

TABLE FOR MOMENTS IN DIFFERENT STATES

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EXAMPLEFOR A TOTALSTRESSSTATE