Aerospace Structural Design

MAE 4281

Stiffened Shear Webs

David FlemingAssociate Professor

Aerospace Engineering

Reference

Curtis, H.D., Fundamentals of Aircraft Structural Analysis, Irwin, 1997, §2.5, “Stiffened Shear Webs”

Sarafin, § 15.3

Megson, § 9.7

In this section, we will review some Statics and Mechanics of Materials, and introduce analysis relevant to semimonocoque structures

NASA SP-8088Megson, Aircraft Structures for Engineering Students

Beams support bending moments and shear resultants

“Internal Resultants” are total forces and moments supported by a cross-section (Statics)

Load is actually distributed across the plane of the cross-section (Mechanics of Materials)

• Normal stress [assumes symmetric cross-section beam]

• Shear stress

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max

)(;y

y

ydAyQIt

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“I-beam” is effective geometry for resisting bending loading

http://www.2719.com/pages/2719photo41.html

mcmaster.com

stainless-structurals.thomasnet.com

FlangesWeb

In a wide-flange I-beam, the flanges carry most of the moment resultant while the web supports most of the shear resultant

Consider axial stress based on the simple linear bending formula. Total axial load carried by the web is small compared to the flange (Pw << Pf)

Because vertical stress resultants are low near upper and lower boundaries, magnitude of vertical shear stress resultant is small in flanges.

Shear stress is approximately uniform in the web τ ≈ V/Aweb

Consider cantilevered load supported by truss geometry shown below

Truss:• Straight members joined by

pinned connections• Individual members carry

only axial loads (no shear)• Loads applied only at joints

The geometry is obviously unstable due because it cannot support the shear resultant

Adding diagonal truss members would provide shear support

Considering these trusses as a beam:

Horizontal members provide bending support (similar to flanges in an I-beam)

Diagonal members provide shear support

Hibbeler, Statics

Megson, Aircraft Structures for Engineering Students NASA

Shear web is an alternate approach to provide support of shear loading

• Visualize the original rectangular “truss” frame to which a thin panel has been attached along all four sides

• Called a “stiffened shear web” structure

Shear web structures in AE: “built-up” spar structure, and similar structures

J.-C. Flabel, Practical Stress Analysis for Design Engineers, Lake City Publishing, 1997.

“Built-Up” Spar Beam, assembled from metal extrusions etc forming spar caps, vertical stiffeners, plus thin plate material forming the webs

If area of the stiffeners is large compared to the C/S area of the web, response is similar to that of a wide-flange I-beam

J.-C. Flabel, Practical Stress Analysis for Design Engineers, Lake City Publishing, 1997.

“Structural idealization” is a simplified representation of a structure for easier analysis

Idealization of a stiffened shear web structure:

Stiffeners (“flanges”) support only axial loads, no shear

Shear webs support only shear loads, no axial loads

Use idealization to improve FDB of beam cross-section

Bending moment will result from axial forces in the upper and lower stiffeners

PVF

PxhFM

FFF

:0

0:0

:0

1

21

PVh

xPF

h

xPF

2

1 Stiffener forces are functions of x

Variation in stiffener forces along their length directly rates to shear loading in the web

FBD of a portion of one of the horizontal stiffeners

“Shear flow”, q, is shear force per unit length

Often a more convenient quantity for structural analysis than shear stress

Shear flow will be the same for stiffened shear webs of different thickness loaded in the same way

In our simple shear web: For web of uniform thickness t: h

Vq

in

lb,

m

N e.g.

][

][][

length

Forceq

ht

V

t

q

Simple relationship between shear flow in web and axial force in stiffener

h

Pq

dxqh

Pdx

qdxh

dxxP

h

xPF

0

0:0

In a single rectangular shear web, the shear flow q is constant (not function of x or y)

• Already showed it’s constant in x:

• Constancy in y illustrated with a simple FBD:

h

Pq

Rectangular shear panels (shear webs) are “constant shear flow panels”

Parallelogram shaped shear webs also have constant shear flow, along planes parallel to the edges

Curtis, H.D., Fundamentals of Aircraft Structural Analysis, Irwin, 1997, §2.5

constant),( yxq

Tapered shear panels have more complex internal stress states (not constant shear flow)

Flabel

Curtis, H.D., Fundamentals of Aircraft Structural Analysis, Irwin, 1997, §2.5

Axial loads in stiffeners in a stiffened shear web structure will vary linearly along each segment

Curtis, H.D., Fundamentals of Aircraft Structural Analysis, Irwin, 1997, §2.5

xqN

q

xqN

xqxxNxNF

constant is r web,rectangulaFor

0

0)()(:0

Knowing the axial force in the stiffener at each of the endpoints, for example, is therefore sufficient to describe the axial force throughout the stiffener

Examples of analysis of rectangular shear web structures

Objectives are to:• Determine the shear flow in each web panel

(this will enable the webs to be sized (determine the necessary thickness) –more next class period.

• Describe the axial forces in each stiffener segment (this will enable the stiffeners to be sized)

Example: simple two-panel cantilevered shear web

Results: Stiffener forces are illustrated graphically

• Note that it makes a huge difference in structural behavior whether a stiffener is loaded in tension or in compression and therefore it should be clearly noted– Use sign convention:

• positive axial loads: tension

• negative axial loads: compression

Example from handout

In more complex cases, it may be necessary to use systems of equations to solve.

Analysis of individual stiffener segments may be key to the solution process.

Curtis, H.D., Fundamentals of Aircraft Structural Analysis, Irwin, 1997, §2.5

Exploded view of results gives clear idea of the loads carried by each part

Curtis, H.D., Fundamentals of Aircraft Structural Analysis, Irwin, 1997, §2.5