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Table of Contents

Transverse Shear of Thin-Walled Beams.......................................................................................... 11Introduction....................................................................................................................................12Objective........................................................................................................................................23Proof that the Shear Stresses are li!ned "ith the #edian $ine.................................................. 2%Sam&les of 'elevant reas in (alculatin! the Shear Stress..........................................................))Si!n (onvention for the Shear Stress ......................................................................................... *

).1+a, luid nalo!..................................................................................................................*).2+b, Si!n (onvention.............................................................................................................. *

*'es&onse of Beams "hen / is li!ned "ith an 0is of Smmetr............................................. *.10am&le.................................................................................................................................

*.1.1(omments.......................................................................................................................*.20am&le................................................................................................................................. 4

Beams "ith / Per&endicular to an 0is of Smmetr 5 Shear (enter....................................... 12.10am&le............................................................................................................................... 12

.1.1(omments.....................................................................................................................1%.20am&le............................................................................................................................... 1).30am&le............................................................................................................................... 1

.3.1(omments.....................................................................................................................26(ross- Sections "ithout Smmetr..............................................................................................214Tabulated $ocations of the Shear (enter.....................................................................................2116'eferences..................................................................................................................................22

Transverse Shear of Thin-Walled Beams1

1 Introduction

Beams are subjected to shear stresses !iven b =VQ

z

Iz

t. major difference bet"een a thic7

and a thin "all cross-section8 is that the shear stresses for thin-"alled beams are al"as ali!ned"ith the median line of the cross-section8 see the fi!ure belo".

1 hmad #ansour

#echanics of Structures 5 Second 9ear 5 #echanical n!ineerin! - (airo :niversit 1;22

vis notvertical

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The shearin! force / causes &ure bendin! "ithout t"istin! moment8 onl "hen its line of action&asses throu!h a s&ecific &oint named the shear center +S,. The shear centers for t"o cross-sectionsare sho"n belo" +the centroids ( are also de&icted to sho" that S and ( are in !eneral t"o different

&oints,,. In addition8 the force actin! alon! theflan!e de e?uals f8 refer to fi!ure.

#echanics of Structures 5 Second 9ear 5 #echanical n!ineerin! - (airo :niversit 13;22

4@

:4 mm4@

c

y

)

A 8s

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4@ mm

c

y

)

e

#V$

Ff .1@* V

Ff

V

The fi!ure sho"s also the resultant +or e?uivalent, force +/,. Since8 the resultant is e?uivalent tothe "all forces8 its moment about an &oint in the &lane e?uals to that of the three "all forces.

Ta7e the moment about &oint &8 refer to the &revious fi!ure.

+/, e D 6.13 / 0 4% D 1.*6 /Cence8

e D 1.* mm

This is the distance from +/, to the median line of the "eb. The distance to the left "all of the "ebis +1.* - 3, D 1%.* mm.

The shear center is at the intersection of the line of action of +/, and the =-a0is. It is a !eometric&ro&ert "hich is not function of /.

+*1*1 #oents

+1, If the transverse force does not &ass throu!h the shear center8 then the shear stress is com&osedof transverse and torsional stresses8 refer to the fi!ures belo".

The ma0imum shear stress is at the inner "all of the "eb at D 6 +&oint &,

#echanics of Structures 5 Second 9ear 5 #echanical n!ineerin! - (airo :niversit 1%;22

V

c

y

)SC

a

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max

=V Q

max

Iz

t

T

1

3a *3

8 "here a D +% F 4% F%, D 1 mm

Gote at D6 and at the outer "all of the "eb8 the stresses should be subtracted.

+2, We a&&l the transverse force at the shear center b usin! an attachment as sho"n belo"