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Unformatted text preview: 1 CHAPTER 6 SHEAR FLOWS IN THINWALLED SECTIONS Once resultant shear forces are given over a crosssection of a thinwalled slender structure, shear flows over the crosssection can be determined by considering the force equilibrium in the axial direction and using what we have learned in the previous chapters. We will consider the following under the assumption that the crosssection is uniform in the axial direction. Force Equilibrium in the Axial Direction Shear Flows in ThinWalled Open Sections Shear Center of Thinwalled Open Sections Shear Flows in ThinWalled Closed Sections 2 1. Equilibrium in the Axial Direction Open sections For a slender structure with an open crosssection as shown in the sketch, lets introduce cuts at , x x dx + and b to create a free body. * ( , ) [ ( ) ( ) ] xx xx A q x b dx x dx x dA + + + = Since dx x x dx x xx xx xx + = + ) ( ) ( , then, * [ ( , ) ] xx A q x b dA dx x + + = * ( , ) xx A q x b dA x + + = A s , , ( , ) ( , ) dx q x b q x b + * ( , ) xx A q x b dA x + = * ( , ) xx A q x b dA x = xx x dx ( ) + b b xx x ( ) Area A* ( , ) q x b + x+dx x ( ) dx Stressfree surface q b b A * 3 Singlecell closed sections Lets introduce cuts at , , x x dx a + and b to create a free body. * ( , ) ( , ) [ ( ) ( ) ] xx xx A q x a dx q x b dx x dx x dA + + + + = Since dx x x dx x xx xx xx + = + ) ( ) ( , then, * [ ( , ) ( , ) ] xx A q x a q x b dA dx x + + + = * ( , ) ( , ) xx A q x a q x b dA x + + + = As 0, 0, ( , ) ( , ) ( , ) ( , ) dx q x a q x a q x b q x b + + * ( , ) ( , ) xx A q x a q x b dA x + = * ( , ) ( , ) xx A q x a q x b dA x = Note : The above equation can be applied to an opencell sections if a or b is placed at the stressfree surface. a a q xx x dx ( ) + a b b b xx x ( ) Area A* ( , ) q x b + ( , ) q x a + x+dx x ( ) dx a b A * 4 Twocell closed sections (forces in the direction) = x 0 * ( , ) [ ( , ) ( , )] xx A q x a q x b q x c dA x + = where, ( , ) out q x a q = and in [ ( , ) ( , )] q x b q x c q + = In general, dA x q q A xx = * in out ( * ) The above equation is general as it applies to singlecell and multicell closed sections as well as opencell sections....
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 Spring '11

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