Chapter 6_Shear Flows

# Chapter 6_Shear Flows - 1 CHAPTER 6 SHEAR FLOWS IN...

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Unformatted text preview: 1 CHAPTER 6 SHEAR FLOWS IN THIN-WALLED SECTIONS Once resultant shear forces are given over a cross-section of a thin-walled slender structure, shear flows over the cross-section 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 cross-section is uniform in the axial direction. Force Equilibrium in the Axial Direction Shear Flows in Thin-Walled Open Sections Shear Center of Thin-walled Open Sections Shear Flows in Thin-Walled Closed Sections 2 1. Equilibrium in the Axial Direction Open sections For a slender structure with an open cross-section as shown in the sketch, let’s introduce cuts at , x x dx + and ˆ b to create a free body. * ˆ ( , ) [ ( ) ( ) ] xx xx A q x b dx x dx x dA ε σ σ − + + + − = ∫ Since dx x x dx x xx xx xx ∂ ∂ + = + σ σ σ ) ( ) ( , then, * ˆ [ ( , ) ] xx A q x b dA dx x σ ε ∂ − + + = ∫ ∂ * ˆ ( , ) xx A q x b dA x σ ε ∂ → − + + = ∫ ∂ ˆ ˆ A s , , ( , ) ( , ) dx q x b q x b ε ε → → + → * ˆ ( , ) xx A q x b dA x σ ∂ − + = ∫ ∂ * ˆ ( , ) xx A q x b dA x σ ∂ = ∫ ∂ σ xx x dx ( ) + ˆ b ˆ b σ xx x ( ) Area A* ˆ ( , ) q x b ε + x+dx x ( ) ≤ ≤ ε dx Stress-free surface q ˆ b ˆ b A * 3 Single-cell closed sections Let’s introduce cuts at ˆ , , x x dx a + and ˆ b to create a free body. * ˆ ˆ ( , ) ( , ) [ ( ) ( ) ] xx xx A q x a dx q x b dx x dx x dA ε ε σ σ + − + + + − = ∫ Since dx x x dx x xx xx xx ∂ ∂ + = + σ σ σ ) ( ) ( , then, * ˆ ˆ [ ( , ) ( , ) ] xx A q x a q x b dA dx x σ ε ε ∂ + − + + = ∫ ∂ * ˆ ˆ ( , ) ( , ) xx A q x a q x b dA x σ ε ε ∂ → + − + + = ∫ ∂ ˆ ˆ As 0, 0, ( , ) ( , ) ˆ ˆ ( , ) ( , ) dx q x a q x a q x b q x b ε ε ε → → + → + → * ˆ ˆ ( , ) ( , ) xx A q x a q x b dA x σ ∂ − + = ∫ ∂ * ˆ ˆ ( , ) ( , ) xx A q x a q x b dA x σ ∂ − = − ∫ ∂ Note : The above equation can be applied to an open-cell sections if ˆ a or ˆ b is placed at the stress-free surface. ˆ a ˆ a q σ xx x dx ( ) + ˆ a ˆ b ˆ b ˆ b σ xx x ( ) Area A* ˆ ( , ) q x b ε + ˆ ( , ) q x a ε + x+dx x ( ) ≤ ≤ ε dx ˆ a ˆ b A * 4 Two-cell closed sections (forces in the direction) ∑ = x 0 * ˆ ˆ ˆ ( , ) [ ( , ) ( , )] xx A q x a q x b q x c dA x σ ∂ → − + = − ∫ ∂ where, ˆ ( , ) out q x a q = and in ˆ ˆ [ ( , ) ( , )] q x b q x c q + = ∑ In general, dA x q q A xx ∫ ∑ ∑ ∂ ∂ − = − * in out σ ( * ) The above equation is general as it applies to single-cell and multi-cell closed sections as well as open-cell sections....
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Chapter 6_Shear Flows - 1 CHAPTER 6 SHEAR FLOWS IN...

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