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1 CHAPTER 2 INTERNAL LOADS IN AEROSPACE STRUCTURES 2.1 Force and Moment Distributions Slender body under axial force Slender body under torque Slender body under lateral loads 2.2 Inertia Loads Load factor Examples

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2 2.1 Force and Moment Distributions Aircraft with high aspect ratio wings and rockets can be modeled as a slender structures subjected to external loads in the form of axial forces, lateral forces and moments. These external loads in turn induce internal forces and moments. In the following sections, we will look at the distributions of forces and moments along slender structures. We will consider only statically determinate cases. Recall: 1) For statically determinate structures , force and moment distributions can be determined considering only equilibrium equations. 2) For statically indeterminate structures , it is necessary to consider the deformation under applied load to determine force and moment distributions.
3 2.1.1 Resultant Forces and Moments Consider a slender body with the x-axis placed along the longest dimension. Now let’s introduce an imaginary cut normal to the x axis and consider the stress components acting over a cross-section located at coordinate x as shown below. Then three resultant forces and three resultant moments acting over the cross-section are defined as follows : = dA x F xx σ ) ( : axial force in the x direction = dA x V xy y τ ) ( : (transverse) shear force in the y direction = dA x V xz z ) ( : (transverse) shear force in the z direction = dA z x M xx y ) ( : moment around the y axis = dA y x M xx z ) ( : moment around the z axis = dA z y x T xy xz ) ( ) ( : torque or moment around the x axis xz xy y z dA y z x y z x V x z () V x y T(x) M x y M x z z y x xx This figure shows positive forces and moments on the positive x -surface. F(x) positive x -surface

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4 ( ) 1000 Fx l b = ( ) 1000 l b =− x x 1000 1000 x x 1000 1000
5 5000 z Vl b = ( ) 5000 z Vx l b =− x x 5000 5000 z x x 5000 5000 z

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6 5000 y M lb in =− 5000 y M lb in x x 5000 5000 z x x 5000 5000 z
7 2.1.2 Slender Body under Axial Force A rocket or a helicopter blade can be modeled as a slender body under axial force. f x () : applied force per unit length, e.g. gravity A x : cross-sectional area To look at equilibrium, let’s create a free body by introducing imaginary cut(s). Introduce a cut at x and consider the free body on the right hand side of the cut. (Axial forces) = 0 for the free body. P lb x =0 f lb/in x = L x P x ξ d + P lb 0 x = x f d f ) ( F(x) Fx x 0 0 x = = d

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8 () 0 L x Fx P f d ξ ξξ = = −− + = ( 1 ) L x Fx P f d = = =− + ( 2 ) Example : Consider a rocket on a launch pad modeled as a slender body under its own weight. Mg : payload weight, m ( x ): mass per length, g : gravity Introducing a cut at x , ( ) LL xx f d M g m gd M g == =− = ∫∫ (1) For constant m , ( ) L x m g M g m gL x M g →= ( 2 ) x =0 f lb/in x = L x payload Mg
9 2.1.3 Slender Body under Torque Consider a high aspect ratio wing subject to aerodynamic moment and, possibly, wing tip moment due to a wing tip fuel tank or an engine. The x -axis is along the wingspan.

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