lec8 - M S = x ( e x ) dS S x e x , y e y , z e z , | | h b...

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1.050 Engineering Mechanics II. Stresses and Strength Application in Structural Mechanics
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Program 8 th Lecture 1-050 CONTENT I. Dimensional Analysis: II. Stresses & Strength 2. Stresses and Equilibrium 1. Discrete Model 2. Continuum Model 3. Beam Model 3. Strength Models III. Deformation and Strain 4. How Strain Gages work? IV. Elasticity 5. Elastic Model 6. Variational Methods in Elasticity V. How Things Fail? And How to avoid it. TODAY: 1. Scales of Structural mechanics: Section vs. Beam structure 2. Link between stresses and forces and moments 3. Beam Equilibrium Conditions 4. Example Goal: Construct a Force-Moment Beam Model Appreciate the link between Continuum Model and Beam Model
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Three Scale Approach Beam Scale defined by beam length Cross-section scale (height, width) ( h , b ) << l Continuum scale 1/ 3 O ( d Ω ) << ( h , b ) << l
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From the Continuum Scale to the Cross Section Scale • Continuum Quantity: – Stress vector T ( n = e x ) = σ e x • Section Quantities: –Fo rces F S = σ e dS x S
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Unformatted text preview: M S = x ( e x ) dS S x e x , y e y , z e z , | | h b dS n d From the Cross Section Scale to the Beam Length Scale Differential Force equilibrium e z M z + dM z dx e z dx e d F S ext y M y + dx e y dM y + f = dx dx M e x x f ext F x e x e e dx x x F y e y M x + dM x dx e x dx F e z z Differential Moment M e y y e z z y M e e equilibrium e z d M S + e x F S = dx Formulation of a Beam Boundary Value Problem Example Force and Moment Boundary Conditions Sum of all forces and Moments along x is zero x Differential Equilibrium of Section forces Section moments z ext e gS f = R y z...
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lec8 - M S = x ( e x ) dS S x e x , y e y , z e z , | | h b...

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