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Analyzing Axial Deformation in Statically Indeterminate Structural Systems

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This lecture elaborates on the principles of axial deformation in systems of axially loaded bars, focusing on both statically determinate and indeterminate systems. Key techniques for analyzing deformations, equilibrium equations, and force-deformation relationships are discussed. The lecture also explores the geometry of deformation and compatibility equations necessary for solving complex structures. Practical problems are included to illustrate these concepts, enhancing understanding of how to calculate normal stress and deflection in various structural members.

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Analyzing Axial Deformation in Statically Indeterminate Structural Systems

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  1. Mechanics of Materials Engr 350 - Lecture 9 Axial Deformation 2 - Systems of Axially Deformed Members • “We cannot solve our problems with the same thinking we used when we created them.” • -Albert Einstein

  2. Deformations in a system of axially loaded bars • Consider deformations in more than one direction • Useful technique to analyze complex systems of homogeneous uniaxial bars loaded by forces at the end • We will assume some bars are perfectly rigid for now • 2 Types • Statically determinate (covered in Philpot 5.4) • Statically indeterminate (section 5.5) • What’s the difference?

  3. Statically Determinate Systems 1) Equilibrium equations 2) Find forces in members 3) Find deformations in members 4) Determine how structure as a whole deforms based on deformations in members - as necessary

  4. Statically indeterminate systems Solution procedure 1) Equilibrium Equations – Express in terms of unknown forces 2) Geometry of Deformation – Geometric relationship between various members (step unique to statically indeterminate system) 3) Force-Deformation Relationships – FLEA equation for each member 4) Compatibility Equation – Combine equations from Step 3 with Step 2, expressed in terms of unknown forces. Should have a closed set of equations after this step (equal number of knowns and unknowns) 5) Solve the Equations – by hand or via software

  5. Practice problem P5.15 • Write equilibrium equations • Equations from geometry of deformation • Force-deflection relationships • Create compatibility relationships: 2➞3

  6. Example: Philpot P5.24 • Write equilibrium equations • Equations from geometry of deformation • Force-deflection relationships • Create compatibility relationships: 2➞3 • Solve 1 & 4 simultaneously • 𝜎1= • 𝜎2= • Deflection at B: • Given: • E=200GPa • A=1475mm2 • Find: a) normal stress in the upper and lower pipes • b) the deflection of flange B

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