This section will introduce how to solve problems of axially loaded members such as stepped and tapered rods loaded in tension. The concept of strain energy will also be introduced.

1. Section 6 Axially Loaded Structural Members This section will introduce how to solve problems of axially loaded members such as stepped and tapered rods loaded in tension. The concept of strain energy will also be introduced. © Loughborough University 2010. This work is licensed under a Creative Commons Attribution 2.0 Licence .

9. Two-Step Steel Rod: FBD Red lines show cuts to establish internal forces 2 1 3 200kN P 1 Internal force P 1 = 200kN (tension) 200kN 300kN P 2 P 2 = -100kN (compression) 200kN 300kN 500kN P 3 P 3 = 400kN (tension) Cut 1 1 Cut 2 2 Cut 3 3 2 1 3 200kN 300kN B A C R CX R CY M CZ 500kN

10. Solution: Two-Step Steel Rod (Units used are kN and mm) 300 200kN 300kN B A C R CX R CY M CZ 300 400 500kN

11. Axial Member with Tapered Cross-Section  x L A B x d 1 d 2 d i P

12. Axial Member with Tapered Cross-Section (Circular Cross-Section) Force equilibrium at any cross section shows P is constant along length Taper is linear so diameter d of rod at distance x is d i Area A of rod at distance x is A i  x L A B x d 1 d 2 d i P

13. Axial Member with Tapered Cross-Section (Circular Cross-Section) Elongation over entire length  x L A B x d 1 d 2 d i P

15. Example: Flat Bar of Rectangular Cross-Section  x L x b 1 b 2 b i P P

16. Example: Flat Bar of Rectangular Cross-Section  x L x b 1 b 2 b i P P

18. Strain Energy in Tension and Compression Strain energy U is stored internally in the bar during the loading process. If the bar behaves elastically, it is called elastic strain energy. For energy conservation: Internal strain energy = external work W required to deform bar U = W=P  /2 Units for strain energy: Nm or Joule (J). P   P

19. Strain Energy – Axial Loading

20. Strain Energy – Shear Loading

22. Example: Strain Energy L/4 L 2d 2d d P

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