LECTURE+6+COE-3001-A

# Displacement of a gently tapered bar

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: cross- sec>onal area as a func>on of distance (x) (2.7) Apply analy>c geometry and calculus principles Analysis con>nues 1/17/13 M. Mello/Georgia Tech Aerospace 8 y EXAMPLE 2- 4 (CONT.) dA 0, 2 L, r = RL N (x)dx 0 EA(x) dB 2 x (2.7) ²་  Line: ²་  Slope: y = r(x) = mx + b m= dB 2 dA 2 L dA ²་  Intercept: b = 2 ✓ dB ²་  Radius: r(x) = dA 2L ✓ dB ²་  Area: A(x) = ⇡ ◆ dA 2L x+ ◆ dA 2 dA x+ 2 2 NOTE: Lower limit of integra>on is zero in fundamental formula ( x) = Let, and, RL 0 ⇡E u( x) = ⇥ du(x) = 1/17/13 P dx dB dA 2L dB dA 2L dB dA 2L dA x+ 2 x+ dx ⇤2 dA 2 R dB 2 dA 2 = P 2L ⇡ E dB dA = P 2L [u 1 ] d2 A ⇡ E dB dA 2 u du dB 2 = 4P L ⇡ EdA db M. Mello/Georgia Tech Aerospace (2-8) Axial displacement of gently tapered bar (BLUE REGION ABOVE) 9 STATICALLY INDETERMINATE STRUCTURES ²་  StaMcally determinate conﬁguraMon: reac>ons and internal forces can be determined solely from free- body diagrams and equa>ons of equilibrium ²་  Forces are determined without knowledge of material proper>es, such as elas>c modulus (E) P Fvert = 0 (reac>on force easily determined) R = P1 + P2 Rigid constraints 1/17/13 ²་  StaMcally indeterminate conﬁguraMon: reac>ons and internal forces cannot be determined solely from free- body diagrams and equa>ons of equilibrium! RA P + RB = 0 ²་  Equilibrium problem must be complemented by compa>bility equa>ons which describe the the constraint of the structure (support condi>ons). =0 AB ²་  Compa>bility equa>on must then be expressed in terms of Ra and Rb using the linear elas>c force- displacement rela>on Pi Li (calculate for each sec>on and combine together) i= Ei Ai M. Mello/Georgia Tech Aerospace 10 ANALYSIS OF A STATICALLY INDETERMINATE BAR RA GIVEN: ²་  Consider a bar of cross- sec>onal area (A) and modulus (E) ²་  A force P is applied at point C as shown ²་  The bar is constrained and prevented from displacing at the ﬁxed ends A and B. DETERMINE: ²་  Reac>on forces RA and RB ²་  Displacement δc of point C ²་  Stresses within the bar (above and below point C) SOLUTION STRATEGY: ²་  EQUILIBRIUM EQUATION: P Fvert = 0 RA P + RB = 0 (ONE EQUATION AND TWO UNKNOWNS) ²་  COMPATIBILITY CONDITON: AB = AC + C B = 0 ²་  FORCE- DISPLACEMENT RELATION: i 1/17/13 = Pi L i E i Ai RA RB RB...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online