Chap7 slides fm M Vable Mechanics of Materials Chapter 7 Pr in te d fr om h ttp // w w w m e m tu e du /~ m av ab le /M oM 2n d ht m Deflection of Symmetric Beams Learning objective • Learn to formula[.]
M Vable Mechanics of Materials: Chapter Deflection of Symmetric Beams Learning objective Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm • Learn to formulate and solve the boundary-value problem for the deflection of a beam at any point August 2012 7-1 M Vable Mechanics of Materials: Chapter Second-Order Boundary Value Problem v v • The deflected curve represented by v(x) is called the Elastic Curve Differential equation: M z = EI zz d v dx • The mathematical statement listing all the differential equations and all the conditions necessary for solving for v(x) is called the Boundary Value Problem for the beam deflection Boundary Conditions Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm v ( xA ) = v ( xA ) = dv (x ) = dx A August 2012 7-2 dv (x ) = dx A M Vable Mechanics of Materials: Chapter C7.1 In terms of w, P, L, E, and I determine (a) equation of the elastic curve (b) the deflection of the beam at point A y A x Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm L August 2012 7-3 M Vable Mechanics of Materials: Chapter Class Problem 7.1 Write the boundary value problem to determine the elastic curve Note the reaction force at the support has been calculated for you y PL x A B L/2 P Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm P August 2012 7-4 M Vable Mechanics of Materials: Chapter Continuity Conditions • The internal moment Mz will change with change in applied loading • Each change in Mz represents a new differential equation, hence new integration constants v(x) Broken beam v2(x) v1(x) Kinked beam v(x) Discontinuous Slope Discontinuous Displacement v2(x) v1(x) x xj x xj v1 ( xj ) = v2 ( xj ) dv dv Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm (x ) = (x ) dx j dx j • ‘continuity conditions’, also known as ‘compatibility conditions’ or ‘matching conditions’ August 2012 7-5 M Vable Mechanics of Materials: Chapter C7.2 In terms of w, L, E, and I, determine (a) the equation of the elastic curve (b) the deflection at x = L Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm Fig C1.2 August 2012 7-6 M Vable Mechanics of Materials: Chapter Class Problem 7.2 C7.3 Write the boundary value problem for determining the deflection of the beam at any point x Assume EI is constant Do not integrate or solve The internal moments are: 2 wLx wL In BC: M = + -6 wx In AB: M = – 2 Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm 7wLx 14wL In CD: M = – -3 August 2012 7-7 M Vable Mechanics of Materials: Chapter Class Problem 7.3 v1 and v2 represents the deflection in segment AB and BC For the beams shown, identify all the conditions from the table needed to solve for the deflection v(x) at any point on the beam (a) v ( ) = (e) v ( 2L ) = (i) v ( L ) = v ( L ) (b) v ( L ) = (f) v ( 3L ) = (j) v ( 2L ) = v ( 2L ) (c) v ( L ) = (d) v ( 2L ) = (g) dv (0) = dx (k) dv dv (L) = (L) dx dx (h) dv ( 3L ) = dx (l) dv dv ( 2L ) = ( 2L ) dx dx w Beam A B x C 2L L w Beam A x L Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm C B 2L w Beam A x B L August 2012 2L 7-8 C ... http://www.me.mtu.edu/~mavable/MoM2nd.htm 7wLx 14wL In CD: M = – -3 August 2012 7- 7 M Vable Mechanics of Materials: Chapter Class Problem 7. 3 v1 and v2 represents the deflection in segment AB and BC For the beams. .. http://www.me.mtu.edu/~mavable/MoM2nd.htm Fig C1.2 August 2012 7- 6 M Vable Mechanics of Materials: Chapter Class Problem 7. 2 C7.3 Write the boundary value problem for determining the deflection of the beam at any point x Assume... ‘matching conditions’ August 2012 7- 5 M Vable Mechanics of Materials: Chapter C7.2 In terms of w, L, E, and I, determine (a) the equation of the elastic curve (b) the deflection at x = L Printed from: