MT-1620 Fall 2002 Figure 18.4 Representation of region of highest stress in cross-section of beam-column ighest compressive stress Thus, this outer part is the first part to yield As the material yields, the modulus decreases Figure 18.5 Representation of tangent modulus angent modulus AE This changes the location of the centroid Paul A Lagace @2001 Unit 18-6
MIT - 16.20 Fall, 2002 Figure 18.4 Representation of region of highest stress in cross-section of beam-column highest compressive stress Thus, this outer part is the first part to yield. As the material yields, the modulus decreases. Figure 18.5 Representation of tangent modulus tangent modulus This changes the location of the centroid… Paul A. Lagace © 2001 Unit 18 - 6
MT-1620 Fall 2002 Figure 18.6 Representation of change in location of centroid of cross section due to local yielding lower modulus, E<E E This continues and it may eventually squash" or buckle(or a combination See rivello 14.6 c)Nonuniform Beam-Columns Have looked only at beams with uniform cross-sectional property El. Now let this vary with x(most likely I, not E) EXample: Tapered section Paul A Lagace @2001 Unit 18-7
MIT - 16.20 Fall, 2002 Figure 18.6 Representation of change in location of centroid of crosssection due to local yielding lower modulus, ET < E E This continues and it may eventually “squash” or buckle (or a combination) --> See Rivello 14.6 (c) Nonuniform Beam-Columns Have looked only at beams with uniform cross-sectional property EI. Now let this vary with x (most likely I, not E). Example: Tapered section Paul A. Lagace © 2001 Unit 18 - 7
MT-16.20 al.2002 Figure 18.7 Representation of beam-column with tapered cross-section EIE EI I(x) Thus, the governing equation is El P dx(a dx dx must keep this inside the derivative Solve this via numerical techniques Energy Methods Galerkin Finite Element method · Finite Difference Rayleigh-Ritz -->See rive/lo 14.3 Paul A Lagace @2001 Unit 18-8
MIT - 16.20 Fall, 2002 Figure 18.7 Representation of beam-column with tapered cross-section EI = EI(x) Thus, the governing equation is: 2 2 d 2 EI dw + P dw = 0 dx2 dx2 dx2 must keep this “inside” the derivative Solve this via numerical techniques: • Energy Methods • Galerkin • Finite Element Method • Finite Difference • Rayleigh-Ritz --> See Rivello 14.3 Paul A. Lagace © 2001 Unit 18 - 8