six beam introduction & internal forces Beams Beams Beams span horizontally

Transcription

1 EEMENTS OF ARCHITECTURA STRUCTURES: FORM, BEHAIOR, AND DESIGN DR. ANNE NICHOS SPRING 2014 lecture six beam introduction & internal forces Beams span horizontally floors bridges roofs loaded transversely by gravity loads may have internal axial force will have internal shear force will have internal moment (bending) R M Internal Beam Forces 1 ecture 6 S2009abn Internal Beam Forces 4 Beams transverse loading sees: bending shear deflection torsion bearing behavior depends on cross section shape Beams bending bowing of beam with loads one edge surface stretches other edge surface squishes Internal Beam Forces 5 Internal Beam Forces 6 1

2 Beam Stresses stress = relative force over an area tensile compressive bending tension and compression +... Beam Stresses Internal Beam Forces 7 Internal Beam Forces 8 Beam Stresses tension and compression causes moments Beam Stresses prestress or post-tensioning put stresses in tension area to pre-compress Copyright Kirk Martini. Internal Beam Forces 9 Internal Beam Forces 8 S2006abn 2

3 Beam Stresses shear horizontal & vertical Beam Stresses shear horizontal & vertical Internal Beam Forces 11 Internal Beam Forces 12 Beam Stresses shear horizontal Beam Deflections depends on load section material Internal Beam Forces 13 Internal Beam Forces 14 3

4 Beam Deflections moment of inertia Beam Styles vierendeel open web joists manufactured nisee.berkeley.edu/godden Internal Beam Forces 15 Internal Beam Forces 16 Internal Forces trusses axial only, (compression & tension) F F A A B F F F B F Beam oading concentrated force concentrated moment spandrel beams in general axial force shear force, bending moment, M T T T Internal Beam Forces 17 Internal Beam Forces 18 4

5 Beam oading uniformly distributed load (line load) non-uniformly distributed load hydrostatic pressure = h wind loads Beam Supports statically determinate simply supported (most common) overhang cantilever statically indeterminate continuous (most common case when 1= 2) Propped Restrained Internal Beam Forces 19 Internal Beam Forces 20 Beam Supports in the real world, modeled type Internal Forces in Beams like method of sections / joints no axial forces section must be in equilibrium want to know where biggest internal forces and moments are for designing M R Internal Beam Forces 21 Internal Beam Forces 22 5

6 & M Diagrams tool to locate max and M max necessary for designing M max occurs when = 0 Sign Convention shear force, : cut section to EFT if F y is positive by statics, acts down and is POSITIE beam has to resist shearing apart by (+) (+) (+)M (+)M R R Internal Beam Forces 23 Internal Beam Forces 24 Shear Sign Convention Sign Convention bending moment, M: cut section to EFT if M cut is clockwise, M acts ccw and is POSITIE flexes into a smiley beam has to resist bending apart by M (+) (+)M R Internal Beam Forces 25 Internal Beam Forces 26 6

7 Bending Moment Sign Convention Deflected Shape positive bending moment tension in bottom, compression in top negative bending moment tension in top, compression in bottom zero bending moment inflection point Internal Beam Forces 27 Internal Beam Forces 28 Constructing & M Diagrams along the beam length, plot, plot M Mathematical Method cut sections with x as width write functions of (x) and M(x) load diagram - M + M + x Internal Beam Forces 29 Internal Beam Forces 30 7

8 Equilibrium Method cut sections at important places plot & M + - M + /2 Equilibrium Method important places supports concentrated loads start and end of distributed loads concentrated moments free ends zero forces Internal Beam Forces 31 Internal Beam Forces 32 Equilibrium Method relationships Basic Procedure 1. Find reaction forces & moments : Plot axes, underneath beam load diagram 2. Starting at left 3. Shear is 0 at free ends 4. Shear has 2 values at point loads 5. Sum vertical forces at each section Internal Beam Forces 32 S2006abn Internal Beam Forces 37 S2006abn 8

9 Basic Procedure Shear Through Zero M: slope of is w (-w:1) 6. Starting at left w (force/length) 7. Moment is 0 at free ends 8. Moment has 2 values at moments 9. Sum moments at each section 10.Maximum moment is where shear = 0! load height = A x w A x A w shear A width = x Internal Beam Forces 38 S2006abn Internal Beam Forces 39 S2006abn Tools software & spreadsheets help Tools Multiframe in computer lab & M Diagrams 14 ecture 13 & M Diagrams 18 ecture 13 S2005abn 9

10 Tools Multiframe frame window define beam member select points, assign supports select members, assign section load window select point or member, add point or distributed loads Tools Multiframe to run analysis choose Analyze menu inear plot choose options double click (all) results choose options & M Diagrams 19 ecture 13 S2005abn & M Diagrams 20 ecture 13 S2005abn 10