six beams internal forces & diagrams Beams Beams Beams span horizontally

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1 ARCHITECTURA STRUCTURES: FOR, BEHAIOR, AND DESIGN DR. ANNE NICHOS SUER 2013 lecture si beams internal forces & diagrams nisee.berkeley.edu/godden Beams span horizontally floors bridges roofs loaded transversely by gravity loads may have internal aial force will have internal shear force will have internal moment (bending) R Internal Beam Forces 1 ecture 6 Architectural Structures F2009abn Internal Beam Forces 2 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 3 Internal Beam Forces 4 1

2 Beam Stresses stress = relative force over an area tensile compressive bending tension and compression... Beam Stresses Internal Beam Forces 5 Internal Beam Forces 6 Beam Stresses tension and compression causes moments Beam Stresses prestress or post-tensioning put stresses in tension area to pre-compress Copyright Kirk artini. Internal Beam Forces 7 Internal Beam Forces 8 2

3 Beam Stresses shear horizontal & vertical Beam Stresses shear horizontal & vertical Internal Beam Forces 9 Internal Beam Forces 10 Beam Stresses shear horizontal Beam Deflections depends on load section material Internal Beam Forces 11 Internal Beam Forces 12 3

4 Beam Deflections moment of inertia Beam Styles vierendeel open web joists nisee.berkeley.edu/godden manufactured Internal Beam Forces 13 Internal Beam Forces 14 Internal Forces trusses aial only, (compression & tension) F F A A B F F F B F Beam oading concentrated force concentrated moment spandrel beams in general aial force shear force, bending moment, T T T Internal Beam Forces 15 Internal Beam Forces 16 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 17 Internal Beam Forces 18 Beam Supports in the real world, modeled type Internal Forces in Beams like method of sections / joints no aial forces section must be in equilibrium want to know where biggest internal forces and moments are for designing R Internal Beam Forces 19 Internal Beam Forces 20 5

6 & Diagrams tool to locate ma and ma (at = 0) necessary for designing have a different sign convention than eternal forces, moments, and reactions R () () 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 R () () Internal Beam Forces 21 Internal Beam Forces 22 Shear Sign Convention Sign Convention bending moment, : cut section to EFT if cut is clockwise, acts ccw and is POSITIE flees into a smiley beam has to resist bending apart by () () R Internal Beam Forces 23 Internal Beam Forces 24 6

7 Bending oment 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 25 Internal Beam Forces 26 Constructing & Diagrams along the beam length, plot, plot athematical ethod cut sections with as width write functions of () and () - load diagram - Internal Beam Forces 27 Internal Beam Forces 28 7

8 ethod 1: Equilibrium cut sections at important places plot & - /2 ethod 1: Equilibrium important places supports concentrated loads start and end of distributed loads concentrated moments free ends zero forces Internal Beam Forces 29 Internal Beam Forces 30 ethod 2: Semigraphical by knowing area under loading curve = change in area under shear curve = change in concentrated forces cause jump in concentrated moments cause jump in D C D C wd D C D C d ethod 2 relationships Internal Beam Forces 31 Internal Beam Forces 32 8

9 ethod 2: Semigraphical ma occurs where = 0 (calculus) Curve Relationships integration of functions line with 0 slope, integrates to sloped - no area y e: load to shear, shear to moment y Internal Beam Forces 33 Internal Beam Forces 34 Curve Relationships Curve Relationships line with slope, integrates to parabola parabola, integrates to 3 rd order curve y y y y e: load to shear, shear to moment e: load to shear, shear to moment Internal Beam Forces 35 Internal Beam Forces 36 9

10 Basic Procedure with Sections 1. Find reaction forces & moments : Plot aes, 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 Basic Procedure with Sections : 6. Starting at left 7. oment is 0 at free ends 8. oment has 2 values at moments 9. Sum moments at each section 10.aimum moment is where shear = 0! (locate where = 0) Internal Beam Forces 37 Internal Beam Forces 38 Basic Procedure by Curves 1. Find reaction forces & moments : Plot aes, underneath beam load diagram 2. Starting at left 3. Shear is 0 at free ends 4. Shear jumps with concentrated load 5. Shear changes with area under load Basic Procedure by Curves : 6. Starting at left 7. oment is 0 at free ends 8. oment jumps with moment 9. oment changes with area under 10.aimum moment is where shear = 0! (locate where = 0) & Diagrams 9 & Diagrams 10 10

11 Shear Through Zero slope of is w (-w:1) Parabolic Shapes cases w (force/length) - - load height = A w A A w shear A width = up fast, then slow up slow, then fast down fast, then slow down slow, then fast Internal Beam Forces 39 Internal Beam Forces 40 Deflected Shape & () -() gives shape indication boundary conditions must be met Boundary Conditions at pins, rollers, fied supports: y = 0 at fied supports: = 0 at inflection points from symmetry: = 0 y ma at dy d 0 & Diagrams 13 & Diagrams 14 11

12 Tabulated Beam Formulas how to read charts Tools software & spreadsheets help & Diagrams 15 & Diagrams 16 Tools ultiframe in computer lab Tools ultiframe frame window define beam members select points, assign supports select members, assign section load window select point or member, add point or distributed loads & Diagrams 17 & Diagrams 18 12

13 Tools ultiframe to run analysis choose Analyze menu inear plot choose options double click (all) results choose options & Diagrams 19 13