seven beams internal forces ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SPRING 2016 lecture

Transcription

1 ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SPRING 2016 lecture seven beams nisee.berkeley.edu/godden internal forces Internal Beam Forces 1

2 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 V M Internal Beam Forces 2

3 Beams transverse loading sees: bending shear deflection torsion bearing behavior depends on cross section shape Internal Beam Forces 3

4 Beams bending bowing of beam with loads one edge surface stretches other edge surface squishes Internal Beam Forces 4

5 Beam Stresses stress = relative force over an area tensile compressive bending tension and compression +... Internal Beam Forces 5

6 Beam Stresses Internal Beam Forces 6

7 Beam Stresses tension and compression causes moments Copyright Kirk Martini. Internal Beam Forces 7

8 Beam Stresses prestress or post-tensioning put stresses in tension area to pre-compress Internal Beam Forces 8

9 Beam Stresses shear horizontal & vertical Internal Beam Forces 9

10 Beam Stresses shear horizontal & vertical Internal Beam Forces 10

11 Beam Stresses shear horizontal Internal Beam Forces 11

12 Beam Deflections depends on load section material Internal Beam Forces 12

13 Beam Deflections moment of inertia Internal Beam Forces 13

14 Beam Styles vierendeel open web joists manufactured nisee.berkeley.edu/godden Internal Beam Forces 14

15 Internal Forces trusses axial only, (compression & tension) F A B F F A F F B F in general axial force shear force, V bending moment, M T T V T Internal Beam Forces 15

16 Beam Loading concentrated force concentrated moment spandrel beams Internal Beam Forces 16

17 Beam Loading uniformly distributed load (line load) non-uniformly distributed load hydrostatic pressure = h wind loads Internal Beam Forces 17

18 Beam Supports statically determinate L L L simply supported (most common) overhang cantilever statically indeterminate L continuous (most common case when L 1 =L 2 ) L L L Propped Restrained Internal Beam Forces 18

19 Beam Supports in the real world, modeled type Internal Beam Forces 19

20 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 V M R Internal Beam Forces 20

21 V & M Diagrams tool to locate V max and M max (at V = 0) necessary for designing have a different sign convention than external forces, moments, and reactions (+)V (+)M R Internal Beam Forces 21

22 Sign Convention shear force, V: cut section to LEFT if F y is positive by statics, V acts down and is POSITIVE beam has to resist shearing apart by V (+)V (+)M R Internal Beam Forces 22

23 Shear Sign Convention Internal Beam Forces 23

24 Sign Convention bending moment, M: cut section to LEFT if M cut is clockwise, M acts ccw and is POSITIVE flexes into a smiley beam has to resist bending apart by M (+)V (+)M R Internal Beam Forces 24

25 Bending Moment Sign Convention Internal Beam Forces 25

26 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 26

27 Constructing V & M Diagrams along the beam length, plot V, plot M V + - L load diagram M + L Internal Beam Forces 27

28 Mathematical Method cut sections with x as width write functions of V(x) and M(x) V + L - M + L x Internal Beam Forces 28

29 Method 1: Equilibrium cut sections at important places plot V & M V + L - M + L L/2 Internal Beam Forces 29

30 Method 1: Equilibrium important places supports concentrated loads start and end of distributed loads concentrated moments free ends zero forces Internal Beam Forces 30

31 Method 2: Semigraphical by knowing V area under loading curve = change in V area under shear curve = change in M concentrated forces cause jump in V concentrated moments cause jump in M D V C x x D C wdx M D M C x x D C Vdx Internal Beam Forces 31

32 Method 2 relationships Internal Beam Forces 32

33 Method 2: Semigraphical M max occurs where V = 0 (calculus) V + L no area - M + L Internal Beam Forces 33

34 Curve Relationships integration of functions line with 0 slope, integrates to sloped y y x x ex: load to shear, shear to moment Internal Beam Forces 34

35 Curve Relationships line with slope, integrates to parabola y y x x ex: load to shear, shear to moment Internal Beam Forces 35

36 Curve Relationships parabola, integrates to 3 rd order curve y y x x ex: load to shear, shear to moment Internal Beam Forces 36

37 Basic Procedure 1. Find reaction forces & moments V: 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 37

38 Basic Procedure M: 6. Starting at left 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! (locate where V = 0) Internal Beam Forces 38

39 Shear Through Zero slope of V is w (-w:1) w (force/length) load height = V A x w V A x V A w shear A width = x Internal Beam Forces 39

40 Parabolic Shapes cases up fast, then slow up slow, then fast down fast, then slow down slow, then fast Internal Beam Forces 40