How Beams Work, I. Statics and Strength of Materials

Transcription

1 How Beams Work, I Statics and Strength of Materials

2 HOW BEAMS WORK Beams work by transferring transverse loads along their length to their supports primarily by resisting a force called internal bending moment.

3 BEAM TYPES : FUNCTIONAL DISTINCTION joists closely spaced beams supporting floors and roofs lintels beams over masonry wall openings lintel spandrel beams supporting exterior walls and sometimes the floors stringers bridge beams parallel to the roadway floor beams large beams perpendicular to the roadway that transfer loads from the stringers to the supporting trusses or girders spandrel girder large beam into which other beams are framed transfer beam/girder

4 BENDING OERIEW What are the forces the member must resist? Primarily bending, which is a combination of: 1) A force couple of tension and compression 2) Shear (vertical and horizontal) 3) Torsion 4) Bearing Factors that influence beam strength GEOMETRY 1) length ( span ) of beam 2) depth of beam 3) shape of cross section MATERIAL 1) modulus of elasticity 2) failure/yield stress DESIGN FACTORS 1) Internal Bending Moment 2) Horizontal Shear 3) ertical Shear 4) Bearing 5) Torsion 6) Loading conditions 7) Support and bracing conditions

5 BEAM THEORY P Transfer transverse loads to supports... d = depth of the beam L = span of the beam

6 BEAM THEORY Take a simply supported beam under load... neutral axis One model of describing beam action is to divide the beam into segments and discuss the motion of a single segment as the beam deforms. As we observe the deformation of a single segment, we notice that the top side of each segment is squeezed and the bottom side spreads. Compressive stresses Tensile stresses Section rotates Neutral axis = no tensile or compressive stresses Because, for the elastic range of the material, stress and strain are directly related, we can conclude that where deformation (strain) is present, the analogous stress is also present. Thus, the top is in compression, and the bottom in tension.

7 BEAM THEORY C T C T As a beam bends, the force couple formed by compression in the top fibers and tension in the bottom fibers creates what is called internal bending moment. The ability of a beam to resist this force is the primary determining factor for the strength of the beam.

8 BENDING DETAILS compression (shortens) ƒ C b C C neutral axis F C T T F T tension (lengthens) original element ƒ T

9 INTERNAL SHEAR P vertical shear P vertical shear horizontal shear

10 MODES OF FAILURE What are the modes of failure for each one of those forces? 1) Bending failure (local buckling, tension failure) 2) Shear (horizontal shear failure) 3) Torsion (lateral buckling) 4) Bearing failure (at support or point loads) Lateral buckling occurs when the compression flange buckles like a column. The effect is the beam twisting on its axis. compression flange tension flange

11 SHEAR AND MOMENT DIAGRAMS Graphic representations of the magnitudes of internal forces (vertical shear and internal bending moment) along the length of a beam lbs d ft lb M

12 DRAWING THE SHEAR DIAGRAM 1000 lbs d F = 500lbs

13 DRAWING THE SHEAR DIAGRAM 1000 lbs d F = 500lbs

14 DRAWING THE SHEAR DIAGRAM 1000 lbs d F = 500lbs

15 DRAWING THE SHEAR DIAGRAM 1000 lbs d F = 500lbs

16 DRAWING THE SHEAR DIAGRAM d lbs F = -500lbs -

17 DRAWING THE SHEAR DIAGRAM d lbs F = -500lbs -

18 DRAWING THE SHEAR DIAGRAM d lbs F = 0 0 lbs

19 WHAT ABOUT THIS? 2 k / ft?

20 WHAT ABOUT THIS? 2 k / ft 30 k 30 k

21 DRAWING THE MOMENT DIAGRAM 1000 lbs d - M = Fd 2 M M = Fd = = 1000 ft lb 2

22 DRAWING THE MOMENT DIAGRAM 1000 lbs d - M = Fd 4 M M = Fd = = 2000 ft lb 4

23 DRAWING THE MOMENT DIAGRAM 1000 lbs d - M = Fd 15 M = Fd = = 7500 ft lb M 15

24 DRAWING THE MOMENT DIAGRAM 1000 lbs d lbs M = Fd 17 2 M M = Fd = = 8500 ft lb ft lb = 6500 ft lb 4

25 FINAL SHEAR AND MOMENT DIAGRAMS 1000 lbs d ft lb M

26 WHAT ABOUT THIS? 2 k / ft 30 k 30 k M?

27 WHAT ABOUT THIS? 2 k / ft 30 k 30 k M

28 RULES FOR DRAWING SHEAR AND MOMENT DIAGRAMS Start by drawing the free body diagram of the beam, with all loads and reactions. Drop lines at each. Generally draw from left to right. Always draw the complete shear diagram first. The value of the shear diagram is the slope of the moment diagram at that point. Point loads make the shear diagram jump up or down. M run rise The moment diagram flattens out where the shear diagram is 0. The shear diagram is flat when there is no increase in load from left to right. Uniformly distributed loads make parabolas in the moment diagram and downward sloping lines in the shear diagram. Point loads make peaks in the moment diagram.