Presentation on theme: "Beams Beams: Comparison with trusses, plates t"— Presentation transcript:
1 Beams Beams: Comparison with trusses, plates t L, W, t: L >> W and L >> tLWComparison with trusses, platesExamples:1. simply supported beams2. cantilever beams
2 Beams - loads and internal loads Loads: concentrated loads, distributed loads, couples (moments)Internal loads: shear force and bending moments
3 Shear Forces, Bending Moments - Sign Conventions right sectionleft sectionShear forces:positive shear:negative shear:Bending moments:positive momentnegative moment
4 Shear Forces, Bending Moments - Static Equilibrium Approach Procedure:1. find reactions;2. cut the beam at a certain cross section, draw F.B.D. of one piece of the beam;3. set up equations;4. solve for shear force and bending moment at that cross section;5. draw shear and bending moment diagrams.Example 1: Find the shear force and bending diagram at any cross section ofthe beam shown below.
5 Relationship between Loads, Shear Forces, and Bending Diagram
6 Beam - Normal Strain M M no transverse load Pure bending problem no axial loadno torqueObservations of the deformed beam under pure bendingLength of the longitudinal elementsVertical plane remains plane after deformationBeam deforms like an arc
7 Normal Strain - Analysis neutral axis (N.A.):radius of curvature:Coordinate system:qlongitudinal strain:ryN.A.
8 Beam - Normal Stress Hooke’s Law: y M M M x Maximum stresses: Neutral axis:
9 Flexure Formula y Moment balance: M x Comparison: Axially loaded members Torsional shafts:
10 Moment of Inertia - IExample 2:hwExample 3:hww4hw
11 Design of Beams for Bending Stresses Design Criteria:1.2. cost as low as possibleDesign Question:Given the loading and material, how to choose the shape and the sizeof the beam so that the two design criteria are satisfied?
12 Design of Beams for Bending Stresses Procedure:Find MmaxCalculate the required section modulusPick a beam with the least cross-sectional area or weightCheck your answer
13 Design of Beams for Bending Stresses Example 4: A beam needs to support a uniform loading with density of200 lb /ft. The allowable stress is 16,000 psi. Select the shape and the sizeof the beam if the height of the beam has to be 2 in and only rectangular andcircular shapes are allowed.6 ft
15 Shear Stresses inside Beams Relationship between the horizontal shear stresses and the vertical shear stresses:yh1y1xh2Shear stresses - force balanceV: shear force at the transverse cross sectionQ: first moment of the cross sectional area above the level at whichthe shear stress is being evaluatedw: width of the beam at the point at which the shear stress is beingevaluatedI: second moment of inertial of the cross section
16 Shear Stresses inside Beams Example 5: Find shear stresses at points A, O and B located at cross sectiona-a.PaAOaBw
17 Shear Stress Formula - Limitations - elementary shear stress theoryAssumptions:1. Linearly elastic material, small deformation2. The edge of the cross section must be parallel to y axis, not applicable fortriangular or semi-circular shape3. Shear stress must be uniform across the width4. For rectangular shape, w should not be too large
18 Shear Stresses inside Beams Example 6: The transverse shear V is 6000 N. Determine the vertical shear stressat the web.
19 Beams - Examples Example 7: For the beam and loading shown, determine (1) the largest normal stress(2) the largest shearing stress(3) the shearing stress at point a
20 Deflections of BeamDeflection curve of the beam: deflection of the neutral axis of the beam.yPyxxDerivation:Moment-curvature relationship:Curvature of the deflection curve:(1)Small deflection:(2)(3)Equations (1), (2) and (3) are totally equivalent.
21 Deflections by Integration of the Moment Differential Equation Example 8 (approach 1):
22 Deflections by Integration of the Load Differential Equation Example 8 (approach 2):
23 Method of Superposition qDeflection: yPDeflection: y1Deflection: y2