AE3160 Experimental Fluid and Solid Mechanics Cantilever Beam Bending Claudio Di Leo 1
Learning Objectives 1. On Structural Mechanics: a) Mechanics of Slender Beams b) Strain Transformation Theory c) Principal stresses and strains. On experimental methods: a) Micrometer measurements for beam deflection b) Learn about load control vs. displacement control
Experimental set-up Cantilever beam with load or displacement control Displacement control through micrometer Load control using hanging weights 3
Load vs. Displacement Control Depending on the material/structure we might want to control either the applied load or applied displacement Consider the stress-strain curve for Polycarbonate below taken under displacement control conditions What would the curve look like under load control? Under load control, there would have been a displacement jump when the material softens! 4
Load vs. Displacement Control Depending on the material/structure we might want to control either the applied load or applied displacement Consider the buckling problem demonstration below under load control Sanjay Govindjee UC Berkley https://www.youtub e.com/user/sanjayg 0/about 5
Experimental set-up Cantilever beam with load or displacement control Misaligned strain gauge rosette 6
Beam reference frames We have a global reference frame --Z aligned with the beam, and a local reference frame *-*-Z* aligned with the strain gauge Frames share a common Z-axis and frame *-*-Z* is rotated about the Z-axis by an angle Θ. * Z * P 7
Plane stress and Hook s Law Z P What components of stress/strain are non-zero? Assume Euler Bernoulli beam theory: = z M() I All other components of stress are assumed zero! 8
Strains Z P The non-zero stress is = zm(x)/i. The non-zero strain components are = E, = E, and Z = If we measure on the surface (z = t/), and we know the applied moment M and the beam geometry I we can compute the material properties of the beam. E. 9
Strains * Z * P The non-zero stress is = zm(x)/i. Matrix Form 0 0 0 0 0 0 Z 10
Strain transformation * Z * P The non-zero stress is = zm(x)/i. We measure the strains in the *-*-Z* frame 0 0 0 0 Z 0 0 0 0 0 0 Z Note: The engineering shear strain relates to the tensorial shear strain through = / 11
Strain transformation * Z * P Convert strains from the --Z frame to the *-*-Z* frame = + + cos( )+ sin( ) = + cos( ) sin( ) = sin( )+ cos( ) 1
Strain Gauge Rosette Want to measure * * Misaligned strain gauge rosette C B 45 A Strain gauges measure axial strains in a single direction! 13
Strain gauge Rosette Want to measure * C * B 45 Assume * is aligned with A and * is aligned with C A = A = C Consider a frame aligned with B B = + + cos( 45 ) + sin( 45 ) 14
Strain gauge Rosette Want to measure * C * B 45 A Solving for the desired strain components we have = A = C = B A + C 15
Strain transformation * unknown misalignment angle Z * P Convert strains from the --Z frame to the *-*-Z* frame = = + + + cos( )+ sin( ) cos( ) sin( ) 3 equations 4 unknowns! = sin( )+ cos( ) 16
Strain transformation * unknown misalignment angle Z * P We measure the strains in the *-*-Z* frame and need to convert to the beam frame --Z This frame is 0 0 0 the principal 0 0 0 strain frame, 0 0 Z 0 0 Z where all shear strains are zero! 17
Strain transformation * unknown misalignment angle Z * P Convert strains from the --Z frame to the *-*-Z* frame = = + + + cos( )+ sin( ) cos( ) sin( ) 3 equations 3 unknowns! = sin( )+ cos( ) 18
Strain transformation * unknown misalignment angle Z * P The principal strains are found with respect to any arbitrary reference frame through 1 = + = + + +( ) +( ) 1/ 1/ and tan( p )= 19
Mohr s circle for strain 0
General strain transformations For D strain cases, we can also express the transformation of strain components as follows: x y xy/ = T( ) / This is often written as follows, in terms of the matrix R = diag{1, 1, } x y xy = RT( )R 1 The transformation matrix is given as follows T( )= cos sin sin cos sin cos sin cos sin cos sin cos cos sin 1
Orthotropic materials In this class, we will stick to isotropic materials, which is often a good model for metals For orthotropic materials, like composites, the properties vary strongly with direction Obtain the stress-strain relationship in the Z frame = T( ) 1 C mat RT( )R 1 = C( ) For isotropic materials C( )=C mat Isotropic C mat = E 1 1 0 1 0 0 0 1 (1 ) Orthotropic C mat = Q 11 Q 1 0 Q 1 Q 0 0 0 Q 66
Overview 3