Lecture 7: The Beam Element Equations.
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1 4.1 Beam Stiffness. A Beam: A long slender structural component generally subjected to transverse loading that produces significant bending effects as opposed to twisting or axial effects. MECH 40: Finite Element Applications
2 MECH 0 and 0 leave us with... The bending moment, shear force, and distributed load are internal loads. Their sense is defined according to specific sign conventions ŷ 1 M = EI = EIκ ρ dm V = dv w = Based on the assumption that plane sections remain plane M V x MECH 40: Finite Element Applications
3 Governing kinematic relationships (displacement and motion knowledge irrespective of causal forces). d v x 1 M( x) ( ) = κ( x) = ; κ( x) = beam curvature. ρ( x ) EI Give the differential relationships that exist between bending moment, shear force, and distributed transverse load (from static equilibrium) M V w EI EI = EI ( ) ( ) ( ) dvx dvx 4 dvx 4 MECH 40: Finite Element Applications = = By definition, these quantities must to be interpreted according to the civil engineering sign convention.
4 So to form our element equations we must form our own approximation to the beam mechanics. This approximate beam (or beam element) should be applicable in all types of structural problems involving beams. f f f ix iy iz i = 1.. N K (the stiffness matrix) d d d ix iy iz i = 1.. N Defined in terms of a Cartesian reference frame MECH 40: Finite Element Applications
5 Step 1: Set the element type: An important step as we are identifying the state variables of the beam element. M M MECH 40: Finite Element Applications
6 We have introduced some limitations by requiring that only nodal loads exist. Comparing Figures 4.1 and 4.: ( ) dvx 1y =+ = f V EI ( ) dvx 1z = = m M EI ( ) dvx y = = f V EI ( ) dvx z =+ = m M EI x= 0 x= 0 x = x = Since we have not included distributed loads in the analysis, we only see up to third order differentials of the displacement function v. MECH 40: Finite Element Applications
7 Step : Select a displacement function for vx ( ). Given our assumption/limitation that 4 dvx ( ) w= EI = 4 d x 0 A conforming element is one that will ensure: Compatibility/Continuity: a continuous displacement and rotation exist within a single beam element and across element boundaries. That is, a smooth first derivative of v(x) exists. Completeness: a constant shear force can exist within the element. That is v(x) can be differentiated up to times. MECH 40: Finite Element Applications
8 So we choose a cubic function to approximate vx ( ) : vx ( ) = a + ax+ ax + ax 4 1 The displacement function must interpolate the nodal generalized displacements. This ensures compatibility. v(0) = a = d 4 1y dv = a = tan( φ 1) φ1 x= 0 v ( ) = a+ a+ a+ a = d dv x = 1 4 y = a + a+ a = tan( φ ) φ 1 1 Assuming small transverse displacements MECH 40: Finite Element Applications
9 In a matrix form [ ] vx ( ) N N N N d 1y φ d y φ 1 = N1 = ( x x + ) 1 N = x x + x 1 N = ( x + x ) 1 N4 = ( x x ) ( ) Shape functions that blend the nodal displacements over the element domain. Eq. (4.1.7) of ogan. MECH 40: Finite Element Applications
10 Step : Define the stress-strain relationships. Also a means to recover any stress-strain knowledge after obtaining the state variables of the model. dv ux ( ) = y dx du ε ( x x) = dv ε ( x x) = y dx dv σ ( x x) = Ey dx MECH 40: Finite Element Applications
11 Step 4: Derive the element equations Can we express the nodal forces in terms of the displacement function? ( φ φ ) f V EI d d x= 0 dvx ( ) EI 1y =+ = = 1 1y y + 6 ( φ φ ) m M EI 6d 4 6d x= 0 dvx ( ) EI 1z = = = 1y + 1 y + ( φ φ ) f V EI d d x = dvx ( ) EI y = = = 1 1y y 6 dvx ( ) EI z = = = ( 6d 1y + φ1 6dy + 4 φ) m M EI x = MECH 40: Finite Element Applications
12 And in matrix form f 1y d 1y m 1z EI 4 6 φ1 = f 1 6 d y y m SYM 4 z φ k MECH 40: Finite Element Applications
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