CHAP 4 FINITE ELEMENT ANALYSIS OF BEAMS AND

CHAP 4 FINITE ELEMENT ANALYSIS OF BEAMS AND FRAMES FINITE ELEMENT ANALYSIS AND DESIGN Nam-Ho Kim Audio: Raphael Haftka 1

INTRODUCTION • We learned Direct Stiffness Method in Chapter 2 – Limited to simple elements such as 1 D bars • we will learn Energy Method to build beam finite elements – Structure is in equilibrium when the potential energy is minimum • Potential energy: Sum of strain energy and potential of applied loads • Interpolation scheme: Potential of applied loads Strain energy Beam Interpolation Nodal deflection function DOF 2

BEAM THEORY • Assumptions for our plane beam element – – – carries transverse loads slope can change along the span (x-axis) Cross-section is symmetric w. r. t. xy-plane The y-axis passes through the centroid Loads are applied in xy-plane (plane of loading) y y Neutral axis Plane section x L F z F A 3

BEAM THEORY cont. • Euler-Bernoulli Beam Theory – Plane sections normal to the beam axis remain plane and normal to the axis after deformation (no shear strain) – Transverse deflection (deflection curve) is function of x only: v(x) – Displacement in x-dir is function of x and y: u(x, y) y y(dv/dx) Neutral axis x L F y q = dv/dx v(x) 4

Quiz-like questions • What are the assumptions of the Euler-Bernoulli beam theory geometrically? • What are the implications in terms of the displacements? • The assumptions lead to zero shear strains, but obviously we calculate the shear stresses in beams due to transverse loads. How do we reconcile this contradiction? • Answers in notes page 5

BEAM STRESSES AND FORCE RESULTANTS • Stresses – – Strain along the beam axis: Strain exx varies linearly w. r. t. y; Strain eyy = 0 Curvature: Can assume plane stress in z-dir basically uniaxial status • Axial force resultant and bending moment Moment of inertia I(x) EA: axial rigidity EI: flexural rigidity 6

BEAM LOADING • Beam constitutive relation – We assume P = 0 (We will consider non-zero P in the frame element) – Moment-curvature relation: Moment is proportional to curvature • Sign convention +M +Vy y x +P +M +P +Vy – Positive directions for applied loads y p(x) x C 1 F 1 C 2 F 2 C 3 F 3 7

BEAM EQUILIBRIUM EQUATIONS – Combining three equations together: – Fourth-order differential equation 8

STRESS AND STRAIN • Bending stress – This is only non-zero stress component for Euler-Bernoulli beam • Transverse shear strain – Euler beam predicts zero shear strain (approximation) – Traditional beam theory says the transverse shear stress is – The approximation that first neglects shear strains and then calculates them from equilibrium is accurate enough for slender beams unless shear modulus is small. 9

POTENTIAL ENERGY • Potential energy • Strain energy – Strain energy density – Strain energy per unit length Moment of inertia – Strain energy 10

POTENTIAL ENERGY cont. • Potential energy of applied loads • Potential energy – Potential energy is a function of v(x) and slope – The beam is in equilibrium when P has its minimum value P v* v 11

Quiz-like questions • • What are the assumptions made for the beam constitutive equation? For which case will the fourth order beam equilibrium equation be insufficient to describe exactly the beam curvature? • • The bending stress approximation holds good for slender beams with large shear modulus. Why? Answers in notes page 12