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

2. INTRODUCTION We learned Direct Stiffness Method in Chapter 2
Limited to simple elements such as 1D 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 deflection
Interpolation function
Nodal DOF

3. 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) L F x y
Plane section z
Neutral axis A

4. 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)
q = dv/dx
v(x) L F x
Neutral axis
Jacob and Daniel Bernoulli, and Leonard Euler, Swiss mathematicians of the 18th century. Theory matured around Today we often use a more advanced theory that permits shear deformation, Timoshekno beam (Stephen Timoshenko , Russian-Ukranidan).

5. 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
The assumptions is that plane sections remain plane and normals to these planes remain normal.
These lead to v depending only on x and u(x,y) varying linearly as function of y.
The shear stresses are indeed non-zero. However, for slender beams, their contribution to the elastic energy are negligible, and so the assumptions do not lead to significant errors.

6. BEAM STRESSES AND FORCE RESULTANTS
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
EA: axial rigidity
EI: flexural rigidity
Moment of inertia I(x)

7. Moment is proportional to curvature
BEAM LOADING
Beam constitutive relation
We assume P = 0 (We will consider non-zero P in the frame element)
Moment-curvature relation:
Sign convention
Positive directions for applied loads
Moment is proportional to curvature +P +M
+Vy y x
p(x) F1 F2 F3 C1 C2 C3 y x

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

9. STRESS AND STRAIN Bending stress Transverse shear strain
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.
Bending stress

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

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

12. 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
Assumptions: plane stress and the centroid of beam cross section is at y=0
When bending moments are applied along the span, the equation is invalid. When concentrated shear loads are applied along the span, this corresponds to infinite p, and the equation is valid provided that the loads are described as delta functions. Alternatively the equation applies only between the concentrated loads.
For slender beam with large shear modulus, the energy associated with shear deformation is small compared to the energy associated with bending.