Recall governing equations: Boundary Value Problem …aprakas/CE570/Ch5-BVPs.pdfChapter 5: Boundary value problems in solid mechanics Recall governing equations: Example: (pages 250-255,

Chapter 5: Boundary value problems in solid mechanics

Recall governing equations:

Example:

(pages 250-255, Timoshenko & Goodier)Pure bending of a prismatic cantilever beam:

Note: For a cross section at z = c:

Note: For the lateral surfaces of the beam:

Boundary Value Problem

Ch5-BVPs Page 1

Simplifying Assumptions:

These systems of PDEs, cannot be solved analytically, in general. We use numerical methods and use simplifying assumptions.

Read Example 27 from the textbook.(See. 101 from Timoshenko & Goodier)

Example: 2D Plane Problems

Ch5-BVPs Page 2

To understand how to obtain numerical solutions to complicated 2D/3D problems in general, let's first study some 1-D problems where we can usually obtain exact solutions.

One-dimensional (1D) little BVP

Alternative derivation:

Ch5-BVPs Page 3

Example. Column under self weight:

Ch5-BVPs Page 4

Strong Forms and Weak forms

Weak forms

Weak form of the Problem Statement.

This relies on the fundamental theorem of Calculus of Variations:

Ch5-BVPs Page 5

Proof of Fundamental theorem of calculus of variations

Restriction on the choice of Function spaces

Ch5-BVPs Page 6

Possible choices for function spaces:

Example of Approximate solution to Weak form:

Ch5-BVPs Page 7

Weak Form: Method of weighted Residuals (MWR):

Weak Form: Principle of Virtual Work (PVW):

Substituting the material model:

Weak form problem statements:

Ch5-BVPs Page 8

Boundary Conditions

Ch5-BVPs Page 9

Examples of Boundary Conditions & Approximation Function Spaces

Ch5-BVPs Page 10

Variational (Energy-based) methods

Ch5-BVPs Page 11

Energy weak form:

Ch5-BVPs Page 12

Vainberg's Theorem: Existence of an Energy form (Variational form)

Vainberg's Theorem

Ch5-BVPs Page 13

Example:

Ch5-BVPs Page 14

Hamilton's Principle (for Dynamics)

Ch5-BVPs Page 15

Weak form in 3D

One can show that the Vainberg's Theorem is satisfied for the above weak form.Energy form in 3D

Ch5-BVPs Page 16