ME 221 Statics Lecture #14 Sections 7.1 – 7.4

Homework #6 Chapter 7 problems: Chapter 6 problems 6, 19 & 26 Chapter 6 problems 3 & 6 Due Monday, June 28 MatLab Group Problems 7.19, 7.26 & 6.15 ME221 Lecture 14

Quiz #6 Friday, June 25 ME221 Lecture 14

Chapter 7: Internal Forces in Structures Review internal/external forces How to find internal forces Sample problems ME221 Lecture 14

External/Internal Forces External forces arise from contact or gravitational attraction Point and distributed loading Weight Internal forces are forces arising to hold bodies together Internal stress is a form of an internal force ME221 Lecture 14

Exposing Internal Forces To analyze the stress at a given location in a part, we need to know the forces at that particular section. 100 lb At any given section a-a, b-b, or c-c, there is an internal force arising from the 100 lb external force. c b a ME221 Lecture 14

Method for Finding Internal Forces Determine reaction forces Use equilibrium equations Section and solve second equilibrium problem to find internal forces 100 lb Ay=100 lb Ax=0 Mz=0 100 lb a 100 lb Fy Fx Mz ME221 Lecture 14

General Internal Forces In general, there is a force and moment component for each coordinate direction at a given section 6 possible unknowns Sample problems: ME221 Lecture 14

Example: x y z O l h P Determine the internal forces and moments in the bar built into the foundation as shown in the figure. ME221 Lecture 14

x y z O l h P x (l-x) P Horizontal Portion ME221 Lecture 14

x y z O l h P l (h-y) P y Vertical Portion ME221 Lecture 14

Shear and Bending Diagrams (Secs. 7.3, 7.4) Topic is also called transversely loaded beams Beam classifications and boundary conditions Internal forces and the components’ specific rolls Relation between shear and bending Generation of shear and bending diagrams ME221 Lecture 14

Types of Beams by Supports Transversely loaded beams have several standard configurations Determinate beams have the same number of reactions as nontrivial equilibrium eqns. Determinate Indeterminate Simple Overhanging Cantilever ME221 Lecture 14

Internal Force Component Rolls Force components Shearing forces are transverse components Vy Vz Axial is along beam P Moment components Bending for transverse components My Mz Torsion along beam T ME221 Lecture 14

Shear and Moment Diagrams -Sectioning Method -Integration -Singularity Functions ME221 Lecture 14

What is expected for shear and bending diagrams? 1. Show FBD and statics for each section 2. Determine equation for V(x) and M(x) 3. Draw shear and bending diagrams indicating linear or parabolic 4. Label end points of diagram as well as every region endpoint ME221 Lecture 14

Shear and Moment Diagrams using Sectioning Method Generate a shear / bending diagram as follows: 1. Find reaction forces 2. Take a section on each side of an applied force or moment and inside a distributed load (take a new section whenever there is a change in the load or shape of the beam) - draw a FBD and sum forces / moments 3. Repeat 2 along the length of the beam. w(x) distributed load V(x) shear force M(x) moment ME221 Lecture 14

Positive Shear and Positive Moment Sign Convention V M V M Positive Shear and Positive Moment ME221 Lecture 14

Effect of External Forces Positive Shear M Positive Moment ME221 Lecture 14

+ve -ve 125 lb 20 lb/in 9 in. 12 in. 12 in. 12in. V x M x tension up Tension down ME221 Lecture 14