ME221Lecture 251 ME 221 Statics Lecture #25 Sections 7.1 – 7.4

ME221Lecture 252 Homework #9 Chapter 5 problems: –53, 54, 56, 62, 64, 69, 71 & 73 Due Friday, October 31

ME221Lecture 253 Homework #10 Chapter 7 problems: –2, 5, 6, 8, 19, 21, 24, 26 & 35 Due Friday, November 7

ME221Lecture 254 Chapter 7: Internal Forces in Structures Review internal/external forces How to find internal forces Sample problems

ME221Lecture 255 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

ME221Lecture 256 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 aa bb cc At any given section a-a, b-b, or c-c, there is an internal force arising from the 100 lb external force.

ME221Lecture 257 Method for Finding Internal Forces Determine reaction forces 100 lb A y =100 lb A x =0 M z =0 –Use equilibrium equations Section and solve second equilibrium problem to find internal forces aa 100 lb FyFy FxFx MzMz

ME221Lecture 258 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:

ME221Lecture 259 Example: Determine the internal forces and moments in the bar built into the foundation as shown in the figure. x y z O l h P

ME221Lecture 2510 (l-x) P x y z O l h P x Horizontal Portion

ME221Lecture 2511 x y z O l h P y l (h-y) P Vertical Portion

ME221Lecture 2512 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 Sample problems

ME221Lecture 2513 Types of Beams by Supports Transversely loaded beams have several standard configurations Determinate beams have the same number of reactions as nontrivial equilibrium eqns. Indeterminate Determinate Simple Overhanging Cantilever

ME221Lecture 2514 Internal Force Component Rolls Force components Moment components –Axial is along beam P –Shearing forces are transverse components VyVy VzVz –Torsion along beam T –Bending for transverse components MyMy MzMz

ME221Lecture 2515 Shear and Moment Diagrams -Sectioning Method -Integration -Singularity Functions

ME221Lecture 2516 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

ME221Lecture 2517 Generate a shear / bending diagram as follows: 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. 1. Find reaction forces w(x) distributed load V(x) shear force M(x) moment Shear and Moment Diagrams using Sectioning Method

ME221Lecture 2518 V M V M Sign Convention Positive Shear and Positive Moment

ME221Lecture 2519 Positive Shear M M Positive Moment Effect of External Forces

ME221Lecture lb/in 125 lb 9 in. 12 in. 12 in. 12in. V x M x -ve tension up +ve Tension down

ME221Lecture 2521