Beams – Internal Effects The external load applied to a beam can cause changes in the shape of the beam, it can bend for example. We do not want.

Slides:



Advertisements
Similar presentations
BENDING MOMENTS AND SHEARING FORCES IN BEAMS
Advertisements

M 1 and M 2 – Masses of the two objects [kg] G – Universal gravitational constant G = 6.67x N m 2 /kg 2 or G = 3.439x10 -8 ft 4 /(lb s 4 ) r – distance.
Chapter 6 Bending.
Shear Force and Bending Moment
Strength of Material-5 Torsion Dr. Attaullah Shah.
CHAPTER 6 BENDING.
Equilibrium Equilibrium refers to a condition in which an object is at rest originally at rest (static equilibrium) or has a constant velocity if originaly.
CHAPTER 7 TRANSVERSE SHEAR.
Fluid Statics Pascal’s Law tells us that “Pressure at any point in a fluid is the same in all directions”. This means that any object submerged in a fluid.
CTC / MTC 222 Strength of Materials
AERSP 301 Shear of beams (Open Cross-section)
ENGR 220 Section 6.1~6.2 BENDING.
4 Pure Bending.
AERSP 301 Structural Idealization
Professor Joe Greene CSU, CHICO
Strength of Materials I EGCE201 กำลังวัสดุ 1
ENGR 225 Section
CTC / MTC 222 Strength of Materials
Beams Beams: Comparison with trusses, plates t
SHEAR AND MOMENT DIAGRAMS WITH APPLICATIONS IN TWO ORTHOGONAL PLANES
4.6 Moment due to Force Couples
10 Pure Bending.
Mechanics of Materials – MAE 243 (Section 002) Spring 2008 Dr. Konstantinos A. Sierros.
Chapter 1 Stress.
Engineering Mechanics: Statics
Bending Shear and Moment Diagram, Graphical method to construct shear
Shear Forces & Bending Moments Shear & Moment Diagrams
7.2 Shear and Moment Equations and Diagrams
Beams Session Subject: S1014 / MECHANICS of MATERIALS Year: 2008.
ERT 348 Controlled Environment Design 1
7.4 Cables Flexible cables and chains are used to support and transmit loads from one member to another In suspension bridges and trolley wheels, they.
SHEAR AND BENDING MOMENT DIAGRAMS IN HORIZONTAL BEAMS WITH
Copyright © 2010 Pearson Education South Asia Pte Ltd
Civil Engineering Materials – CIVE 2110
Engineering Mechanics: Statics
Mechanics of Materials – MAE 243 (Section 002) Spring 2008 Dr. Konstantinos A. Sierros.
Introduction Stress: When some external system of forces act on a body, the internal forces are set up at various sections of the body, which resist the.
CHAPTER OBJECTIVES To show how to transform the stress components that are associated with a particular coordinate system into components associated with.
Bölüm 7 Çubuk ve Kablolarda Kuvvetler
Beams - structural members supporting loads at various points along the member. Transverse loadings of beams are classified as concentrated loads or distributed.
Chapter 6: Bending.
Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 1 ENT 450 MECHANICS OF MATERIALS (MoM) RC. Hibbler Lecture: DR. HAFTIRMAN Teaching.
1. PLANE–STRESS TRANSFORMATION
Analysis and Design of Beams for Bending
Analysis and Design of Beams for Bending
Shear in Straight Members Shear Formula Shear Stresses in Beams
Shear Force and Bending Moment
Mechanics of Materials Dr. Konstantinos A. Sierros
Pure Bending.
shear force and bending moment diagram
INTERNAL FORCES AND FORCES IN BEAMS
Shear Force and Bending Moment
4 Pure Bending.
Structure I Course Code: ARCH 208 Dr. Aeid A. Abdulrazeg
Theory of Simple Bending
Chapter 7 FORCES IN BEAMS AND CABLES
STATICS (ENGINEERING MECHANICS-I)
Analysis and Design of Beams for Bending
Chapter 5 Torsion.
Chapter 6 Bending.
3 Torsion.
Analysis and Design of Beams for Bending
Eng Ship Structures 1 Hull Girder Response Analysis
Shear Force and Bending Moment
Beam relationships Moment sign conventions: Deflection: ν
Forging new generations of engineers
4 Pure Bending.
Engineering Mechanics: Statics
CE Statics Chapter 7 – Lecture 3.
Presentation transcript:

Beams – Internal Effects The external load applied to a beam can cause changes in the shape of the beam, it can bend for example. We do not want to examine changes in the shape, but we do want to know something about the forces inside the beam that could result in these deformations. In order to do this we must examine how the load affects the internal forces in a beam. There are three sets of forces we can examine – Shear, Bending and Torsion. Shear (V) is a translational force perpendicular to the long axis of the beam. Bending is described using the Bending Moment (M), which is a rotational force where the ends of the beam are rotated in opposite directions towards the center of the beam. Torsion is described using the Torsional Moment (T), which is a rotational force where the ends of the beam are rotated around the long axis of the beam.

To examine the internal shear, bending moments or torsional moments of a beam we analyze the situation as if the beam was cut and therefore make the internal forces external. The direction of the vectors representing these forces in one half of a cut end of the beam is shown in the image to the right. We will primarily be working with shear forces and bending moments, therefore there will be no further discussion of torsion. We do need to define a sign convention because the force and moment directions are reversed when discussing each half of the cut end of the beam. This is because the sum of the internal forces or moments must be zero, since they are not observed external effects. The standard sign convention is shown below. Positive directions for shear force and bending moment at the cut section of the beam.

The I-beam is a common construction material that is used because of its nearly equal strength and reduced weight as compared to a solid bar. The flanges on the top and bottom of the I-beam support tension or compression (for the image below compression on top and tension on the bottom). This pair of forces is a couple, which results in a bending moment. The thin center webbing provides a link between the two flanges but provides minimal additional support. Any object subjected to the forces described above is under the influence of a bending moment. The distribution along the center would have a different profile.

Shear Force and Bending Moment Relationships To examine the internal forces of a beam we will begin by analyzing a small segment of the beam. When we extract this segment the internal forces become external and we can analyze the situation using known techniques. Analyzing the forces on the segment, assuming w is constant over the small segment: 𝐹 =𝑉−𝑤𝑑𝑥− 𝑉+𝑑𝑉 =0 →𝑤=− 𝑑𝑉 𝑑𝑥 →−𝑤𝑑𝑥−𝑑𝑉=0 Does not hold at the location of the concentrated load! The shear force can then be determined by integrating w: 𝑉 0 𝑉 𝑑𝑉 =− 𝑥 0 𝑥 𝑤𝑑𝑥 𝑉− 𝑉 0 =− 𝑥 0 𝑥 𝑤𝑑𝑥 It is necessary to know the function w(x) to complete the integration.

Similarly, we can analyze the moments relative to position x: 𝑀 =𝑀+𝑤𝑑𝑥 𝑑𝑥 2 + 𝑉+𝑑𝑉 𝑑𝑥− 𝑀+𝑑𝑀 =0 → 𝑤𝑑𝑥 𝑑𝑥 2 + 𝑉+𝑑𝑉 𝑑𝑥−𝑑𝑀=0 Second order differential terms, therefore they can be ignored. →𝑉𝑑𝑥−𝑑𝑀=0 →𝑉= 𝑑𝑀 𝑑𝑥 Solving for M: 𝑀 0 𝑀 𝑑𝑀 = 𝑥 0 𝑥 𝑉𝑑𝑥 →𝑀− 𝑀 0 =− 𝑥 0 𝑥 𝑉𝑑𝑥 The bending moment can also be determined directly from w: 𝑤=− 𝑑𝑉 𝑑𝑥 →𝑤=− 𝑑 𝑑𝑥 𝑑𝑀 𝑑𝑥 →𝑤=− 𝑑 2 𝑀 𝑑𝑥 2 If w is continuous in x, M can be determined from two integrations. If bending occurs in multiple planes it is possible to look at each plane independently and add the results vectorally.