chapt 04a beams
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Internal Forces7
Engineering Mechanics: Statics in SI Units, 12e
Copyright © 2010 Pearson Education South Asia Pte Ltd
Copyright © 2010 Pearson Education South Asia Pte Ltd
Chapter Objectives
• Method of sections for determining the internal loadings in a member
• Develop procedure by formulating equations that describe the internal shear and moment throughout a member
Copyright © 2010 Pearson Education South Asia Pte Ltd
Chapter Outline
1. Internal Forces Developed in Structural Members2. Shear and Moment Equations and Diagrams3. Relations between Distributed Load, Shear and
Moment
Copyright © 2010 Pearson Education South Asia Pte Ltd
7.1 Internal Forces Developed in Structural Members
• The design of any structural or mechanical member requires the material to be used to be able to resist the loading acting on the member
• These internal loadings can be determined by the method of sections
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7.1 Internal Forces Developed in Structural Members
• Force component N, acting normal to the beam at the cut session
• V, acting tangent to the session are normal or axial force and the shear force
• Couple moment M is referred as the bending moment
Copyright © 2010 Pearson Education South Asia Pte Ltd
7.1 Internal Forces Developed in Structural Members
• For 3D, a general internal force and couple moment resultant will act at the section
• Ny is the normal force, and Vx and Vz are the shear components
• My is the torisonal or twisting moment, and Mx and Mzare the bending moment components
Copyright © 2010 Pearson Education South Asia Pte Ltd
7.1 Internal Forces Developed in Structural Members
Procedure for AnalysisSupport Reactions• Before cut, determine the member’s support reactions• Equilibrium equations used to solve internal loadings
during sectioning
Free-Body Diagrams• Keep all distributed loadings, couple moments and
forces acting on the member in their exact locations• After section, draw FBD of the segment having the
least loads
Copyright © 2010 Pearson Education South Asia Pte Ltd
7.1 Internal Forces Developed in Structural Members
Procedure for AnalysisFree-Body Diagrams (Continue)• Indicate the z, y, z components of the force, couple
moments and resultant couple moments on FBD• Only N, V and M act at the section• Determine the sense by inspection
Equations of Equilibrium• Moments should be summed at the section • If negative result, the sense is opposite
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Example 7.3
Determine the internal force, shear force and the bending moment acting at point B of the two-member frame.
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Solution
Support ReactionsFBD of each memberMember AC∑ MA = 0;-400kN(4m) + (3/5)FDC(8m)= 0
FDC = 333.3kN (C)+→∑ Fx = 0;-Ax + (4/5)(333.3kN) = 0
Ax = 266.7kN+↑∑ Fy = 0;Ay – 400kN + 3/5(333.3kN) = 0
Ay = 200kN
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Solution
Support ReactionsMember AB+→∑ Fx = 0; NB – 266.7kN = 0
NB = 266.7kN+↑∑ Fy = 0; 200kN – 200kN - VB = 0
VB = 0∑ MB = 0; MB – 200kN(4m) + 200kN(2m) = 0
MB = 400kN.m
Copyright © 2010 Pearson Education South Asia Pte Ltd
7.2 Shear and Moment Equations and Diagrams
• Beams – structural members designed to support loadings perpendicular to their axes
• A simply supported beam is pinned at one end and roller supported at the other
• A cantilevered beam is fixed at one end and free at the other
Copyright © 2010 Pearson Education South Asia Pte Ltd
7.2 Shear and Moment Equations and Diagrams
Procedure for AnalysisSupport Reactions• Find all reactive forces and couple moments acting on
the beam• Resolve them into components
Shear and Moment Reactions• Specify separate coordinates x• Section the beam perpendicular to its axis• V obtained by summing the forces perpendicular to
the beam• M obtained by summing moments about the sectioned
end
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7.2 Shear and Moment Equations and Diagrams
Procedure for AnalysisShear and Moment Reactions (Continue)• Plot (V versus x) and (M versus x)• Convenient to plot the shear and the bending moment
diagrams below the FBD of the beam
Copyright © 2010 Pearson Education South Asia Pte Ltd
Example 7.7
Draw the shear and bending moments diagrams for the shaft. The support at A is a thrust bearing and the support at C is a journal bearing.
Copyright © 2010 Pearson Education South Asia Pte Ltd
Solution
Support ReactionsFBD of the shaft
mxkNMMkNVFy
.5.2;0
5.2;0
mkNxMxkNmxkNMM
kNVVkNkNFy
.)5.210(0)(5.2)2(5 ;0
5.2055.2 ;0
Copyright © 2010 Pearson Education South Asia Pte Ltd
Solution
Shear diagramInternal shear force is always positive within the shaft AB.
Just to the right of B, the shear force changes sign and remains at constant value for segment BC.
Moment diagram Starts at zero, increases linearly to B and therefore decreases to zero.