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ENGR 225 Section 1.3 – 1.6. Internal Loadings Resultant Force and Moment.
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Transcript of ENGR 225 Section 1.3 – 1.6. Internal Loadings Resultant Force and Moment.
Stress Derivation
• A very small finite force ΔF is acting on an associated area ΔA.
• Replace this force with its directional components, ΔFx, ΔFy, and ΔFz.
• Take the limit as ΔA approaches zero of the quotient of the component forces and the areas
• This quotient is called, stress. It represents the intensity of the force on a specified plane.
Shear Stress, τzx τzy
• The intensity of force per unit area acting tangent to ΔA.
A
FxAxz
0lim
A
FyA
zy
0
lim
Average Normal Stress Distribution
A
P
If we assume is constant throughout the area or
Averaged over the cross sectional area, then
P = dA
Assumptions in using relation for average normal stress
1. Uniform cross section throughout the length
2. Uniform deformation.
3. Applied load is along Centroidal axis
4. Homogenous and Isotropic material
5. Only axial load applied.
6. Weight of the bar is neglected.
Uniaxial Stress – Tensile or Compressive
Example 1.7: The 80 kg lamp is supported by two rods AB and BC. If AB has a diameter of 10 mm and BC has a diameter of 8 mm determine the average normal stress in each rod.
Lecture Example: Determine the stress on the floor of a 135 lb. woman standing still wearing the following shoes that she purchased on sale. Assume that her weight is equally distributed between the heels and toes of her feet.
Lecture Example: Determine the average normal and average shear stress developed in the wood fibers that are oriented along the section a-a.
Lecture Example: Determine the average compressive stress along the smooth areas of contact defined by AB and BC.
Complementary Property of Shear Stress
zy
yz
’zy
’yz
Force balance giveszy = ’zy
Moment balance about x axis giveszy = yz
Hence
zy = ’zy = yz = ’yz
Two- dimensional case
y
z
Allowable Stress
Failure Load : Material testing
To ensure safe working : Allowable Load < Failure Load
F.S. = Ffailure / Fallowable
Factor of Safety : accounts for
• Unknown natural factors • Errors in manufacturing and assembly• Errors in load estimation• Material weathering
Rods A and B are made of steel having failure stress 510 MPa. Use Factor of Safety 1.75, determine the smallest diameter to support the given load. Beam is assumed to be pin connected at A and C.