Shear Centre and Pressure Vessels
Shear Centre
The shear centre is defined as the point about which the external load has to be applied so that it produces no twisting moment.
- What happens when loads act in a plane that is not a plane of symmetry?
Loads must be applied at particular point in the cross section, called shear center, if the beam is to bend without twisting.
Shear Stress distribution
- Constrained by the shape of the cross section
- Its resultant acts at the shear center
- Not necessarily the centroid
Shear Center
- A lateral load acting on a beam will produce bending without twisting only if it acts through the shear center
For a doubly symmetric section Shear centre and Centroid coincide
Locating Shear Centre
Unsymmetric Loading of Thin-Walled Members
Beam loaded in a vertical plane of symmetry deforms in the symmetry plane without twisting.
Beam without a vertical plane of symmetry bends and twists under loading.
Unsymmetric Loading of Thin-Walled Members
If the shear load is applied such that the beam does not twist, then the shear stress distribution satisfies
F and F’ indicate a couple Fh and the need for the application of a torque as well as the shear load.
When the force P is applied at a distance e to the left of the web centerline, the member bends in a vertical plane without twisting.
Shear stress distribution strategy
- Determine the location of centroid and Iyy, Izz and Iyz as needed - (symmetric sections subject to Vy needs only Izz)
- Divide section into elements according to geometry (change in slope)
- Start with a vector s following element centre line from a free end
- Calculate first moment of area(s). This determines the shear flow distribution
- Negative shear value indicate the direction of shear flow opposite to assumed vector s
- Calculate the first moment of area(s). This determines the shear flow distribution
For symmetric sections subject to bending about one axis
· Elements parallel to bending axis-Linear distribution
· Elements normal to bending axis-Parabolic distribution
For unsymmetric sections shear flow in all elements is parabolic
When moving from one element to another the end value of shear in one element equals the initial value for the subsequent element (from equilibrium)
Shear Centres for Some Other Sections
Shear Center
How to locate Shear Center?
Doubly symmetric cross sections- Coincides with centroid
Singly symmetric cross sections- Lies on the axis of symmetry
· Unsymmetric Cross sections
Thin-walled open sections
Opposite side of open part
· Doubly or singly symmetric section
Pressure Vessels
Types of Pressure Vessels Pressure vessels are mainly of two type:
- Thin shells
If the thickness of the wall of the shell is less than 1/10 to 1/15 of its diameter, then shell is called Thin shells.
- Thick shells
If the thickness of the wall of the shell is greater than 1/10 to 1/15 of its diameter, then shell is called Thick shells.
where Nature of stress in thin cylindrical shell subjected to internal pressure
- Hoop stress/circumferential stress will be tensile in nature.
- Longitudinal stress/axial stress will be tensile in nature
- Radial stress will be compressive in nature.
Stresses in Thin Cylindrical Shell
- Circumferential Stress /Hoop Stress
Where, p = Intensity of internal pressure
d = Diameter of the shell
t = Thickness of shell
η = Efficiency of joint
- Longitudinal Stress
- Hoop Strain
- Longitudinal Strain
- Ratio of Hoop Strain to Longitudinal Strain
- Volumetric Strain of Cylinder
Stresses in Thin Spherical Shell
- Hoop stress/longitudinal stress
- Hoop stress/longitudinal strain
- Volumetric strain of sphere
Lame’s Theory/Analysis of Thick Cylinders Lame’s theory is based on the following assumptions
Assumptions
- Homogeneous, isotropic and linearly elastic material.
- Plane section of cylinder, perpendicular to longitudinal axis remains plane.
- Hoop stress at any section
- Radial pressure
Subjected to Internal Pressure (p)
- At
- At
- At
- At
Note: Radial and hoop stresses vary hyperbolically.
You can avail of BYJU’S Exam Prep Online classroom program for all AE & JE Exams:
BYJU’S Exam Prep Online Classroom Program for AE & JE Exams (12+ Structured LIVE Courses)
You can avail of BYJU’S Exam Prep Test series specially designed for all AE & JE Exams:
BYJU’S Exam Prep Test Series AE & JE Get Unlimited Access to all (160+ Mock Tests)
Thanks
Team BYJU’S Exam Prep
Download BYJU’S Exam Prep APP, for the best Exam Preparation, Free Mock tests, Live Classes.
Comments
write a comment