Structures: Bending Flashcards
(22 cards)
What are the two types of stresses?
- Normal Stresses: stress in xx, yy, zz directions
- Shear Stresses: in xy, xz, yz directions - cause changes in the angles
Notes:
* Stresses is an internal force per unit area on a specific plane
* Stress is a point variable: defined at each point and not for the whole object
What is strain?
- Strain: displacement per unit length
In 1D: strain = change in length / length (technically just extension, not strain)
In 3D ‘small strain’ definition is shown in picture below:
Note:
* Strain is a point-variable too
What is poisson’s effect?
When a material is transversely free to deform, the ratio of negative transverse strain (i.e. negative strain in yy direction) to applied longitudinal strain (in xx direction) is equal to poissons ratio.
What is Hooke’s law in 2D and 3D?
Try name any assumptions.
In 1D: stress = strain x young’s Modulus (not considering poissons ratio)
In 3D: shown below
Assumptions:
* Linear = relationship between stress and strain is a straight line
* Elastic = returns to original shape after deformation
* Isotropic = material properties (elastic modulus ‘E’ and poissons ratio ‘v’) are the same in all directions.
What is the condition for equilibrium?
- Sum of applied external forces and moments to any object or part of an object should always be equal to zero to keep the object stationary
- Free body digrams are used to work out support reactions or internal forces.
- Use Positive shear force sign when Q at the cut surface produces a clockwise rotation of the segment.
- Use positive bending moment when at a cut face it produces sagging, or curve up. (generally positive when the moment creates a positive stress in the positive side of the plane its in)
True or False:
The netural axis is always through the beams centroid.
True: it is, however it’s angle compared to the axis will vary depending on if its an assymetric or symmetric beam.
What are the steps for finding the axial stress (xx) for an arbitrary section with moments in multiple axis (My and Mz)?
Do you know the equation and assumptions?
- Seperate the moments in y and the moments in z into two scenarios, which you can then combine via superposition.
- You must consider the curvature ratius and then using engineers bending theory, you can relate the axial strains and stresses and calculate the moments in each Mz and My direction for both seperate curvature scenarios (Ry and Rz).
- Then you can combine the two, and simplify by using moments rather than curvature radius’ (because it’s easier to find) to find the final solution for axial stresses in an arbitrary section with multiple moments.
Note:
* the equation can be simplified if the beam is symmetric as Iyz = 0 (they cancel out). This means on ly Mz, My and Iz and Iy need to be known.
* Also the equation means that the stresses won’t necessarily be a maximum on upper and lower surfaces. It now matters how far it is from the neutral axis.
How do you find the Nuetral axis line?
- Neutral axis point will always be on the centroid, which is usually when you change the coordinate system to. However it won’t always align with the axis so.
- By definition neutral axis is where the bending stress is zero, so stress (xx) is zero from bending stress equation. This leaves the equation below, and s omaximum stresses occur at furthest distances from NA.
How do you work out deflection in the y direction (v) and z direction (w)?
- Using engineers beam theory and the knowledge of the equation of the second deriviative of defelction being equal to 1/R (1/curvature radius)
- Relate the Radius of curvature for an arbitrary section to the the second deriviative of deflection.
- Then solve by integrating twice and making things one constant, and then apply BCs where v(0) = 0 and V(0)’ = 0. etc.
What are the equations for second moment of areas?
How to find the centroid and therefore the neutral axis point for complex shapes?
- Set a coordinate system (i.e. origin in the top right)
- Divide the shape into less complex subsections such as rectanbles.
- Work out the areas of each subsection and their centroid (y and z) cooridinates with respect to the chosen coordinate system.
- Finally, multiply each y coordinate with the area it associates with, add them up and divide by the total area of each section, to work out the centroid Y position relative to your chosen coordinate, this is the same process for z direaction.
What is the principle axes?
The set of mutually perpendicular axis, where the product of inertia, Iyz equals zero. This means that they are balanced and mass is evenly distributed. This means that the maximum and minimum second moments of area can also be found along these axis.
What is the equation for principle axes angle?
How do you work out the minimum and maximum second moments of area?
What equation?
What is the equation fo shear stress in a symmetric section?
We must assume:
* The shear stress in the yx and xy direction cancel, otherwise the beam rotates.
* Also on the outer edges the shear stresses must be zero, because there is no material there.
To Do:
* equate the shear force in the yx with the shear stress in that direction.
* This can then be equated to the shear stress in the xy direction.
* The equation derivation is shown below:
What are the two equations for shear stress in a rectangular cross section of a beam?
Can you derive them? What are their features?
Features:
* It is a parabolic curve that is maximum in the middle of the beams cross section
Why is bending shear stress important?
The bending shear stress and the axial stress need to be compared for the material, to know if it will fail a certain way.
In general if the length of the beam is about 10x greater than its height, the bending axial stress will be dominant stress.
What is shear flow, q?
Shear flow (in N/m units) is the shear stress per unit thickness of an arbitraty cross section, when the thin wall assumption holds true.
What is the equation for shear flow, q?
Consider how it changes for a section without a free edge?
What is the Shear Centre, SC?
The Shear Centre is the point where there will be no twisting if a load is applied through it.
How do you find the Shear Centre?
At the Shear Centre, the moments of the applied shear forces (Qy and Qz) and the moments of the shear flow in the cross section must be equal and opposite to cancel.
Therefore, the shear Centre will always lie on an axis of symmetry.
What if the load isn’t applied through the shear centre? How is shear stresses calculated?
Need to add the bending and the torsional stresses
