MindMap Gallery Engineering Mechanics
The material mechanics part of engineering mechanics includes the geometric properties of plane figures, bending, Torsion, axial tension and compression, Shearing, extrusion, etc.
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Mechanics of Materials
Axial tension and compression
deformation
Axial deformation and axial strain
Lateral deformation and lateral strain
Poisson's Ratio and Hooke's Law
Poisson's ratio: The ratio of transverse strain to axial strain is constant μ
Hooke's law: δ=F/EA
Supplementary equations for statically indeterminate problems
Geometric relations
Hooke's theorem
A: Cross-sectional area
E: elastic modulus
δ: axial strain
Internal forces and stress
internal force
Axial force
Definition: Internal force on the cross section of the axial tension and compression rod
Calculation method: cross-section method, column equilibrium equation
Positive and negative regulations: consistent with the direction of the outer normal line
Axis force diagram
The abscissa is the cross-sectional position, and the ordinate is the axial force.
stress
cross section
Plane assumption: the internal forces on the cross section are uniformly distributed
Stress = axial force/area
oblique section
The axial force on the oblique section is along the direction of the central axis but the area becomes larger
The total stress p on the inclined section is the COSa of the stress on the cross section, and a is the angle between the cross section and the inclined section.
Material tensile and compression properties
Tensile mechanical properties
Low-carbon steel
surrender stage
The minimum value is the yield stage
For materials that do not have an obvious yield stage, the stress that produces a plastic strain of 0.2% is used as the elastic limit.
The stress remains basically unchanged and the strain increases significantly.
Resilience stage
Proportional stage
The maximum value is the proportional limit
According to Hooke's theorem
Reinforcement phase
The maximum value is the strength limit
Restore ability to resist strain
necking stage
The lateral size of a certain section becomes smaller rapidly
rate of reduction in area
Elongation
cast iron
The stress law approximately satisfies Hooke's law
Compression mechanical properties
Low-carbon steel
The first half is basically the same
In the second half, due to the increasing area, EA becomes larger and the pressure resistance becomes larger.
cast iron
The compressive capacity is 4-5 times the tensile capacity, and the curve is basically the same as the tensile one.
Strength calculation
Fmax/A<=[δ]
Three types of questions
Saint-Venant's Theorem
shear extrusion
cut
Related concepts
Definition: The two sides of a certain section of a component are deformed by relative displacement under the action of two forces of equal magnitude and opposite directions.
shear plane
Shear force
Shear stress
Shear strength conditions
extrusion
Related concepts
extrusion surface
squeezing force
extrusion stress
Extrusion strength conditions
calculate
Think of it as a uniform force
twist
Torsional deformation of circular shaft
ψ=Tl/GIp (obtained by integrating the general equation)
Note the conversion into degrees/meters
Torsional internal forces and stresses
internal force
Calculation of external couple moment
Me=9550*P/n
Torque and Torque Diagrams
Calculate the external couple first, cross-section method: the direction consistent with the external normal line is positive.
stress
Geometric relations
shear strain
Offset of right angle side γ
γ=rΨ/l
physical relationship
Shear Hooke's Law
t=Gγ
G: shear modulus
G=E/2(1 μ)
static relations
Me=∫tρda
Torsional section coefficient: Wt Polar moment of inertia: Pt
Maximum shear stress: τmax=T/Wt
Shear Stress Reciprocal Law and Shear Hooke's Law
Shear stress during torsion of thin-walled cylinder
The force is uniformly distributed and can be calculated by integral t=Me/(2πr*rδ)
Shear stress equality theorem
Shear stresses must exist in pairs, pointing toward or away from the intersection of the two planes at the same time.
Strength and stiffness of circular shaft when torsion
Strength condition
The maximum shear stress is not greater than the allowable shear stress
stiffness condition
The maximum torsion angle is not greater than the allowable torsion angle (note the unit conversion)
bending
Classification of statically determined beams
Simply supported beam
One section is a fixed hinge and one end is a movable hinge.
Outrigger
One end is a fixed hinge, one end is a free end, and the middle position is a movable hinge.
cantilever beam
One end is a fixed end and one end is a free end.
Bending internal forces and stresses
Bending related concepts
Shear force
The force on the cross section that is tangent to the cross section is called shear force
bending moment
The moment on the cross section that is balanced by the stressed part is called the bending moment
Shear force and bending moment
Calculation of shear forces and bending moments
Section method: The force on the section obtained from the equilibrium equation is the shear force. The moment obtained from the plane equilibrium equation is the bending moment
Summation method: Add up all the forces on the left half to get the interface force. Positive and negative regulations: upper left and lower right, left and right in reverse
Shear force diagram and bending moment diagram
Shear force and bending moment as a function of x
Differential relations between load concentration, shear force and bending moment
Distribution concentration is the derivative of shear force, and shear force is the derivative of bending moment.
bending normal stress
Pure bending: only bending moment and no shear force
Geometric relations
The strain of a certain layer is directly proportional to the distance between this layer and the neutral layer, and inversely proportional to the radius of curvature of the neutral layer (note the sign, positive and negative)
physical relationship
hooke's law
static relations
It can be seen from the force analysis that only the bending moment M on the cross section is not zero, and the resultant moment on the y and z axes is zero.
General calculation formula for bending stress
σ=My/Iz
The size of the normal stress is proportional to the Y-axis coordinate and the product of the total external moment, and is affected by the polar moment of inertia of the Z-axis.
Strength condition
Strength Check
Ways to increase intensity
Reasonably arrange the stress of the beam
Adopt reasonable cross-sectional shape
Use variable cross-section beams
transverse force bending
Slender rod hooke's law
Approximately regarded as pure bending
Bending deformation
Basic equation of bending deformation
Deflection: Deviation from the axial symmetry plane with respect to the Y-axis
Angle: the angle between the tangent line of the neutral layer and the X-axis
Under small deformation θ≈tanθ=dw/dx
Integral method to solve bending deformation
The second derivative is M(x)/EIz
Determine conditions
Boundary conditions
Deflection and angle of hinge support
continuous condition
The left and right turning angles are the same
Superposition method to solve bending deformation
Beam stiffness conditions
Geometric properties of plane figures
static distance and centroid
Moment of inertia and radius of inertia
moment of inertia
polar moment of inertia
radius of inertia
product of inertia
Product of inertia - the product integral of two coordinates
Translation axis theorem