Supplementary equations for statically indeterminate problems

A: Cross-sectional area

E: elastic modulus

δ: axial strain

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

The abscissa is the cross-sectional position, and the ordinate is the axial force.

Plane assumption: the internal forces on the cross section are uniformly distributed

Stress = axial force/area

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.

surrender stage

Resilience stage

Reinforcement phase

necking stage

rate of reduction in area

Elongation

The stress law approximately satisfies Hooke's law

The first half is basically the same

The compressive capacity is 4-5 times the tensile capacity, and the curve is basically the same as the tensile one.

Three types of questions

Shear stress

Me=9550*P/n

Calculate the external couple first, cross-section method: the direction consistent with the external normal line is positive.

shear strain

Shear Hooke's Law

Me=∫tρda

The force is uniformly distributed and can be calculated by integral t=Me/(2πr*rδ)

Shear stresses must exist in pairs, pointing toward or away from the intersection of the two planes at the same time.

The force on the cross section that is tangent to the cross section is called shear force

The moment on the cross section that is balanced by the stressed part is called the bending moment

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 and bending moment as a function of x

Differential relations between load concentration, shear force and bending moment

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)

hooke's law

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.

σ＝My/Iz

Strength Check

Ways to increase intensity

Slender rod hooke's law

Determine conditions