MECHANICS OF MATERIALS - UPT OF MATERIALS 1. What represent the strength modulus? Explain for a rectangular, respectively I cross section. The strength modulus is a geometrical characteristic

MECHANICS OF MATERIALS

1. What represent the strength modulus? Explain for a rectangular, respectively I cross section.

The strength modulus is a geometrical characteristic of the cross section, about a principal axis:

,

2. Define the 2 specific deformations. Sign conventions.

The elongation represents the total lengthening of the unit length:

l

lx

The specific elongation εx is a non-dimensional notion and the above relation is valid only if Δl

is uniformly distributed along the entire length l, so only if εx = const. Otherwise εx , εy and εz are

expressed by Cauchy’s relations:

x

u

x

, y

v

y

, z

w

z

The sliding is defined as the modification of the initial straight angle. The sliding (shear strain) is

positive if decreases the initial straight angle (it becomes an acute angle), respectively negative if

it increases this (becomes an obtuse angle).

The sliding is expressed by the other 3 Cauchy’s relations:

z

u

x

wxz

,

xy

u v

y x

, zy

v w

z y

3. The unit stresses from a point of a cross section. Represent them on a figure.

The unit stress from a point of a cross section is decomposed into 2 components:

- a normal component , called normal unit stress or direct stress

- a tangent component , called tangential unit stress or shear stress

4. Define the axial force, bending moment, shear force and torsional moment, from a strength

calculation (from interior). The relations will be accompanied by figures.

The axial force: ∫

The bending moments: ∫

; ∫

The shear forces: ∫

; ∫

The torsional moment (torque): ∫ –

5. Write Navier’s formula for one of the 2 cross sections from figure: a section with minimum a

symmetry axis and a non-symmetrical section. Explain the factors from formula.

a.1)

My : the bending moment about the neutral axis Gy

Iy : the second moment of area about the neutral axis Gy

z: the coordinate about the centroid G

a.2)

My : the bending moment about the principal axis Gy

Mz : the bending moment about the principal axis Gz

Iy : the second moment of area about the principal axis Gy

Iz : the second moment of area about the principal axis Gz

z : the coordinate of the current point in the principal system of

axis about the centroid G

y : the coordinate of the current point in the principal system of

axis about the centroid G

6. Define Juravski’s formula, explaining the factors from formula, and represent the shear

stresses diagrams for one of the cross section subjected by the shear force from figure. Indicate

(graphically) which is the area used to write the static moment to compute the shear stress τx in

points K, respectively L.

a.) and b.)

c.) and d.)

Tz and Ty: the shear forces about the principal axis Gz, respectively Gy

Sy(z) and Sz(y): the first moment of the area which tends to slide about the principal axis Gy,

respectively Gz

bz and by: the cross section width (thickness) at the calculus level z, respectively y

Iy and Iz: the second moment of area about the principal axis Gy, respectively Gz

7. Write the relation to compute the normal stress σx for one of the cross section from figure.

Explain the terms from relation. Represent in cross section σx diagram (eventually diagrams),

indicating the extreme points subjected to compression, respectively tension.

N: the axial force from section

A: the cross section area

My : the bending moment about the principal axis Gy

Mz : the bending moment about the principal axis Gz

Iy : the second moment of area about the principal axis Gy

Iz : the second moment of area about the principal axis Gz

z : the coordinate of the current point in the principal system of axis about the centroid G

y : the coordinate of the current point in the principal system of axis about the centroid G

8. What is the neutral axis? Indicate the neutral axis (n.a.) and the normal stress diagram for one

of the cross section from figure.

The neutral axis is the straight line in which the normal (direct) stress is 0 ( the intersection

between the neutral longitudinal strip and the cross section)

9. Define the relation of the shear stress in case of pure torsion. Explain the terms from relation

for 2 types of sections (simple connex section and double connex section).

Mt : the torsional moment from section

Wt : the strength modulus in torsion, which is:

- for simple connex sections:

It : the second moment of area in torsion, which for simple connex sections is:

- for double connex sections:

Ω: is the area of the surface limited by the median line of the section

10. For one of the cross section subjected to eccentric compression by the axial force N acting

like in figure, represent graphically the central core. Which is the limit condition which must be

written to have in section only compressive unit stresses? Represent σx diagram for this situation,

specifying the design relation of the normal stress σx.

To have only compressive stresses in section:

- for a.), b.) and e.) : e = yv1

- for c.) and d.) : e = zv1

The unit stresses are computed with the relations:

- for a.), b.) and e.) :

- for c.) and d.) :

11. The types of plane problems of elasticity. Explain.

The types of plane problems of elasticity:

- problems of plane state of stresses, characteristic to lamellas: the gusset of a truss, the plate

with a circular hole, the wall beam of infinite height

- problems of plane state of deformations, characteristic to massives: the dams, the foundation

soil under a continuous foundation

12. How many unit stresses (on unit width) characterize a slab? Enumerate and explain them as

resultants of the unit stresses σ and τ, isolating a corner from a slab.

A slab is characterized by 5 unit stresses (on unit width):

- 2 bending moments mx şi mz

- 1 torsional moment mxz

- 2 shear forces tx şi tz