Depending on the purpose of the structure and the conditions of its operation, requirements for certain properties are imposed on its material: corrosion, magnetic, heat-resistant, etc.
However, for almost all structures the most important requirement is strength.
What is meant by strength?
Strength in the broad (engineering) sense of the word is understood as the ability of a material or structural element to resist not only destruction, but also the onset of yield, loss of stability, crack propagation, etc.
In a narrower, scientific sense of the word, strength is understood not only as resistance to destruction.
In accordance with these two concepts, hypotheses are created that explain the occurrence of any limiting states of a metal or part.
There are currently many engineering theories of strength put forward (1st, 2nd, 3rd, 4th strength theories). For example, according to the 4th (energy) theory, “The plastic state (or destruction) occurs when specific energy changes in shape reaches a certain limiting value” (Huber-Mises-Genki hypothesis). Then the condition for the onset of yield will be
If as limit state any element to accept the onset of yield, then the corresponding calculation formula it will look like this
Usually they don't take 
Then 
According to almost all engineering theories of strength, the strength condition for a given type of loading will be written in the form
Does this mean that in the case of, for example
(i.e., in an engineering sense, a loss of strength has occurred) the structure collapses. Therefore, one should not equate the loss of strength in the engineering sense with the onset of destruction of the part.
Modern technical materials have a complex, heterogeneous structure. Materials are usually divided into ductile (or plastic) and brittle. Ductile fractures occur at large, and brittle fractures at relatively small deformations. Due to differences in material properties, we may receive different kinds destruction.
BASIC POINTS1. What is meant by strength?
2. What is hardness?
3. What is meant by sustainability?
4 What property of bodies is called elasticity?
5 What are the simplest types from the point of view of form? various elements designs?
6 What objects are called rods?
8. What objects are called plates and shells? What is the difference between plates and shells?
9. What bodies are called volumetric?
10. What are the main problems solved in the course on strength of materials?
11. List the main assumptions regarding the properties of structural materials that are accepted in the strength of materials.
12. What does the property of homogeneity mean?
13. What is meant by continuity?
14. Why is wood considered an anisotropic material?
15. What is the principle of independence of the action of forces?
17. Which forces are called static and which dynamic?
18. What is volumetric force, its dimension? Give examples of body forces?
22. What systems are called statically indeterminate?
23. What systems are called statically determinate?
24. Support reactions– external or internal forces?
26. What method is used to determine internal forces?
27. How many internal forces arise in the cross sections of the rod in the general case of loading? Name them.
28. By what criteria are types of rod deformation classified?
29. What cases of simple deformation do you know?
30. What is called stress at a point and what is its dimension?
31. Which stress is called normal and which is tangential?
32. What voltages are called dangerous (maximum)?
33. What is the safety factor?
34. How is the permissible stress determined?
35. What is deformation? What are the simplest deformations you know?
36. How are the concepts of “relative elongation” and “relative shift” introduced?
37. What is the calculation for rigidity?
^ STENSION AND COMPRESSION
38. What type of loading is called axial deformation?
39. What hypothesis underlies the theory of tension (compression) of straight rods and what law of stress distribution follows from it?
40. Write down the static equivalence condition for the normal force.
41. How are stresses calculated in the cross section of a rod during axial deformation?
42. How will the force in a statically determinate rod change under axial deformation if: a) the area is doubled cross section; b) replace the material from which the rod is made?
43. How will the stress in a statically determinate rod change during axial deformation if: a) the cross-sectional area is doubled; b) replace the material from which the rod is made?
44. In what parts of a stretched rod is the stress distribution not uniform?
45. What is stress concentration and how is it assessed in the elastic stage of the material?
46. Does the distribution of stresses during axial deformation depend on the method of application? external forces?
47. What is the Saint-Venant principle?
48. How is the strength condition for axial deformation written? What problems can be solved using this condition?
49. How is the elongation of a rod calculated if the normal force is constant?
50. How is the elongation of a rod calculated if the normal force changes according to a linear law?
51. How many times will the absolute elongation of a round rod, stretched by a certain force, change if its length and diameter are halved?
52. How is Hooke’s law written for tension (compression)?
53. What is the physical meaning of Young's modulus?
54. What is Poisson's ratio? Within what limits does it vary for isotropic materials?
55. Which linear tensile strain is greater: longitudinal or transverse?
56. Which of the given values of Poisson’s ratio (0.12; 0.00; 0.52; 0.35; 0.50) cannot be for an isotropic material?
57. What properties of a material are characterized by Young’s modulus and Poisson’s ratio?
^ STRESS THEORY
75. What is the state of stress at a point and how is it quantified?
76. How many significantly different components does the stress tensor have?
77. Formulate the law of pairing of tangential stresses (verbally).
78. On the faces of an elementary parallelepiped parallel to the xOz plane, show the positive directions of the stresses acting on them.
79. What stresses are called the main ones?
80. At what sites are there no shear stresses?
82. How many main areas can be drawn through a point of a deformable body, how are they oriented in relation to each other?
84. At what sites do normal stresses reach extreme values?
85. What is the relationship between the principal stresses?
86. What quantities are called invariant?
87. What is the first invariant of the stress tensor?
88. What does the stress tensor look like if the coordinate axes coincide in direction with the principal stresses?
89. What is the maximum tangential stress at a point on the body and on what areas does it act?
90. Give a classification of stress states at a point on the body.
91. On which areas of a stretched rod do the largest normal stresses occur and on which areas do the largest tangential stresses occur?
92. What stress state is called pure shear? What are the main stresses in this case and how are the main areas oriented?
93. What is the deformed state at a point on the body and how is it quantified?
94. What axes are called the main axes of deformation?
95. What does the strain tensor look like if the coordinate axes coincide in direction with the main strain axes?
98. What quantities are related by Hooke’s generalized law?
^ STRENGTH HYPOTHESES
99. Why are strength hypotheses (theories) needed?
100. What classical hypotheses of brittle fracture do you know (list)?
101. What classical hypotheses of plasticity do you know (list)?
102. What is equivalent (calculated) voltage?
103. What condition is considered dangerous according to the first strength hypothesis?
104. How is the equivalent (calculated) stress determined according to the first strength hypothesis?
105. Which condition is considered dangerous according to the II strength hypothesis?
106. How is the equivalent (calculated) stress determined according to the II strength hypothesis?
107. What condition is considered dangerous according to the III strength hypothesis?
108. How is the equivalent (calculated) stress determined according to the III strength hypothesis?
109. Which condition is considered dangerous according to the IV strength hypothesis?
110. How is the equivalent (calculated) stress determined according to the IV strength hypothesis?
TORSION
113. What type of deformation of a rod is called torsion?
114. What is called torque and how is its sign determined?
116. How is Hooke's law expressed during a shift?
117. What properties of a material are characterized by the shear modulus? What is the relationship between the elastic constants of an isotropic material?
118. According to what law are tangential stresses distributed in the cross sections of a round shaft in the region of elastic deformations?
119. How are the tangential stresses directed in relation to the vector connecting the center of gravity of the section and the point under consideration?
120. Write down the static equivalence condition for torque.
121. At what points of the cross section of a round shaft do the greatest tangential stresses occur and how are they determined?
122. What are the polar moment of inertia and the polar moment of resistance? How are they calculated and what is the dimension of these quantities?
123. How is the strength condition for a round shaft written and what problems does it allow to solve?
124. What benefits are achieved by using hollow shafts?
127. What formula is used to determine the angle of twist of a round shaft with a constant torque along the length and constant cross-sectional rigidity?
128. What value is called the torsional stiffness of the cross section and what is its dimension?
129. How is the torsional rigidity condition for a round shaft formulated?
130. What stress state occurs when a round shaft rotates? On which areas are the maximum tangential stresses and on which are the maximum normal stresses?
^ GEOMETRICAL CHARACTERISTICS OF CROSS SECTIONS OF THE ROD
132. What is the static moment of a section about a certain axis and in what units is it measured?
133. Which axis is called the central one?
134. What is the static moment about the central axis?
135. How are the concepts of axial and centrifugal moment of inertia for a plane figure and their dimensions introduced?
136. Let the moment of inertia of a figure with area A relative to the central axis x be known. How to determine the moment of inertia about an axis parallel to a given one?
137. Let the moment of inertia of a figure with area A relative to an arbitrary axis x be known. How to determine the moment of inertia about an axis parallel to a given one?
138. Relative to which of all possible parallel axes does the axial moment take on the smallest value?
139. How is the moment of inertia of a rectangle with base b and height h relative to the central axis parallel to the base calculated?
140. What is the moment of inertia of a circle with diameter D relative to the central axis?
142. How are polar and axial moments of inertia related?
143. Which axes are called the main axes of inertia?
144. Relative to which axes do axial moments reach extreme values?
145. In what case is it possible to determine the position of the main axes of inertia of a section without calculations?
^ FLAT BEND
147. What type of deformation of a rod is called bending?
148. What is a beam?
149. How is the load applied, under the influence of which the rod is in plane bending conditions?
150. What internal force factors arise in the cross sections of beams?
151. Which bend is called pure?
152. When does transverse bending occur?
153. What dependencies exist between distributed load, shear force and bending moment?
154. Why are diagrams made? shear forces and bending moments?
155. Write down the conditions of static equivalence for the bending moment and shear force.
157. What is the neutral line of the cross section of a beam?
159. What value is called the cross-sectional stiffness of a beam?
160. How do normal stresses during bending change along the height of the cross section of a beam?
161. What quantity is called the moment of resistance of a section during bending and what is its dimension?
162. What is the axial moment of resistance for beams of rectangular and circular sections?
163. How is the normal stress strength condition written for beams made of plastic materials?
164. How are the normal stress strength conditions written for beams made of brittle materials?
166. Brittle material tested for compression and obtained the ultimate strength. Is this enough to calculate a bending structure, and why?
167. How many times will the stresses and deflections of the beam increase if the load is increased by 5 times?
168. How are normal stresses distributed across the width of the beam section?
170. How are shear stresses distributed along the height of a beam of rectangular cross-section?
^ BENDING MOVEMENTS
171. What is deflection, angle of rotation?
172. How are deflection and rotation angle related to each other in any section of a beam?
173. What does approximate look like? differential equation beam bending?
174. What is the geometric meaning of the parameters v0, 0 in the universal equation of the curved axis of a beam (method of initial parameters)?
175. What are boundary conditions?
176. How are boundary conditions for a hinged support written?
177. How are boundary conditions for embedding written?
178. What technique is used to take into account a uniformly distributed load when writing the universal equation for the curved axis of a beam?
^ ENERGY METHODS FOR STATATICALLY INDETERMINATE SYSTEMS
179. State Clapeyron’s theorem.
180. Why does the factor 0.5 appear in Clapeyron’s theorem?
181. What is generalized force?
182. What is generalized movement?
183. What concepts are related to generalized force and generalized displacement?
185. How are linear and angular displacements of beams determined by Mohr’s method?
187. What techniques (methods) for calculating the Mohr integral do you know?
188. What systems are called statically indeterminate? What is the degree of static indetermination?
191. What is meant by the main system?
192. What is the physical meaning of the canonical equations of the method of forces?
193. What are the coefficients of the canonical equations of the force method and how are they determined?
197. What is characteristic of diagrams of bending moments of statically indeterminate beams?
^ COMPLEX RESISTANCE
198. What is called complex resistance (complex deformation)?
199. Which bend is called spatial (complex)?
200. How are stresses during spatial bending calculated?
201. How are stresses distributed during spatial bending?
202. What is a neutral (zero line)?
203. Write down the strength condition for spatial bending of a rod of rectangular cross-section.
205. Under what conditions is oblique bending realized?
206. How are normal stresses distributed during oblique bending?
207. How does the neutral line go during an oblique bend?
208. What is the relative position of the force and neutral lines during oblique bending?
209. Can a beam of circular cross-section experience oblique bending?
210. What is equal normal voltage at the center of gravity of the cross section during oblique bending?
211. At what points of the cross section do normal stresses during oblique bending reach their maximum values?
212. What form do the strength conditions for oblique bending have for a section of arbitrary shape?
213. What are the strength conditions for oblique bending for beams of rectangular cross-section?
214. How are displacements during oblique bending calculated?
215. What is the direction of the displacement vector during oblique bending?
216. What is the normal stress at the center of gravity of the cross section under eccentric tension (compression)?
217. How is the position of the neutral line determined during eccentric tension (compression)?
218. How does the neutral line pass if the force is applied at the boundary of the core of the section?
219. What type of section kernel does it have for a rectangle and a circle?
220. What points are dangerous under eccentric tension (compression) loading?
222. How is the condition for strength in bending with torsion of a round rod written according to the III strength hypothesis?
223. How is the condition for strength in bending with torsion of a round rod written according to the IV strength hypothesis?
^ STABILITY OF COMPRESSED RODS
224. What form of equilibrium of a structure is called stable?
225. What is critical force?
226. How is the critical force determined if the resulting stresses do not exceed the proportionality limit?
227. How will the critical force for a compressed strut change if the diameter of the strut is simultaneously increased by 2 times and the length of the strut by 4 times? Euler's formula is considered applicable.
228. How is the critical force determined if the resulting stresses go beyond the limit of proportionality?
229. What is the flexibility of a rod?
231. At what stresses do highly flexible rods lose stability? What formula is used to determine the critical force for them?
232. At what stresses do moderately flexible rods lose stability? What formula is used to determine the critical force for them?
233. Is it possible to use Euler’s formula beyond the proportionality limit of the material?
234. How is the stability condition for a compressed rod written and what problems does it allow to solve?
^ DYNAMIC TASKS
235. On what principle is the strength calculation of moving structural elements based?
236. What types of blows do you know?
237. What assumptions are made when calculating impact?
238. What is the dynamic coefficient for a longitudinal impact?
239. What is the value of the dynamic coefficient when a load falls from zero height?
240. How are stresses and displacements upon impact determined?
^ VARIABLE VOLTAGES
241. What is called fatigue?
242. What is called the endurance of a material?
243. What is a stress cycle?
244. List the main parameters of the cycle.
245. What is the cycle asymmetry coefficient?
246. Which cycle is called symmetrical (illustrate with a graph)?
247. Which cycle is called constant sign (illustrate with a graph)?
248. Which cycle is called alternating (illustrate with a graph)?
249. Which cycle is called zero (illustrate with a graph)?
252. What is a fatigue curve?
253. Draw a time diagram of a cycle with an asymmetry coefficient equal to -1.
255. What is called the endurance limit of a material?
256. Can there be a limit to endurance? equal to the limit fluidity, tensile strength?
257. What factors influence the value of the endurance limit?
258. How do the absolute dimensions of the cross-section of a part affect the value of the endurance limit?
259. How does the quality of surface treatment affect the fatigue limit of a part?
1. The main objectives of the discipline “Strength of Materials” What is meant by strength, rigidity and stability of the body?
2. What is called a rod (beam), shell (plate), massive body? What is the axis of the rod?
3. By what criteria and how are loads classified? How are concentrated forces and moments expressed, as well as the intensity of distributed force loads, and in what units are they expressed?
4. What are the main types of support fastenings? What reactions occur in them and how are they determined?
5. What are they? internal forces? What internal forces (internal force factors) can arise in the cross sections of the rods (their names and designations) and what types of deformation (loads) are associated with them?
6. What is the essence of the section method?
7. What are the sign rules for longitudinal and transverse forces, torsional and bending moments?
8. Differential relationships between shear force, bending moment and distributed load intensity.
9. What is called voltage? What are the types of stresses, their designations and dimensions?
10. Basic hypotheses and assumptions accepted in resistance
materials.
STENSION AND COMPRESSION
1. What stresses and deformations occur during tension and compression (names, designations, dimensions)?
2. How is Hooke’s law written for tension and compression? What is the modulus of elasticity?
3. What is called the transverse deformation coefficient (Poisson’s ratio) and what values does it have for various materials?
4. What is called the limit of proportionality, elastic limit, yield strength and tensile strength (tensile strength)? Their designation and dimension.
5. What is the permissible stress? How is it selected for ductile and brittle materials?
6. What is called the safety factor and what main factors does its value depend on?
7. What rod structures are called statically indeterminate? The procedure for calculating such structures.
8. Temperature stresses in statically indeterminate structures.
9. Condition of tensile and compressive strength. Types of strength calculations.
10. Condition of rigidity in tension and compression. Types of calculations for stiffness.
SHEAR AND TORSION
1. Which case of plane state voltage is called pure
2. What stresses and strains occur during shear?
3. Hooke's law for pure shear. What kind of addiction exists?
between elastic moduli of the first and second kind?
4. How are the permissible tangential stresses selected?
5. Shear strength conditions. Shear calculations
6. Under what load does a straight beam experience deformation?
torsion?
7. What stresses and deformations occur during torsion?
Name, designation, dimension.
8. What state of stress occurs at each point of the round
timber in torsion?
9. Condition for strength and torsional rigidity of a round rod
cross section. Types of calculations.
10.Statically indeterminate problems in torsion.
STRAIGHT BEND.
1. Which bend is called pure? Which bend is called straight?
2. What is the neutral layer and neutral line, and how are they located?
3. What is a line of force called?
4. How are normal stresses determined in the cross section of a beam at pure bend and how do they change along the height of the section?
5. How are normal and shear stresses determined during transverse bending?
6. What are the diagrams of normal and shear stresses during bending?
7. What beams are called statically indeterminate? What are the basic and equivalent systems?
8. What is the essence of the force method for solving statically indeterminate beams? How are canonical equations composed?
9. What beams are called continuous (multi-span)? What is the equation of the three moments?
10. Condition for bending strength. Types of calculations.
COMPLEX RESISTANCE.
1. What kind of bend is called oblique? What types of bending is it a combination of?
2. What is the position of the neutral line during oblique bending?
3. For which sections is oblique bending impossible and why?
4. Strength condition for oblique bending. Types of calculations.
5. What complex resistance is called eccentric tension or compression?
6. How is the position of the neutral line determined during eccentric tension or compression? What is the kernel of a section called?
7. Condition of strength under eccentric tension or compression. Types of calculations.
8. What state stress occurs at dangerous points of the section during bending with torsion?
9. How is the equivalent moment determined according to various theories of strength in bending with torsion of a round rod?
10. Condition for bending strength with torsion of round rods. Types of calculations.
1BASIC POINTS
1. What is meant by strength?
2. What is hardness?
3. What is meant by sustainability?
4 What property of bodies is called elasticity?
5 What are the simplest types in terms of form that various structural elements are reduced to?
6 What objects are called rods?
8. What objects are called plates and shells? What is the difference between plates and shells?
9. What bodies are called volumetric?
10. What are the main problems solved in the course on strength of materials?
11. List the main assumptions regarding the properties of structural materials that are accepted in the strength of materials.
12. What does the property of homogeneity mean?
13. What is meant by continuity?
14. Why is wood considered an anisotropic material?
15. What is the principle of independence of the action of forces?
17. Which forces are called static and which dynamic?
18. What is volumetric force, its dimension? Give examples of body forces?
22. What systems are called statically indeterminate?
23. What systems are called statically determinate?
24. Support reactions – external or internal forces?
26. What method is used to determine internal forces?
27. How many internal forces arise in the cross sections of the rod in the general case of loading? Name them.
28. By what criteria are types of rod deformation classified?
29. What cases of simple deformation do you know?
30. What is called stress at a point and what is its dimension?
31. Which stress is called normal and which is tangential?
32. What voltages are called dangerous (maximum)?
33. What is the safety factor?
34. How is the permissible stress determined?
35. What is deformation? What are the simplest deformations you know?
36. How are the concepts of “relative elongation” and “relative shift” introduced?
37. What is the calculation for rigidity?
STENSION AND COMPRESSION
38. What type of loading is called axial deformation?
39. What hypothesis underlies the theory of tension (compression) of straight rods and what law of stress distribution follows from it?
40. Write down the static equivalence condition for the normal force.
41. How are stresses calculated in the cross section of a rod during axial deformation?
42. How will the force in a statically determinate rod change during axial deformation if: a) the cross-sectional area is doubled; b) replace the material
43. How will the stress in a statically determinate rod change during axial deformation if: a) the cross-sectional area is doubled; b) replace the material from which the rod is made?
44. In what parts of a stretched rod is the stress distribution not uniform?
45. What is stress concentration and how is it assessed in the elastic stage of the material?
46. Does the distribution of stresses during axial deformation depend on the method of applying external forces?
47. What is the Saint-Venant principle?
48. How is the strength condition for axial deformation written? What problems can be solved using this condition?
49. How is the elongation of a rod calculated if the normal force is constant?
50. How is the elongation of a rod calculated if the normal force changes according to a linear law?
51. How many times will the absolute elongation of a round rod, stretched by a certain force, change if its length and diameter are halved?
52. How is Hooke’s law written for tension (compression)?
53. What is the physical meaning of Young's modulus?
54. What is Poisson's ratio? Within what limits does it vary for isotropic materials?
55. Which linear tensile strain is greater: longitudinal or transverse?
56. Which of the given values of Poisson’s ratio (0.12; 0.00; 0.52; 0.35; 0.50) cannot be for an isotropic material?
57. What properties of a material are characterized by Young’s modulus and Poisson’s ratio?
STRESS THEORY
75. What is the state of stress at a point and how is it quantified?
76. How many significantly different components does the stress tensor have?
77. Formulate the law of pairing of tangential stresses (verbally).
78. On the faces of an elementary parallelepiped parallel to the xOz plane, show the positive directions of the stresses acting on them.
79. What stresses are called the main ones?
80. At what sites are there no shear stresses?
82. How many main areas can be drawn through a point of a deformable body, how are they oriented in relation to each other?
84. At what sites do normal stresses reach extreme values?
85. What is the relationship between the principal stresses?
86. What quantities are called invariant?
87. What is the first invariant of the stress tensor?
88. What does the stress tensor look like if the coordinate axes coincide in direction with the principal stresses?
89. What is the maximum tangential stress at a point on the body and on what areas does it act?
90. Give a classification of stress states at a point on the body.
91. On which areas of a stretched rod do the largest normal stresses occur and on which areas do the largest tangential stresses occur?
92. What stress state is called pure shear? What are the main stresses in this case and how are the main areas oriented?
93. What is the deformed state at a point on the body and how is it quantified?
94. What axes are called the main axes of deformation?
95. What does the strain tensor look like if the coordinate axes coincide in direction with the main strain axes?
98. What quantities are related by Hooke’s generalized law?
STRENGTH HYPOTHESES
99. Why are strength hypotheses (theories) needed?
100. What classical hypotheses of brittle fracture do you know (list)?
101. What classical hypotheses of plasticity do you know (list)?
102. What is equivalent (calculated) voltage?
103. What condition is considered dangerous according to the first strength hypothesis?
104. How is the equivalent (calculated) stress determined according to the first strength hypothesis?
105. Which condition is considered dangerous according to the II strength hypothesis?
106. How is the equivalent (calculated) stress determined according to the II strength hypothesis?
107. What condition is considered dangerous according to the III strength hypothesis?
108. How is the equivalent (calculated) stress determined according to the III strength hypothesis?
109. Which condition is considered dangerous according to the IV strength hypothesis?
110. How is the equivalent (calculated) stress determined according to the IV strength hypothesis?
TORSION
113. What type of deformation of a rod is called torsion?
114. What is called torque and how is its sign determined?
116. How is Hooke's law expressed during a shift?
117. What properties of a material are characterized by the shear modulus? What is the relationship between the elastic constants of an isotropic material?
118. According to what law are tangential stresses distributed in the cross sections of a round shaft in the region of elastic deformations?
119. How are the tangential stresses directed in relation to the vector connecting the center of gravity of the section and the point under consideration?
120. Write down the static equivalence condition for torque.
121. At what points of the cross section of a round shaft do the greatest tangential stresses occur and how are they determined?
122. What are the polar moment of inertia and the polar moment of resistance? How are they calculated and what is the dimension of these quantities?
123. How is the strength condition for a round shaft written and what problems does it allow to solve?
124. What benefits are achieved by using hollow shafts?
127. What formula is used to determine the angle of twist of a round shaft with a constant torque along the length and constant cross-sectional rigidity?
128. What value is called the torsional stiffness of the cross section and what is its dimension?
129. How is the torsional rigidity condition for a round shaft formulated?
130. What stress state occurs when a round shaft rotates? On which areas are the maximum tangential stresses and on which are the maximum normal stresses?
GEOMETRICAL CHARACTERISTICS OF CROSS SECTIONS OF THE ROD
132. What is the static moment of a section about a certain axis and in what units is it measured?
133. Which axis is called the central one?
134. What is the static moment about the central axis?
135. How are the concepts of axial and centrifugal moment of inertia for a plane figure and their dimensions introduced?
136. Let the moment of inertia of a figure with area A relative to the central axis x be known. How to determine the moment of inertia about an axis parallel to a given one?
137. Let the moment of inertia of a figure with area A relative to an arbitrary axis x be known. How to determine the moment of inertia about an axis parallel to a given one?
138. Relative to which of all possible parallel axes does the axial moment take on the smallest value?
139. How is the moment of inertia of a rectangle with base b and height h relative to the central axis parallel to the base calculated?
140. What is the moment of inertia of a circle with diameter D relative to the central axis?
142. How are polar and axial moments of inertia related?
143. Which axes are called the main axes of inertia?
144. Relative to which axes do axial moments reach extreme values?
145. In what case is it possible to determine the position of the main axes of inertia of a section without calculations?
FLAT BEND
147. What type of deformation of a rod is called bending?
148. What is a beam?
149. How is the load applied, under the influence of which the rod is in plane bending conditions?
150. What internal force factors arise in the cross sections of beams?
151. Which bend is called pure?
152. When does transverse bending occur?
153. What are the relationships between distributed load, shear force and bending moment?
154. Why are diagrams of shear forces and bending moments constructed?
155. Write down the conditions of static equivalence for the bending moment and shear force.
157. What is the neutral line of the cross section of a beam?
159. What value is called the cross-sectional stiffness of a beam?
160. How do normal stresses during bending change along the height of the cross section of a beam?
161. What quantity is called the moment of resistance of a section during bending and what is its dimension?
162. What is the axial moment of resistance for beams of rectangular and circular sections?
163. How is the normal stress strength condition written for beams made of plastic materials?
164. How are the normal stress strength conditions written for beams made of brittle materials?
166. A brittle material was tested in compression and the ultimate strength was obtained. Is this enough to calculate a bending structure, and why?
167. How many times will the stresses and deflections of the beam increase if the load is increased by 5 times?
168. How are normal stresses distributed across the width of the beam section?
170. How are shear stresses distributed along the height of a beam of rectangular cross-section?
BENDING MOVEMENTS
171. What is deflection, angle of rotation?
172. How are deflection and rotation angle related to each other in any section of a beam?
173. What is the form of the approximate differential equation for bending beams?
174. What is the geometric meaning of the parameters v0, 0 in the universal equation of the curved axis of a beam (method of initial parameters)?
175. What are boundary conditions?
176. How are boundary conditions for a hinged support written?
177. How are boundary conditions for embedding written?
178. What technique is used to take into account a uniformly distributed load when writing the universal equation for the curved axis of a beam?
ENERGY METHODS FOR STATATICALLY INDETERMINATE SYSTEMS
179. State Clapeyron’s theorem.
180. Why does the factor 0.5 appear in Clapeyron’s theorem?
181. What is generalized force?
182. What is generalized movement?
183. What concepts are related to generalized force and generalized displacement?
185. How are linear and angular displacements of beams determined by Mohr’s method?
187. What techniques (methods) for calculating the Mohr integral do you know?
188. What systems are called statically indeterminate? What is the degree of static indetermination?
191. What is meant by the main system?
192. What is the physical meaning of the canonical equations of the method of forces?
193. What are the coefficients of the canonical equations of the force method and how are they determined?
197. What is characteristic of diagrams of bending moments of statically indeterminate beams?
COMPLEX RESISTANCE
198. What is called complex resistance (complex deformation)?
199. Which bend is called spatial (complex)?
200. How are stresses during spatial bending calculated?
201. How are stresses distributed during spatial bending?
202. What is a neutral (zero line)?
203. Write down the strength condition for spatial bending of a rod of rectangular cross-section.
205. Under what conditions is oblique bending realized?
206. How are normal stresses distributed during oblique bending?
207. How does the neutral line go during an oblique bend?
208. What is the relative position of the force and neutral lines during oblique bending?
209. Can a beam of circular cross-section experience oblique bending?
210. What is the normal stress at the center of gravity of the cross section during oblique bending?
211. At what points of the cross section do normal stresses during oblique bending reach their maximum values?
212. What form do the strength conditions for oblique bending have for a section of arbitrary shape?
213. What are the strength conditions for oblique bending for beams of rectangular cross-section?
214. How are displacements during oblique bending calculated?
215. What is the direction of the displacement vector during oblique bending?
216. What is the normal stress at the center of gravity of the cross section under eccentric tension (compression)?
217. How is the position of the neutral line determined during eccentric tension (compression)?
218. How does the neutral line pass if the force is applied at the boundary of the core of the section?
219. What type of section kernel does it have for a rectangle and a circle?
220. What points are dangerous under eccentric tension (compression) loading?
222. How is the condition for strength in bending with torsion of a round rod written according to the III strength hypothesis?
223. How is the condition for strength in bending with torsion of a round rod written according to the IV strength hypothesis?
STABILITY OF COMPRESSED RODS
224. What form of equilibrium of a structure is called stable?
225. What is critical force?
226. How is the critical force determined if the resulting stresses do not exceed the proportionality limit?
227. How will the critical force for a compressed strut change if the diameter of the strut is simultaneously increased by 2 times and the length of the strut by 4 times? Euler's formula is considered applicable.
228. How is the critical force determined if the resulting stresses go beyond the proportionality limit?
229. What is the flexibility of a rod?
231. At what stresses do highly flexible rods lose stability? What formula is used to determine the critical force for them?
232. At what stresses do moderately flexible rods lose stability? What formula is used to determine the critical force for them?
233. Is it possible to use Euler’s formula beyond the proportionality limit of the material?
234. How is the stability condition for a compressed rod written and what problems does it allow to solve?
DYNAMIC TASKS
235. On what principle is the strength calculation of moving structural elements based?
236. What types of blows do you know?
237. What assumptions are made when calculating impact?
238. What is the dynamic coefficient for a longitudinal impact?
239. What is the value of the dynamic coefficient when a load falls from zero height?
240. How are stresses and displacements upon impact determined?
VARIABLE VOLTAGES
241. What is called fatigue?
242. What is called the endurance of a material?
243. What is a stress cycle?
244. List the main parameters of the cycle.
245. What is the cycle asymmetry coefficient?
246. Which cycle is called symmetrical (illustrate with a graph)?
247. Which cycle is called constant sign (illustrate with a graph)?
248. Which cycle is called alternating (illustrate with a graph)?
249. Which cycle is called zero (illustrate with a graph)?
252. What is a fatigue curve?
253. Draw a time diagram of a cycle with an asymmetry coefficient equal to -1.
255. What is called the endurance limit of a material?
256. Can the endurance limit be equal to the yield limit, the tensile strength?
257. What factors influence the value of the endurance limit?
258. How do the absolute dimensions of the cross-section of a part affect the value of the endurance limit?
259. How does the quality of surface treatment affect the fatigue limit of a part?
Strength of materials the science of strength, rigidity and reliability of engineering structural elements. Using the methods of strength of materials, practical calculations are carried out and the necessary, as they say, reliable dimensions of machine parts are determined, various designs and structures.
Basic concepts of strength of materials are based on laws and theorems general mechanics and first of all on the laws of statics, without knowledge of which the study of this subject becomes almost impossible.
Unlike theoretical mechanics Strength of Materials considers problems where the properties of deformable bodies are the most important, and the laws of motion of the body as a rigid whole not only recede into the background, but in some cases are simply unimportant.
Strength of materials aims to create practically acceptable simple methods for calculating typical, most frequently encountered structural elements. The need to bring the solution of each practical problem to a certain numerical result forces in a number of cases to resort to simplifying hypotheses - assumptions that are further justified by comparing calculated data with experiment.
It should be noted that the first notes on strength are mentioned in the notes of the famous artist LEONARDO De VINCI, and the beginning of the science of strength of materials is associated with the name of the famous physicist, mathematician and astronomer GALILEO GALILEO. In 1660, R.GUK formulated a law establishing the relationship between load and deformation: “ What is the force such is the action" In the 18th century, it is necessary to note the work of L. EULER on the stability of structures. XIX XX centuries are the time of the most intensive development of science in connection with the general rapid growth construction and industrial production with the undoubtedly enormous contribution of Russian mechanical scientists.
So we'll do solid deformed bodies with the study of their physical properties.
Let us introduce the basic concepts adopted when studying the discipline.
Strength This is the ability of a structure to withstand a given load without collapsing.
Rigidity the ability of the structure to deform in accordance with the specified regulatory regulations.
Deformation the ability of a structure to change its geometric dimensions and shape under the influence of external forces
Sustainability the property of a structure to maintain a given form of equilibrium under the action of external forces.
Reliability property of a design to perform specified functions, keeping your performance indicators within certain regulatory limits for the required period of time.
Resource permissible service life of the product. Indicated in the form of total operating time or the number of loading cycles of the structure.
Refusal disruption of the structure.
Based on the above, we can give a definition of strength reliability.
Strength reliability called the absence of failures associated with destruction or unacceptable deformations of structural elements.
Figure 1 shows the structure of the strength reliability model. It includes famous models or restrictions that are a priori imposed on the properties of materials, geometry, product shapes, loading methods, as well as the fracture model. Engineering models of continuum consider the material as a continuous and homogeneous body endowed with the property of homogeneous structure. The material model is endowed with the properties of elasticity, plasticity and creep.
Fig.1. Structure of the strength reliability model of structural elements
Elasticity called the property of the body to restore its shape after removal external loads.
Plasticity is the property of a body to retain, after the load has ceased, or the deformation partially obtained during loading.
Creep is the property of a body to increase deformation under constant external loads.
The main shape models in strength reliability models, as is known, are: rods, plates, shells and spatial bodies (arrays), Fig. 2. Models
Fig.2. Basic shape models in strength reliability models: a) rod, b) plate, c) shell
loads contain a schematization of external loads by magnitude, nature of distribution (concentrated or distributed force or moment), as well as the influence of external fields and environments.
External forces acting on a structural element are divided into 3 groups: 1) concentrated forces, 2) distributed forces, 3) volumetric or mass forces.
Concentrated forces forces acting on small areas of the surface of a part (for example, the pressure of a ball bearing on a shaft, the pressure of a wheel on rails, etc.)
Distributed forces are applied to significant areas of the surface (for example, steam pressure in a steam line, pipeline, boiler, air pressure on an airplane wing, etc.
Volumetric or mass forces applied to each particle of material (for example, gravity, inertial forces)
After a reasonable choice of models of shape, material, and loading, they proceed to direct assessment of reliability using fracture models. Fracture models are equations that connect the performance parameters of a structural element at the moment of destruction with the parameters that ensure strength. These equations (conditions) are called strength conditions. Four failure models are usually considered depending on loading conditions:
- static destruction,
- long-term static destruction,
- low-cycle static destruction,
- fatigue failure.
With a small number of cycles (N<10 2) развиваются значительные пластические деформации (статическое разрушение), при большом числе циклов (N>10 5) there are no plastic deformations (fatigue failure). In the intermediate region (10 2
As already noted, studying the discipline is impossible without knowledge of the fundamentals of theoretical mechanics. Therefore, I recommend checking your remaining knowledge resource in the “Statics” section using the system of entrance tests.
Since the study of the resistance of materials is based primarily on such well-known concepts as force, a pair of forces, bonds, reactions in bonds, the resultant system of external forces, then
