Statics Flashcards
(234 cards)
1
Q
What is the definition of statics?
A
Statics is the study of systems where the overall force acting is zero and systems are in equilibrium.
2
Q
Who was Archimedes and what was his contribution to statics?
A
Archimedes was a Greek mathematician and scientist who contributed to statics by explaining the principle of levers and equilibrium.
3
Q
What is the first law of motion according to Newton?
A
An object remains at rest or continues to move at a constant velocity unless acted upon by an external force.
4
Q
What are scalars?
A
Scalars are quantities that have only magnitude.
5
Q
Give examples of scalars.
A
- Mass (3 kg)
- Time (1 s)
- Volume (1.2 m³)
- Temperature (°C)
6
Q
What are vectors?
A
Vectors are quantities that have both magnitude and direction.
7
Q
Provide examples of vectors.
A
- Velocity (3 m/s)
- Force (6 N)
- Position (4 m)
8
Q
What is the formula for resolving a force vector into its components?
A
The components can be calculated as:
* (F_x = F cos heta)
* (F_y = F sin heta)
9
Q
What method can be used to add vector components?
A
Trigonometry can be used to add vector components to recreate the original vector.
10
Q
What is the moment of a force about a point?
A
The moment is the product of the force and the perpendicular distance from the line of action to the point.
11
Q
What is the formula for calculating the moment?
A
The formula is (M = Fd).
12
Q
How do levers work in terms of moments?
A
Levers work by creating moments about a pivot that oppose each other.
13
Q
What is a couple in mechanics?
A
A couple is a pair of equal and opposite forces separated by a distance, creating a pure moment.
14
Q
What is the significance of equilibrium in statics?
A
A body is in equilibrium when all forces and moments sum to zero.
15
Q
What are Newton’s three laws of motion?
A
- 1st Law: An object remains at rest or in uniform motion unless acted upon by an external force.
- 2nd Law: The sum of forces equals mass times acceleration.
- 3rd Law: For every action, there is an equal and opposite reaction.
16
Q
Fill in the blank: A body is in equilibrium when _______.
A
the sum of the forces and moments acting on the body is zero.
17
Q
What does it mean when forces form a closed loop in a force diagram?
A
It indicates that the body is in equilibrium.
18
Q
What is the component method for vector addition?
A
The component method involves breaking vectors into their components and then adding these components.
19
Q
What is the significance of direction in moments?
A
Moments can be clockwise or anticlockwise, and the direction determines the sign of the moment.
20
Q
What is meant by resolving vectors?
A
Resolving vectors means breaking a vector down into its horizontal and vertical components.
21
Q
True or False: The addition of vectors is commutative.
A
True.
22
Q
What do you need to ensure for a particle to be in equilibrium?
A
The magnitude of the forces must sum to zero.
23
Q
What is the relationship between moments in a lever system?
A
The clockwise moment is equal and opposite to the anticlockwise moment.
24
Q
What is the condition for a body to be in equilibrium?
A
All forces and moments acting on the body must sum to zero
This is expressed as ∑ F = 0 and ∑ M = 0
25
What are the two general equations for equilibrium in two dimensions?
∑ F_x = 0, ∑ F_y = 0
## Footnote
Additionally, ∑ M_z = 0 for moments
26
When considering a particle in equilibrium, what must be true about the forces?
The sum of the forces must equal zero and the forces must be concurrent
## Footnote
This means no moments can arise
27
What are the three equations used in three-dimensional equilibrium?
∑ F_x = 0, ∑ F_y = 0, ∑ F_z = 0, ∑ M_x = 0, ∑ M_y = 0, ∑ M_z = 0
## Footnote
These equations ensure equilibrium in 3D space
28
What does Newton's third law state?
For every action, there is an equal and opposite reaction
## Footnote
This applies to forces exerted between two bodies
29
What is a Free Body Diagram (FBD)?
A diagram that shows all external forces acting on a single free body
## Footnote
It helps visualize forces for analysis
30
What is the relationship between normal reaction force and frictional force?
F_friction = μR
## Footnote
Where μ is the coefficient of friction and R is the normal force
31
What is the principle of force transmissibility?
A force can be moved along its line of action without affecting equilibrium
## Footnote
This is useful in calculating moments
32
What conditions must be met for a two-force body to be in equilibrium?
The forces must be equal, opposite, and collinear
## Footnote
This ensures ∑ F = 0 and ∑ M = 0
33
What is the method of joints in truss analysis?
A method to determine forces at each joint using equilibrium conditions
## Footnote
Forces in x and y directions must sum to zero
34
What does it mean for a structure to be pin-jointed?
Joints are pinned in position but free to rotate, carrying only axial forces
## Footnote
No moments are transmitted at these joints
35
What are the types of supports commonly encountered in engineering problems?
* Built-in supports
* Fixed pin supports
* Roller or slider supports
## Footnote
Each provides specific reactions based on constraints
36
How do you calculate external forces in a truss structure?
Identify reaction forces at supports and apply equilibrium equations
## Footnote
Analyze the structure as a whole free body
37
What is the method of sections in truss analysis?
Cutting the structure into sections to analyze forces in specific members
## Footnote
Useful for finding forces in only a few members
38
What happens when a bar in a truss is loaded mid-way?
It is treated as a 3-force member for analysis
## Footnote
This is necessary to determine end reactions
39
What is the convention for tension and compression in bar forces?
Tension is positive and compression is negative
## Footnote
This helps in drawing free body diagrams
40
What is buckling in the context of structural analysis?
The tendency of struts to bend when subjected to compressive forces
## Footnote
Buckling is not considered in basic statics problems
41
What should be included in a good quality sketch for solving problems?
* Clear labeling of forces
* Dimensions
* Relevant points
* Coordinate system
## Footnote
This aids in visualizing the problem and applying equations
42
What are the necessary conditions for equilibrium in a three-force body?
The forces must form a closed force triangle and all pass through a single point
## Footnote
This ensures both force and moment conditions are satisfied
43
What is buckling in the context of structural mechanics?
The tendency of struts to bend if the line of action of the force moves outside of the material.
44
What is the first step in calculating forces within the bars connected to joints A-D in a bridge?
Draw the free body diagram for joint A.
45
What equation represents the equilibrium at joint A?
F(A) = P2 + T_A sin 45 = 0.
46
What is the value of T_A after solving the equilibrium equation at joint A?
T_A = -P/√2.
47
What is the next joint to analyze after joint A?
Joint B.
48
What equations are used to analyze joint B?
F(B) = -(-P√2) sin 45 - T_B = 0.
49
What is the value of T_B after solving the equilibrium equation at joint B?
T_B = P/2.
50
What is the method of sections in structural analysis?
A technique based on the principle that if an entire structure is in equilibrium, so are discrete parts of the same structure.
51
What is required to apply the method of sections?
Calculate the reaction forces acting on the structure.
52
What is the significance of drawing a free body diagram of the cut section in the method of sections?
It helps to represent the forces where the cut bars provide the required forces to maintain equilibrium.
53
What equilibrium equations can be used for substructures?
* Horizontal and vertical equilibrium and 1 moment equation
* Either horizontal or vertical equilibrium and 2 moment equations
* 3 moment equations
54
What is the purpose of labeling the forces in all the bars after analyzing a joint?
To keep track of all the forces acting in the structure.
55
What is a zero-force member in a structure?
A member that does not carry a load, typically found at junctions of three bars where two are collinear unless there is an external force or reaction at the joint.
56
What condition must be satisfied for bar forces to be symmetric in a pin-jointed frame?
* Symmetry of the structure
* Symmetry of the loading
57
How are externally applied moments treated in structural analysis?
They are dealt with in the same way as external forces but only contribute to moment equations.
58
What is a distributed load?
A load applied uniformly to a bar or section of bar, represented by a single force through its midpoint.
59
What is the equivalent load concept in structural analysis?
Forces at the nodes that produce the same net force and moment equations as the original loading.
60
What is the formula for calculating the reaction forces at supports given a distributed load?
R_A + R_B = total load.
61
What is the significance of calculating moments about a point in structural analysis?
It helps to eliminate certain unknown forces from the equations.
62
What is the method to find reaction forces using equilibrium?
Apply three equilibrium equations to find three unknowns (horizontal, vertical, and moment equations).
63
What is the graphical/trigonometric method used for in structural analysis?
To check the reaction forces by creating a triangle of forces and ensuring all forces pass through a single point.
64
What is the significance of using trigonometry in calculating distances in structural analysis?
To accurately determine relationships between lengths and angles in the structure.
65
What is the formula used to find forces in the force diagram?
𝑅�� = sin 6.73° ∗ 90𝑘𝑁/sin128.27°
## Footnote
This uses the sine rule to resolve forces.
66
What is the value of 𝑅�� calculated using the sine rule?
13.4𝑘𝑁
67
What is the value of 𝑅� calculated using the sine rule?
81.1𝑘𝑁
68
What is the resolved value of 𝑅� in the vertical direction?
63.6 𝑘𝑁
69
What is the resolved value of 𝑅� in the horizontal direction?
50.2 𝑘𝑁
70
What do you observe about the force in BC by inspection?
To be determined through analysis.
71
What is the method used to find reactions at supports and the force in bar EF?
Equilibrium equations
72
What equilibrium equation represents the sum of vertical forces?
𝑅�� + 𝑅�� − 2𝑃 − 2𝑃 = 0
73
What is the value of 𝑅�� calculated from the moment equation?
10𝑃
74
How can the force in EF be found?
Using the method of equilibrium at the joints or method of sections.
75
What is the method of equilibrium at the joints used for?
To find bar forces
76
What is the significance of zero-force members in a structure?
They do not carry any load and can be identified through certain conditions.
77
What is a simple supported beam?
A beam held up using simple supports at either end.
78
What is a cantilevered beam?
A beam secured at only one end by a built-in support.
79
What are the two equilibrium equations required to calculate reaction forces?
∑ 𝐹� = 0 and ∑ 𝑀 = 0
80
What are shear forces and bending moments in a beam?
Internal forces and moments developed to support and transmit loads.
81
What does a positive bending moment indicate?
The beam is sagging.
82
What does a negative bending moment indicate?
The beam is hogging.
83
What is the sign convention for shear forces?
Positive shear causes anticlockwise rotation; negative shear causes clockwise rotation.
84
What happens at a pin joint in terms of bending moments?
No moment is held, so the bending moment value is zero.
85
What occurs at the point where a force is applied in a shear force diagram?
There is a discontinuous change in the shearing force.
86
How do you handle externally applied moments in beam analysis?
Consider them in the moment equilibrium equations according to the sign convention.
87
What effect does a clockwise external moment have on the bending moment diagram?
Produces a +ve jump in the BM diagram.
88
What is the relationship between forces around an internal pin?
Forces either side of the pin must be equal and opposite.
89
What is the first step in calculating internal forces and moments in a beam?
Calculate the reaction forces.
90
What is the formula for the shear force at a specific distance?
S = -R�
91
What is the formula for the bending moment at a specific distance?
M = R�x - M
92
What is the relationship between the forces on either side of a pin?
They have to be equal and opposite.
93
How do you start calculating reaction forces?
Start with the right-hand side, as it is easiest.
94
What is the equation used to calculate the reaction force at D?
𝑀(𝐷) = 1 * 1 - 𝑅� * 2 = 0.
95
What is the value of 𝑅� at point D?
0.5 kN.
96
What is the equation for the force F at point D?
𝐹 = −𝑆� − 1 + 𝑅� = 0.
97
What is the calculated value of 𝑆�?
−0.5 kN.
98
What is the equation for the moment at point A?
𝑀(𝐴) = 0.5 * 1 - 𝑅� * 2 - (−0.5) * 3 = 0.
99
What is the value of 𝑅� at point A?
1 kN.
100
What is the equilibrium equation for the whole beam?
𝑓 = 𝑅� − 0.5 + 𝑅� − 1 + 𝑅� = 0.
101
True or False: Internal forces like SD are included in the whole structure equilibrium.
False.
102
What is the shear force at point D?
-0.5.
103
What is the bending moment at point D?
Zero.
104
What must be calculated when a load is applied through a lever arm?
Reactions at the point of contact with the beam.
105
What should be done when forces are applied at an angle?
Resolve the force into horizontal and vertical components.
106
List the steps for solving beam problems.
* Determine all reaction forces using equilibrium
* Draw FBD for each side of internal pins
* Draw FBD for sections of beams
* Write equilibrium equations to find S and M
* Draw SF and BM diagrams.
107
What is the equation for the load W along the beam?
W(x) = −dS/dx.
108
How does the shear force relate to the load function?
S(x) = −dM/dx.
109
What is the significance of the negative gradient of the shear force graph?
It is equal to the load function.
110
What is the first step in the worked example for sketching shear force and bending moment diagrams?
Label and find reaction forces.
111
What is the equation for equilibrium at point A in the worked example?
𝑓 = 𝑅� − 6 * 1 + 𝑅� − 2 = 0.
112
What is the calculated value of 𝑅� in the worked example?
6 kN.
113
Fill in the blank: The shear force at point x is ______.
S = x - 2.
114
What is the maximum point of M found using its derivative?
M=2 at x=2.
115
What equation is used to find internal forces S and M in a section of the beam?
Write two equilibrium equations.
116
What is the relationship between shear force and bending moment?
The shear force is the derivative of the bending moment.
117
What is the method of sections used for?
To find generalized equations to calculate how shear forces and bending moments change.
118
What should you be aware of regarding engineering situations?
Situations that result in bending.
119
What is the notation and sign convention in bending analysis?
Positive for tension and negative for compression.
120
What can you determine and sketch regarding beams?
Simple shear force and bending moment diagrams.
121
What external factors should you know how to deal with in beam problems?
External moments and internal pins.
122
What is the formula for direct stress?
σ = F/A, where σ is stress, F is force, and A is cross-sectional area.
123
What is Hooke's Law?
F ∝ x (or ΔL), relating force and extension in materials.
124
What does Young's modulus (E) represent?
The constant of proportionality between stress and strain.
125
What is Poisson's ratio?
The ratio of lateral strain to axial strain, symbolized as ν.
126
What is the range of Poisson's ratio for most normal materials?
0 to 0.5, with metals around 0.3.
127
What is shear stress defined as?
τ = F/A, where τ is shear stress.
128
What is the relationship between shear stress and shear strain?
τ = Gγ, where G is shear modulus and γ is shear strain.
129
What are statically determinate problems?
Problems where the majority of variables are known and do not depend on material interaction.
130
What is the procedure for tackling statically determinate problems?
* Calculate forces from equilibrium * Determine stresses * Use Hooke's law for strains * Calculate extensions if necessary * Use Poisson's ratio for transverse strains
131
What is the significance of stress-strain curves?
They illustrate the relationship between stress and strain for materials.
132
How is compressive stress defined?
The stress experienced by a material when it is being squeezed.
133
What is direct tensile strain?
The increase in length per unit length of a bar when it is stretched.
134
What is the unit of Young's modulus?
N/m² or Pa.
135
What does a high Poisson's ratio indicate about a material?
The material is very resistant to volume change.
136
What is the formula for calculating direct strain?
ε = ΔL/L, where ε is strain, ΔL is change in length, and L is original length.
137
What materials are typically compared using Young's modulus and Poisson's ratio?
* Steel: E = 210 GPa, ν = 0.29 * Aluminium: E = 69 GPa, ν = 0.34 * Concrete: E = 14 GPa, ν = 0.1 * Nylon: E = 3 GPa, ν = 0.4 * Rubber: E = 0.01 GPa, ν = 0.495
138
What is the significance of statically indeterminate problems?
They require additional information in the form of compatibility equations to solve.
139
What happens to two materials constrained to expand by the same length?
The division of force depends on their Young's moduli.
140
What is the equation relating elastic moduli and Poisson's ratio for isotropic materials?
2(1 + ν) = E/G.
141
What is shear strain represented by?
γ (gamma), defined as the shear angle.
142
How does the behavior of composite materials differ under stress?
Internal stresses can arise due to mismatched thermal expansion properties.
143
What is the significance of the gradient of the stress-strain curve?
It represents the Young's modulus of the material.
144
What is the typical behavior of materials in the linear elastic region?
It is generally small (<1% strain).
145
What is the effect of an axial tensile force on a bar?
It causes the bar to stretch and may change its cross-sectional area.
146
What does the term 'statical determinacy' refer to?
The ability to solve for forces and displacements without needing compatibility equations.
147
What are the compatibility equations for a three-material problem?
𝐿� = 𝐿� = 𝐿� = 𝐿 and Δ𝐿� = Δ𝐿� = Δ𝐿� = Δ𝐿
## Footnote
These equations indicate that the lengths and changes in length for each material are equal.
148
What happens to the bars when a force is applied if they are pinned in place?
They must all expand by the same ΔL
## Footnote
This occurs despite each bar having different strain equations.
149
What is Hooke's law equation for material stress and strain?
𝐸 = 𝜎/𝜀
## Footnote
This equation relates stress (σ) to strain (ε) through Young's modulus (E).
150
What is the diameter and length of the copper tube in the worked example?
15 mm internal diameter, 25 mm external diameter, 1.2 m length
## Footnote
This is part of a composite structure with a steel rod inside.
151
How is the change in length related to the tightening of the nut in the example?
Δ𝐿� + Δ𝐿� = 0.24 mm
## Footnote
This equation balances the changes in length of the steel and copper components.
152
What is the formula for the stress-strain equations for steel and copper?
𝐹 = 𝐹� − 𝐹� and 𝐸 = 𝜎/𝜀
## Footnote
These equations help to relate the forces and material properties.
153
What is the relationship between elongation and the stiffness of materials in a composite?
K = (E_steel * A_steel * L_steel)/(E_copper * A_copper * L_copper)
## Footnote
This ratio helps to find the change in length for each material.
154
What is Young's modulus for steel and copper in the worked example?
Esteel = 207 GPa and Ecopper = 103 GPa
## Footnote
These values are used to calculate stresses in the materials.
155
What are the key concepts learned in this section regarding material behavior?
* Direct stress and strain
* Shear stress and strain
* Young’s modulus
* Poisson’s ratio
* Shear modulus
* Compatibility equations
## Footnote
Understanding these concepts is vital for solving stress-strain problems.
156
What does the neutral axis in a beam indicate?
It is the section where stress is zero
## Footnote
The neutral axis separates the compressive and tensile stresses in a bent beam.
157
What is the second moment of area and its significance?
I = ∫ y² dA
## Footnote
This term measures the resistance of a beam to bending; larger values indicate greater resistance.
158
What happens to the stress distribution in a beam when it is bent?
Stress varies from negative (compression) to positive (tension)
## Footnote
This variation occurs across the cross-section of the beam.
159
Fill in the blank: The length of a beam segment A'1B'1 after bending can be calculated as _______.
(R + y)θ
## Footnote
This relates the geometry of bending to the change in length.
160
What does the curvature of a beam relate to?
Curvature (𝜅) = 1/R
## Footnote
The curvature is inversely proportional to the radius of curvature.
161
True or False: The stress at the neutral axis is always positive.
False
## Footnote
The stress at the neutral axis is zero, indicating no strain at that point.
162
What is the relationship between bending moment and the second moment of area?
M = E𝜅I
## Footnote
This equation shows how the bending moment is proportional to the second moment of area.
163
What happens to the length of the beam at the bottom and top when bending occurs?
At the bottom the length increases – tension. At the top the length decreases – compression.
164
Where is 𝜎 largest in a beam during bending?
At the top and bottom surfaces.
165
What is the relationship between R, I, and E when a fixed bending moment is applied?
R = I * E * M.
166
What does curvature (1/R) indicate about the bending of a beam?
The smaller R is, the greater the curve in the beam.
167
If I is large, what can be said about R and the bending of the beam?
R is large, resulting in not much bending.
168
What happens to bending when mass is distributed far from the neutral axis (NA)?
y gets large, I gets large, hard to bend, R is large (little bending).
169
What happens to bending when mass is distributed close to the neutral axis (NA)?
y stays small, I stays small, easy to bend, R is small (large bend achieved).
170
What is the second moment of area for a rectangle about its own neutral axis?
I = b * d^3 / 12.
171
What is the second moment of area for a circular cross-section about its own neutral axis?
I = π * r^4 / 4.
172
What is the combined general beam bending equation?
σ = (M * E) / (I * κ).
173
What factors increase the stress in a beam when bending occurs?
The smaller R is, the tighter the bend, thus greater stress induced.
174
What assumptions underpin the beam bending equations?
* The beam is initially straight.
* Curvature k is small (R is large).
* Plane transverse sections remain plane after bending.
* Material remains linear elastic during bending.
* Bending stress is significantly larger than all other stresses.
175
What theorem is used to calculate the second moment of area for complex shapes?
The parallel axis theorem.
176
What is the formula for the second moment of area about a different axis using the parallel axis theorem?
I' = I + A * h^2.
177
What is the significance of distributing mass away from the neutral axis in beams?
It increases a beam's resistance to bending.
178
How do you calculate the second moment of area for a right angle section?
By calculating the centroid and the second moment of areas for each part.
179
What information is needed to use the beam bending equation for complex beam sections?
* Position of the centroid for neutral axis
* IXX, the second moment of area
* Dimensions (for maximum y).
180
What is the maximum bending moment for a simply supported beam of length L centrally loaded by a force F?
M = F * L / 4.
181
What is the maximum stress in a beam given the bending moment and second moment of area?
σ = (M * y) / I.
182
What does the bending equation σ = (M * y) / I signify?
It relates moments to stresses in beams.
183
How does the second moment of area affect stress in a beam?
The larger the second moment of area, the smaller the stress in the bar will be.
184
What is the second moment of area for an I-beam calculated using the parallel axis theorem?
I = I1 + A * d^2 for each section of the I-beam.
185
Calculate the total second moment of area for a complex shape.
Sum of the second moments of each section.
186
What is the effect of a distributed load on shear forces in a beam?
Shear forces vary across the span, affecting maximum bending moments.
187
What is the significance of the neutral axis in beam bending?
It's the axis about which the beam bends and where stress is zero.
188
What is the relationship between maximum stress and beam span under a distributed load?
L can be calculated to ensure that maximum stress does not exceed a limit.
189
What are the dimensions for maximum y in beam bending calculations?
The distance from the neutral axis to the farthest edge of the beam.
190
What is the formula for calculating the maximum span of a girder under a distributed load?
L = √(8 * σ * I / (w * y)).
191
What is direct stress?
A stress whose ‘force’ line acts normal to the surface.
192
What is shear stress?
A stress whose ‘force’ line acts tangential to the surface.
193
In a 2-dimensional stress state, what forces can act on a material?
Forces acting in the x and y direction, and shear stresses acting across the x and y surfaces.
194
What condition must be met for equilibrium regarding shear stresses?
The shear stresses across the face of the x plane and the y plane must sum to zero: 𝜏�� + 𝜏�� = 0.
195
What does the equation 𝜏�� = −𝜏�� represent?
It indicates the relationship between shear stresses acting on different planes.
196
What is the superposition of stress states?
If elements of stress are aligned and the material response is linear, we can superpose the stress states and the strain fields.
197
What is the effect of a tensile force acting in the x direction on an object?
It elongates the object in the x direction while causing a decrease in length in the perpendicular directions (y and z).
198
What is the general equation for calculating strain in a particular direction?
Strain can be calculated using the summation of competing forces acting in different directions.
199
What is the definition of a thin wall in the context of vessels?
A thin wall is when the thickness of the shell is significantly less than the radius of the vessel, typically t < 1/20th r.
200
What is the longitudinal stress in a thin-walled cylinder under internal pressure?
𝜎� = 𝑝𝑟/2𝑡.
201
What is the hoop stress in a thin-walled cylinder under internal pressure?
𝜎� = 𝑝𝑟/t.
202
What is Mohr's Circle used for?
To visualize how direct and shear stress changes with angle and to find principal stresses.
203
What is the relationship between principal stresses and maximum shear stress?
The principal stress directions are the directions with no shear stress, while maximum shear stress occurs at 45 degrees to the principal stress directions.
204
What does the equation 𝜏���� = 𝜏�� sin 2𝜃 + 𝜏�� cos 2𝜃 represent?
It describes the transformation of shear stress in different coordinate systems.
205
What are the steps to draw Mohr's Circle?
1. Draw the normal stress axis horizontal and shear stress axis vertical.
2. Identify and plot points for the stress state.
3. Join points to form a circle.
206
What is the significance of the principal stress directions?
They are important for design considerations and determining yield criteria.
207
What is the equation for the spherical stress in a thin-walled sphere?
𝜎� = 𝑝𝑟/2𝑡.
208
True or False: The stress in a thin-walled sphere is the same in all directions.
True.
209
Fill in the blank: When the thickness of a shell is significantly less than the radius, it is considered a _______.
[thin wall].
210
What is the relationship between longitudinal stress and hoop stress in thin-walled cylinders?
The hoop stress is typically twice the longitudinal stress.
211
What is the method of sections?
A technique used to analyze forces and stresses by making imaginary cuts through a structure.
212
What is the equilibrium force equation for a cylinder under internal pressure?
𝑝𝜋𝑟^2 − 𝜎�2𝜋𝑟𝑡 = 0.
213
What does the term 'equibaxial' refer to in the context of spheres?
It means that the stress within the wall is the same in all directions.
214
What is the significance of the radius in Mohr's Circle?
The radius represents the maximum shear stress.
215
What is the angle to the horizontal in Mohr's Circle used for?
It is used to find the angle of rotation corresponding to the stress state.
216
What is the formula for spherical stress?
𝜎� = 𝑝𝑟2𝑡
## Footnote
This formula represents the relationship between pressure, radius, and wall thickness in spherical structures.
217
What is the minimum wall thickness for a spherical gas container with an internal radius of 1.5 m and an internal pressure of 200 kPa if the tensile stress for the material is 40 MPa?
Use the formula 𝜎� = 𝑝𝑟2𝑡 to calculate wall thickness.
218
What are the concepts to understand regarding superposition of stress?
Superposition of stress and plane stress
## Footnote
These concepts are fundamental in analyzing stress in materials.
219
What does generalized Hooke’s law help to find?
Stresses and strains in 2D and 3D
## Footnote
Generalized Hooke's law applies to isotropic materials and relates stress and strain.
220
What is Mohr’s circle used for?
To find principal stresses and maximum shear stress
## Footnote
Mohr's circle is a graphical representation of stress transformations.
221
What is the first step to sketch a rotated stress element?
Understand the stress state and apply rotation angles.
222
What is the normal stress on a plane at 60° to the x-axis if a 2D stress element has a tensile stress of 40 MPa in the x direction and a compressive stress of 20 MPa in the y direction?
Calculate using transformation equations for stress.
223
What is the area of a bar with a rectangular cross-section of 20 mm x 20 mm?
400 mm²
## Footnote
Area is calculated as length x width.
224
What normal and shear stresses occur on a plane at an angle of 30° to the transverse plane through a bar subject to a tensile load of 10 kN?
Calculate using stress transformation equations.
225
What stresses are acting on a plane element subjected to 𝜎� = 100 MPa, 𝜎� = 40 MPa, and 𝜏�� = 25 MPa at an angle of 45° to the y-axis?
Calculate using transformation equations for stress.
226
What stresses are acting on a plane element if 𝜎� = 20 MPa, 𝜎� = 0 MPa, 𝜏�� = 30 MPa at an angle of 45°?
Calculate using Mohr’s Circle.
227
What stresses are acting on a plane element if 𝜎� = 50 MPa, 𝜎� = 30 MPa, 𝜏�� = 20 MPa at an angle of 45°?
Calculate using Mohr’s Circle.
228
What stresses are acting on a plane element if 𝜎� = 40 MPa, 𝜎� = 20 MPa, 𝜏�� = 30 MPa at an angle of 30°?
Calculate using Mohr’s Circle.
229
What stresses are acting on a plane element subjected to 𝜎� = 6 MPa, 𝜎� = 0 MPa, and 𝜏�� = -4 MPa at an angle of -26.6° to the y-axis?
Calculate using transformation equations for stress.
230
What is the required wall thickness for a cylinder with an internal diameter of 0.15 m if the hoop stress is not to exceed 125 MPa with an internal pressure of 40 MPa?
Calculate using hoop stress formula.
231
What are the hoop and longitudinal stresses in a thin pipe with an inner diameter of 200 mm and wall thickness of 4 mm under an internal pressure of 2.4 MPa?
Calculate using stress formulas for thin-walled cylinders.
232
What is the maximum circumferential stress for a vertical cylinder with an inner diameter of 3 m, wall thickness of 3 mm, and height of 20 m filled with water?
Calculate using circumferential stress formula.
233
What will be the % change in the internal volume of a closed steel tube with an inner diameter of 0.1 m and wall thickness of 0.015 m under an internal pressure of 2 MPa?
Calculate using volumetric strain formula.
234
What should you do to check your answers for stress element rotations using Mohr’s Circle?
Verify calculations and results against expected outcomes.
