Mechanics of Materials: Stresses and Compression

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Questions and Answers

What is the ratio of the failure stress to the working stress called?

Stress ratio

Ratio of safety

Factor of safety (correct)

Factor of design

What condition must be satisfied to avoid failure?

Maximum induced stress ≤ Allowable stress (correct)

Average induced stress = Allowable stress

Minimum induced stress ≥ Allowable stress

Induced stress > Allowable stress

How is the allowable stress defined for ductile materials with a well-defined yield stress?

Yield stress / Factor of safety (correct)

Ultimate stress / Factor of safety

Proof stress / Factor of safety

Working stress / Factor of safety

What type of stresses are tensile and compressive stresses referred to as?

Direct or Normal stresses Signup and view all the answers

What law describes the linear relationship between stress and strain within the elastic region?

Hooke's law Signup and view all the answers

What is the direction of compressive stresses in a bar?

Towards the bar Signup and view all the answers

What determines the maximum stress in a bar with concentrated loads or changes in shape?

The bar's geometry and the type of discontinuity Signup and view all the answers

What is the purpose of determining the maximum permissible stress?

To ensure safe design and avoid failure Signup and view all the answers

What is the purpose of stress concentration factor K?

To determine the maximum stress Signup and view all the answers

How is the allowable stress defined for brittle materials?

Ultimate stress / Factor of safety Signup and view all the answers

What is considered when determining the working stress?

Both social and economic consequences of failure Signup and view all the answers

What is the definition of tensile stress?

A stress that stretches the material Signup and view all the answers

What is the unit of measurement for stress?

Pa Signup and view all the answers

For ductile materials with no well-defined yield stress, how is the allowable stress defined?

Proof stress / Factor of safety Signup and view all the answers

What is the significance of the average stress equation?

It gives the average normal stress acting on the cross-sectional area Signup and view all the answers

What is the significance of sign convention in stress?

Tensile stresses are positive, compressive stresses are negative Signup and view all the answers

What is the purpose of solving equations (1) and (3) simultaneously?

To find the reactions at the ends of the bar Signup and view all the answers

What is the relationship between the cross-sectional areas from A to C and from C to D?

The cross-sectional area from A to C is half the cross-sectional area from C to D Signup and view all the answers

What can be found directly from the internal axial forces?

The stresses in the two segments of the bar Signup and view all the answers

What type of structures involves setting up and solving equations of equilibrium and equations of compatibility?

Indeterminate structures Signup and view all the answers

What is the formula for δAB in terms of δAC and δCB?

δAB = δAC + δCB Signup and view all the answers

What is the formula for RA in terms of Pb and L?

RA = Pb / L Signup and view all the answers

What is a condition for using the equation δ= ∑(Pi Li) / (Ei Ai) for a bar?

The bar consists of several prismatic segments with different materials. Signup and view all the answers

What is the meaning of Pi in the equation δ= ∑(Pi Li) / (Ei Ai)?

Internal axial force in segment i Signup and view all the answers

Why is the equation δ= ∑(Pi Li) / (Ei Ai) not suitable for a tapering bar?

The bar has a continuous axial force and cross-sectional area. Signup and view all the answers

What is the purpose of considering a differential element of the bar in finding the elongation?

To find the elongation of the bar when the axial force or cross-sectional area varies continuously Signup and view all the answers

What type of load can produce a continuous axial force along a bar?

Centrifugal forces, magnetic forces or the weight of a bar hanging vertically Signup and view all the answers

How can the internal axial force P(x) acting at a cross-section C be determined?

By using static equilibrium using part of the bar AC or CB as a free body Signup and view all the answers

What is the relationship between the cross-sectional area A(x) and the distance x from the left-hand end of the bar?

A(x) can be expressed as a function of x. Signup and view all the answers

What is the significance of the subscript i in the equation δ= ∑(Pi Li) / (Ei Ai)?

It represents the numbering index for the various segments of the bar. Signup and view all the answers

What is the formula for the reaction RA at support A due to load P?

b1 A2 P / (b1 A2 + b2 A1) Signup and view all the answers

What is the formula for the downward displacement δC of point C?

P / (E (b1 A2 + b2 A1)) Signup and view all the answers

What is the formula for the strain energy U of the bar in problem 2.4?

23P 2 L / 12EA Signup and view all the answers

If P = 21 kN, L = 1.2 m, A = 1,780 mm 2, and the material is aluminium with E = 72 GPa, what is the strain energy U?

8 kJ Signup and view all the answers

What is the strain energy in a bar suspended from one end if it supports a load P at the lower end?

P 2 L / 8EA Signup and view all the answers

What is the strain energy U1 of the truss when the load P1 acts alone (P2 = 0)?

P1 2 L / 12EA Signup and view all the answers

What is the strain energy U2 when the load P2 acts alone (P1 = 0)?

P2 2 L / 12EA Signup and view all the answers

What is the strain energy U when both loads act simultaneously?

U1 + U2 Signup and view all the answers

Study Notes

Types of Stresses

Compression occurs when forces are directed towards the bar, resulting in compressive stresses.
Tension occurs when forces are directed away from the bar, resulting in tensile stresses.
Direct or normal stresses act perpendicular to the cut surface.

Stress Calculation

The stress equation gives the average normal stress acting on the cross-sectional area.
The maximum stress may vary depending on the bar's geometry and type of discontinuity.
Stress concentration factor K can be used to determine the maximum stress.

Factor of Safety

The factor of safety is defined as the ratio between the failure stress and the working stress.
To avoid failure, the maximum induced stress must be less than or equal to the allowable stress.

Allowable Stress

For ductile materials, the allowable stress is often defined as the yield stress divided by the factor of safety.
For ductile materials with no well-defined yield stress, the allowable stress is defined as the proof stress divided by the factor of safety.
For brittle materials, the allowable stress is defined as the ultimate stress divided by the factor of safety.

Relationship between Stress and Strain

Within the elastic region, there is a linear relationship between stress and strain, as described by Hooke's Law.
The change in length of a bar can be calculated using the equation δ = ∑(Pi Li) / (EAi).

Analysis of Tapered Bars

A tapered bar subjected to a continuously distributed load can be analyzed by considering a differential element of the bar.
The internal axial force P(x) and cross-sectional area A(x) can be expressed as functions of x.
The elongation of the bar can be found by integrating the expression for the elongation of the differential element over the length of the bar.

Analysis of Structurally Indeterminate Structures

The analysis involves setting up and solving equations of equilibrium and equations of compatibility.
The reactions, forces, and displacement equations can be found by solving the equations simultaneously.

Strain Energy

The strain energy of a bar can be calculated using the equation U = (P^2 L) / (2EA).
The strain energy of a bar suspended from one end and supporting a load P at the lower end can be calculated using a similar equation.
The strain energy of a truss can be calculated by considering the strain energy of each bar separately.

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Description

This quiz covers the concept of compressive stresses in a bar when forces are reversed in direction at each end, resulting in a state of compression. It discusses normal stresses, including tensile and compressive stresses.