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Resistance of Materials Exercises Solved with 7 Russian, Hibeler, Singer, and Mosto Techniques: A Practical Guide

Resistance of materials is a branch of mechanics that studies the behavior of solid bodies under the action of external forces. It is also known as strength of materials or mechanics of materials. Resistance of materials is important for designing and analyzing structures such as buildings, bridges, machines, vehicles, and more.

One of the main objectives of resistance of materials is to determine the internal stresses and strains that occur in a body when it is subjected to external loads. These stresses and strains can affect the strength, stability, and deformation of the body. To calculate them, we need to apply some principles and methods that are based on the properties and geometry of the material and the shape of the body.

In this article, we will show you how to solve some resistance of materials problems using 7 different techniques that are derived from the works of 7 Russian authors (Timoshenko, Popov, Gere, Ugural, Shames, Beer, and Johnston), as well as Hibeler, Singer, and Mosto. These techniques are widely used and recognized in the field of resistance of materials and can help you understand and apply the concepts and formulas involved.

What are the 7 Russian techniques for resistance of materials?

The 7 Russian techniques for resistance of materials are:

• The method of sections: This technique consists of dividing a body into two parts by a section plane and applying the equilibrium equations to one of the parts. This allows us to find the internal forces (normal force, shear force, and bending moment) at any point along the section plane.
• The method of superposition: This technique consists of decomposing a complex load into simpler components (such as concentrated forces, distributed forces, moments, etc.) and finding the internal forces and displacements due to each component separately. Then, we can add them up to obtain the total internal forces and displacements.
• The method of virtual work: This technique consists of applying a small virtual displacement to a body and equating the virtual work done by the external forces to the virtual work done by the internal forces. This allows us to find the displacements or rotations at any point or section of the body.
• The method of conjugate beam: This technique consists of transforming a beam with variable cross-section or support conditions into an equivalent beam with constant cross-section and simple supports. Then, we can apply the formulas for deflection and slope for a simple beam to find the deflection and slope at any point or section of the original beam.
• The method of strain energy: This technique consists of calculating the strain energy stored in a body due to external loads and equating it to the work done by the external loads. This allows us to find the displacements or rotations at any point or section of the body.
• The method of Castigliano’s theorem: This technique consists of applying Castigliano’s theorem, which states that the partial derivative of the strain energy with respect to an external load gives the displacement or rotation in the direction of that load. This allows us to find the displacements or rotations at any point or section of the body.
• The method of Mohr’s circle: This technique consists of using Mohr’s circle, which is a graphical representation of the state of stress at a point in a body. It shows the relationship between normal stress, shear stress, principal stress, maximum shear stress, angle of inclination, and angle of rotation. This allows us to find any stress component or angle at any point in a body.

These 7 Russian techniques are useful and powerful tools for solving resistance of materials problems. However, they are not always applicable or convenient for every situation. Therefore, we will also show you how to use other techniques that are derived from the works of Hibeler, Singer, and Mosto.

What are the Hibeler, Singer, and Mosto techniques for resistance of materials?

The Hibeler, Singer, and Mosto techniques for resistance of materials are:

• The method of integration: This technique consists of using the differential equations of equilibrium and compatibility to find the internal forces and displacements in a body. It involves integrating the equations with respect to the coordinate axes and applying the boundary conditions to find the constants of integration.
• The method of singularity functions: This technique consists of using singularity functions, which are mathematical functions that have discontinuities or singularities at certain points. They can represent concentrated forces, distributed forces, moments, or other types of loads. By using singularity functions, we can simplify the expressions for internal forces and displacements and avoid piecewise integration.
• The method of influence lines: This technique consists of using influence lines, which are graphs that show the variation of a response (such as internal force, displacement, or reaction) at a specific point or section of a body due to a unit load moving along the body. By using influence lines, we can find the maximum or minimum values of the response for any position or combination of loads.
• The method of elastic curve: This technique consists of using the elastic curve, which is the shape of the deformed body due to external loads. It shows the relationship between deflection, slope, bending moment, and curvature. By using the elastic curve, we can find the deflection and slope at any point or section of a body.
• The method of column buckling: This technique consists of using the Euler’s formula or other empirical formulas to find the critical load that causes a slender column to buckle or lose its stability. It involves finding the effective length and slenderness ratio of the column and applying the appropriate end conditions.

These Hibeler, Singer, and Mosto techniques are also useful and powerful tools for solving resistance of materials problems. However, they are not always applicable or convenient for every situation either. Therefore, we will show you how to use a combination of different techniques to solve some examples of resistance of materials problems.

How to solve resistance of materials problems using different techniques?

To solve resistance of materials problems using different techniques, we need to follow some steps that are common to most problems. These steps are:

1. Draw a free body diagram of the body or the part of the body that we are interested in. This involves identifying and labeling the external forces, reactions, dimensions, and coordinates.
2. Apply the equilibrium equations to find the unknown reactions or internal forces. This involves summing up the forces and moments in the horizontal, vertical, and rotational directions and setting them equal to zero.
3. Apply the compatibility equations to find the unknown displacements or rotations. This involves relating the displacements or rotations of different points or sections of the body using geometry or kinematics.
4. Apply the constitutive equations to find the unknown stresses or strains. This involves relating the stresses or strains to the internal forces or displacements using the material properties such as modulus of elasticity, shear modulus, Poisson’s ratio, etc.
5. Apply the failure criteria to find the safety factor or the allowable load. This involves comparing the actual stress or strain to the allowable stress or strain using a suitable failure theory such as maximum normal stress, maximum shear stress, distortion energy, etc.

Depending on the type and complexity of the problem, we might need to use one or more techniques to apply these steps. For example, we might need to use the method of sections to find the internal forces at a point, the method of superposition to find the displacements due to multiple loads, the method of Mohr’s circle to find the principal stresses at a point, etc.

To illustrate how to use different techniques to solve resistance of materials problems, we will show you some examples of problems and their solutions. We will also explain why we chose a certain technique and how we applied it.

< x \leq 6 $$
– Bending moment: $$M(x) = -5x + 25x – 60 \text{ for } 0 \leq x \leq 3 $$
$$M(x) = -5x + 25x – 60 – 10(6-x)^2 \text{ for } 3 < x \leq 6 $$

We can plot these expressions to see the variation of the internal forces along the beam. See Figure 6.

Apply the method of integration to find the deflection and slope at any point along the beam. This involves using the differential equations of equilibrium and compatibility to find the internal forces and displacements in a body. It involves integrating the equations with respect to the coordinate axes and applying the boundary conditions to find the constants of integration.

We can use the method of elastic curve to relate the deflection, slope, bending moment, and curvature. The elastic curve is the shape of the deformed body due to external loads. It shows the relationship between deflection, slope, bending moment, and curvature. The equation of the elastic curve is:

$$EI\frac{d^2y}{dx^2} = M(x) $$

where $E$ is the modulus of elasticity, $I$ is the moment of inertia, $y$ is the deflection, $x$ is the horizontal coordinate, and $M$ is the bending moment.

We can integrate this equation twice to get:

$$EI\frac{dy}{dx} = \int M(x) dx + C_1 $$
$$EIy = \int (\int M(x) dx + C_1) dx + C_2 $$

where $C_1$ and $C_2$ are constants of integration.

We can use the method of singularity functions to simplify the integration. Singularity functions are mathematical functions that have discontinuities or singularities at certain points. They can represent concentrated forces, distributed forces, moments, or other types of loads. By using singularity functions, we can simplify

Conclusion

In this article, we have shown you how to solve resistance of materials problems using different techniques that are derived from the works of 7 Russian authors (Timoshenko, Popov, Gere, Ugural, Shames, Beer, and Johnston), as well as Hibeler, Singer, and Mosto. These techniques are widely used and recognized in the field of resistance of materials and can help you understand and apply the concepts and formulas involved.

We have also shown you some examples of problems and their solutions using these techniques. We have explained why we chose a certain technique and how we applied it. We have also used singularity functions to simplify the expressions for internal forces and displacements and avoid piecewise integration.

We hope this article has helped you learn how to solve resistance of materials problems using different techniques. Thank you for reading.

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