In continuum mechanicsstress is a physical quantity that expresses the internal forces that neighbouring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material.

For example, when a solid vertical bar is supporting an overhead weighteach particle in the bar pushes on the particles immediately below it. When a liquid is in a closed container under pressureeach particle gets pushed against by all the surrounding particles. The container walls and the pressure -inducing surface such as a piston push against them in Newtonian reaction. These macroscopic forces are actually the net result of a very large sendkeys alternative of intermolecular forces and collisions between the particles in those molecules.

Strain inside a material may arise by various mechanisms, such as stress as applied by external forces to the bulk material like gravity or to its surface like contact forcesexternal pressure, or friction. Any strain deformation of a solid material generates an internal elastic stressanalogous to the reaction force of a springthat tends to restore the material to its original non-deformed state. In liquids and gasesonly deformations that change the volume generate persistent elastic stress.

However, if the deformation changes gradually with time, even in fluids there will usually be some viscous stressopposing that change. Elastic and viscous stresses are usually combined under the name mechanical stress. Significant stress may exist even when deformation is negligible or non-existent a common assumption when modeling the flow of water.

Stress may exist in the absence of external forces; such built-in stress is important, for example, in prestressed concrete and tempered glass. Stress may also be imposed on a material without the application of net forcesfor example by changes in temperature or chemical composition, or by external electromagnetic fields as in piezoelectric and magnetostrictive materials. The relation between mechanical stress, deformation, and the rate of change of deformation can be quite complicated, although a linear approximation may be adequate in practice if the quantities are sufficiently small.

Stress that exceeds certain strength limits of the material will result in permanent deformation such as plastic flowfracturecavitation or even change its crystal structure and chemical composition.

In some branches of engineeringthe term stress is occasionally used in a looser sense as a synonym of "internal force". For example, in the analysis of trussesit may refer to the total traction or compression force acting on a beam, rather than the force divided by the area of its cross-section.

Since ancient times humans have been consciously aware of stress inside materials. Until the 17th century, the understanding of stress was largely intuitive and empirical; and yet, it resulted in some surprisingly sophisticated technology, like the composite bow and glass blowing.

Over several millennia, architects and builders in particular, learned how to put together carefully shaped wood beams and stone blocks to withstand, transmit, and distribute stress in the most effective manner, with ingenious devices such as the capitalsarchescupolastrusses and the flying buttresses of Gothic cathedrals.

The understanding of stress in liquids started with Newton, who provided a differential formula for friction forces shear stress in parallel laminar flow. Stress is defined as the force across a "small" boundary per unit area of that boundary, for all orientations of the boundary.

Following the basic premises of continuum mechanics, stress is a macroscopic concept. Namely, the particles considered in its definition and analysis should be just small enough to be treated as homogeneous in composition and state, but still large enough to ignore quantum effects and the detailed motions of molecules. Thus, the force between two particles is actually the average of a very large number of atomic forces between their molecules; and physical quantities like mass, velocity, and forces that act through the bulk of three-dimensional bodies, like gravity, are assumed to be smoothly distributed over them.When a force is applied to a structural member, that member will develop both stress and strain as a result of the force.

The applied force will cause the structural member to deform by some length, in proportion to its stiffness. Strain is the ratio of the deformation to the original length of the part:. There are different types of loading which result in different types of stress, as outlined in the table below:.

In the equations for axial stress and transverse shear stressF is the force and A is the cross-sectional area of the member. In the equation for bending stressM is the bending moment, y is the distance between the centroidal axis and the outer surface, and I c is the centroidal moment of inertia of the cross section about the appropriate axis. In the equation for torsional stress, T is the torsion, r is the radius, and J is the polar moment of inertia of the cross section.

In the case of axial stress over a straight section, the stress is distributed uniformly over the entire area.

Strength of Material (SOM ) part 1. Stress strain concept, Static Equilibrium equation.

More discussion can be found in the section on shear stresses in beams. In the case of bending stress and torsional stress, the maximum stress occurs at the outer surface.

More discussion can be found in the section on bending stresses in beams. We have a number of structural calculators to choose from. Here are just a few:. Just as the primary types of stress are normal and shear stress, the primary types of strain are normal strain and shear strain. In the case of normal strain, the deformation is normal to the area carrying the force:.

In the case of transverse shear strain, the deformation is parallel to the area carrying the force:. The maximum shear strain occurs on the outer surface. In the case of a round bar, the maximum shear strain is given by:.

The shear strains are proportional through the interior of the bar, and are related to the max shear strain at the surface by:. Stress is proportional to strain in the elastic region of the material's stress-strain curve below the proportionality limit, where the curve is linear. The elastic modulus and the shear modulus are related by:. Essentially, everything can be treated as a spring.

Hooke's Law can be rearranged to give the deformation elongation in the material:. When force is applied to a structural member, that member deforms and stores potential energy, just like a spring. The strain energy i. The total strain energy corresponds to the area under the load deflection curve, and has units of in-lbf in US Customary units and N-m in SI units.

The elastic strain energy can be recovered, so if the deformation remains within the elastic limit, then all of the strain energy can be recovered.Equations of Static Equilibrium: Consider a case where a book is lying on a frictionless table surface. Now, if we apply a force F 1 horizontally as shown in the Fig. However, if we apply the force perpendicular to the book as in Fig. This question was answered by Newton when he formulated his famous second law of motion.

In a simple vector equation it may be stated as follows:. However, if the body is in the state of static equilibrium then the right hand of equation 1 must be zero.

Strength of Materials

Hence, the book lying on the table subjected to external force as shown in Fig. A vector in 3-dimensions can be resolved into three orthogonal directions viz. Also, if the resultant force vector is zero then its components in three mutually perpendicular directions also vanish.

Hence, the above two equations may also be written in three co-ordinate axes directions as follows:. For such structures we could have forces acting only in x and y directions. Also the only external moment that could act on the structure would be the one about the z -axis. For planar structures, the resultant of all forces may be a force, a couple or both.

The static equilibrium condition along x -direction requires that there is no net unbalanced force acting along that direction. For such structures we could express equilibrium equations as follows:.

Using the above three equations we could find out the reactions at the supports in the beam shown in Fig. After evaluating reactions, one could evaluate internal stress resultants in the beam. Admissible or correct solution for reaction and internal stresses must satisfy the equations of static equilibrium for the entire structure. They must also satisfy equilibrium equations for any part of the structure taken as a free body.

If the number of unknown reactions is more than the number of equilibrium equations as in the case of the beam shown in Fig. Such structures are known as the statically indeterminate structures. In such cases we need to obtain extra equations compatibility equations in addition to equilibrium equations. Self-compacting Concrete: Fresh Properties. Wednesday 15th April Civil Engineering Forum.

Equations of Static Equilibrium Equations of Static Equilibrium: Consider a case where a book is lying on a frictionless table surface. In a simple vector equation it may be stated as follows: Eq. Fig 1 a. Fig 1 b.

equilibrium equations in strength of material

Fig 2 Statically Determinate Beam. Fig 3 Statically Indeterminate Beam. Share this:. Recycled Concrete Aggregate. Sorry, your blog cannot share posts by email.Mechanics of Materials.

When a metal is subjected to a load forceit is distorted or deformed, no matter how strong the metal or light the load. If the load is small, the distortion will probably disappear when the load is removed. The intensity, or degree, of distortion is known as strain. If the distortion disappears and the metal returns to its original dimensions upon removal of the load, the strain is called elastic strain. If the distortion disappears and the metal remains distorted, the strain type is called plastic strain.

Strain will be discussed in more detail in the next chapter. When a load is applied to metal, the atomic structure itself is strained, being compressed, warped or extended in the process. The atoms comprising a metal are arranged in a certain geometric pattern, specific for that particular metal or alloy, and are maintained in that pattern by interatomic forces.

When so arranged, the atoms are in their state of minimum energy and tend to remain in that arrangement. Work must be done on the metal that is, energy must be added to distort the atomic pattern. Work is equal to force times the distance the force moves. Stress is the internal resistance, or counterforce, of a material to the distorting effects of an external force or load. These counterforces tend to return the atoms to their normal positions.

The total resistance developed is equal to the external load. This resistance is known as stress. Although it is impossible to measure the intensity of this stress, the external load and the area to which it is applied can be measured.

Stress s can be equated to the load per unit area or the force F applied per cross-sectional area A perpendicular to the force as shown in the Equation below.

equilibrium equations in strength of material

Types of Stress Stresses occur in any material that is subject to a load or any applied force. There are many types of stresses, but they can all be generally classified in one of six categories: residual stresses, structural stresses, pressure stresses, flow stresses, thermal stresses, and fatigue stresses.

Residual stresses are due to the manufacturing processes that leave stresses in a material. Welding leaves residual stresses in the metals welded. Structural stresses are stresses produced in structural members because of the weights they support.

The weights provide the loadings. These stresses are found in building foundations and frameworks, as well as in machinery parts. Pressure stresses are stresses induced in vessels containing pressurized materials. The loading is provided by the same force producing the pressure. Flow stresses occur when a mass of flowing fluid induces a dynamic pressure on a conduit wall.


The force of the fluid striking the wall acts as the load. This type of stress may be applied in an unsteady fashion when flow rates fluctuate. Water hammer is an example of a transient flow stress. Thermal stresses exist whenever temperature gradients are present in a material. Different temperatures produce different expansions and subject materials to internal stress.

This type of stress is particularly noticeable in mechanisms operating at high temperatures that are cooled by a cold fluid.Often times, like in the case of the pressure vessels that we studied in the previous lesson, the stress in one direction is really small compared with the other two. These two states of stress, the 3D stress and plane stress, are often discussed in a matrix, or tensorform.

Now that we've reduced our state of stress to two dimensions, we can learn how to transform the coordinates along which these stress components act into any coordinate frame we are interested in.

Why would we want to do that? Well, take a look at the image below. Two pieces of wood, cut at an angle, and glued together. How do we know if the glued joint can sustain the resultant stress that this force produces? We need to calculate the normal and shear stresses perpendicular and parallel to the joint.

Therefore, we need to rotate, or transform, the coordinates associated with the force P to the direction associated with the angle of the glued joint. Then, we can evaluate the stresses along these new directions, x' and y'. Once we've rotated the coordinate system, we need to transform the forces acting in the old coordinate frame to this new coordinate frame. That means, we must draw a detailed free body diagram. If we take the a differential element near the origin of the new coordinate system, we can get the forces acting on each surface from the stress times the differential area.

As you'll notice from the free body diagram, most of the forces aren't pointing in the directions we're interested in, that is x' and y'. So, we need to break these forces into their components, and sum the resulting forces in the directions of our new coordinate system.

This results in a long, but straightforward, calculation given below. We can stop here if we'd like. However, if we make use of a couple trigonometric identities, we can make these equations a bit more friendly looking. Using these identities, we can get rid of all those squared sines and cosines.

The final result for the normal and shear stresses in our new coordinate system denoted by theta, which is a counterclockwise rotation from the x axis to the x' axis is given by. We have just shown that the magnitude of the normal and shear stress will depend on the coordinate system you choose. If you rotate the coordinate system by some angle, the magnitude of these stresses will change. As you may suspect, certain angles will correspond to maximum and minimum values of these stresses.The concept of equilibrium is introduced to describe a body which is stationary or which is moving with a constant velocity.

A body under such a state is acted upon by balanced forces and balanced couples only. There is no unbalanced force or unbalanced couple acting on it. The concept must be really understood by every student. The size of a particle is very small compared to the size of the system being analysed. Rigid body. A body is formed by a group of particles. The size of a body affects the results of any mechanical analysis on it.

Most bodies encountered in engineering work can be considered rigid from the mechanical analysis point of view becase the deformations that take place within these bodies under the action of loads can be neglected when compared to other effects produced by the loads.

Springs undergo deformations that cannot be neglected when acted upon by forces or moments. For the analyses ini this book, only the effects of the deformations of springs on a rigid body interacting with the springs are considered but the springs themselves will not be analysed as a body.

In general, a force acting on a particle tends to cause the particle to translate. Also, a force on on a body not only tends to cause the body to translate as in the case of the particle but also tends to cause the body to rotate about any axis which does not intersect with or is not parallel to the line of action of the force. To see what actually happens to any particular part of a structure, that part has to be isolated from the other parts of the body. A mechanical system is defined as a body system that can be isolated from other bodies.

The system can be formed by a single body, part of a body, or a group of connected bodies. The bodies forming the system can either be rigid or non-rigid. A mechanical system can be solid, fluid, or even a combination of solild and fluid. The isolation of a mechanical system is achieved by cutting and isolating the system from its surroundings.

The isolation enables us to see the interactions between the isolated part and the other parts. The part which has been cut imaginarilyforms a free body. A diagram which portrays the free body, complete with the system of external forces acting on it due to its interaction with the parts which have been removed, is called the free-body diagram FBD of the isolated part.

The FBD of of a body system shows all loads acting on the external boundary of the isolated body. Assume that an analysis is to be carried-out on the whole structure of the arm when it is carrying a load as shown, where the weight of the component members of the arm can be nglected compared to the weight of the load.

Assume also that all joints of the arm do not prevent rotation around the respective joints, i. Because the direction and the sense of every reactive force are not known, the direction nad sense shall be assumed.

The arm can be isolated from the body of the lift truck at point A where it is pinned to the body of the lorry and at point C where it is acted upon by the active forceof the hydraulic piston rod.

The isolated arm is shown in Figure 3.The application of Newton's second law to a system gives:. Where bold font indicates a vector that has magnitude and direction.

The summation of forces will give the direction and the magnitude of the acceleration and will be inversely proportional to the mass.

The summation of forces, one of which might be unknown, allows that unknown to be found. So when in static equilibrium, the acceleration of the system is zero and the system is either at rest, or its center of mass moves at constant velocity. Likewise the application of the assumption of zero acceleration to the summation of moments acting on the system leads to:.

The summation of moments, one of which might be unknown, allows that unknown to be found. These two equations together, can be applied to solve for as many as two loads forces and moments acting on the system. From Newton's first lawthis implies that the net force and net torque on every part of the system is zero.

The net forces equaling zero is known as the first condition for equilibrium, and the net torque equaling zero is known as the second condition for equilibrium. See statically indeterminate. Archimedes c. A scalar is a quantity which only has a magnitudesuch as mass or temperature. A vector has a magnitude and a direction. There are several notations to identify a vectorincluding:. Vectors are added using the parallelogram law or the triangle law.

Vectors contain components in orthogonal bases. Unit vectors ijand k are, by convention, along the x, y, and z axesrespectively. Force is the action of one body on another. A force is either a push or a pull, and it tends to move a body in the direction of its action. The action of a force is characterized by its magnitude, by the direction of its action, and by its point of application. Thus, force is a vector quantity, because its effect depends on the direction as well as on the magnitude of the action.

Forces are classified as either contact or body forces. A contact force is produced by direct physical contact; an example is the force exerted on a body by a supporting surface. A body force is generated by virtue of the position of a body within a force field such as a gravitational, electric, or magnetic field and is independent of contact with any other body.

An example of a body force is the weight of a body in the Earth's gravitational field. In addition to the tendency to move a body in the direction of its application, a force can also tend to rotate a body about an axis.

The axis may be any line which neither intersects nor is parallel to the line of action of the force.

equilibrium equations in strength of material

This rotational tendency is known as the moment M of the force. Moment is also referred to as torque.