Taking into account that this theory assumes the Poisson's Ratio is negligible for the calculation of normal stresses but not for tangential stresses, we can express the stress field with the help of Lamé's equations as follows:

And that the tangential forces are written as:.

Where we have that is the Young's Modulus, is the Shear Modulus or Shear Modulus, is one of the Lamé Parameters and is the cubic dilation, which is calculated as .

We can define the bending moment acting on the element as the sum of a pair of forces acting on the cross section, represented by the normal forces acting on the cross section area of ​​the beam. We can express this mathematically as:.

In the same way we can define the shear force acting on the beam, as the sum of the forces acting on the cross section of the beam, that is, the shear forces that act on the cross section area of ​​the element. Mathematically it is expressed as:

Replacing in these expressions the value of the normal stresses () and the shear stresses () obtained in the previous section, we have:

Where is the bending deformation of the element. It is of interest to appreciate that and depend only on the coordinate axis.

The expression for the normal stresses (), tells us that the variation through the thickness of the element is linear, which we can consider exact, according to the classical theory of beams.[2]​ In contrast, the expression for the tangential stresses (), shows us that their distribution in the cross section is uniform (independent of the coordinates and ). This result can be considered as an "improvement" compared to the Euler-Bernoulli theory, which predicts and even if the shear force is non-zero.

Even so, the result of Timoshenko's theory of uniformly distributed in the cross section cannot be considered exact either, since it is in clear contradiction, for example, with the parabolic distribution present in elements of rectangular section[2]​ that is deduced exactly using Collignon's formula. In order to counteract this problem, a shape or distortion coefficient is adopted, which seeks to match the constant deformation work with the exact one. In this way we can express the tangential stress () and the shear force as:.

Where it is called reduced area of ​​the cross section. The shape coefficient depends on the shape of the cross section and the Poisson's ratio and is in the range . The precise determination of this coefficient is not simple in the general case and several authors have provided estimates for different cross sections; For example, Cowper (1966)[3]​ provided expressions for rectangular, circular and thin-walled tube sections, among others.

The following table shows the shear correction factor for beams with isotropic elastic materials, according to Cowper, for the most common sections.

where is Poisson's ratio.

Next, we are interested in finding the field of transverse displacements, which gives us the shape of the deformed beam in terms of the external actions, which can be forces perpendicular to the axis of the beam (punctual or distributed), as well as concentrated torques. Eventually, it might be necessary to also obtain the twist of the cross section. We will begin by stating the balance equations for a differential beam element like the one shown in the figure, which also explains the sign convention that will be used in this development.

From the balance of forces in direction for the differential element, we obtain:

From the equilibrium of moments, we obtain:

On the other hand, combining the expression for the bending moment obtained in the previous section, with the expression for provided above (in the section on Displacement Field) and using , we obtain:

Alternative 1.

If the two previous expressions are combined and the balance of vertical forces is also considered, a second-order differential equation is obtained in unknown, which can be solved by double integration if two boundary conditions are known and if the distribution of the bending moment is known. (It is noted that the distributed force is usually a fact of the problem). This equation is:

Alternative 2.

If it is derived with respect to the moment balance equation and the relationship obtained in the section on Internal Forces is considered, we obtain:

If these two expressions are combined, again considering the relationship between and the shear and assuming a constant cross section, a differential equation is obtained in which the derivative of the shear force with respect to appears. This derivative can be eliminated using the balance of forces expressed above, to obtain a fourth-order differential equation in unknown that can be solved in terms of the distributed load, provided that four boundary conditions are known. This equation is:

Alternative 3.

Alternatively, it is possible to find the deformation of the beam by solving two first-order differential equations in a stepwise manner: the first of them is simply the relationship already mentioned, while the second comes from rewriting the expression for given in the section on Displacement Field, taking into consideration (1), the relationship ; (2), the relationship between and shear and (3), the moment balance equation.

The two differential equations mentioned are the following:.

The first equation can be solved for in terms of the bending moment if a boundary condition for the rotation of the cross section of the beam is known. Once found, the second equation can be solved in terms of shear if a boundary condition for the deformed is known.

Taking into account that this theory assumes the Poisson's Ratio is negligible for the calculation of normal stresses but not for tangential stresses, we can express the stress field with the help of Lamé's equations as follows:

And that the tangential forces are written as:.

Where we have that is the Young's Modulus, is the Shear Modulus or Shear Modulus, is one of the Lamé Parameters and is the cubic dilation, which is calculated as .

We can define the bending moment acting on the element as the sum of a pair of forces acting on the cross section, represented by the normal forces acting on the cross section area of ​​the beam. We can express this mathematically as:.

In the same way we can define the shear force acting on the beam, as the sum of the forces acting on the cross section of the beam, that is, the shear forces that act on the cross section area of ​​the element. Mathematically it is expressed as:

Replacing in these expressions the value of the normal stresses () and the shear stresses () obtained in the previous section, we have:

Where is the bending deformation of the element. It is of interest to appreciate that and depend only on the coordinate axis.

The expression for the normal stresses (), tells us that the variation through the thickness of the element is linear, which we can consider exact, according to the classical theory of beams.[2]​ In contrast, the expression for the tangential stresses (), shows us that their distribution in the cross section is uniform (independent of the coordinates and ). This result can be considered as an "improvement" compared to the Euler-Bernoulli theory, which predicts and even if the shear force is non-zero.

Even so, the result of Timoshenko's theory of uniformly distributed in the cross section cannot be considered exact either, since it is in clear contradiction, for example, with the parabolic distribution present in elements of rectangular section[2]​ that is deduced exactly using Collignon's formula. In order to counteract this problem, a shape or distortion coefficient is adopted, which seeks to match the constant deformation work with the exact one. In this way we can express the tangential stress () and the shear force as:.

Where it is called reduced area of ​​the cross section. The shape coefficient depends on the shape of the cross section and the Poisson's ratio and is in the range . The precise determination of this coefficient is not simple in the general case and several authors have provided estimates for different cross sections; For example, Cowper (1966)[3]​ provided expressions for rectangular, circular and thin-walled tube sections, among others.

The following table shows the shear correction factor for beams with isotropic elastic materials, according to Cowper, for the most common sections.

where is Poisson's ratio.

Next, we are interested in finding the field of transverse displacements, which gives us the shape of the deformed beam in terms of the external actions, which can be forces perpendicular to the axis of the beam (punctual or distributed), as well as concentrated torques. Eventually, it might be necessary to also obtain the twist of the cross section. We will begin by stating the balance equations for a differential beam element like the one shown in the figure, which also explains the sign convention that will be used in this development.

From the balance of forces in direction for the differential element, we obtain:

From the equilibrium of moments, we obtain:

On the other hand, combining the expression for the bending moment obtained in the previous section, with the expression for provided above (in the section on Displacement Field) and using , we obtain:

Alternative 1.

If the two previous expressions are combined and the balance of vertical forces is also considered, a second-order differential equation is obtained in unknown, which can be solved by double integration if two boundary conditions are known and if the distribution of the bending moment is known. (It is noted that the distributed force is usually a fact of the problem). This equation is:

Alternative 2.

If it is derived with respect to the moment balance equation and the relationship obtained in the section on Internal Forces is considered, we obtain:

If these two expressions are combined, again considering the relationship between and the shear and assuming a constant cross section, a differential equation is obtained in which the derivative of the shear force with respect to appears. This derivative can be eliminated using the balance of forces expressed above, to obtain a fourth-order differential equation in unknown that can be solved in terms of the distributed load, provided that four boundary conditions are known. This equation is:

Alternative 3.

Alternatively, it is possible to find the deformation of the beam by solving two first-order differential equations in a stepwise manner: the first of them is simply the relationship already mentioned, while the second comes from rewriting the expression for given in the section on Displacement Field, taking into consideration (1), the relationship ; (2), the relationship between and shear and (3), the moment balance equation.

The two differential equations mentioned are the following:.

The first equation can be solved for in terms of the bending moment if a boundary condition for the rotation of the cross section of the beam is known. Once found, the second equation can be solved in terms of shear if a boundary condition for the deformed is known.