Analysis of Statically Determinate Structures

We will restrict our discussions to primarily one-dimensional members (in reality these are three-

dimensional structural members, but the other two dimensions are relatively much smaller). When the

loading on such a member is on a plane same as the member itself, we call it a two-dimensional (planar)

case (see Figure 2.3a for example). In such cases, the internal forces also lie on the same plane. The

internal forces on any cross-section can be expressed with two orthogonal force components and one

moment in the plane of loading ( , , M in Figure 2.3b). We can align x -axis along the centroidal axis

of the member and we can also align one of the forces, let's say , along this centroidal axis (along the

primary dimension). Then this internal force will be known as the axial force (Figure 2.3c). In general, we

consider an internal surface perpendicular to the centroidal axis (transverse cross-section, also called the yz-

plane or x-plane ). Then the other force component acts tangentially to this surface and it is known as

the shear force . The internal moment, which is acting on the transverse cross-section, is known as the

bending moment .

Axial force, shear force and bending moment on a cross-section of a two-dimensional

(planar) system

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 For a three-dimensional case, that is when the loading is not restricted to one plane, we have three
orthogonal force components and three orthogonal moment components on an internal surface (Figures 2.4
a & b).
We align the centroidal axis of the member along, say, x -axis, and consider an internal surface
perpendicular to it (Figures 2.4c). Then is the axial force and and are the two shear forces .
Moments and are the two bending moments . The moment is known as torsion . This set of
forces is the most generic case of internal forces for such structural members.

Axial force, two shear forces and two bending moments for three-dimensional

systems

Note that these internal forces are defined according to their orientation respective to the structural

member. The axial force acts along the centroidal axis of the member. The shear forces act in a plane which

is perpendicular to this centroidal axis and the bending moments act along directions perpendicular to this

axis as well. The torsion acts along the centroidal axis.

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The sign convention for internal forces depends on the internal surface on which these forces are being

considered. So, we need to define the internal surface first. Let us assume that the centroidal axis of a

longitudinal structural member is aligned along the x -axis, and we consider the internal forces on an x-

plane (or x-surface ). If this cross-section is facing positive x -direction, then it is called a positive x -

surface, and vice-versa. Figures 2.5a & b show the positive internal forces on positive and negative x -

surfaces, respectively.

Direction of positive internal forces on a positive x -surface (a) and a negative x -

surface (b)

This sign convention is followed throughout this course and relations involving these forces (and other

parameters, such as stresses and deformations) are derived based on this sign convention. It will not be

illogical to adopt any other sign convention for internal forces. However, in that case one will have to develop the equations involving these forces independently following the new sign convention.

Let us restrict our discussion to planer loading (that is, two-dimensional) cases with no torsion. For an

internal segment of a member, we can show the internal forces both on the positive and the negative

internal surfaces. Positive directions for each internal force (an axial force, a shear force and a bending

moment) are shown individually in Figure 2.6. This is an easy and standard way of defining sign conventions

for two-dimensional cases, and students are encouraged to define (for each specific case) their adopted sign

convention similarly.

Notations that we follow for these internal forces are: P for axial force, V for shear force, and M for bending

moment. However, please note that in a three-dimensional case, we need suffixes to distinguish between

the two shear forces and also between the two bending moments. General notation for torsion is T .

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