In statics, a structure is statically indeterminate when the static equilibrium equations are not sufficient for determining the internal forces and reactions on that structure.

Based on Newton’s laws of motion, the equilibrium equations available for a two-dimensional body are

* Σ F = 0 : the vectorial sum of the forces acting on the body equals zero. This translates to

Σ H = 0: the sum of the horizontal components of the forces equals zero;
Σ V = 0: the sum of the vertical components of forces equals zero;

* Σ M = 0 : the sum of the moments (about an arbitrary point) of all forces equals zero.

In the beam construction on the right, the four unknown reactions are VA, VB, VC and HA. The equilibrium equations are:

Σ V = 0:

VA − Fv + VB + VC = 0

Σ H = 0:

HA − Fh = 0

Σ MA = 0:

Fv · a − VB · (a + b) – VC · (a + b + c) = 0.

Since there are four unknown forces (or variables) (VA, VB, VC and HA) but only three equilibrium equations, this system of simultaneous equations cannot be solved. The structure is therefore classified statically indeterminate. Considerations in the material properties and compatibility in deformations are taken to solve statically indeterminate systems or structures.

The Euler-Bernoulli beam equation defines the behaviour of a beam element (see below). It is based on five assumptions:

(1) continuum mechanics is valid for a bending beam
(2) the stress at a cross section varies linearly in the direction of bending, and is zero at the centroid of every cross section.
(3) the bending moment at a particular cross section varies linearly with the second derivative of the deflected shape at that location.
(4) the beam is composed of an isotropic material
(5) the applied load is orthogonal to the beam’s neutral axis and acts in a unique plane.

The most important natural laws for structural engineering are Newton’s Laws of Motion

Newton’s first law states that every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.

Newton’s second law states that the rate of change of momentum of a body is proportional to the resultant force acting on the body and is in the same direction. Mathematically, F=ma (force = mass x acceleration).

Newton’s third law states that all forces occur in pairs, and these two forces are equal in magnitude and opposite in direction.

With these laws it is possible to understand the forces on a structure and how that structure will resist them. The Third Law requires that for a structure to be stable all the internal and external forces must be in equilibrium. This means that the sum of all internal and external forces on a free-body diagram must be zero.

A structural engineer must understand the internal and external forces of a structural system consisting of structural elements and nodes at their intersections.

A statically determinate structure can be fully analysed using only consideration of equilibrium, from Newton’s Laws of Motion.

A statically indeterminate structure has more unknowns than equilibrium considerations can supply equations for (see simultaneous equations). Such a system can be solved using consideration of equations of compatibility between geometry and deflections in addition to equilibrium equations, or by using virtual work.

If a system is made up of b bars, j pin joints and r support reactions, then it cannot be statically determinate if the following relationship does not hold:

r + b = 2j

It should be noted that even if this relationship does hold, a structure can be arranged in such a way as to be statically indeterminate.

A load case is a combination of different types of loads with safety factors applied to them. A structure is checked for strength and serviceability against all the load cases it is likely to experience during its lifetime.

Typical load cases for design for strength (ultimate load cases; ULS) are:

1.4 x Dead Load + 1.6 x Live Load
1.2 x Dead Load + 1.2 x Live Load + 1.2 x Wind Load

A typical load case for design for serviceability (characteristic load cases; SLS) is:

1.0 x Dead Load + 1.0 x Live Load

Different load cases would be used for different loading conditions. For example, in the case of design for fire a load case of 1.0 x Dead Load + 0.8 x Live Load may be used, as it is reasonable to assume everyone has left the building if there is a fire.

In multi-story buildings it is normal to reduce the total live load depending on the number of storys being supported, as the probability of maximum load being applied to all floors simultaneously is negligibly small.

It is not uncommon for large buildings to require hundreds of different load cases to be considered in the design.

Some Structural loads on structures can be classified as live (imposed) loads, dead loads, earthquake (seismic) loads, wind loads, soil pressure loads, fluid pressure loads, impact loads, and vibratory loads.

Live loads are transitory or temporary loads, and are relatively unpredictable in magnitude. They may include the weight of a building’s occupants and furniture, and temporary loads the structure is subjected to during construction.

Dead loads are permanent, and may include the weight of the structure itself and all major permanent components. Dead load may also include the weight of the structure itself supported in a way it wouldn’t normally be supported, for example during construction.

Structural engineering depends upon a detailed knowledge of loads, physics and materials to understand and predict how structures support and resist self-weight and imposed loads. To apply the knowledge successfully a structural engineer will need a detailed knowledge of mathematics and of relevant empirical and theoretical design codes.

The criteria which govern the design of a structure are either serviceability (criteria which define whether the structure is able to adequately fulfill its function) or strength (criteria which define whether a structure is able to safely support and resist its design loads). A structural engineer designs a structure to have sufficient strength and stiffness to meet these criteria.

Loads imposed on structures are supported by means of forces transmitted through structural elements. These forces can manifest themselves as:

* tension (axial force)
* compression (axial force)
* shear
* bending, or flexure (a bending moment is a force multiplied by a distance, or lever arm, hence producing a turning effect or torque)

Structural engineering dates back to at least 270 BC when the step pyramid for Pharoah Djoser was built by Imhotep, the first engineer in history known by name. Pyramids were the most common major structures built by ancient civilisations because the structural form of a pyramid is inherently stable and can be almost infinitely scaled (as opposed to most other structural forms, which cannot be linearly increased in size in proportion to increased loads).

Throughout ancient and medieval history most architectural design and construction was carried out by artisans, such as stone masons and carpenters, rising to the role of master builder. No theory of structures existed and understanding of how structures stood up was extremely limited, and based almost entirely on empirical evidence of ‘what had worked before’. Knowledge was retained by guilds and seldom supplanted by advances. Structures were repetitive, and increases in scale were incremental.

No record exists of the first calculation of the strength of structural members or the behaviour of structural material, but the profession of structural engineer only really took shape with the industrial revolution and the re-invention of concrete. The physical sciences underlying structural engineering began to be understood in the Renaissance and have been developing ever since.

Structural engineers are responsible for engineering design and analysis. Entry-level structural engineers may design the individual structural elements of a structure, for example the beams, column, and floor of a building. More experienced engineers would be responsible for the structural design and integrity of an entire system, such as a building.

Structural engineers often specialize in particular fields, such as bridge engineering, building engineering, pipeline engineering, industrial structures or special structures such as vehicles or aircraft.

Structural engineering has existed since humans first started to construct their own structures. It became a more defined and formalised profession with the emergence of the architecture profession as distinct from the engineering profession during the industrial revolution in the late 19th Century. Until then, the architect and the structural engineer were often one and the same – the master builder. Only with the understanding of structural theories that emerged during the 19th and 20th century did the professional structural engineer come into existence.

The role of a structural engineer today involves a significant understanding of both static and dynamic loading, and the structures that are available to reist them. The complexity of modern structures often requires a great deal of creativity from the engineer in order to ensure the structures support and resist the loads they are subjected to. A structural engineer will typically have a four or five year undergrad degree, followed by a minimium of three years of professional practice before being considered fully qualified.

Structural engineers are licensed or accredited by different learned societies and regulatory bodie around the world (for example, the Institution of Structural Engineers in the UK). Depending on the degree course they have studied and/or the jurisdiction they are seeking licensure in, they may be accredited (or licensed) as just structural engineers, or as civil engineers, or as both civil and structural engineers.

Structural engineering is a field of engineering dealing with the design of structures that support or resist loads. Structural engineering is usually considered a speciality within civil engineering, but it can also be studied in its own right.

Structural engineers are most commonly involved in the design of buildings and large nonbuilding structures but they can also be involved in the design of machinery, medical equipment, vehicles or any item where structural integrity affects the item’s function or safety. Structural engineers must ensure their designs satisfy given design criteria, predicated on safety (e.g. structures must not collapse without due warning) or serviceability and performance (e.g. building sway must not cause discomfort to the occupants).

Structural engineering theory is based upon physical laws and empirical knowledge of the structural performance of different geometries and materials. Structural engineering design utilises a relatively small number of basic structural elements to build up structural systems than can be very complete. Structural engineers are responsible for making creative and efficient use of funds, structural elements and materials to achieve these goals.