Loading and Distortion

At a glance: A meaning of “loading” and “distortion” is set out here. Basically a loading on a cross-section causes a distortion.

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“distortion”

“Strain” and “curvature” defined

It is usually accepted that a meaning of the word “strain” is the slight change in a dimension of a body caused by stress in that body. This meaning is extended to include slight changes in dimension from other causes such as:

• drying shrinkage of concrete,
• change in temperature,
• creep of concrete, and
• relaxation of prestressing steel.

Strain is expressed as the proportional change in dimension. That is the change in dimension divided by the dimension before the change.

Curvature of a structural member (from any cause) implies strains on a cross-section that are normal to the plane of that cross-section and vary across that plane. This application uses the plane-sections-remain-plane assumption. This is a usual assumption. The typical design code includes something like “Strain in reinforcement and concrete shall be assumed directly proportional to the distance from the neutral axis”.

In accordance with this assumption at any instant in time:

• there are parallel lines across the plane of the cross-section along each of which the strain is constant
• and there is a constant rate of change of the strain with respect to distance at right angles to those parallel lines.

Often “neutral axis” means one of those parallel lines on which the strain is zero.

The rate of change of strain with respect to distance is a measure of the curvature magnitude.

Curvature has an orientation being the orientation of the parallel lines and a magnitude and so can be viewed as a vector in the plane of the cross-section. Relative to given orthogonal axes in the plane of the cross-section a curvature can be expressed in polar coordinates being the magnitude and the angle of the orientation or alternatively in cartesian coordinates as the components of the curvature about each of the axes.

The strain at the origin of the axes is called the “axial strain” although in some contexts “axial strain” simply means a strain normal to the plane of the cross-section.

“Distortion” defined

The word “distortion” means the combination of an axial strain and a curvature. Just three numbers completely describe a distortion relative to a given orthogonal axes; – the strain at the origin and the curvature magnitude about each of the two axes.

Further, the three numbers that describe a distortion that is the combined affect of more than one distortion are each the sum of the respective numbers of those contributing distortions. This gives meaning to the language that implies distortions can be added and subtracted.

Where a description of a distortion of a shape component (usually concrete) is required there is provision for the three numbers. However, the distortion of one member of a point component (usually a steel bar) can be described by just one number; the strain.

Sign convention

A strain that is an increase in a dimension is represented by a positive number and a strain that is a decrease in a dimension is represented by a negative number.

The right hand rule is used for curvatures. Thus a curvature aligned with the positive direction of the X axis has strains increasing with increasing Y.

That is strains become more tensile with increasing Y which is contrary to the usual convention of positive bending having compression at the top. The default value of the orientation angle of the reference axes being 180 degrees is intended to resolve this inconsistency. (For the meaning of References axes see Load cases).

“Stage” distortion, “other” distortion

The expressions Stage distortion and Other distortion are used in facilities intended to take into account the affects of prestressing, composite construction, staged construction, drying shrinkage, creep, relaxation and thermal expansion and contraction.

Where these facilities are not required all the variables labeled “stage” and all the variables labeled “other” should be set to zero.

The default value of all these variables is zero.

Stage, Other, Stress and Load-case distortion defined

Stage distortion of a component is the distortion that exists in an initial state. For an explanation of “initial state” see Total Distortion.

Other distortion of a component is the combined affect of all causes other than stress.

Stress distortion of a component is the distortion caused by stress. It is used with the stress/strain relationships of the material to compute the stresses and thus the contribution to the loading.

Load-case distortion is a change from an initial state. It applies equally to all components and is the subject of the trial and adjustment processes. For an explanation of Load-case Distortion see Total Distortion.

Where the Stage Distortion and the Other Distortion of a component are all zeros the Stress Distortion of that component is the same as the Load-case Distortion.

Total distortion

The concept of total distortion of a component of a cross-section is justified by the idea that changes in such a total distortion can be related to changes in the distortion of the whole structural member.

Other Distortion is the combined affect of all causes other than stress. The total distortion is thus the sum of the Stress Distortion and the Other Distortion.

Stage Distortions are used in the analysis of a structural member resulting from the joining of more than one structural member. The envisaged contributing members are parallel and the joining is a means of transferring shear loads.

A simple example of this is the situation that exists immediately after in-situ topping concrete sets on a precast supporting slab. The precast slab is one contributing member and the topping concrete another.

The Stage Distortion of each component is the distortion that exists at the instant of the joining. At that instant each of the contributing members has distortions corresponding to the loading it is sustaining. Immediately after that instant the resulting member is sustaining a loading that is the combined affect of the loadings on the contributing members and the distortion of each component is the same as it was before that instant being the stage distortion. This is referred to as the “initial state” of the new combined member.

In the example of the precast slab and topping before the topping sets the precast slab is supporting itself and supporting the wet topping concrete and has a corresponding distortion. At the instant the topping sets the topping has no stress and no distortion (it is reasonable to assume), although it is part of the combined member. The topping has a stage distortion that is zeros.

The analysis of a structural member resulting from the joining of more than one structural member usually involves loadings different from the loading in the initial state. Distortions different from the distortion in the initial state are involved.

Load-case distortion is the change in distortion from the distortion in the initial state. In accordance with the assumption of no slip between components a load-case distortion applies equally to all components. Thus, for each component the total distortion is the Stage Distortion plus the Load-case distortion.

To recap, for each component the total distortion is the Stage Distortion plus the Load-case Distortion and also that total distortion is the Other Distortion plus the Stress Distortion.

These two relationships imply that the Stress Distortion equals the Load-case Distortion plus the Stage Distortion minus the Other Distortion. During trial and adjustment the Load-case distortion is changed in each cycle and this relationship is used to obtain the stress distortion in each component that is used with the material properties to compute the loading.

For an explanation of this approach to total distortion using mathematical notation and expressions see Thompson, P. J. “Computer design of concrete member”, Transaction, Institution of Professional Engineers New Zealand, v 21 1/CE, Nov. 1994, pp. 25-29.

“Loading”

A force transmitted through a cross-section

This application works with stresses that are normal to the plane of a cross-section. The summation of such stresses is a force transmitted through the cross-section. That is it is a force normal to the plane of the cross-section exerted by the fabric of the structural member on one side of the cross-section on the fabric on the other side.

The “loading” is the force that is the summation of the stresses over the whole cross-section.

Often that force can be described by just three numbers; the magnitude and the X and Y of the location relative to given orthogonal axes in the plane of the cross-section.

Bending moments and axial load

Stresses can be both tensile and compressive and there can be a balance so that the loading force has zero magnitude and the corresponding X and Y have no meaning.

In an alternative description of a loading rather than using the X and Y use the product of the force magnitude and X and the product of the force magnitude and Y. These products are called “bending moments” and referred to as “the bending moment about the Y axis” and “the bending moment about the X axis” respectively.

The loading is still described by just three numbers being; the force magnitude and the bending moments about the X and Y axes.

That force is called the “axial load”. It is usually defined as being at the origin of the axes where it is independent of the bending moments

When the axial load is zero the bending moments can have meaning. In mathematical terms “as the axial load tends towards zero the X and the Y tend towards infinity”.

In general just three numbers completely describe a loading relative to a given orthogonal axes; – the axial load at the origin and the bending moments about each of the two axes.

Sign convention

The sign convention for loadings follows the sign convention for strain and distortion.

The sign of a stress is the same as the sign of the strain it causes.

Thus a tensile axial load is positive and a compressive axial load negative.

Note that where there is an axial load and the loading is described as a force at X, Y the right hand rule requires that the bending moment about the Y axis is minus the magnitude of the force multiplied by X.