Geometrical description in the side view

The definition of the tendon’s vertical profile is given by the supports or by specifying height attributes at the tendon input points. The so−called folded view in the vertical section, which is permitted by the program, corresponds, as may be seen in the follow$ ing pages of the manual, to the development of a curved surface in vertical section through the respective tendon. Therefore, the abscissa designated by x’ represents the projection of the tendon onto the plan view.

In view of the deviation forces one often wants to work with special generating el$ ements (e.g. 2straight line−parabola", 2parabola−parabola"). By means of the following example it is shown how in most cases met with in practice the support attributes can be controlled.

A tendon can be given the attribute 2Standard elements" or Polynomial segment of 3rd order". Normally one works with standard elements (described below). The program tries in this case to describe the profile of the tendons between the support points with (quadratic) parabolic sections and straight lines, based on the attributes specified at the supports or at the intermediate points. The advantage of (quadratic) par$ abolas lies in the practically constant deviation forces:

u[ P
8f l2

with:

Par.min Kr.fix

Support and point attributes

The following support and point attributes can be input:

Fixed point:

Height z:

Explicit height information or with reference to the boundary of the slab (highest and lowest points). The z−axis points upwards.

Slope z’:

The slope input is optional. For the highest or lowest point input z’=0, otherwise (without explicit informa$ tion) z’ is determined by the program: z’=slope of the bisecting angle.

Minimum radius conditions: free : no special condition

parabola :condition is that tendon at support or at the indicated polygon point

should exhibit the minimum radius (prerequisite: 2standard elements"). circ. arc : instruction that here a circular arc

with the minimum radius and half chord length l_k should be introduced. Point of inflexion:

This attribute is used to fix the position of the point of inflexion of two adjoining parabolas (left and right curve).

Examples of generated tendon profiles

In the following some standard cases are put together to show how they are created by the program on the basis of the attributes in the section (support and polygon points, respectively). As mentioned already in the introduction, the tendon profile is first drawn in the plan view using the Graphics Editor and then brought into the desired vertical position by inputting the supports. It is also possible to assign height attributes directly to the support points. In all internal calculations (e.g. of friction losses) the 3 D profile of the tendon is used.

It could happen that some conditions in a certain section are in conflict with each other or that the input values do not allow reasonable results to be obtained (e.g. chord lengths that are too long for circular arcs). In this case the program may have to ignore some conditions to still be able to draw a curve. All input height attributes (height z), however, are always used! In the vicinity of a support one should not assign height at$ tributes to the polygon points, to avoid defining them twice. At the ends of the tendons a height input is always required by the program otherwise the start and end points are specified in a standard way at the middle axis of the slab (reason: avoiding tolerance problems with the calculation of the points of intersection between the tendons and the supports). The required user input is demonstrated by means of the two span beam shown in Fig. A−2 .

E 2 Basics Part E Prestressing Module

T H, Par.min T G Par .

Par Par Par G

Calculated sections: G = straight line, Par = parabola, Par’ = parabola with minimum radius at the vertex

W

Polygon points: A, E = start and end points (elevation z defined by default in middle axis of slab, can be changed)

Supports: H,T = highest and lowest points

Par.min = extra condition: parabola with minimum radius at vertex

W

A E

Fig. A–2 User input for a prestressed two span beam Par’

Further examples of single sections:

If, in addition to the height at the start, the slope is specified, the program uses at the end of the in itial straight line a polynomial of 3rd order.

Fig. A–3 Geometry at start of tendon: straight line – parabola

z z (

z’)

z (z’) z, z’

parabola (2nd order) polynomial of 3rd order straight line deviation forces u+ deviation forces u+ straight line

To avoid the start and end sections being geo metrically redundant, at the anchorages only the height is specified. Then, optionally, at the end of the first section also the slope of the tangent z’ can be specified.

If there are two inflexion points, then the part in between is a straight line. Further inflexion points lying in between do not influence the shape of the curve, i.e. only the two outermost inflexion points are effective.

w w w

If, in the section under consideration, a point of inflexion is introduced between the highest and lowest points, then this defines the subdivision of the two parabolic sections. If no extra information is supplied the point of inflexion is placed in the

z ( z’) z ( z’) z ( z’) z ( z’)

parabola (2nd order) parabola (2nd

order) v str. line parabola (2.ord.)

deviation forces u+ deviation forces u+ deviation forces u− deviation forces u− Points of inflex ion Point of inflex ion

If, in the highest and lowest points, the condition Pmin is input, a trapezium can be produced. If a point of inflexion is moved closer to an end

point, the radius of the shorter parabola section is always smaller. In the extreme, just permissible, case the minimum radius at the vertex of the curve is reached. This corresponds to the condi tion Par.min (parabola with minimum radius at vertex), which can be input as a support attribute, i.e. in this case the corresponding length of the parabola lp1 is determined by the program.

u–

u+

u–

u+

Fig. A–5 Parabolas with minimum radius at vertex

z, Par.min z, Par.min z, Par.min z straight line parabola (2. ord.) parabola (2. order) parabola (2. order) parabola (2. order) lp1 u– u+ u– u+ In contrast to the condition Par.min (see above)

for which the minimum radius is just reached at the vertex, with the condition Cir.min a circular arc segment of half chord length lk is produced. Purely geometrically therefore, at a given dis tance l_k the steepest variation (max. shear forces due to prestressing !!) is enforced. The program of course takes no account of constructional con straints (stiffness of the duct, etc.). If the length l_k is too short, the point of inflexion of the connect ing 3rd order polynomial may not occur at the de sired position.

With a higher value of l_k the second section is ap proximated by several polynomials of 3rd order, in order to produce as smooth a curve as possible (approximation with Bezier curve).

Fig. A–6 Circle instead of parabola at vertex z, Cir.min

circular arc

z, Cir.min

z z

Bezier curve (approximated by pol. of 3.ord)

circle polynomial of 3rd

order

E 2 Basics Part E Prestressing Module

u–

u+

u–

u+

Section with circular arc conditions at both ends. For smaller values of l_k the transition is with a polynomial of 3rd order.

Circular arc section with very large value of l_k. In the extreme case the curve passes directly from one circle to another.

Fig. A–7 Further possibilities with circular arcs at vertex

z, Cir.min z, Cir.min z, Cir.min (lk=large) z

3rd order poly nomial circle

circle

circle circle straight line str.l . lk lk lk lk u– u+

Fig. A–8 Cubic polynomials

For +Non−Standard" input of tendons the pro gram uses cubic polynomials. If the tangent slope is not explicitly given, it is calculated from the two adjoining polygon sides (bisector).

This input possibility is needed above all to re calculate existing objects, by inputting sequen tially the tendon heights measured from the plan.

z, z’ z, z’ 3rd order poly nomial 3rd order poly nomial