Building Construction & Drawing -Bricks

 

Limit State Design in Torsion: Equilibrium Torsion and Compatibility Torsion

Introduction

In reinforced concrete structures, torsion often occurs in combination with flexure shear. While pure torsion, as seen in metal shafts, is rare in reinforced concrete, the interaction of torsion with bending moments and flexural shear in concrete beams is complex. To simplify design, codes provide streamlined procedures, blending theory and experimentation. This chapter explores the general behavior of reinforced concrete beams under torsion and elucidates the concepts of equilibrium torsion and compatibility torsion.

Equilibrium Torsion and Compatibility Torsion

Torsion can manifest in various ways during load transfer in structural systems. In reinforced concrete design, two terms, namely "equilibrium torsion" and "compatibility torsion," describe different torsion-inducing situations. Equilibrium torsion arises from eccentric loading, relying solely on equilibrium conditions to determine twisting moments. Compatibility torsion, on the other hand, is induced by an angle of twist, and the resulting twisting moment depends on the torsional stiffness of the member.

In certain situations, both equilibrium and compatibility torsion may coexist, such as in circular beams supported on multiple columns.

Equilibrium Torsion

Equilibrium torsion involves twisting moments developed in a structural member to maintain static equilibrium with external loads. This torsion is independent of the torsional stiffness of the member. The magnitude of the twisting moment is determined by statics alone, and the member must be designed to resist this full torsion. Common scenarios for equilibrium torsion include beams supporting lateral overhanging projections or beams with curved plans subjected to gravity loads.

Ends of the member must be suitably restrained to effectively resist induced torsion.

Compatibility Torsion

Compatibility torsion is induced by rotations applied at one or more points along the length of the member. The twisting moments induced are directly dependent on the torsional stiffness of the member. Analysis involves compatibility conditions due to rotational deformations. Torsional stiffness is significantly reduced by torsional cracking, allowing designers to simplify structural analysis by neglecting torsional stiffness. However, to control cracking and enhance ductility, minimum torsional reinforcement is recommended.

Estimation of Torsional Stiffness

Torsional stiffness in reinforced concrete members is influenced by the amount of torsional reinforcement. In the linear elastic phase, torsional stiffness is similar to that of the plain concrete section. However, once torsional cracking occurs, there is a drastic reduction in stiffness, emphasizing the importance of proper torsional reinforcement.

In conclusion, understanding equilibrium and compatibility torsion is crucial for designing safe and resilient reinforced concrete structures. Equilibrium torsion relies on static equilibrium conditions, while compatibility torsion considers deformations induced by twists, requiring an accurate estimation of torsional stiffness. Striking a balance between these torsional considerations ensures the integrity and durability of reinforced concrete members under various loading conditions.

Important Civil Engineering All Formulas

 Important Civil Engineering All Formulas

Spire test in Theodolite

Spire test in Theodolite 

Condition- To make the horizontal axis perpendicular to vertical axis.

Necessity- By mean of the 2nd and 3rd adjustment, we ensure that line of sight will revolve in vertical plane. The adjustment become essential in all work necessitating motion of the telescope in altitude.

Spire test-

1. Set up the instrument over high building or other object on which there is defined point at a considerable altitude such as flag pole , lighting etc.
2. And mark a well defines point A at a considerable height.
3. Level the instrument accurately thus making vertical axis truly vertical.
4. Sight the point as shown in fig and horizontal motion clamped depress the telescope and set a point P on or near the ground.
5. Unclamped and transit the telescope and swing through 1800 with telescope inverted again sight on P.
6. Depress the telescope as before, if the light of sight fall on P, the horizontal axis is perpendicular to vertical axis.

Adjustment

1. It not marks another point Q in the line of sight on the wall at the same level as P.
2. Mark point R midway between P and Q sight on point R.
3. Clamp the upper motion.
4. Raise the telescope.
5. The line of sight will now strike the point A.

Permanent Adjustments of A Level

 Permanent Adjustments of A Level

 
The permanent adjustments of different level are made to establish the fixed relationships between its fundamental lines. It indicates the rectification of instrumental errors In a dumpy level, there are only two adjustments as the telescope is rigidly fixed to the spindle. 1. The axis of the bubble tube should be perpendicular to the vertical axis 2. The line of collimation should be parallel to the axis of the bubble tube.

First Adjustment: To make the axis of the bubble tube perpendicular to the vertical axis. Object: The object of this adjustment is to ensure that if the instrument is once levelled up, the bubble remains in the centre of its run for all positions of the telescope. Necessity: The adjustment is made only for the convenience of taking readings quickly. Since it is necessary that the bubble should be central while taking any reading, much time is wasted if this adjustment is not made as in that case the bubble has to be brought in centre every time for each pointing of telescope. Test: (i) Set-up the level on firm ground and level it carefully by tripod-legs and foot-screws. The bubble will now be central in two positions at right angles to each other, one being parallel to a pair of foot-screw and the other over the third foot-screw. (ii) Bring the telescope over a pair of foot-screws or over the third foot-screw and turn it through 180 in the horizontal plane. If the bubble still remains central, the adjustment is correct. 

Adjustment: (i) If the bubble does not remain in the centre, note down the deviations of the bubble from the centre, say it is ‘2n’ division over the bubble half way back i.e., ‘n’ divisions by raising or lowering end of the bubble tube by means of capstan headed must and the remaining half with the pair of foot-screws beneath the telescope at its present position. (ii) Turn the telescope through 90° so that it lies over the single foot- screw below the telescope or parallel to a pair of this screw or pair of foot -screws and bring the bubble in the centre of its run by means of this screw pair of foot-screws. (iii) Rotate the telescope and see if the bubble remains central for all positions of the telescope. If not repeat the whole process until the adjustment is correct.

Second Adjustment: To make the Line of collimation parallel to the axis of the bubble tube Object: The object of this adjustment is to set the line of collimation parallel to the bubble axis so that when the bubble is centered, the line of collimation should become exactly horizontal and not remain inclined as otherwise it would be. Necessity: The whole function of a level is to furnish a horizontal line of collimation, which is possible only if the above condition is satisfied. Test and Adjustments: The collimation error may be tested by any of the following three methods and then the necessary adjustments are made (concentrate on Two-Peg Method)

Two-Peg Method.: Test: (i) Drive two pegs A and B at a distance of (D) metres say 60 to 100 metres on a fairly level ground. Drive another peg at O exactly midway between A and B (ii) Set up and level the instrument at O and take the staff readings on A and B. The bubble must be in the centre while the readings are being taken. Let the staff readings on A and B, be a and b respectively. (iii) Shift the level and set it up a point O1 , d metres away from A (or B) and along the same line BA (Fig. 7.37). levels the instrument accurately and take staff readings on A and B with the bubble central. Let the readings be a1 and b1 respectively. (The level may also be set up at a point between A and B, d metres away from A or B)

 (iv) Find the difference between the staff readings a and b, and that between the staff readings a1 and b1 . The difference of staff readings a and b gives the true difference in elevation between A and B as the instrument was set up exactly midway between A and B and that the back and for sight distances were exactly difference, whereas the difference between a1 and b1 gives the apparent difference. If the two differences are equal, the line of collimation is in adjustment, otherwise it is inclined and needs adjustment. Adjustment: (i) Find out whether the difference is a rise or a fall from the peg A to B. If a is greater than b, the peg A is lower than peg B and the ground is rising from A to B. If b is greater than a, the ground is falling from A to B. (ii) Find out the reading on the far peg B at the same level are of a1 by adding the true difference to a1 if it is a fall, or by subtracting the true difference from a1 if it is a rise. Let the reading be b2

(iii) If b1 is greater than b2 , the line of collimation is Inclined upwards and if b1 is smaller than b2 , the line of collimation is inclined downwards. b1 – b2 (difference of b1 and b2 ) is the collimation error in the distance “D”. ∴ the collimation error for unit distance:

(iv) The corrections to be applied for readings on the pegs A and B may be found out as under:

These corrections are additive if the of collimation is inclined downward and subtractive if the same is inclined upwards.

 

Measurement of Magnetic Bearing and Vertical Angles by using Theodolite.

 

Surveying-Levelling

 

Study of Theodolite

 

Measurement of Horizontal Angles by Different Methods by using theodolite.

 

Experiment using Total Station