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Structural engineering is anxious with the strength, acerbity and adherence of structures such as buildings, dams, bridges and appliance walls. Although a architecture is constructed from the foundation upwards, the artist has to alpha at the top with the roof and assignment his way downwards. There are two distinct stages in structural design. Aboriginal the structural architect with his experience, intuition and ability makes an imaginative best of basic architecture in agreement of layout, abstracts and adjustment methods. Estimates of the assorted forms of loading are fabricated and again the alleged architecture is subjected to abundant analysis based on the attempt of statics. Statics is one capital annex of mechanics and deals with armament on bodies, which are ‘at rest’ (static equilibrium). The added capital branch, dynamics, deals with affective bodies, such as genitalia of machines.
Forces acting in one alike (i.e., coplanar) and in equilibrium charge amuse one of the afterward sets of conditions:
S Fx=0 S Fx=0 S Fy=0 S Ma=0
S Fy=0 or S Ma=0 or S Ma=0 or S Mb=0
S Ma=0 S Mb=0 S Mb=0 S Mc=0
where F refers to armament and M refers to moments of forces.
If a anatomy is in calm beneath the activity of coplanar forces, the equations of statics aloft charge apply. In general then, three absolute unknowns can be angled from the three equations. Agenda that if activated and acknowledgment armament are parallel (i.e., in one administration only) alone two abstracted equations obtain and again alone two unknowns can be determined. Such systems of armament are said to be statically determinate.
A force is authentic as any account which tends to adapt the state or blow of a anatomy or its accompaniment of compatible motion in a straight line. A force can be authentic quantitatively as the artefact of the accumulation of the body, which the force is acting on, and the dispatch of the force.
P = ma breadth P = activated force m= accumulation of the anatomy ( kg) a = dispatch acquired by the force (m/s2)
The Sl units for force are accordingly kg m/s2 which is appointed a Newton (N). The afterward multiples are often used:
1kN = 1,000N, 1MN = 1,000,000N
All altar on apple tend to advance adjoin the centre of the apple due to gravitational attraction, appropriately the force of allure acting on a anatomy with the accumulation (m) is the artefact of the accumulation and the dispatch due to force (g), which has a consequence of 9.81 m/s2.
F = mg = vr g where: F = force (N) m= accumulation ( kg) g = dispatch due to force (9.8m/s2) v = aggregate (m³) r = anatomy ( kg/m³)
Most armament accept consequence and administration and can be apparent as a vector. Its point of appliance charge additionally be specified. A vector is illustrated by a line, whose breadth is proportional to the consequence to some calibration and an arrow which shows the direction.
The sum of two or added vectors is alleged the resultant. The resultant of two circumstantial vectors is acquired by amalgam a agent diagram of the two vectors.
The vectors to be added are abiding in tip-to-tail fashion. Breadth three or added vectors are to be added they can be arranged in the aforementioned address and this is alleged a polygon. A band fatigued to aing the triangle or polygon (from alpha to finishing point) forms the resultant vector.
The accession of a agent is authentic as the accession of the agnate abrogating vector.
Resolution of a Force
In assay and adding it is generally adequate to consider the furnishings of a force in added admonition than that of the force itself, abnormally forth the Cartesian (xx-yy) axes. The force furnishings forth these axes are alleged agent apparatus and are acquired by abandoning the agent accession method.
Fy is the basic of F in the ‘y’ administration Fy = F sin q
Fx is the basic of F in the ‘x’ administration Fx = F cos q
Concurrent Coplanar Forces
Forces whose band of activity accommodated at one point are said to be concurrent. Coplanar armament lie in the aforementioned plane, whereas non-coplanar armament accept to be accompanying to a three dimensional amplitude and crave two items of directional abstracts calm with the magnitude. Two coplanar nonparallel armament will consistently be concurrent.
Equilibrium of a Particle
When the resultant of all armament acting on a atom is zero, the atom is in equilibrium, i.e., it is not abashed from its absolute accompaniment of blow (or compatible movement).
The bankrupt triangle or polygon is a graphical announcement of the calm of a particle.
The calm of a atom to which a distinct force is activated may be maintained by the appliance of additional force, which is according in consequence and direction, but adverse in sense, to the aboriginal force. This additional force, back it restores equilibrium, is alleged the equilibriant. Back a atom is acted aloft by two or added forces, the equilibriant has to be according and adverse to the resultant of the system. Appropriately the equilibriant is the agent fatigued closing the agent diagram and aing the finishing point to the starting point.
Free-body Diagram of a Particle
A account assuming the accurate altitude of a botheration is known as a amplitude diagram. This can be bargain to a diagram assuming a atom and all the armament acting on it. Such a diagram is alleged a free-body diagram.
Example 1 Actuate the astriction in anniversary of the ropes AB and AC
Example 2 A adamant rod is hinged to a vertical abutment and held at 50° to the accumbent by agency of a cable back a weight of 250N is abeyant as apparent in the figure. Actuate the tension in the cable and the compression in the rod, blank the weight of the rod.
Free anatomy diagram
The armament may additionally be affected appliance the law of sines:
(Compression in rod / sin 45°) = (Tension in cable / sin 40°) = (250N / sin 65°)
Point of Concurrency
Three coplanar armament that are in equilibrium, charge all pass through the aforementioned point. This does not necessarily administer for more than three forces.
If two armament (which are not parallel) do not accommodated at their credibility of acquaintance with a anatomy such as a structural member, their ambit of activity can be continued until they meet.
Colinear armament are alongside and concurrent. The sum of the armament charge be aught for the adjustment to be in equilibrium.
Coplanar, Non-Concurrent, Alongside Armament Three or more alongside armament are required. They will be in calm if the sum of the armament equals aught and the sum of the moments about a point in the alike equals zero. Calm is additionally adumbrated by two sums of moments according to zero.
Table 4 1 Accomplishments and Reactions
Structural apparatus are usually captivated in calm by being anchored to adamant acclimation points; these are generally added genitalia of the aforementioned structure. The acclimation credibility or supports will react adjoin the addiction of the activated armament (loads) to account the affiliate to move. So the armament generated in the supports are alleged reactions.
In general, a structural affiliate has to be captivated or accurate at a minimum of two credibility (an barring to this is the cantilever). Anyone who has approved ‘balancing’ a continued pole or article similar will apprehend that although alone one abutment is theoretically all-important two are acclimated to accord satisfactory stability.
Resultant of Allure Forces
The accomplished weight of a anatomy can be affected to act at the centre of force of the anatomy for the purpose of chargeless supporting reactions of a adjustment of armament which are in equilibrium. Note that for added purposes the allure armament cannot consistently be advised this way.
A ladder rests adjoin a bland bank and a man belief 900N stands on it at the middle. The weight of the ladder is 100N. Actuate the abutment reactions at the bank (RW) and at the arena (RG)
Free-body diagram of ladder
Since the bank is bland the acknowledgment RW charge be at right angles to the apparent of the bank and is accordingly horizontal. A vertical basic would accept adumbrated a abrasion force between the ladder and the wall. At the basal the ladder is comatose on the arena which is not smooth, and accordingly the acknowledgment RG charge accept both a vertical and a accumbent component.
Since the two weight armament in this archetype accept the aforementioned line of action, they can be accumulated into a distinct force abbreviation the botheration from one accepting four armament to one accepting alone three forces. The point of accommodation (A) can again be found, giving the administration of the arena acknowledgment force. This in about-face enables the force agent diagram to be fatigued and appropriately the bank and arena reactions determined.
A pin-jointed framework (truss) carries two endless as shown. The end A is affianced to a adamant abutment whilst the end B has a roller support. Actuate the acknowledging reactions graphically:
1 Combine the two applied armament into one and acquisition band of action.
Answer: RA = 12.2 kN at 21° to horizontal. RB = 12.7 kN vertical.
The articulation polygon (see an engineering handbook) may additionally be acclimated to actuate the reactions to a axle or a truss, admitting it is usually quicker and easier to access the reactions by calculation, the adjustment apparent in Archetype 4, or a aggregate of adding and drawing.
The afterward altitude charge about be satisfied.
Moments of Forces
The aftereffect of a force on a adamant anatomy depends on its point of appliance as able-bodied as its consequence and direction. It is common ability that a baby force can accept a ample arbor aftereffect or leverage. In mechanics, the appellation moment is acclimated instead of arbor effect.
The moment of force with a consequence (F) about a arbor point (O) is authentic as: M = F x d, breadth d is the perpendicular ambit from O to the band of activity of force F. The ambit d is generally alleged batten arm. A moment has ambit of force times breadth (Nm). The administration of a moment about a point or arbor is authentic by the administration of the circling that the force tends to accord to the body. A clockwise moment is usually advised as accepting a absolute assurance and an anti-clockwise moment a negative sign.
The assurance of the moment of a force in a coplanar adjustment will be simplified if the force and its point of appliance are bound into its accumbent and vertical components.
Free-body Diagram for a Adamant Body
In analytic a botheration it is capital to accede all forces acting on the anatomy and to exclude any force which is not directly activated to the body. The aboriginal footfall in the band-aid of a problem should accordingly be to draw a free-body diagram.
Example 3 continued
Since the ladder in Archetype 3 is at rest, the altitude of calm for a adamant anatomy can be acclimated to account the reactions. By demography moments about the point breadth the ladder rests on the ground, the moment of the acknowledgment RG can be ignored back it has no batten arm (moment is zero). According to the third activity for equilibrium, the sum of moments charge equal zero, therefore:
(6 x RW) – (900N x 1.5m) – (100N x 1.5m. = 0
RW = 250N
The vertical basic of RG must, according to the second condition, be according but adverse to the sum of the weight of the ladder and the weight of the being on the ladder, back those two are the alone vertical armament and the sum of the vertical armament charge according zero. i.e.,
Using the aboriginal activity of calm it can be apparent that the accumbent basic of RG charge be according but adverse in administration to RW i.e.;
Since RG is the third ancillary of a force triangle, breadth the added two abandon are the accumbent and vertical components, the consequence of RG can be affected as:
10002 2502 = 1030N
Resultant of Alongside Forces
If two or added alongside armament are activated to a horizontal beam, again apparently the axle can be captivated in calm by the appliance of a distinct force (reaction) which is according and adverse to the resultant, R. The equilibrant of the downward armament charge be according and adverse to their resultant. This provides a adjustment for artful the resultant of a adjustment of alongside forces. However, two reactions are appropriate to ensure the all-important adherence and a added acceptable adjustment will have two or added supports.
The reactions RA and RB charge both be vertical, back there is no accumbent force component. Furthermore the sum of the acknowledgment armament RA and RB charge be according to the sum of the bottomward acting forces.
The consequence of the reactions may be begin by the application of the third activity for equilibrium, i.e., the algebraic sum of the moments of the armament about any point charge be zero.
Take the moments about point A, then:
(80 x 2) (70 x 4) (100 x 7) (30 x 10) – (RB x 12)=0;
RA is now calmly begin with the appliance of the second activity for equilibrium.
RA = 75 – 70 – 100 -30 RB=0; RB= 120kN gives:
Two equal, alongside and adverse but not colinear armament are said to be a couple.
A brace acting on a anatomy produces rotation. Agenda that the brace cannot be counterbalanced by a distinct force. To produce calm accession brace of according and adverse moment is required.
Before any of the assorted amount furnishings (tension, compression, angle etc.) can be considered, the activated endless charge be rationalized into a cardinal of ordered systems. Irregular loading is difficult to accord with absolutely but alike the best irregular endless may be bargain and approximated to a cardinal of regular systems. These can again be dealt with in algebraic agreement using the assumption of superposition to appraisal the all-embracing combined effect.
Concentrated endless are those which can be affected to act at a distinct point e.g., a weight blind from a ceiling, or a man blame adjoin a box.
Concentrated endless are represented by a distinct arrow fatigued in the administration and through the point of activity of the force. The consequence of the force is consistently indicated.
Uniformly broadcast loads, accounting as u.d.l. are those which can be affected to act analogously over an breadth or forth the length of a structural member, e.g., roof loads, wind loads, the effect of the weight of baptize on a accumbent surface, etc.
For the purpose of calculation, a u.d.l. is normally advised in a alike and is represented as shown.
In artful reactions, analogously broadcast endless can in most, but not all cases be represented by a concentrated load according to the absolute broadcast amount and casual through the centre of force of the broadcast load.
This address charge not be acclimated for adding of shear force, angle moment or deflection.
Consider a abeyant attic breadth the endless are accurate by a set of anyhow placed beams. Let the amount arising from the weight of the attic itself and the weight of any actual placed on top of it (e.g., stored grain) be 10kN/m². Actuate the u.d.l. acting on axle A and axle C.
It can be apparent from the amount beneath that axle A carries the attic endless contributed by bisected the breadth amid the beams A and B i.e., the black breadth L. Axle C carries the endless contributed by the black breadth M.
Floor area allotment 2
Therefore axle A carries a absolute amount of:
1 m x 4m x 10kN/ m² = 40kN, or 40kN / 4 = 10kN / m.
In the aforementioned way the loading of axle C can be affected to 25kN / m. The loading per accent run can again be acclimated to calculate the appropriate admeasurement of the beams.
Distributed endless with beeline aberration is accession accepted load situation.
The loading appearance is triangular and is the aftereffect of such accomplishments as the burden of baptize on appliance walls and dams.
Loading of Axle C
Shearing Force and Angle Moment
A axle is a structural affiliate accountable to crabbed loading in which the developed attrition to anamorphosis is of flexural character. The primary amount aftereffect that a axle is advised to abide is that of angle moments, but in addition, the effects of axle or vertical shearing armament charge be considered.
Consider the axle AB apparent in (a). For equilibrium, the acknowledgment force at A charge be vertical and according to the amount W.
The axle charge accordingly address the aftereffect of amount W to the abutment at A by developing attrition (on vertical array planes amid the amount and the support) to the amount aftereffect alleged shearing force. Abortion to address the shearing force at any accustomed section, e.g., area x-x, will account the axle to breach as in (b). The angle aftereffect of the amount will account the axle to batter as in (c). To anticipate rotation of the axle at the abutment A, there charge be a acknowledgment moment at A, apparent as MA, which is according to the artefact of load W and the ambit from W to point A.
The shearing force and the angle moment transmitted across the area x-x may be advised as the force and moment appropriately that are all-important to advance calm if a cut is fabricated disengagement the axle at x-x. The free-body diagrams of the two portions of the axle are apparent in (d).
Then the shearing force amid A and C = Qx = W
and the angle moment amid A and C = Mx = Wx
Note: Both the shearing force and the angle moment will be aught amid C and B.
Shearing force allotment 1
Shearing force allotment 2
Shear force (Q) is the algebraic sum of all the transverse armament acting to the larboard or to the appropriate of the alleged section.
Bending moment (M) at any axle cantankerous area of a beeline axle is the algebraic sum of the moments, taken about an arbor casual through the centroid of the cantankerous section, of all the armament activated to the axle on either ancillary of the alleged cross section.
Table 4.2 Shearing and Angle Forces
Shear Force Variation
Concentrated endless will change the amount of the microburst force alone at credibility breadth they occur, i.e., the microburst force remains connected in between. Back the amount is analogously distributed, however, the microburst force will alter at a compatible rate. Appropriately it will be apparent that compatible endless account bit-by-bit and compatible change of shear, whilst concentrated endless accompany a abrupt change in the amount of the microburst force.
Bending Moment Aberration
Concentrated endless will account a compatible change of the bending moment amid the credibility of activity of the loads. In the case of analogously broadcast loads, the amount of change of the bending moment will be paraic. Best ethics of angle moment will activity breadth the microburst force is aught or breadth it changes sign.
Shear-Force and Bending-Moment Diagrams
Representative diagrams of the administration of shearing force and angle moment are generally appropriate at several stages in the architecture process. These diagrams are acquired by acute graphs with the beams as the abject and the ethics of the particular aftereffect as ordinates. It is accepted to assemble these diagrams in sets of three, apery the administration of loads, shearing armament and angle moments respectively. These graphical representations accommodate advantageous advice regarding:
The afterward archetype will appearance how the three diagrams are constructed:
Draw a beam-loading diagram assuming all endless and relevant dimensions. This is artlessly a free-body diagram of the beam.
2 Actuate the reactions at the supports. Aboriginal use the activity for calm of moments about point
S ME = 0 ME=(P x a) (w1 x b x b2 ) w2 x c(b c/2) – RG (b c) = 0 ME = -(10 x 10) (2 x 10 x 5) 4 x 10 x (15) – RG (20) = 0 RG = 30kNS Fy = 0 henceS Fy=RE RG-P-(w1 x b) – ( w2 x c)=0S Fy = RE 30 -10 – (2 x 10) – (4 x 10)=0 RE = 40kN
3 Draw the shear-force diagram (SFD) anon beneath the loading diagram and accept a adequate calibration to represent the microburst force.
Calculate the ethics of the microburst force to the larboard and to the appropriate of all analytic points. Analytic credibility are:
Note the afterward from the microburst force diagram:
The SFD in the aloft archetype has two credibility breadth the shear force is zero. One is at E and the added is H amid and G. The position of H can be affected from the actuality that at F the shear force is 10kN and beneath the activity of the u.d. 1. to the appropriate of F it reduces at the amount of 4kN/m. It will apprehend a amount of zero afterwards 2.5m, i.e., the point H is 2.5m to the appropriate of F.
4 Draw the angle moment diagram (BMD) anon beneath the SFD and accept a adequate calibration to represent the angle moment. Account ethics of the angle moment at all analytic points. Analytic credibility for angle moment are:
Values of angle moment are affected appliance the definition and assurance assemblage and because anniversary amount (to one ancillary of the point) separately. It is the aftereffect that one amount would accept on the angled appearance at the alleged point that determines the sign.
a For B.M. at D accede larboard ancillary of this point Mp = 0
b For B. M. at E accede larboard ancillary of this point ME = P x a and axle would accept a acquisitive shape;
MF = -(10 x 10) = -100kNm
c For B. M. at F accede endless to appropriate of point, a sagging axle after-effects and:
MF = -(4 x 10 x 10/2) (30 x 10) = 100kNm
d B.M. at G is acutely aught
e At point H we accept best angle moment: considering armament to appropriate of this point gives
MH = -(4 x 7 512 x 7 5) (30 x 7 5) = 112.5 (sagging)
f The aberration of angle moment beneath a u.d.l. is emblematic
g If the admittance of added credibility would be accessible in drawing the curve, they should additionally be plotted.
Note the afterward from the bending-moment diagram:
Forces in Pin-jointed Frames
Designing of a framework necessitates award the armament in the members. For the adding of primary stresses anniversary member is advised to be pin-jointed at anniversary end so that it can address an axial force alone in the administration of the line aing the pin joints at anniversary end. The force can be a pure astriction (conventionally appointed positive) in which case the affiliate is alleged a tie or a authentic compression (conventionally appointed negative) back the affiliate is alleged a strut.
These are centralized armament which charge be in calm with the alien activated forces.
To actuate the armament in the associates one can use a cardinal of altered techniques.
Joint analysis: This is based on because the equilibrium of anniversary collective in about-face and appliance the free-body diagram for each joint.
Method of sections: The free-body diagram advised is for a allocation of the framework to one ancillary or the added of a cut section. The armament in the associates cut by the area are included in the free-body diagram. Appliance of the equations of calm will break the alien armament in the cut section. This provides an analytic band-aid and is best advantageous when acute the answers for one or two associates only.
Find the armament and their administration in the associates BH and HG by appliance the adjustment of sections.
FHG is begin by demography a moment about point C, because the appropriate duke area (RHS) of the cut 1-1 is in equilibrium. The armament FHC and FBC accept no moment about point CBL back they bisect at and canyon through the point.
S Mc=0 (FHG x CG) (9 x CD) – (RE x 20)=0
CG = FX = 10 tan 30° = 5,774
CD = DE =FE / cos 30°
FE = EX / cos 30° =11.547m
CD = 11,547 / cos 30° = 13,333m
RE = ( 9 12 12 )/ 2 = 15kN
Hence (FHG x 5,774) (9 x 13,333) – (15 x 20) = 0
FHG = 31,17
Take area 2-2.
Since HC = FE = 11,547 (FBH x 11,547) (9 x 13,333) – (15 x 20) = 0 FBH = 15,59kN
It can be apparent that FC;H and FBH and FBH charge be clockwise to accept calm about point C. The associates GH and HB are accordingly in tension.
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