The Analysis Of Beams Frames And Trusses Engineering Essay

Structural elements are used to divide a complex construction into simple elements in structural analysis. An construction component can non be broken decomposed into parts of different sorts. A construction is something that will. A construction supports an object or a burden and must be strong plenty to back up its ain weight and appliedA burden.

2.1 Radio beams

A beam is a long slender structural component that is capable of defying tonss chiefly by defying bending. The flexing force induced into the stuff of the beam as a consequence of the external tonss, ain weight, span and external reactions to these tonss.The applied tonss can be shears, flexing minute and tortuosity Beams must hold an equal safety border against other types of failure, some of which may be more unsafe than flexural failure.

The term flexural failure with respects to concrete beams can hold two distinguishable mechanisms. Concrete beams are reinforced with reenforcing steel bars located in the tenseness zones of the beam. Concrete is assumed non to take any tenseness, but is good at defying compaction. The current American Concrete Institute Code ( ACI ) requires that the beam be reinforced such that failure is initiated by tenseness giving up of the rebar instead than the explosive suppression of the concrete in compaction. The ground for this codification demand root from the fact that it is a more benign manner for the beam to neglect. It will be noticed as a sagging or debaring beam component good before the steel rebar fails wholly, whereas the devastating failure of concrete will give no warning and could be ruinous in footings of life and safety of the public.Shear failure of strengthened concrete, more decently called “ diagonal tenseness failure ” is one illustration.

2.1.2 Types of Beams

deep beam

RC deep beams, which fail with shear compaction, are the structural members holding a shear span ( orangish line ) to effectual deepness ( brack line ) ratio, a/d, non transcending 1.

Normally the shear action which is really important in RC deep beams leads to the compaction failure along compressive arch linking between point of burden and supports, particularly for the beam failed in shear compaction manner. The uniaxial compressive stress-strain curve obtained from trials presuming the unvarying distortion throughout a concrete specimen is by and large used as the stuff theoretical account for concrete in compaction. However, this is non suited in patterning the falling subdivision of the curve since it was found that the distortion after the peak emphasis is localized within certain zones and depends on the gage length and place of the gage, in other words, it is size-dependent. Tension emphasiss, which are of peculiar concern in the position of the lowtensile capacity of the concrete are non confined merely due to the horizontal bending emphasiss f which are caused by flexing alone.Tension emphasiss of assorted magnitude and dispositions, ensuing from iˆ shear entirely ( at the impersonal axis ) ; or iˆ the combined action of shear and flexing exist in all parts of a beam and if non taken attention of suitably will ensue in failure of the beam. It is for this ground that the inclined tenseness emphasiss, known as diagonal tenseness, must be carefully considered in strengthened concrete design. For longer shear spans in apparent concrete beams, clefts due to flexural tensile emphasiss would happen long earlier clefts due to diagnol tenseness


At present, there are no recommendations in codifications such as ACI 318 and BS 8110 to gauge the critical buckling minute in slender concrete beams. It is assumed that if the slenderness ratios of the beams are limited to the values prescribed by the codifications, the failure minute of the beam will be dictated by flection and non by clasping. With respect to slender RC beams, nevertheless, there are no comprehensive design commissariats at nowadays in the design codifications. Some bounds on slenderness have been prescribed in some codifications. But no such design commissariats soon exist in the instance of slender concrete beams. To develop a suited design footing for such slender RC beams, it is necessary to foretell the critical buckling minutes of RC slender beams and to foretell the conditions under which instability failure governs the ultimate burden transporting

2.2 Columns

2.2.1 Definition

A column is a perpendicular structural component that supports axial compressive burden coming from constructions above it. Harmonizing to ACI codification a structural component is referred as columns holding a ratio of tallness to least sidelong dimension of 3 or greater. For the intent of air current or temblor technology, columns may be designed to defy sidelong forces. A column either crushes ( a strength falure ) or it buckles ( a stableness failure ) . Both manners of failure must be considered for every column. The exact manner of failure is greatly dependent upon the manner in which the column ‘s cross-sectional country is distributed with regard to its centroid

A column either crushes ( a strength falure ) or it buckles ( a stableness failure ) . Both manners of failure must be considered for every column. The exact manner of failure is greatly dependent upon the manner in which the column ‘s cross-sectional country is distributed with regard to its centroid. The undermentioned simple construct must be satisfied at all times:

Stress due to lading & lt ; Resistance potency of the column

This states that the emphasis within the column due to all of the applied tonss must be less than the allowable emphasis of the stuff. This is a logical statement that is the kernel of the structural analysis of a column. The existent failure mechanism could be due to a combination of two or more tonss. These combinations must be carefully considered.

Column Failure Modes


Relatively short columns are more disposed to neglect by the stuff suppression. Every edifice material stuff can defy a distinguishable sum of compressive emphasis before it crushes. This value has been determined by research lab trials and is known as the compressive strength of a stuff. This strength is dependent upon the internal construction of the stuff and/or its constituents. Steel has a really homogenous, finely crystalled internal construction and has a comparatively high compressive strength. Hard forests are powdered and have a higher compressive strength than soft-woods. Wood is alone in that it has two compressive strengths ; one when loaded parallel to the grain and another when loaded perpendicular to the grain. Why is that? When a wood column crushes the fibres of the wood really disconnected apart. In every instance, suppression is a strength failure and does non depend upon the form of the subdivision.


Relatively slight columns are more disposed to neglect by clasping. A column is slender when it has a “ little ” cross-section compared to its effectual length. Small is placed in quotation marks due to the fact that the of import information about the cross subdivision is both the existent size and more significantly, the form of the cross-section. This is so compared to the effectual length to find whether or non the column is slender. If it is, this means that the column will likely neglect in flexing! As a column is loaded, it is likely to flex about the weak axis of the cross-section ( the 1 with the lowest Moment of Inertia ) . A column buckles when it bends about an axis. This is a stableness failure.


There are by and large two types of column burden: axial and bizarre. Axially laden columns may neglect either by oppressing or clasping. Eccentrically laden columns normally fail by clasping. The figure below illustrates the emphasis that a column experiences as a burden, N, is applied with increasing eccentricity. Note how the signifier of the emphasis prism alterations from an even distribution to a really uneven distribution.

consequence of bizarre lading upon the emphasis prism

The instance at the left illustrates how the axial burden creates a compressive emphasis which is equally distributed across the column ‘s subdivision. The burden on each column to the right has an increasing eccentricity. As the burden moves off from the centroidal axis, it introduces a bending minute which the column ‘s cross-section must besides defy. Therefore, one can see that one side of the column receives more compaction than the other. Equally long as the applied burden remains within what is known as the nucleus ( the middle 3rd ) of the subdivision, the column cross-section will merely hold compressive emphasiss. Tesion emphasiss are introduced every bit shortly as the applied burden moves out of the nucleus. The magnitude of the bending minute that the subdivision must besides defy additions as the eccentricity increases. The utmost instance is an infinite eccentricity ensuing in the pure flexing minute emphasis prism ( like that found in a beam ) that is seen in the instance furthest to the right.

Knowledge of the magnitude and distribution of the internal emphasis is of import for the size of the column. A little sum of tensile emphasis has small consequence on a wood or steel column, but jobs could get down to happen if the column is concrete or masonry.

2.2.2 Types of Columns

2.2.3 Types of Columns Harmonizing to Slenderness

1.Short Column

A column in which both compaction and bending is important, whose comparative dimensions guarantee that when it is overloaded it fails by oppressing, instead than clasping by and large holding a slenderness ratio between 30 and 120-150

2.Long Columns

A column so slight whose comparative dimensions guarantee that when it is overloaded it fails by clasping, instead than oppressing, by and large holding a slenderness ratio greater than 120-150.

For frame braced against side sway:

For Frame non braced against side sway:

Long column if klu/r & gt ; 34-12 ( M1/M2 ) or 40

Long column ifA klu/r & gt ; 22

Where K is slenderness factor, Lu is unsupported length, A and R is radius of gyration.A M1 and M2 are the smaller and larger terminal minutes. The value, ( M1/M2 ) is positiveA if the member is dead set in individual curve, negative if the member is dead set in dual curve.

2.2.4 Types of Columns Harmonizing to Loading conditions

1.Axially loaded columns.

When applied tonss act axially on columns.It may neglect either by oppressing or buckling.

Eccentrically loaded columns

2.Eccentrically loaded columns.

When minutes besides act with axial tonss on column, so the column normally fails by clasping.


2.3 Frames

2.3.1 Definition

A structural frame system is a combination of chiefly perpendicular and horizontal members that are designed to convey applied tonss to the land. The major constituents of the frame system are horizontal members, perpendicular members, and some kind of foundation. These members work together to defy both vertically and horizontally applied tonss. Vertical tonss are typically the consequence of the applied unrecorded tonss that the edifice is designed to incorporate every bit good as climactic tonss such as air current, snow, and seismal activity. Horizontal tonss are most normally applied by air current and seismal activity.


Portal frame is made chiefly utilizing steel or steel-reinforced precast concrete although they can besides be constructed utilizing laminated lumber such as glulam.. The connexions between the columns and the balks are designed to be moment-resistant, i.e. they can transport flexing forces.The top member is horizontal. Gable framePortal frame is made chiefly utilizing steel or steel-reinforced precast concrete although they can besides be constructed utilizing laminated lumber such as glulam.. The connexions between the columns and the balks are designed to be moment-resistant, i.e. they can transport flexing forces.The top member is inclined.

2.4 Trusss

A truss is a construction consisting one or more triangular units constructed with consecutive members whose terminals are connected at articulations referred to as nodes. External forces and reactions to those forces are considered to move merely at the nodes and consequence in forces in the members which are either tensile or compressive forces. Moments ( torsions ) are explicitly excluded because, and merely because, all the articulations in a truss are treated as revolutes. Structural analysis of trusses of any type can readily be carried out utilizing a matrix method such as the direct stiffness method, the flexibleness method or the finite component method.

The diagonal and perpendicular members form the truss web, and carry the shear force. Individually, they are besides in tenseness and compaction, the exact agreement of forces is depending on the type of truss and once more on the way of flexing. In the truss shown supra right, the perpendicular members are in tenseness, and the diagonals are in compression.Ideally, the members see no flexing minute or tortuosity. In world, the members in a truss do see a little sum of bending and distortion, nevertheless these tonss are little and the axial tonss ( tenseness or compaction ) are important.

2.4.2 Types of Truss.

The pitched truss, or common truss, is characterized by its triangular form. It is most frequently used for roof building. Some common trusses are named harmonizing to their web constellation. The chord size and web constellation are determined by span, burden and spacing. The parallel chord truss, or level truss, gets its name from its parallel top and underside chords. It is frequently used for floor building. A combination of the two is a abbreviated truss, used in hip roof building. A metal plate-connected wood truss is a roof or floor truss whose wood members are connected with metal connection home bases.

1.king station truss

The male monarch station truss is the simplest type ; the male monarch station consists of two angled supports tilting into a common perpendicular support.

2.queen station

The queen station truss similar to a male monarch station truss, truss adds a horizontal top chord to accomplish a longer span.

The most representative trusses are the Warren truss, the Pratt truss, and the Howe truss.

3.Warren truss

The most common truss for both simple and uninterrupted trusses. A Warren truss, patented by James Warren and Willoughby Monzoni of Great Britain in 1848, can be identified by the presence many equilateral or isoceles trigons formed by the web members which connect the top and bottom chords. These trigons may besides be farther subdivided. Warren truss may besides be found in covered span designs..

4.PRATT Truss

The Pratt truss is a really common type, but has many fluctuations. Originally designed by Thomas and Caleb Pratt in. The basic identifying characteristics are the diagonal web members which form a V-shape. The halfway subdivision normally has traversing diagonal members. Extra counter braces may be used and can do designation more hard, nevertheless the Pratt and its fluctuations are the most common type of all trusses. . Except for those diagonal members near the centre, all the diagonal members are capable to tenseness forces merely while the shorter perpendicular members handle the compressive forces. This allows for thinner diagonal members ensuing in a more economic design

5.Howe truss

A A Howe truss at first appears similar to a Pratt truss, but the Howe diagonal web members are inclined toward the centre of the span to organize A-shapes. The perpendicular members are in tenseness while the diagonal members are in compaction, precisely opposite the construction of a Pratt truss

2.5 Tonss

2.5.1 General

Structural tonss are forces applied to a constituent of a construction or to the construction as a unit.they are those forces for which a given construction should be proportioned.

2.5.2 Types of Loads

2.5.3. Gravity tonss

In common pattern, if the sidelong tonss were undistinguished the edifice elements are chiefly designed for gravitation tonss. Gravity tonss are usally perpendicular and inactive in nature.For high rise constructing sidelong burden design is besides performed. Gravity burden design is acceptable for low or mid rise edifices and short columns.They are of two types.The dead burden includes tonss that are changeless over clip, including the weight of the construction moving with gravitation on the foundations at a lower place, every bit good as other lasting tonss, including weight of walls.Live tonss, or imposed tonss, are impermanent, of short continuance, or traveling. Examples include snow, air current, temblor, traffic, motions, H2O force per unit areas in armored combat vehicles, and occupancied tonss. For certain specialised constructions, vibro-acoustic tonss may be considered.


For high rise edifice elements sidelong tonss design is performed with gravitation burden design.They are dynamic and horizontal in nature.The common types are air current and seismal loads.Wind burden is caused by the blowing air current and depends upon tallness and exposure of structure.Siesmic burden is due to weight of construction and varies with height.UBC codification gives equations for the transmutation of these tonss into tantamount inactive tonss.

Scope of our undertaking is limited to gravitation burden design merely, seismal burden design in non included.


For accurate construction analysis structural tonss, geometry, support conditions, and stuffs belongingss of import information required.stresses and supplantings are the consequences of these analysis. Advanced structural analysis may analyze dynamic response, stableness and non-linear behaviour.

There are three attacks to the analysis: the mechanics of stuffs attack ( besides known as strength of stuffs ) , the snap theory attack ( which is really a particular instance of the more general field of continuum mechanics ) , and the finite component approachFew names of structural analysis methods are:

1.Matrix Force method ( Force method analysis & A ; three minute Equations )

2.Direct stiffness method

3.displacement method ( slope warp & A ; Moment distribution method )

4. Energy Methods ( practical work & A ; Castigiliano ‘s theorem )

Finite component attack is used for the analysis of beams, frames and truss.

2.6 Finite Element Method

2.6.1 Introduction

Finite component method is a tool for foretelling the response of certain technology systems and mathematical natural philosophies. For jobs affecting complicated geometrics, burdens and stuff belongingss. It is by and large non possible to obtained analytical mathematical solutions. Analytic solutions are those given by a mathematical looks that yields the value of the coveted unknown measures at any location. In a organic structure and are therefore valid for an nfinite Numberss of locations in a organic structure. These analytical solutions by and large require the solution of ordinary or partial differential equations. , which because of the complicated geometrics, lading and material belongingss are non normally gettable.

2.6.2 Finite Element Analysis

The finite component analysis involves patterning the construction utilizing little interrelated elements called finite elements. Our displacement maps is associated with each finite component. Every inter connected component is linked straight or indirectly to every other component through common ( or shared ) interfaces, including nodes and/or boundary lines and/or surfaces. By utilizing known emphasis strain belongingss for the stuff doing up the construction, one can find the behaviour of given node in footings of belongingss of every other elements in the construction. The typical set of equations depicting the behaviour of each node consequences in a series of algebraic equations best expressed in matrix notation.


The finite component method can be used to analyse both structural and nonstructural problems.Typical structural countries include:

emphasis analysis, including truss and frame analysis and stress concentration jobs typically with holes, filets or other alterations in geometry in a organic structure


Vibration analysis

Non-structural countries include:

heat transportation

fluid flow

distribution of electric or magnetic potency


Finite component method has the ability to work out many of the technology jobs, it may include:

Model irregularly shaped organic structures rather easy

Handle general burden conditions without trouble

Handle limitless Numberss and boundary conditions

Individually rating of element equations can hold a theoretical account organic structure composed of several different


Vary the size of elements to do possible to utilize little elements where necessary.

2.7 Finite Element Analysis stairss of Beam

2.7.1 Discretize & A ; choice component type

Divide the beam into an tantamount system of finite elements with associated nodes.Fig 2.7.1 ( a ) represent the beam by labeling nodes at each terminal in general by labeling the component

2.7.2 Select Displacement map

Choose a supplanting map utilizing the nodal values with each component. additive, quadratic and three-dimensional multinomials

( Eq. 2.7.1 )

2.7.3 Establish Element Stiffness matrix & A ; Equation

The stiffnes s matrix and elements equations associating nodal forces to nodal supplantings are obtained utilizing force equilibrium conditions for a basic component, along with force/deformation relationships.

( Eq. 2.7.2 )

( Eq. 2.7.1 )

( Eq. 2.7.1 )

2.7.4 Assemble Component equations to obtain Global Stiffness matrix & A ; introduce Boundary Conditions

The single component equations generated in 2.7.3 can now be added together utilizing a method for ace place ( called straight stiffness method ) , whose footing is to obtained planetary equations for the whole construction based upon nodal force equilibrium. implicit in the direct stiffness method is the construct of continuity or compatibility which requires that the construction remain together.

2.7.5 Solve for Nodal supplanting and Element forces.

A set of coincident algebraic equation can be written by set uping relation between external burden and planetary nodal supplantings, these equation can be solved for nodal supplanting and component forces

( Equation 2.7.3 )

2.7.6 Solve for component Strain and Stress.

For the structural emphasis analysis job, of import secondary measures of strain and emphasis ( or minute and shear force ) can be obtained because they can be straight expressed in footings of supplanting.

2.7.7 interpret the consequences

The concluding end is to construe and analyse the consequences for usage in design/analysis procedure.

2.8 Finite Element Analysis stairss of Frames

2.8.1 Discretize & A ; choice component type

See Section 2.7.1

2.8.2 Select Displacement map

See Section 2.7.2

2.8.3 Establish Element Stiffness matrix & A ; Equation

See Section 2.7.3.

( Equation 2.7.4 )

2.8.4 Assemble Component equations to obtain Global Stiffness matrix & A ;

introduce Boundary Conditionss

See Section 2.7.4


( Equation 2.7.5 )

( Equation 2.7.6 )

2.8.5 Solve for Nodal supplanting and Element forces.

See Section 2.7.5

2.8.6 Solve for component Strain and Stress.

See Section 2.7.6

2.8.7 interpret the consequences

See Section 2.7.7

2.9 Finite Element Analysis stairss of Trusss.

2.9.1 Discretize & A ; choice component type

See Section 2.7.1

2.9.2 Select Displacement map

See Section 2.7.2

2.9.3 Establish Element Stiffness matrix & A ; Equation

See Section 2.7.3

( Equation 2.7.7 )

2.9.4 Assemble Component equations to obtain Global Stiffness matrix & A ; introduce Boundary Conditions

See Section 2.7.4

( Equation 2.7.8 )

2.9.5 Solve for Nodal supplanting and Element forces.

See Section 2.7.5

2.9.6 Solve for component Strain and Stress.

See Section 2.7.6

2.9.7 interpret the consequences

See Section 2.7.7