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Type of connection Idealized symbol Reaction of constraints (1) light cable The reaction is a force that acts i

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Type of connection

Idealized symbol

Reaction of constraints

(1)

light cable

The reaction is a force that acts in the direction of the cable or link

F

smooth contacting surface

One Unknown The reaction is a force that acts perpendicular to the surface at the point of contact.

(3) Rollers F

Rocker

One unknown. The reaction is a force that acts perpendicular to the surface at the point of contact.

(4)

F

Smooth pin-connected collar

Two unknowns. The reaction are two force components

Fy Fx

smooth pin or hinge

Two unknowns. The reaction are a force & a moment.

(6) M F slider

B

M Fx

Fy

Static Inteterminancy: DS DSE DSi

No. of additional reactions reuired to analyse a structure is called static indeterminancy. Ds for 2D truss : m + re – 2j Ds for 3D truss : m + re – 3j Ds for 2D rigid frames : Ds = 3C – r where C – no. of cuts required to produce open stable tree

M BA M FBA

2EI 3 2B A l l

Loading Diagram MAB

Where j no. of joints. re reactions released rr reactions available at supports. nr no. of members axially rigid. Maxwell’s reciprocal theorem: It is a special cases of Bettis law. If only two froce P & Q are acting & magnitude of P and Q unit load at P PQ = Deflection at P due to unit load at

A

B

B

MAB

Pl 8

Pl 8

Wl 12

A

2

Wl 12

l1 I 1

A

2

l2 I2

D

B

M l3 I3

Wl 2 30

B

Wl 2 20

MAB

Q

l 2

MAB

Q2 P1Q

5 Wl 2 192

MAB

P a

A

b B

l=a+b

Pab l2

2

2

Pa b l2

MAB

P2Q

Q1

11 2 Wl 192

B l 2

PQ = PQ

Betti’s Theorem: In it, the virtual work done by a P-force system in going through deformation of Q - Force system is equal to the virtual work done by the Q-force system in going through the deformation of P-force system

Q2P

4EI1 3EI 2 4EI3 M K1 K 2 K 3 1 2 3

MAB

B

P2

K = M = 3EI L

Joint Stiffness Factor: Considering joint ‘D’ to be rigidly connected Applying moment ‘M’ at D

K

MAB

PQ

L

K = M = 4EI L MBA

M

MAB

Q QP

M

MAB

1 P

MAB

A

P1

Hinged

M = M1 + M2 + M3 ; M = K1 + K2 + K3

P l 2

l 2

are unity, then PQ QP where, QP = Deflection at Q due to

Q1P

Fixed

L

M FBA 3EI A 2 L L

Fixed end Moments.

W m

MAB A

5 2 Wl 96

B l 2

5 Wl 2 96

C

Distribution Factor (D.F.): Sum of DF for all members at a joint is always one Stiffness of a member

DF = Sumof stiffness of all members at that joint DF =

Re lative stiffness of a member Sum of Re lative stiffness of all members at that joint

Relative stiffness (a) when far end is fixed = I / l

3I 4l

(b) when far end is hinged =

MAB

D

Q-system of forces

P1P1Q P2 P2 Q Q1Q1P Q 2 Q 2 P

(7)

2EI 3 2A B l l

MBA

M AB M FAB

D k 6 j re rr n r

P-system of forces

Three unknowns. The reaction are the moment & the two force

M AB M FAB

E,I, l

l 2 fixed connected collar

=

2. When one end is pin supported

For 3D rigid frames:

Q

(5)

MAB

D k 3j re rr n r

P

Member Stiffness: Stiffness of a member AB when farther end is

B A

Refers to degree of freedom at all joint. For 2D rigid frames:

One unknown. The reaction is a force that acts perpendicular to the surface at the point of contact.

F

A

Kinematic Indeterminancy:

weightless link

(2)

1. Continuous Beam

like structure r – no. of retraints added to make structure perfectly rigid Ds for 3D rigid frames: Ds = 6C – r Ds for beam, Ds = Dse + Dsi, Ds = r – s (because beam have Dsi = 0)

Number of unknowns/ constraints One Unknown.

MAB a

Castigliano’s 1st theorem Castigliano’s 2nd theorem (a) The first partial derivative of (a) The first partial derivative total internal energy (strain of total internal energy in a energy) in a structure with structure with respect to the force respect to any particular applied at any point is equal deflection component at a to the deflection at the point point is equal to the force of application of that force applied at that point & in in the direction of its line the direction corresponding of action. to the deflection component. U U U U P or M or P M (b) Castigliano’s 1st theorem is(b) Castigliano’s 2nd theorem is applicable to linearly or nonapplicable to linearly elastic linearly elastic structures in (Hookean material structures which the temperature is with constant temperature constant & the supports & unyielding supports. are unyilding.

Mo b 2a b

b I=a+b

MAB

L2

M o a(2b a) L2

O

C

A

MAB

l 2

MD l 2

MAB

MD 4

MD 4

MAB

6EI l2

6EI l2

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MAB

MAB

B

0

3EI l2

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Carry Over Factor = (COF)

COF

Methods of Analysis of statically determinate truss

Where, HA = Horizontal thrust = HB

Carry over moment Applied moment

wl VB . 2

VA = Vertical reaction at A =

Standard Cases

Method of section It is used for solving the Fx= 0, Fy = 0 unknown force acting on the Analysis should start at members of a truss. The method joint having atleast one involved breaking the truss known force & at most down into individual sections and two unknown forces. analyzing each section as a separate rigid body

M/2 1 2

COF =

h1

M

H A

(ii)

B

M

A a

(iii)

VA

COF = M

Ma b

Note : Pratt truss is better than Howe truss as the diagonal member in Pratt truss carries tension but in Howe truss, diagonal member carries compression. If longer member carries compression, there is likely chance of buckling of truss member.

VB

a b

l1

l h1 h1 h 2

l h2

; l2

h1 h 2

wl 2

HA HB 2

h1 h 2

2

V

w HA HB sin2

o

(At crown 90 , H

w )

C

w R H

A VA

An arch is subjected to thrust, shear force & bending moment. A three hinged arch is subjected to normal thrust & radial shear and bending moment. A linear arch is subjected to normal thrust only.

w unit per run C

A

H

y

A x

x h L

wL VA = 2

B

H B

wL VB = 2

4h x(l x) l2 wl 2 WL HA HB ; VA VB 8h 2

Profile, y =

Wx 2 Hy Moment at only section × from A – Mx = VA x 2 Mx = 0

L

B

H

VB

Various types of Arches : A drop arch is pointed with a span greater than its radii. Two centered arch - The curve surface of these arches makes from two center points. Examples of two centered arches equilateral, pointed lancet and Venetian arches. A lancet arch is pointed, with radii much larger than the span. Three Centred arch - The curve surface of these arches is made from three center points. This arch is the more or less semi-elliptical arch. Ex. -Ogee arch, Drop arch, Semi-elliptical arch An ogee arch is pointed and usually of four arcs, the centers of two inside the arch & two outside; this produces a compound curve . It is also called as keel arch.

King post truss – The spacing of the king post truss is limited to 3m centre to centre. The truss is suitable for spans varying from 5 - 8 meters. Queen post truss – This truss differs from a king post truss having two vertical post, known as queen posts. This truss is suitable for 8-12m spans. Fink Truss – It is used for longer span having high pitch roof. The web members in such truss are Subdivided to obtain shorter members. Howe Truss – This type of truss is a combination of steel & wood, which makes it elegant, while also offering a very appealing design. It has a very wide span, as it can cover anything from 6-30m. Zero-force Members: 1. If three members join at a point & out of them, two are collinear & also no external load acts at joint, the third member is a zero force member. 2. If only two non-collinear members exist at a truss joint & no external force or support reaction is applied to the joint, the members must be zero force member. Deflection of Truss joint 1. Castigliano’s Method

U F

P 2 dx U i i 1 2 AE n

2. Maxwell’s unit load method

External virtual work Internal vitual wrok

Pdx 1 ui i li i ti i Ai Ei

AE L

L3 12EI

12EI L3

L3 3EI

3EI L3

L GIP

GI P L

L 4EI

4EI L

L 3EI

3EI L

L

(ii) Transverse 6EI L

6EI 2

L

(a) With far end hinged

3EI 2

L

T

(iii)Torsional displacement (iv) Flexural displacement 4EI L

(a) With far 2EI L

end fixed

3EI L

(b) With far

end hinged

Truss

Frames

In truss forces act only along the axis of the members. Members are having tension or compression. Each member is acted upon by two equal and opposite

n Pi 2 dx so F i 1 2 AE

L AE

displacement

displacement

H

l2 B

b

Flexibility Stiffness

(i) Axial

2

COF = -1 M

h2

l1

Diagram

Method of joint

w unit per run

(i)

Type of displacement

forces having line of action along the centre of members. (every member of truss is

In frames forces are acting along the axis of the member, in addition to transverse forces.

One or more than one member of frame is subjected to more than two forces (multiple force members).

a two force member.) Forces are applied at the joints only.

Forces may act anywhere on the member.

Member does not bend.

Members may be bend

Used for large loads.

Used for small & medium loads.

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