Chapter 15: Chapter 15: Beam Analysis Using the Stiffness Method Beam Analysis Using the Stiffness Method

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Chapter 15:

Beam Analysis Using the Stiffness Method

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Preliminary remarks Preliminary remarks

• Member & node identification Member & node identification

• In general each element must be free from load & In general each element must be free from load &

have a prismatic cross section have a prismatic cross section

• The nodes of each element are located at a support The nodes of each element are located at a support or at points where members are connected together or at points where members are connected together

or where the vertical or rotational disp at a point is or where the vertical or rotational disp at a point is

to be determined to be determined

Chapter 15: Beam Analysis Using the Stiffness Method

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Preliminary remarks Preliminary remarks

• Global & member coordinates Global & member coordinates

• The global coordinate system will be identified using The global coordinate system will be identified using x, y, z axes that generally have their origin at a

x, y, z axes that generally have their origin at a node & are positioned so that the nodes at other node & are positioned so that the nodes at other

points on the beam are +ve coordinates

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Preliminary remarks Preliminary remarks

• Global & member coordinates Global & member coordinates

• The global coordinate system will be identified using The global coordinate system will be identified using x, y, z axes that generally have their origin at a

x, y, z axes that generally have their origin at a node & are positioned so that the nodes at other node & are positioned so that the nodes at other

points on the beam are +ve coordinates points on the beam are +ve coordinates

• The local or member x’, y’, z’ coordinates have their The local or member x’, y’, z’ coordinates have their origin at the near end of each element

origin at the near end of each element

• The positive x’ axis is directed towards the far end The positive x’ axis is directed towards the far end

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Preliminary remarks Preliminary remarks

• Kinematic indeterminacy Kinematic indeterminacy

• The consider the effects of both bending & shear The consider the effects of both bending & shear

• Each node on the beam can have 2 degrees of Each node on the beam can have 2 degrees of freedom, vertical disp & rotation

freedom, vertical disp & rotation

• These disp will be identified by code numbers These disp will be identified by code numbers

• The lowest code numbers will be used to identify The lowest code numbers will be used to identify the unknown disp & the highest numbers are used the unknown disp & the highest numbers are used

to identify the known disp

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Preliminary remarks Preliminary remarks

• Kinematic indeterminacy Kinematic indeterminacy

• The beam is kinematically indeterminate to the 4 The beam is kinematically indeterminate to the 4

^th^th

degree

• There are 8 degrees of freedom for which code There are 8 degrees of freedom for which code

numbers 1 to 4 rep the unknown disp & 5 to 8 rep numbers 1 to 4 rep the unknown disp & 5 to 8 rep

the known disp the known disp

• Development of the stiffness method for beams Development of the stiffness method for beams follows a similar procedure as that for trusses

follows a similar procedure as that for trusses

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Preliminary remarks Preliminary remarks

• Kinematic indeterminacy Kinematic indeterminacy

• First we must establish the stiffness matrix for each First we must establish the stiffness matrix for each element

element

• These matrices are combined to form the beam or These matrices are combined to form the beam or structure stiffness matrix

structure stiffness matrix

• We can then proceed to determine the unknown We can then proceed to determine the unknown disp at the nodes

disp at the nodes

• This will determine the reactions at the beam & the This will determine the reactions at the beam & the internal shear & moments at the nodes

internal shear & moments at the nodes

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Beam-member stiffness matrix Beam-member stiffness matrix

• We will develop the stiffness matrix for a beam We will develop the stiffness matrix for a beam element or member having a constant cross-

element or member having a constant cross-

sectional area & referenced from the local x’, y’, z’

coordinate system coordinate system

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Beam-member stiffness matrix Beam-member stiffness matrix

• Linear & angular disp associated with these Linear & angular disp associated with these loadings also follow the same sign convention loadings also follow the same sign convention

• We will impose each of these disp separately & We will impose each of these disp separately &

then determine the loadings acting on the member then determine the loadings acting on the member

caused by each disp

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Beam-member stiffness matrix Beam-member stiffness matrix

• y’ displacement y’ displacement

• A positive disp d A positive disp d

_Ny’_Ny’

is imposed while other possible is imposed while other possible disp are prevented

disp are prevented

• The resulting shear forces & bending moments that The resulting shear forces & bending moments that are created are shown

are created are shown

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Beam-member stiffness matrix Beam-member stiffness matrix

• z’ rotation z’ rotation

• A positive rotation d A positive rotation d

_Nz’_Nz’

is imposed while other is imposed while other possible disp are prevented

possible disp are prevented

• The required shear forces & bending moments The required shear forces & bending moments necessary for the deformation are shown

necessary for the deformation are shown

• When d When d

_Nz’_Nz’

is imposed, the resultant loadings are is imposed, the resultant loadings are shown

shown

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Beam-member stiffness matrix Beam-member stiffness matrix

• By superposition, the resulting four load-disp By superposition, the resulting four load-disp relations for the member can be expressed in relations for the member can be expressed in

matrix form as matrix form as

• These eqn can be written as These eqn can be written as

kd

q 

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Beam-structure stiffness matrix Beam-structure stiffness matrix

• Once all the member stiffness matrices have been Once all the member stiffness matrices have been found, we must assemble them into the structure found, we must assemble them into the structure

stiffness matrix, K stiffness matrix, K

• The rows & columns of each k matrix, the eqns are The rows & columns of each k matrix, the eqns are identified by the 2 code numbers at the near end identified by the 2 code numbers at the near end (N (N

_y’_y’

, N , N

_z’_z’

) of the member followed by those at the ) of the member followed by those at the

far end (F

_y’_y’

, F , F

_z’_z’

) )

• When assembling the matrices, each element must When assembling the matrices, each element must be placed in the same location of the K matrix

be placed in the same location of the K matrix

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Beam-structure stiffness matrix Beam-structure stiffness matrix

• K will have an order that will be equal to the K will have an order that will be equal to the highest code number assigned to the beam highest code number assigned to the beam

• Where several members are connected at a node, Where several members are connected at a node, their member stiffness influence coefficients will their member stiffness influence coefficients will

have the same position in the K matrix & therefore have the same position in the K matrix & therefore

must be algebraically added to determine the must be algebraically added to determine the

nodal stiffness influence coefficient for the nodal stiffness influence coefficient for the

structure structure

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Application of the stiffness method for Application of the stiffness method for

beam analysis beam analysis

• Once the stiffness matrix is determined, the loads Once the stiffness matrix is determined, the loads at the nodes of the beam can be related to the

at the nodes of the beam can be related to the disp using the structure stiffness eqn

disp using the structure stiffness eqn

• Partitioning the stiffness matrix into the known & Partitioning the stiffness matrix into the known &

unknown elements of load & disp, we have unknown elements of load & disp, we have

KD

Q 

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Application of the stiffness method for Application of the stiffness method for

beam analysis beam analysis

• This expands into 2 eqn: This expands into 2 eqn:

• The unknown disp D The unknown disp D

_u_u

are determined from the first are determined from the first of these eqn

of these eqn

• Using these values, the support reactions Q Using these values, the support reactions Q

_u_u

are are computed for the second eqn

computed for the second eqn

k u

u

k u

k

D K

Q

D K

Q

22 21

12 11









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Application of the stiffness method for Application of the stiffness method for

beam analysis beam analysis

• Intermediate loadings Intermediate loadings

• It is important that the elements of the beam be It is important that the elements of the beam be free of loading along its length

free of loading along its length

• This is necessary as the stiffness matrix for each This is necessary as the stiffness matrix for each

element was developed for loadings applied only at element was developed for loadings applied only at

its ends its ends

• Consider the beam element of length L which is Consider the beam element of length L which is subjected to uniform distributed load, w

subjected to uniform distributed load, w

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Application of the stiffness method for Application of the stiffness method for

beam analysis beam analysis

• Intermediate loadings Intermediate loadings

• First, we will apply FEM & reactions to the element First, we will apply FEM & reactions to the element which will be used in the stiffness method

which will be used in the stiffness method

• We will refer to these loadings as a column matrix We will refer to these loadings as a column matrix q q

o_o

• The distributed loading within the beam is The distributed loading within the beam is determined by adding these 2 results

determined by adding these 2 results

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Application of the stiffness method for Application of the stiffness method for

beam analysis beam analysis

• Member forces Member forces

• The shear & moment at the ends of each beam The shear & moment at the ends of each beam

element can be determined by adding on any fixed element can be determined by adding on any fixed

end reactions q

_o_o

if the element is subjected to an if the element is subjected to an intermediate loading