CHAPTER 2Slope Deflection Method

• Objectives1. เพื่อใหเ้ขา้ใจวเิคราะห์ของโครงสร้าง Statically Indeterminate ดว้ยวธีิ slope-deflection method

2. เพื่อใหท้ราบสมการ slope-deflection equations

Outline

• Introduction to slope-deflection method. • General procedure of slope-deflection method of analysis. • Derivation of slope-deflection equations. • Work examples on slope-deflection method of analysis: beams and

frames.

Method of Analyzing Indeterminate Structure

• Force Method• In 530314• Primary unknowns Forces and Moments

• Displacement method• The slope-deflection method• Moment distribution method• Primary unknown “Displacment”• Most computer programs used to analyze a wide range of

indeterminate structures.

Displacement method of Analysis• Satisfy equilibrium equations for the structure• Unknown displacements are written in terms of loads by using the

load-displacement relations, then solved for displacements

• เม่ือ Disp สามารถหาได ้unknown loads กส็ามารถหามาได ้compatibility equations ดว้ยการใชค้วามสมัพนัธ์ load-displacement

Degree of Freedom• When a structure is loaded, the node will undergo unknown

displacements.• These displacements are referred to as the “degree of freedom”.

Specify degree of freedom is a necessary 1st step when apply displacement method

• In 2D, each node can have at most 2 linear displacements & 1 rotational displacement.

Degree of Freedom

• The number of these unknowns are referred to as the degree in which the structure is kinematically indeterminate

• Any load applied to the beam will cause node A to rotate.• Node B is completely restricted from moving.

Degree of Freedom

• The beam has nodes at A, B & C. There are 4 degrees of freedom θA, θB, θC, ∆C

• The frame has 3 degrees of freedom θB, θC, ∆B

Slope-Deflection Method

• The slope-deflection method uses displacements as unknowns and is referred to as a displacement method.

• In the slope-deflection method, the moments at the ends of the members are expressed in terms of displacements and end rotations of these ends.

• An important characteristic of the slope-deflection method is that it does not become increasingly complicated to apply as the number of unknowns in the problem increases.

• In the slope-deflection method the individual equations are relatively easy to construct regardless of the number of unknowns

Derivation of Slope-Deflection Eqs

• To derive the general form of the slope-deflection equation, let us consider the typical span AB of the continuous beam shown below when subjected to arbitrary loading.

• The slope-deflection equation can be obtained using the principle of superposition

• By considering separately the moments developed at each support due to θA, θB, ∆, and P

• Assume clockwise is positive

Derivation of Slope-Deflection Eqs

• Moment due to angular displacement @A, θA

• เพื่อท่ีจะหา MAB จะหา disp เราจะใชว้ธีิ conjugate bm method

4

2

AB A

BA A

EIMLEIML

Derivation of Slope-Deflection Eqs

• Moment due to angular displacement @B, θB

• MBA ท่ี apply เพื่อท่ีจะหา angular disp และ reaction MAB ท่ี support เราสามารถเขียนความสมัพนัธ์ไดว้า่

2

4

AB B

BA B

EIMLEIML

Derivation of Slope-Deflection Eqs

• Moment due to relative linear disp, ∆

• @Node B, มี disp เม่ือเทียบกบั A ช้ินส่วนเกิดการหมุนตามเขม็นาฬิกา ทาํใหไ้ด ้disp + ทาํใหท้ั้งสองดา้นไม่เกิดการหมุนและเท่ากนั

• Moment M สามารถหาไดจ้าก ความสมัพนัธ์ disp ดว้ยวธีิ conjugate bm

26

AB BAEIM M M

L

Derivation of Slope-Deflection Eqs

• Moment due to loading (P or w)• These moments are called Fixed End Moment (FEM).• In general, linear and angular displacement of the nodes are

caused by loadings acting on the span of the member.• To develop the slope-deflection equation, we must transform

these span loadings into equivalent moment acting at the nodes and then use the load-displacement relationships just derived

Derivation of Slope-Deflection Eqs

• เราจะไดค่้าลบ @Node A ทวนเขม็นาฬิกา และ ค่าบวก @Node B ตามเขม็นาฬิกา

• The resultant moment @ the end can be written

Derivation of Slope-Deflection Eqs

• เราสามารถเขียนเป็นสมการเดียวไดด้งัน้ี2 (2 3 ) ( )

For Internal span or End span with Far End FixedN N F NM Ek FEM

Derivation of Slope-Deflection Eqs

• Pin supported End Span

3 ( ) ( )Only for End Span with Far End Pinned or Roller Supported

N N NM Ek FEM

Example 1

Draw SFD & BMD, EI คงท่ี

Example 2

Draw SFD & BMD, EI คงท่ี

Example 2 cont’d

Example 3

Draw SFD & BMD, EI คงท่ี

Example 4

จงหา Moment @ A,B เม่ือกาํหนดให ้disp @B = 80 mm (E = 200 GPa, I = 5x106 mm4

Example 4 cont’d

Analysis of Frames: No sidesway

Example 5

Analysis of Frames: Sideway

Example 6

จงหา Moments @ แต่ละ joint

Example 6 cont’d

Example 7

จงหา Moments @ แต่ละ joint. Note: Supports @A, D เป็น fixed และ @C เป็น pinned

Example 7 cont’d

Example 7 cont’d