m4l26

m4l26 - Module 4 Analysis of Statically Indeterminate...

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Version 2 CE IIT, Kharagpur Module 4 Analysis of Statically Indeterminate Structures by the Direct Stiffness Method
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Version 2 CE IIT, Kharagpur Lesson 26 The Direct Stiffness Method: Temperature Changes and Fabrication Errors in Truss Analysis
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Version 2 CE IIT, Kharagpur Instructional Objectives After reading this chapter the student will be able to 1. Compute stresses developed in the truss members due to temperature changes. 2. Compute stresses developed in truss members due to fabrication members. 3. Compute reactions in plane truss due to temperature changes and fabrication errors. 26.1 Introduction In the last four lessons, the direct stiffness method as applied to the truss analysis was discussed. Assembly of member stiffness matrices, imposition of boundary conditions, and the problem of inclined supports were discussed. Due to the change in temperature the truss members either expand or shrink. However, in the case of statically indeterminate trusses, the length of the members is prevented from either expansion or contraction. Thus, the stresses are developed in the members due to changes in temperature. Similarly the error in fabricating truss members also produces additional stresses in the trusses. Both these effects can be easily accounted for in the stiffness analysis. 26.2 Temperature Effects and Fabrication Errors
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Version 2 CE IIT, Kharagpur Consider truss member of length L , area of cross section A as shown in Fig.26.1.The change in length l Δ is given by T L l Δ = Δ α ( 2 6 . 1 ) where is the coefficient of thermal expansion of the material considered. If the member is not allowed to change its length (as in the case of statically indeterminate truss) the change in temperature will induce additional forces in the member. As the truss element is a one dimensional element in the local coordinate system, the thermal load can be easily calculated in global co- ordinate system by 1 () t p AE L (26.2a) 2 t p AE L =− Δ (26.2b) or {} + Δ = 1 1 ' L AE p t (26.3) The equation (26.3) can also be used to calculate forces developed in the truss member in the local coordinate system due to fabrication error. L Δ will be considered positive if the member is too long. The forces in the local coordinate system can be transformed to global coordinate system by using the equation, = t t t t t t p p p p p p ' 2 ' 1 4 3 2 1 sin 0 cos 0 0 sin 0 cos θ (26.4a) where ()() t t p p 2 1 , and t t p p 4 3 , are the forces in the global coordinate system at nodes 1 and 2 of the truss member respectively Using equation (26.3), the equation (26.4a) may be written as, 1 2 3 4 cos sin cos sin t t t t p p AE L p p ⎧⎫ ⎪⎪ ⎨⎬ ⎩⎭ (26.4b)
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Version 2 CE IIT, Kharagpur The force displacement equation for the entire truss may be written as, {} [] {} { } t p u k p ) ( + = (26.5) where , {} p is the vector of external joint loads applied on the truss and () { } t p is the
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This note was uploaded on 06/30/2009 for the course CE 358 taught by Professor Trifunac during the Fall '07 term at USC.

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m4l26 - Module 4 Analysis of Statically Indeterminate...

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