Solve structures subjected to temperature influences

Elements elongate under uniform loading with an additional strain of \(\epsilon^{\rm{T}} = \alpha \ \Delta T\) with \(\alpha\) the linear expansion coefficient. When loaded by a temperature change over the height of an element, fibres elongate individually, leading to bending of elements with an additional curvature of \(\kappa^{\rm{T}} = \alpha \ \cfrac{\Delta T}{h}\), with \(h\) the height of an element. So, in statically determinate structures, this leads to additional stressless strains (leading to displacements) without influencing the force distribution as the force distribution is independent of the displacements.

Displacement can be found by integrating the stressless strains using the differential equations. Alternatively, using an equivalent load which leads to the same curvature allows the use of forget-me-nots. This requires a kinematic equivalent load which doesn’t influence reactions forces and internal forces:

Fig. 40 Kinematic equivalent load leading to the same strain and curvature as strain due to linear expansion

In statically indeterminate structures the displacement and force distribution is coupled, leading to reaction forces and internal stresses due to the (intended) displacement due to the temperature change. These forces can, again, be found by integrating the strains (both the stress-causing strains and stressless temperature strains) using the differential equations. Alternatively, a kinematic equivalent load, as in statically determinate structures, can be applied in combination with the force method: the displacement due to the temperature are included into the displacement constraints.

Fig. 41 Kinematic equivalent load leading to the same strain and curvature as strain due to linear expansion, while statically indeterminate reaction forces cause stresses and reaction forces

In chapter 4.12 of the book Engineering Mechanics: Volume 2: Stresses, strains and displacements (Hartsuijker and Welleman, 2007), the derivation of temperature influences is discussed. In chapter 6.2.1 of the book Mechanica, Statisch onbepaalde constructies en bezwijkanalyse (Hartsuijker and Welleman, 2007) (in Dutch), this is repeated in a simplified manner for statically determinate structures. The approach using the moment-area method is not covered in this course. Additionally, the standard cases for a beam on two supports and a clamped beam are not used. In chapter 6.2.2, statically indeterminate structures are discussed. Here too, the moment-area method is not part of this course.

Exercises

• Problems 6.25 - 6.30, 6.32 - 6.39, 6.41 - 6.43 in chapter 6.3 of the book Mechanica, Statisch onbepaalde constructies en bezwijkanalyse (Hartsuijker and Welleman, 2007) (in Dutch). Unfortunately, no answers are available.