Two-Dimensional Elasticity and Shape Functions
Comprehensive university textbook unit covering fundamental principles, governing relationships, and engineering applications of Two-Dimensional Elasticity and Shape Functions.
§3.1 Plane stress versus plane strain
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `σ = Dε`. **Field study:** Compare the stress state in a thin plate with a very long restrained extrusion.
§3.2 Isoparametric triangles
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `ε = B d`. **Field study:** Compute strain from prescribed triangular-element nodal displacements.
§3.3 Quadrilateral interpolation
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `K_e=∫ Bᵀ D B dΩ`. **Field study:** Evaluate a warped four-node patch under uniform tension.
§3.4 Strain, stress and von Mises measures
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `σ_vm=√(σ_x²-σ_xσ_y+σ_y²+3τ_xy²)`. **Field study:** Compare biaxial stress at a flat panel far from boundary features.
§3.5 Patch tests and element consistency
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `r=K u-F`. **Field study:** Apply a constant linear displacement field to a mixed triangle mesh.
Step-by-Step Finite Element Analysis and Computational Mechanics Solutions
Three original questions connect calculations, definitions and interpretation. Open each solution to follow the reasoning.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **plane-stress von mises**, based on `sigma_vm=sqrt(sx²-sx*sy+sy²+3*txy²)`. Illustrative inputs are **Normal stress x=80 MPa, Normal stress y=20 MPa, In-plane shear=15 MPa**. Converting the displayed units into SI inside the calculation gives **76.64855 MPa**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **plane-stress von mises**, based on `sigma_vm=sqrt(sx²-sx*sy+sy²+3*txy²)`. Illustrative inputs are **Normal stress x=80 MPa, Normal stress y=20 MPa, In-plane shear=15 MPa**. Converting the displayed units into SI inside the calculation gives **76.64855 MPa**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **plane-stress von mises**, based on `sigma_vm=sqrt(sx²-sx*sy+sy²+3*txy²)`. Illustrative inputs are **Normal stress x=80 MPa, Normal stress y=20 MPa, In-plane shear=15 MPa**. Converting the displayed units into SI inside the calculation gives **76.64855 MPa**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **membrane force per width**, based on `N_x=sigma_x*t`. Illustrative inputs are **In-plane stress=120 MPa, Plate thickness=3 mm**. Converting the displayed units into SI inside the calculation gives **360 N/mm**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **membrane force per width**, based on `N_x=sigma_x*t`. Illustrative inputs are **In-plane stress=120 MPa, Plate thickness=3 mm**. Converting the displayed units into SI inside the calculation gives **360 N/mm**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.