By ACI Committee 214

ISBN-10: 0870314238

ISBN-13: 9780870314230

Statistical systems offer necessary instruments for comparing the result of concrete energy assessments. details derived from such strategies is effective in defining layout standards, requirements, and different parameters wanted for structural assessment and service. This consultant discusses diversifications that happen in concrete energy and offers statistical systems invaluable in analyzing those adaptations with appreciate to certain trying out and standards.

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**Extra resources for ACI 214R-11: Guide to Evaluation of Strength Test Results of Concrete**

**Example text**

103), it is known that δ x1 (k) + (1 − δ )x2 (k) ∈ Xk (x(k − 1)), k = 1, 2, · · · , n. This means Xk (x(k − 1)) is a convex set, k = 1, 2, · · · , n. The proof is completed. 23. Let z ∈ R2 , if ∀δ ∈ [0, 1], ∀y1 (k), y2 (k) ∈ Yk (y(k − 1)), then the set Yk (y(k − 1)) ⊂ R2 is called convex in direction z, if and only if there exists γ ≥ 0, such that δ y1 (k) + (1 − δ )y2 (k) − γ z ∈ Yk (y(k − 1)). 9. The expanded state attainable set of the (k + 1)th stage Yk (y(k − 1)), which is derived from y(k − 1), is convex in direction e = (1, 0), k = 1, 2, · · · , n.

4, it is not difficult to prove the following lemma. 5. The point y∗ (k) is on the conditional lower bound surface of the (k + 1)th stage Π k (y(k − 1)), k = 1, · · · , n. Proof. Let y∗ (k) = (Jk∗ , x∗ (k)), where k = 1, 2, · · · , n. According to Def. 4, it is Jk∗ = FOV k (x∗ (k)). 82) On the other hand, from Eq. 79), it is f ovk (x∗ (k)|y∗ (k − 1)) ≥ FOV k (x∗ (k)). Hence, f ovk (x∗ (k)|y∗ (k − 1)) = Jk∗ . 83) The above equation means that point y∗ (k) is on the conditional lower bound surface of the (k + 1)th stage Π k (y(k − 1)), k = 1, 2, · · · , n.

86) k = 0, 1, · · · , n − 1. 5, and the above statements and definitions, the most important lemma can be proved as at below. 7. It is said that Sk+1 (x(k + 1)|y∗ (k)) is the support function for the conditional lower bound surface of the (k + 1)th stage Π k+1 (y∗ (k)) at point x∗ (k + 1), if and only if H k (x∗ (k), u(k), Sk+1 ) arrives at the maximum value at point u∗ (k), where u∗ (k) ∈ Uk (x∗ (k)), k = 0, 1, · · · , n − 1. Proof. Let Sk+1 (x(k + 1)|y∗ (k)) be the support function for the conditional lower bound surface of the (k + 1)th stage Π k+1 (y∗ (k)) at point x∗ (k + 1), then − f ovk+1 (x(k+1)|y∗ (k))+Sk+1 (x(k+1)|y∗ (k)) ≤ α k+1 , ∀x(k+1) ∈ Xk+1 (x∗ (k)).

### ACI 214R-11: Guide to Evaluation of Strength Test Results of Concrete by ACI Committee 214

by Brian

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